crossﬂow stability and transition experiments in swept-wing...

NASA/TP-1999-209344

Crossflow Stability and TransitionExperiments in Swept-Wing Flow

J. Ray DagenhartLangley Research Center, Hampton, Virginia

William S. SaricArizona State University, Tempe, Arizona

July 1999

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NASA/TP-1999-209344

Crossflow Stability and TransitionExperiments in Swept-Wing Flow

J. Ray DagenhartLangley Research Center, Hampton, Virginia

William S. SaricArizona State University, Tempe, Arizona

July 1999

Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)

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Hanover, MD 21076-1320 Springfield, VA 22161-2171

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Acknowledgments

The authors are grateful to Helen L. Reed of Arizona State University who provided theoretical data for comparisonwith the experimental results and W. Pfenninger of the NASA Langley Research Center who continually providedtechnical insight and discussion throughout the project. Jon Hoos, Marc Mousseux, Ronald Radeztsky, and DanClevenger of Arizona State University provided important assistance and valuable discussion of the research at theUnsteady Wind Tunnel. Thanks are extended to Harry L. Morgan and J. P. Stack of the NASA Langley ResearchCenter for assistance in computational and measurement aspects of the research. Finally, technical discussions withmany members of the Experimental Flow Physics Branch contributed significantly to the work.

This report was completed by Ronald D. Joslin of the NASA Langley Research Center and William S. Saric of ArizonaState University, due to the untimely death of J. Ray Dagenhart. Ray will be missed by his many friends and col-leagues. This work took place during 1988 and 1989. It was, in fact, the foundation experiment that spawned 5 doctor’sand 1 master’s degrees in the subsequent 7 years. We now know a great deal more about crossflow instabilities becauseof the vision and hard work of the late Ray Dagenhart.

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Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.2. Instability Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3. Transition Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4. Detailed Theory and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5. Stability Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.6. State of Present Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.7. Present Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.8. Organization of Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. Experimental Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1. Arizona State University Unsteady Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2. New Test Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94. Model and Liner Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1. Airfoil Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.1. Pressure Gradient Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.2. Wind Tunnel Wall Interference Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2. Stability Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.1. Stationary Crossflow Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.2. Tollmien-Schlichting Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.3. Travelling Crossflow Vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.4. Crossflow–Tollmien-Schlichting Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3. Selection of Experimental Test Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4. Reynolds Number Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.5. Test Section Liner Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155. Experimental Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1. Free-Stream Flow Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2. Pressure Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.3. Flow Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4. Transition Locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.5. Boundary-Layer Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.6. Boundary-Layer Hot-Wire Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.1. Streamwise Velocity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.2. Spanwise Variation of Streamwise Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6.3. Disturbance Profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6.4. Streamwise Velocity Contour Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

iv

5.7. Experimental and Theoretical Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.7.1. Theoretical Disturbance Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.7.2. Disturbance Profile Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.7.3. Velocity Contour Plots and Vector Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.7.4. Wavelength Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.7.5. Growth-Rate Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Appendix A—Relationships Between Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Appendix B—Hot-Wire Signal Interpretation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Appendix C—Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

v

List of Tables

Table 1. Crossflow Stability Analysis With SALLY Code forα = −4° andRc = 3.81× 106 . . . . . . . . 45

Table 2. Transition Locations and Wavelengths From Naphthalene Flow Visualization . . . . . . . . . . . 45

Table 3.N-Factors at Transition Computed With SALLY Code forα = −4°, Rc = 2.37× 106, and(x/c)tr = 0.58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45



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List of Figures

Figure 1. Curved streamlines over swept wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 2. Boundary-layer velocity profiles on swept wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 3. Plan view of Arizona State University Unsteady Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 4. New UWT test section with liner under construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 5. NASA NLF(2)-0415 airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 6. NASA NLF(2)-0415 design point pressure distribution atα = 0° andδf = 0° in free air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 7. NASA NLF(2)-0415 pressure distribution forα = −4° andδf = 0° in free air . . . . . . . . . . . 50

Figure 8. NASA NLF(2)-0415 pressure distribution forα = −2° andδf = 0° in free air. . . . . . . . . . . . 50

Figure 9. NASA NLF(2)-0415 pressure distribution forα = 2° andδf = 0° in free air. . . . . . . . . . . . . 51

Figure 10. NASA NLF(2)-0415 pressure distribution forα = 4° andδf = 0° in free air. . . . . . . . . . . . 51

Figure 11. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = −20° in free air. . . . . . . . . . 52

Figure 12. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = −10° in free air. . . . . . . . . . 52

Figure 13. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = 10° in free air. . . . . . . . . . . 53

Figure 14. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = 20° in free air. . . . . . . . . . . 53

Figure 15. NASA NLF(2)-0415 pressure distribution forα = −4° andδf = 0° in UWT. . . . . . . . . . . . 54

Figure 16. NASA NLF(2)-0415 pressure distribution forα = −2° andδf = 0° in UWT. . . . . . . . . . . . 54

Figure 17. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = 0° in UWT . . . . . . . . . . . . . 55

Figure 18. NASA NLF(2)-0415 pressure distribution forα = 2° andδf = 0° in UWT. . . . . . . . . . . . . 55

Figure 19. NASA NLF(2)-0415 pressure distribution forα = 4° andδf = 0° in UWT. . . . . . . . . . . . . 56

Figure 20. Local spatial growth rates for stationary crossflow vortices atα = −4° andδf = 0° in UWT atRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Figure 21. Local spatial growth rates for stationary crossflow vortices atα = −2° andδf = 0° in UWT atRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Figure 22. Local spatial growth rates for stationary crossflow vortices atα = 0° andδf = 0° in UWT atRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57



Figure 25.N-factors for stationary crossflow vortices atα = −4° andδf = 0° in UWTatRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 26.N-factors for stationary crossflow vortices atα = −2° andδf = 0° in UWTatRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 27.N-factors for stationary crossflow vortices atα = 0° andδf = 0° in UWTatRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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Figure 30. MaximumN-factors for stationary crossflow vortices atα = −4° andδf = 0°in UWT atRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 31. MaximumN-factors for stationary crossflow vortices atα = −2° andδf = 0°in UWT atRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 32. MaximumN-factors for stationary crossflow vortices atα = 0° andδf = 0°in UWT atRc = 3.81× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62



Figure 35. MaximumN-factors for Tollmien-Schlichting waves forα = 0°, 2°, and 4°andδf = 0° in UWT atRc = 3.81× 106. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 36. MaximumN-factors for stationary crossflow vortices atα = −4° andδf = 0°in UWT for a range of Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 37. Streamline traces of wind tunnel end liner onXL-ZL plane forα = −4°. . . . . . . . . . . . . . . . 65

Figure 38. Lateral deflections of end-liner surface at various distances fromwing chord plane forα = −4° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 39. End-liner contours inYL-ZL plane at various longitudinal positions forα = −4°. . . . . . . . . 66

Figure 40. Wind tunnel test section with swept-wing model and end liners installed . . . . . . . . . . . . . . 67

Figure 41. Free-stream velocity spectrum forRc = 3.27× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Figure 42. Measured and predicted model pressure coefficients at upper end ofmodel forα = −4° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

Figure 43. Measured and predicted model pressure coefficients at lower end ofmodel forα = −4° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

Figure 44. Naphthalene flow visualization atα = −4° andRc = 1.93× 106 . . . . . . . . . . . . . . . . . . . . . 70





Figure 49. Naphthalene flow visualization with vortex tracks in turbulent regions shown . . . . . . . . . . 75

Figure 50. Liquid-crystal flow visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 51. Transition location versus Reynolds number atα = −4° . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 52. Boundary-layer velocity spectrum atα = −4° andRc = 2.62× 106 atx/c = 0.40. . . . . . . . . 77


ix



Figure 56. Measured and predicted boundary-layer velocity spectra atα = −4° andRc = 2.92× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Figure 57. Streamwise velocity profiles atx/c = 0.20,α = −4°, andRc = 2.62× 106 . . . . . . . . . . . . . . 79








Figure 65. Stationary crossflow disturbance velocity profiles forf = 0 Hz atx/c = 0.20,α = −4°, andRc = 2.37× 106 obtained from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83








Figure 73. Stationary crossflow disturbance velocity profiles forf = 0 Hz atx/c = 0.20,α = −4°, andRc = 2.37× 106 obtained from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87





us us,avg–

us us,avg–

us us,avg–

us us,avg–

us us,avg–

us us,avg–

us us,avg–

us us,avg–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

x




Figure 81. Travelling wave disturbance velocity profiles forf = 100 Hz atx/c = 0.20,α = −4°, andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


Figure 83. Travelling wave disturbance velocity profiles forf = 100 Hz atx/c = 0.30,α = −4°, andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92u






Figure 89. Mean streamwise velocity contours atx/c = 0.20,α = −4°, andRc = 2.37× 106. . . . . . . . . 95








Figure 97. Stationary crossflow vortex velocity contours obtained fromatx/c = 0.20,α = −4°, andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 98. Stationary crossflow vortex velocity contours obtained fromatx/c = 0.25,α = −4°, andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100




us us,ref–

us us,ref–

us us,ref–

us us,avg–

us us,avg–

us us,avg–

us us,avg–

us us,avg–

xi












Figure 113. Travelling wave rms velocity contours forf = 100 Hz atx/c = 0.20,α = −4°,andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107








Figure 121. Theoretical mean chordwise velocity profiles forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

us us,avg–

us us,avg–

us us,avg–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

us us,ref–

xii

Figure 122. Theoretical mean spanwise velocity profiles forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 123. Theoretical stationary crossflow disturbance velocity profiles(chordwise component) forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 124. Theoretical stationary crossflow disturbance velocity profiles(surface normal component) forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Figure 125. Theoretical stationary crossflow disturbance velocity profiles(spanwise component) forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 126. Theoretical mean streamwise velocity profiles forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 127. Theoretical mean cross-stream velocity profiles forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 128. Theoretical stationary crossflow disturbance velocity profiles(streamwise component) forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 129. Theoretical stationary crossflow disturbance velocity profiles(cross-stream component) forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 130. Theoretical mean velocity profiles along vortex axis forαref = −5°andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 131. Theoretical mean velocity profiles perpendicular to vortex axis forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 132. Theoretical stationary crossflow disturbance velocity profiles alongvortex axis forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 133. Theoretical stationary crossflow disturbance velocity profiles perpendicularto vortex axis forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 134. Theoretical stationary crossflow disturbance velocity vectors acrosssingle vortex wavelength forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 135. Theoretical total velocity vectors (disturbance plus mean flow) acrosssingle vortex wavelength forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 136. Theoretical total velocity vectors (disturbance plus mean flow) acrosssingle vortex wavelength with normal velocity components scaled 100 forαref = −5° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 137. Experimental streamwise disturbance velocity profile determined from and theoretical eigenfunction forx/c = 0.20,α = −4°, andRc = 2.37× 106 . . . . . . . . . 120




Figure 141. Experimental streamwise disturbance velocity profile determined fromand theoretical eigenfunction forx/c = 0.40,α = −4°, andRc = 2.37× 106 . . . . . . . . . 122

us us,avg–

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Figure 145. Experimental mean streamwise velocity contours and theoretical vortexvelocity vector field forx/c = 0.20,α = −4°, andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . 124








Figure 153. Experimental stationary crossflow disturbance velocity contours andtheoretical vortex velocity vector field forx/c = 0.20,α = −4°, andRc = 2.37× 106 . . . . . . . . . . . 128








Figure 161. Theoretical and experimental stationary crossflow vortex wavelengthsfor α = −4° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 162. Theoretical and experimental stationary crossflow vortex growth ratesfor α = −4° andRc = 2.37× 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

us us,avg–

us us,avg–

us us,avg–

xv

Symbols

A amplitude

A, B coefficients in King’s law for hot-wire anemometer (see appendix C)

A0 amplitude at initial station

Al, A2, A3 experimental disturbance amplitude functions (see eqs. (7), (8), and (9), respectively)

Cp pressure coefficient

Cp,3 swept-wing pressure coefficient (see eq. (6))

c chord, m

E hot-wire steady state voltage (see appendix B), V

e hot-wire voltage (see appendix B), V

F, g hot-wire voltage functions (see appendix B)

f frequency, l/sec

L/D ratio of length to diameter of wind tunnel contraction

l length, m

N N-factor, ln (A/A0)

p static pressure, mm Hg

q dynamic pressure,

R Reynolds number,

R gas constant

U velocity

u, v, w disturbance velocity component inx, y, andz direction, respectively

mean velocity component inx, y, andz direction, respectively

X, Y, Z axes

x, y, z Cartesian coordinates

α angle of attack, deg

∆ measurement uncertainty (see appendix C)

δ boundary-layer thickness

ρU2

2----------

ρUlµ

----------

u v w, ,

xvi

δf flap deflection angle, deg

ε local boundary-layer edge flow angle measured with respect toZm-axis (seeappendix A), deg

θ disturbance wave orientation angle measured with respect toZm-axis (seeappendix A), deg

Λ wing sweep angle, deg

λ disturbance function wavelength, mm

µ dynamic viscosity, kg-sec/m2

ρ mass density, kg/m3

σ disturbance growth rate measured with respect to local boundary-layer edge velocity vector

ψ wave orientation angle

Subscripts:

avg average

bp blade passing

bps blade passing plus stators

c chord

cf crossflow

cf,max maximum crossflow amplitude

crit critical

e edge

L liner

L0 liner origin

M measured

m model

m0 model reference point

max maximum

r real

ref reference

xvii

s streamwise (i.e., parallel to free-stream velocity vector)

s, e streamwise at boundary-layer edge

T total or stagnation condition

t tangential (i.e., parallel to local boundary-layer edge velocity vector)

t, e tangential at boundary-layer edge

tr transition

v vortex track

w wave

λ disturbance wavelength

0 origin or beginning value

45 at 45° angle

∞ free stream

Superscripts:

' fluctuating quantity

Abbreviations:

AC alternating current

A/D analog to digital

D/A digital to analog

DC direct current

DLR nationale Forschungszentrum fur Luft- und Raumfahrt und zugleich die nationaleRaumfahrtagentur (Germany’s National Aerospace Research Center as well as theNational Space Agency)

LFC laminar flow control

NBS National Bureau of Standards

NLF natural laminar flow

RAM random access memory

rms root mean square

TS Tollmien-Schlichting

UWT Arizona State University Unsteady Wind Tunnel

1. Summary

This report covers an experimental examinationof crossflow instability and transition on a 45°swept wing that was conducted in the Arizona StateUniversity Unsteady Wind Tunnel during the period1988–1989. The stationary vortex pattern and transi-tion location are visualized by using both sublimatingchemical and liquid-crystal coatings. Extensive hot-wire measurements were obtained at several measure-ment stations across a single vortex track. The meanand travelling wave disturbances were measuredsimultaneously. Stationary crossflow disturbance pro-files were determined by subtracting either a referenceor a span-averaged velocity profile from the meanvelocity profiles. Mean, stationary crossflow, and trav-elling wave velocity data were presented as localboundary-layer profiles and contour plots across a sin-gle stationary crossflow vortex track. Disturbance-mode profiles and growth rates were determined. Theexperimental data are compared with predictions fromlinear stability theory.

Comparisons of measured and predicted pressuredistributions showed that a good approximation ofinfinite swept-wing flow was achieved. A fixed wave-length vortex pattern was observed throughout theflow-visualization range with the observed wave-length 20 percent shorter than that predicted by the lin-ear stability theory. Linear stability computations forthe dominant stationary crossflow vortices showedthat theN-factors at transition ranged from 6.4 to 6.8.

The mean velocity profiles varied slightly acrossthe stationary crossflow vortex at the first measure-ment station. The variation across the vortex increasedwith downstream distance until nearly all profilesbecame highly distorted S-shaped curves. Local sta-tionary crossflow disturbance profiles had eitherpurely excess or deficit values develop at the upstreammeasurement stations. Farther downstream, the pro-files took on crossover shapes not predicted by thelinear theory. The maximum streamwise stationarycrossflow velocity disturbances reached 20 percent ofthe edge velocity just prior to transition. The travellingwave disturbances had single lobes at the upstreammeasurement stations as expected, but farther down-stream double-lobed travelling wave profiles devel-oped. The maximum disturbance intensity remainedquite low until just ahead of the transition location,

where it suddenly peaked at 0.7 percent of the edgevelocity and then dropped sharply. The travellingwave intensity was always more than an order of mag-nitude lower than the stationary crossflow vortexstrength.

The mean streamwise velocity contours werenearly flat and parallel to the model surface at the firstmeasurement station. Farther downstream, the con-tours rose up and began to roll over like a wave break-ing on the beach. The stationary crossflow contoursshowed that a plume of low-velocity fluid rose nearthe center of the wavelength while high-velocityregions developed near the surface at each end of thewavelength. No distinct pattern to the low-intensitytravelling wave contours appeared until a short dis-tance upstream of the transition location where thetravelling wave intensity suddenly peaked near thecenter of the vortex and then fell abruptly.

The experimental disturbance-mode profilesagreed quite well with the predicted eigenfunctions forthe forward measurement stations. At the later sta-tions, the experimental mode profiles took on double-lobed shapes with maxima above and below the singlemaximum predicted by the linear theory. The experi-mental growth rates were found to be less than orequal to the predicted growth rates from the lineartheory. Also, the experimental growth-rate curveoscillated over the measurement range, whereasthe theoretically predicted growth rates decreasedmonotonically.

2. Introduction

2.1. Background

The flow over aircraft surfaces can be either lami-nar or turbulent. Laminar flow smoothly follows theaircraft contours and produces much lower local skinfriction drag than the more chaotic turbulent flow.Often both laminar and turbulent flow regions arefound on a given aircraft. The amount of laminar andturbulent flow areas is highly dependent on the size,shape, surface finish, speed, and flight environment ofthe aircraft. The process of minimizing aircraft dragby maintaining laminar flow by using active meanssuch as suction, heating, or cooling is referred to as“laminar flow control (LFC).” LFC technology isbeing considered for applications on new large

2

transonic and supersonic transport aircraft. The goalof this effort is to reduce direct operating costs ofnew aircraft by reducing the drag and, therefore, thefuel consumption. Adequate understanding of theboundary-layer transition process from laminar to tur-bulent flow lies at the heart of LFC technology. Thepresent research effort is aimed at investigating animportant component of the transition process onswept wings, namely the development and growth ofcrossflow vortices.

The boundary-layer transition problem usuallyconsists of three important phases: receptivity, lineardisturbance amplification, and nonlinear interactionand breakdown (ref. 1). The Navier-Stokes equationsmodel the appropriate physics for all these phases.However, techniques to solve these equations for theentire range of the transition problem are only nowbeing developed. Until recently most experimentaland theoretical examinations have focused on the sec-ond phase, namely, linear disturbance growth in a lam-inar boundary layer. For two-dimensional flows theexperimental and theoretical investigations in this lin-ear regime are in general agreement and are consid-ered to be conceptually well understood (ref. 2).However, for three-dimensional flows, several impor-tant phenomena remain unresolved even for the linearstability phase (ref. 1). These phenomena include(1) determination of the dominance of stationary ortravelling crossflow waves, (2) whether the crossflowvortex wavelength remains fixed over the region ofinstability or increases as the boundary layer thickens,and (3) determination of the influence of surfaceroughness and sound on crossflow vortex growth. Theresolution of these uncertainties has broad implica-tions not only for linear stability analyses but also forthe entire transition problem for three-dimensionalflows.

Receptivity is the process by which disturbancesin the external environment enter the boundary layerto begin the transition process (ref. 3). Examples ofexternal disturbance mechanisms include free-streamturbulence (with both vortical and acoustic compo-nents), wing surface irregularities and roughness, andsurface vibrations. These small disturbances providethe initial amplitude conditions for unstable waves.

The sensitivity of the laminar boundary layer tosmall amplitude disturbances can be estimated by

solving a set of linear disturbance equations obtainedfrom the governing nonlinear Navier-Stokes equations(ref. 4). The best known example of this is theOrr-Sommerfeld equation for two-dimensional,incompressible Tollmien-Schlichting waves (ref. 4),but similar equations can be derived for more generalthree-dimensional, compressible, or incompressibleflows. These linear equations are obtained by assum-ing that the complete flow field can be divided into asteady base flow and a disturbance or perturbationflow that varies both spatially and temporally. Thebase flow is assumed to be a known solution of theNavier-Stokes equations. By eliminating the knownbase flow solution from the complete problem, nonlin-ear disturbance equations result. The disturbanceequations can be linearized by assuming that the inputdisturbances are small so that products of disturbancecomponents are neglected. Although the equations arelinear, the disturbances actually grow exponentially ineither time or space, but the linearity of the equationsallows a Fourier decomposition of the problem intomodes where each mode has its own characteristic fre-quency, wavelength, and wave orientation angle. Thelinear equations can be solved locally when the baseflow solution is known by selecting two of the threecharacteristic variables—frequency, wavelength, ororientation. Upon specifying two variables, the localgrowth rate and the third characteristic variable areobtained from the linear equation solution. To esti-mate a transition location by using the so-calledeN

method of Smith and Gamberoni (ref. 5) and VanIngen (ref. 6), the local solutions to the linear equa-tions are integrated over the wing surface subject tosome parametric constraint. The definition of theproper constraint for the three-dimensional swept-wing flow problem is unknown. Examples of theparameter-constraint relation which have been sug-gested (often very arbitrarily) by various researchersinclude maximum local amplification rate, fixedwavelength, and fixed spanwise wave number. Widelydifferent values for the integratedeN solutions (andthus estimated transition locations) are obtained withthe various constraint relations.

The nonlinear interaction and breakdown phase ofthe transition problem begins when the individualmodes attain sufficient magnitude that products of thedisturbance components can no longer be neglected asbeing small when compared with the base flow. Fromthat point, the linear stability method (eN method) isno longer valid. At this stage, the disturbances may

3

have become so large that they begin to severelydistort the base flow either spatially or temporally.Reed’s computations (ref. 7) indicate that the initialdeparture from linearity is characterized by doubleexponential growth of the interacting modes; however,a complete nonlinear analysis is necessary to demon-strate that this is a physically realistic result. Fortu-nately, this phase of the transition process usuallyoccurs over a fairly short distance when comparedwith the total laminar flow extent so that almost all theprebreakdown flow region can be approximated by thelinear equations only.

2.2. Instability Modes

The laminar boundary layer on a swept wing hasfour fundamental instability modes: attachment line,streamwise, crossflow, and centrifugal. These modesmay exist independently or in combinations. Thecurved streamlines of a typical three-dimensional floware illustrated in figure 1, and the tangential and cross-flow velocity profiles are shown in figure 2. (Appen-dix A outlines relationships between the coordinatesystems used in the present experiment.) The stream-wise instability in a three-dimensional boundary layeris similar to the Tollmien-Schlichting waves in two-dimensional flows. Crossflow vortices arise as a resultof a dynamic (or inviscid) instability of the inflectionalcrossflow velocity profile produced by the three-dimensionality of the mean flow field. Both theseinstabilities are governed to first order by theOrr-Sommerfeld eigenvalue problem or its three-dimensional analog. This equation is obtained byassuming a separation of variables solution to the lin-earized Navier-Stokes disturbance equations. Theresults obtained are predictions of the local distur-bance amplification rates subject to the constraintsrequired by the separation of variables assumption.Görtler vortices may develop because of a centrifugalinstability in the concave regions of a wing. Appropri-ate curvature terms must be included in the governingequations to account for this instability. Theattachment-line instability problem may be significanton wings with large leading-edge radii. For the presentexperiment on a model with a small leading-edgeradius and no upper surface concave regions neitherGörtler vortices nor attachment-line contamination areexpected to be present, and the most important effectsare caused by crossflow and Tollmien-Schlichtinginstabilities.

The principal motivation for the study of three-dimensional boundary layers is to understand the tran-sition mechanisms on swept wings. The crossflowinstability was first identified by Gray (ref. 8) when hefound that high-speed swept wings had only minimallaminar flow even though unswept versions of thesame wings had laminar flow to approximately60 percent chord. He used sublimating chemical coat-ings to visualize the stationary crossflow vortex pat-tern in the short laminar flow region near the wingleading edge. These findings were subsequently veri-fied by Owen and Randall (ref. 9) and Stuart (ref. 10).Owen and Randall introduced a crossflow Reynoldsnumber (based on the maximum crossflow velocityand the boundary-layer height where the crossflowvelocity was 10 percent of the maximum) and deter-mined that the minimum critical crossflow Reynoldsnumber near the leading edge of a swept wing wasvery low (Rcf,crit = 96). This work was put on a firmfooting both experimentally and theoretically in theclassic paper of Gregory, Stuart, and Walker (ref. 11),who established the generality of the results for three-dimensional boundary layers and presented the com-plete disturbance-state equations.

Brown (refs. 12–14), working under Pfenninger’sdirection, was the first to integrate the three-dimensional disturbance equations. Brown obtainedresults in agreement with Gray (ref. 8) and Owen andRandall (ref. 9), but, in addition, showed the potentialof suction in controlling the crossflow instability onswept wings. Pfenninger and his coworkers examinedsuction LFC in a series of experiments—Pfenninger,Gross, and Bacon (ref. 15); Bacon, Tucker, andPfenninger (ref. 16); Pfenninger and Bacon (ref. 17);Gault (ref. 18); and Boltz, Kenyon, and Allen (ref. 19).They verified the achievement of full-chord laminarflow to a maximum chord Reynolds number of29 × 106. With this first successful swept-wing LFCprogram, Pfenninger and his group thus establishedthe foundation of future efforts in this area. SeePfenninger (ref. 20) for a collection of references onLFC efforts.

Smith and Gamberoni (ref. 5) and Van Ingen(ref. 6) introduced the so-calledeN linear stabilitymethod by integrating the local growth rates to deter-mine an overall amplification factor at transition fortwo-dimensional and axisymmetric flows. They foundthat transition occurred whenever theN-factor reached

4

about 10 (or a disturbance amplification ofe10). Manyinvestigators including Jaffe, Okamura, and Smith(ref. 21); Mack (refs. 22 to 24); Hefner and Bushnell(ref. 25); Bushnell and Malik (ref. 26); and Berry et al.(ref. 27) verified that similar results applied for thecrossflow instability on swept wings. Recent windtunnel transition studies that added to theN-factortransition database include Arnal, Casalis, and Juillen(ref. 28); Creel, Malik, and Beckwith (ref. 29); andBieler and Redeker (ref. 30). Flight tests involvingnatural laminar flow (NLF) transition studies includeCollier et al. (ref. 31); Parikh et al. (ref. 32); Collieret al. (ref. 33); Obara et al. (ref. 34); Lee, Wusk, andObara (ref. 35); Horstmann et al. (ref. 36); Waggoneret al. (ref. 37); and Obara, Lee, and Vijgen (ref. 38).Suction LFC wind tunnel transition experimentsinclude Berry et al. (ref. 39); Harvey, Harris, andBrooks (ref. 40); Arnal, Juillen, and Casalis (ref. 41);flight tests with suction LFC include Maddalon et al.(ref. 42); and Runyan et al. (ref. 43).TheseN-factortransition studies were facilitated by the use of linearstability codes such as SALLY (ref. 44), MARIA(ref. 45), COSAL (refs. 46 and 47), and Linear-X(ref. 48). Arnal (ref. 49), Saric (refs. 50 and 2), Stetson(ref. 51), Malik (ref. 52), Poll (ref. 53), and Arnal andAupoix (ref. 54) gave general discussions of theapplicability of theeN-transition methods in three-dimensional flows.

The basic equations for the linear stability analysisof compressible parallel flows were derived by Leesand Lin (ref. 55), Lin (ref. 56), Dunn and Lin (ref. 57),and Lees and Reshotko (ref. 58) by using small distur-bance theory. Mack’s numerical results (refs. 59 to 61and 22) have long been heralded as the state of the artin both compressible and incompressible parallel sta-bility analysis. Other investigations of the crossflowinstability in compressible flows include Lekoudis(ref. 62); Mack (refs. 63 and 64); El-Hady (ref. 65);Reed, Stuckert, and Balakumar (ref. 66); andBalakumar and Reed (ref. 67). These investigationsshowed that compressibility reduced the local amplifi-cation rates and changed the most unstable wave ori-entation angles. The largest impact of this stabilizinginfluence, however, is on the streamwise instability,whereas little effect is noted for the crossflowinstability.

Nonparallel flow effects on the crossflow instabil-ity were considered by Padhye and Nayfeh (ref. 68),

Nayfeh (refs. 69 and 70), El-Hady (ref. 65), and Reedand Nayfeh (ref. 71). Malik and Poll (ref. 72) andReed (ref. 7) found that the most highly amplifiedcrossflow disturbances were travelling waves ratherthan stationary waves. Viken et al. (ref. 73); Mueller,Bippes, and Collier (ref. 74); Collier and Malik(ref. 75); and Lin and Reed (ref. 76) investigated theinfluence of streamline and surface curvature on cross-flow vortices. The interaction of various primary dis-turbance modes was considered by Lekoudis (ref. 77);Fischer and Dallmann (ref. 78); El-Hady (ref. 79); andBassom and Hall (refs. 80 to 83). Transition criteriaother than theeN method were considered by Arnal,Coustols, and Juillen (ref. 84); Arnal, Habiballah, andCoustols (ref. 85); Arnal and Coustols (ref. 86);Michel, Arnal, and Coustols (ref. 87); Arnal, Coustols,and Jelliti (ref. 88); Michel, Coustols, and Arnal(ref. 90); Arnal and Juillen (ref. 90); and King(ref. 91).

2.3. Transition Experiments

Many transition experiments involving both NLFand LFC in wind tunnels and flight are discussed insection 2.2 in relation toN-factor correlation studies.Several transition experiments such as Poll (ref. 92);Michel et al. (ref. 93); and Kohama, Ukaku, and Ohta(ref. 94) deserve further discussion.

Poll (ref. 92) studied the crossflow instability on along cylinder at various sweep angles. He found thatincreasing the yaw angle strongly destabilized theflow producing both stationary and travelling wavedisturbances. The fixed disturbance pattern was visu-alized with either surface-evaporation or oil flow tech-niques. These disturbances appeared as regularlyspaced streaks nearly parallel to the inviscid flowdirection and ended at a sawtooth transition line. Theunsteady or travelling disturbances appear as high-frequency (f = 1 kHz) harmonic waves that reachedamplitudes in excess of 20 percent of the local meanvelocity before the laminar flow broke down.

Michel et al. (ref. 93) investigated the crossflowinstability on a swept airfoil model. Surface visualiza-tion studies show the regularly spaced streamwisestreaks and a sawtooth transition pattern found by Poll(ref. 92). Hot-wire probes were used to examine boththe stationary vortex structure and the unsteady wavemotion. Based on their hot-wire studies Michel et al.

5

concluded that the ratio of the spanwise wavelengthto boundary-layer thickness was nearly constant atλ/δ = 4. They also found a small spectral peak near1 kHz, which was attributed to the streamwise insta-bility. Theoretical work included in the paper showedthat the disturbance flow pattern consisted of a layer ofcounterrotating vortices with axes aligned approxi-mately parallel to the local mean flow. But, when themean flow was added to the disturbance pattern thevortices were no longer clearly visible.

Kohama, Ukaku, and Ohta (ref. 94) used hot-wireprobes and smoke to examine the three-dimensionaltransition mechanism on a swept cylinder. A travellingwave disturbance appeared in the final stages of transi-tion that was attributed to an inflectional secondaryinstability of the primary stationary crossflow vorti-ces. The secondary instability consisted of ringlikevortices surrounding the primary vortex. They con-cluded that the high-frequency waves detected by Poll(ref. 92) were actually produced by the secondaryinstability mechanism.

2.4. Detailed Theory and Simulation

Several papers that investigated the developmentand growth of crossflow vortices on swept wings byusing detailed theoretical and simulation techniqueshave recently appeared. Choudhari and Streett(ref. 95) investigated the receptivity of three-dimensional and high-speed boundary layers to sev-eral instability mechanisms. They used both numericaland asymptotic procedures to develop quantitativepredictions of the localized generation of boundary-layer disturbance waves. Both primary and secondaryinstability theories were applied by Fischer andDallmann (refs. 78, 96, and 97) to generate theoreticalresults for comparison with the DLR swept flat-plateexperiments (refs. 98 to 101). They used the Falkner-Skan-Cooke similarity profiles as a model of theundisturbed flow to find that the secondary instabilitymodel yielded good agreement with the experimentalresults, especially the spatial distribution of the root-mean-square velocity fluctuations. Meyer and Kleiser(refs. 102 and 103); Singer, Meyer, and Kleiser(ref. 104); Meyer (ref. 105); and Fischer (ref. 106)used temporal simulations to investigate the nonlinearstages of crossflow vortex growth and the interactionbetween stationary and travelling crossflow vortices.They found generally good agreement between their

numerical solutions and the DLR swept flat-plateexperimental results. A primary stability analysis ofthe nonlinearly distorted, horizontally averaged veloc-ity profiles showed stability characteristics similar tothe undistorted basic flow.

Probably the most relevant computations are thosewhich allow spatial evolution of the flow field espe-cially for the nonlinear interaction problems wherelarge distortions of the mean flow occur. However,these methods require a fixed spanwise periodicity andallow the streamwise pattern to evolve naturally. Thismethod seems to inappropriately eliminate constantwavelength crossflow vortices from computationalconsideration. Spalart (ref. 107) solved the spatialNavier-Stokes equations for the case of sweptHiemenz flow to show the development of both sta-tionary and travelling crossflow vortices with initialinputs consisting of either random noise, single distur-bance waves, or wave packets. He found disturbanceamplification beginning at crossflow Reynolds num-bers of 100 and a smooth nonlinear saturation whenthe vortex strength reached a few percent of the edgevelocity. Also, preliminary evidence of a secondaryinstability was obtained. Reed and Lin (ref. 108) andLin (ref. 109) conducted a direct numerical simulationof the flow over an infinite swept wing similar to thatof the present experiment. Malik and Li (ref. 110) usedboth linear and nonlinear parabolized stability equa-tions (Herbert (ref. 111)) to analyze the sweptHiemenz flow that approximates the flow near theattachment line of a swept wing. Their linear computa-tions agreed with the direct numerical simulations ofSpalart (ref. 107). Malik and Li (ref. 110) showed awall vorticity pattern that they concluded is remark-ably similar to the experimental flow visualizationpatterns seen near a swept-wing leading edge. Thenonlinear growth rate initially agreed with the linearresult, but farther downstream it dropped below thelinear growth rate and oscillated with increasingdownstream distance. When both stationary and trav-elling waves were used as initial conditions, the travel-ling waves were shown to dominate even when thetravelling wave was initially an order of magnitudesmaller than the stationary vortex.

2.5. Stability Experiments

Detailed experimental investigations of the cross-flow instability in three-dimensional boundary layers

6

similar to those on swept wings have been conductedin two ways—with swept flat plates having a chord-wise pressure gradient imposed by an associated windtunnel wall bump or with actual swept wings (or sweptcylinders). Experiments using the flat-plate techniqueinclude Saric and Yeates (ref. 112); the DLR experi-ments of Bippes and coworkers (refs. 98 to 100and 113 to 115) and Kachanov and Tararykin(ref. 116). The swept flat-plate crossflow experimentsoffered the advantage of allowing easy hot-wire probeinvestigation over the flat model surface but sufferedfrom the lack of a properly curved leading edge wherethe boundary-layer crossflow began its development.Arnal and coworkers at ONERA (refs. 84 and 90) andSaric and coworkers (refs. 117 to 120) have conductedexperiments on swept-wing or swept-cylinder models.

Arnal, Coustols, and Juillen (ref. 84) found themean velocity exhibited a wavy pattern along the spandue to the presence of stationary crossflow vortices.The spanwise wavelength of this wavy pattern corre-sponded to the streamwise streaks observed in flowvisualization studies. The crossflow-vortex wave-length increased with downstream distance as somestreaks observed in the flow visualizations coalescewhile others vanish. The ratio of spanwise wavelengthto local boundary-layer thickness remained approxi-mately constant atλ/δ ≈ 4. Low-frequency travellingwaves were observed that reached large amplitudes(±20 percent of the local edge velocity) before transi-tion to turbulence took place. They concluded thatboth stationary and travelling crossflow waves consti-tuted the primary instability of the flow on a sweptwing. Arnal and Juillen (ref. 90) investigated a swept-wing configuration with both negative and positivechordwise pressure gradients. They found that whentransition occured in the accelerated flow region, theircrossflow transition criterion gave good results. In themildly positive pressure gradient regions they foundthat interactions between crossflow vortices andTollmien-Schlichting waves produced a complicatedbreakdown pattern that was not properly characterizedby their crossflow transition criterion.

Saric and Yeates (ref. 112) originated the tech-nique of using contoured wall bumps to force a chord-wise pressure gradient on a separate swept flat plate.This technique sets the foundation for detailed cross-flow instability research that has been repeated byother investigators. They used the naphthalene flow

visualization technique to show a steady crossflowvortex pattern with nearly equally spaced streaksaligned approximately with the inviscid flow direc-tion. The wavelength of these streaks agreed quitewell with the predictions from linear stability theory.Saric and Yeates used straight and slanted hot-wireprobes to measure both streamwise and crossflowvelocity profiles. The probes are moved along themodel span (z direction) at a fixed heighty above themodel surface for a range of locations using two dif-ferent free-stream velocities. Typical results showed asteady vortex structure with vortex spacing half thatpredicted by the linear stability theory and shown bythe surface flow visualization studies. Reed (ref. 7)used her wave-interaction theory to show that theobserved period doubling was apparently due to a res-onance between the dominant vortices predicted bythe linear theory and other vortices of half that wave-length, which were slightly amplified in the farupstream boundary layer. This period doubled patternpersisted for a long distance down the flat plate with-out the subsequent appearance of subharmonics.Unsteady disturbances were observed by Saric andYeates but only in the transition region.

Nitschke-Kowsky (ref. 113) and Nitschke-Kowsky and Bippes (ref. 98) used oil coatings andnaphthalene for flow visualization studies on theswept flat plate. Flow velocities and surface shear dis-turbances were measured with hot-wire and hot-filmprobes. They found a stationary crossflow vortex pat-tern withλ/δ ≈ 4 and travelling waves in a broad fre-quency band. The rms values for the travelling waveswere modulated by the stationary vortex pattern; thismodification indicated disturbance interaction. Thewavelength of the stationary vortices and the frequen-cies of the travelling waves were found to be well pre-dicted by the generalized Orr-Sommerfeld equation.Bippes (ref. 99); Mueller (refs. 100 and 114); Bippesand Mueller (ref. 115); and Bippes, Mueller, andWagner (ref. 115) found that stationary crossflow vor-tices dominated the instability pattern when the free-stream disturbance level was low and that travellingwaves tended to dominate in a high-disturbance envi-ronment. They found that when the swept plate wasmoved laterally in the open-jet wind tunnel flow thestationary vortex pattern remained fixed and movedwith the plate. The most amplified travelling wave fre-quency was observed to differ between wind tunnels.Nonlinear effects were found to dominate although the

7

linear theory adequately predicted the stationary vor-tex wavelengths and the travelling wave frequencyband.

Saric, Dagenhart, and Mousseux (ref. 117) andDagenhart et al. (refs. 118 and 119) used contouredend liners on a 45° swept wing in a closed-return windtunnel to simulate infinite swept-wing flow. Measuredpressure distributions indicated that a good approxi-mation of infinite swept-wing flow was achieved. Thetransition process was believed to be dominated by thecrossflow instability because a favorable (i.e.,negative) pressure gradient existed on the model tox/c = 0.71. Stationary fixed wavelength crossflow vor-tices were observed by flow visualization techniquesat several chord Reynolds numbers. The vortex wave-length, which remained fixed over the entire crossflowinstability region for a given Reynolds number, variedwith Reynolds number approximately as predicted bylinear stability theory but with the predicted wave-lengths about 25 percent larger than those observed.Hot-wire and hot-film measurements indicated travel-ling waves in the frequency range predicted by lineartheory. In addition, higher frequency travelling wavesthat may be harmonics of the primary travelling waveswere observed. Near the transition location a com-plicated flow situation developed with highly dis-torted mean flow and disturbance velocity profiles.Radeztsky et al. (ref. 120) showed that micron-sizedroughness can strongly influence crossflow-dominatedtransition. This effect was confined to roughness nearthe attachment line and was not influenced by sound.They quantify the effects of roughness height anddiameter on transition location.

Kachanov and Tararykin (ref. 116) duplicated theexperiments of Saric and Yeates (ref. 112) with identi-cal swept flat-plate and wall-bump geometries. Theydemonstrated that streamwise slots with alternate suc-tion and blowing could be used to artificially generatestationary crossflow vortices.

2.6. State of Present Knowledge

Few detailed crossflow instability experimentshave been made, yet some significant observationswere made. Both stationary and travelling crossflowwaves were observed. The balance between stationaryand travelling waves was shown to vary with externalenvironmental conditions. Some evidence of nonlinear

developments including disturbance interactions anddisturbance-mode saturation was detected.

Theoretical and computational methods are cur-rently being developed at a rapid pace. Benchmarkexperimental data sets are urgently needed for com-parison with results from these new codes. Manyuncertainties about three-dimensional boundary-layerstability and transition remain to be explained. Sta-tionary crossflow vortices seem to dominate in lowdisturbance environments even though the existingtheories indicate that the travelling waves are morehighly amplified. The stationary vortex flow patternsobserved in different environments are observed tovary. That is, some studies show a fixed stationaryvortex pattern throughout the flow and others show anevolving vortex pattern with vortices occasionallymerging or vanishing. One must determine how toaccurately compute disturbance growth rates and tran-sition locations for engineering applications. Theeffects of compressibility, curvature, nonparallelism,and nonlinearity on disturbance evolution must beproperly accounted for. Three-dimensional flow tran-sition must be compared and contrasted with the situa-tion in two-dimensional mean flow. Information aboutthe transition process is extremely important for thedesign of aircraft ranging from subsonic transports tohypersonic space vehicles. Understanding the instabil-ity mechanisms to be controlled by LFC systems iscentral to their design and optimization.

2.7. Present Experiment

The intent of the present investigation was to iso-late the crossflow instability of the three-dimensionalflow over a 45° swept wing in such a way that it isindependent of the other instabilities. The 45° sweepangle was chosen because the crossflow instability hadmaximum strength at this angle. The wing consisted ofa NASA NLF(2)-0415 airfoil that had its minimumpressure point for its design condition atx/c = 0.71.(See refs. 121 and 122.) The model was tested atangles of attack from−4° to +4°, adjustable in steps of1°. Contoured end liners are used in a closed-return1.37- by 1.37-m wind tunnel test section to simulateinfinite swept-wing flow. When operated atα = −4°,the wing produces a long extent of favorable stream-wise pressure gradient that stabilizes the Tollmien-Schlichting waves while strongly amplifying cross-flow vortices. The streamwise chord of 1.83 m allows

8

the development of a relatively thick boundary layer(≈2 to 4 mm in the measurement region) so thatdetailed velocity profile measurements are possible inthe region of crossflow vortex development. Becausethe wing had a small leading-edge radius and theupper surface had no concave regions, attachment-lineinstability and Görtler vortices were not expected.Thus, this test condition allows the examination of thecrossflow instability in isolation from the other threeinstability modes.

Naphthalene sublimation and liquid-crystal flowvisualization studies were performed at several testconditions to determine both the extent of laminarflow and the stationary vortex wavelengths. Detailedstreamwise velocity profiles were measured with hot-wire anemometers at several spanwise stations acrossa selected vortex track. The evolution of the vortex isanalyzed over this single wavelength and comparedwith theoretical computations. Velocity profiles at thevarious spanwise locations and velocity contoursacross the vortex wavelength for both the mean anddisturbance velocities are presented. Vector plots ofthe theoretical disturbance vortices are shown overlaidon the experimental velocity contour plots. Experi-mental and theoretical growth rates and wavelengthsare compared.

2.8. Organization of Publication

The research philosophy employed for this inves-tigation consists of three steps:

1. Use available computational methods to designthe experiment

2. Conduct the experiment

3. Compare the experimental results with compu-tational predictions

With the exception of the theoretical disturbance pro-files introduced in section 5.7.1, all computations pre-sented were performed by the authors.

The experimental facility is described in section 3.Wind tunnel dimensions and features that produce lowdisturbance flow are discussed along with descriptionsof the instrumentation, hot-wire traverse, and data-acquisition systems. Section 4 gives details of themodel and liner design. Extensive computations

including linear stability analyses are performed forthe highest possible test Reynolds number to ensure,to the extent possible, that the proper parameter rangeis selected for the experiment. The relevant coordinatesystems are introduced in appendix A. The hot-wiredata-acquisition and analysis procedures are outlinedin appendix B. The experimental results are presentedand discussed in section 5. These data include modelpressure distributions, flow visualization photographs,boundary-layer spectra, and detailed hot-wire velocityprofiles and contour plots. Comparisons of the experi-mental results with those from linear stability analysesfor the exact test conditions are also shown. Thesecomparisons require the introduction of computationalresults provided by other researchers. An analysis ofthe experimental measurement errors is discussed inappendix C.

3. Experimental Facility

3.1. Arizona State University Unsteady WindTunnel

The experiments are conducted in the ArizonaState University Unsteady Wind Tunnel (UWT). Thewind tunnel was originally located at the NationalBureau of Standards and was reconstructed at ArizonaState during 1984 to 1988 (ref. 123).

The tunnel is a low-turbulence, closed-returnfacility that is equipped with a 1.4- by 1.4- by 5-m testsection, in which oscillatory flows of air can be gener-ated for the study of unsteady problems in low-speedaerodynamics. It can also be operated as a conven-tional low-turbulence wind tunnel with a steady speedrange of 1 to 36 m/s that is controlled to within0.1 percent. A schematic plan view of the tunnel isshown in figure 3. The facility is powered by a 150-hpvariable-speed DC motor and a single-stage axialblower.

The UWT is actually a major modification of theoriginal NBS facility. A new motor drive with thecapability of continuous speed variation over a 1:20range was purchased. In order to improve the flowquality, the entire length of the facility was extendedby 5 m. On the return leg of the tunnel, the diffuserwas extended to obtain better pressure recovery andto minimize large-scale fluctuations. The leg justupstream of the fan was internally contoured with

9

rigid foam. The contour was shaped to provide asmooth contraction and a smooth square-to-circulartransition at the fan entrance. A large screen wasadded to the old diffuser to prevent flow separationand a nacelle was added to the fan motor. Anotherscreen was added downstream of the diffuser splitterplates. Steel turning vanes with a 50-mm chord,spaced every 40 mm, are placed in each corner of thetunnel.

On the test section leg of the tunnel, the contrac-tion cone was redesigned by using a fifth-degree poly-nomial withL /D = 1.25 and a contraction ratio of 5.33.It was fabricated from 3.2-mm-thick steel sheet. Theprimary duct had seven screens that were uniformlyspaced at 230 mm. The first five screens had an openarea ratio of 0.70 and the last two had an open arearatio of 0.65. This last set of screens was seamless andhad dimensions of 2.74 by 3.66 m with 0.165-mm-diameter stainless steel wire on a 30 wire/inch mesh.Aluminum honeycomb, with a 6.35-mm cell size andL /D of 12, was located upstream of the screens. Thislocation helped to lower the turbulence levels to lessthan 0.02 percent (high pass at 2 Hz) over the entirevelocity range.

Both the test section and the fan housing are com-pletely vibration isolated from the rest of the tunnel bymeans of isolated concrete foundations and flexiblecouplings. The test section is easily removable andeach major project has its own test section.

Static and dynamic pressure measurements aremade with a 1000-torr and a 10-torr temperature-compensated transducers. These are interfaced with14-bit signal conditioners. Real-time data-processingcapabilities are provided by 32-bit wind tunnel com-puters with output via floppy disk, printer, CRTdisplay, and digital plotting. The computers controlboth the experiment and the data acquisition. They arebuilt around a real-time UNIX operating system. Allstatic and instantaneous hot-wire calibrations, mean-flow measurements, proximeter calibration, three-dimensional traverse control, conditional sampling,free-stream turbulence, and boundary-layer distur-bance measurements are interfaced into the data-acquisition system. The facility has a two-dimensionallaser Doppler anemometer system and a low-noisehot-wire anemometer system to measure simulta-neously two velocity components in the neighborhood

of model surfaces. Signal analysis devices include twocomputer-controlled differential filter amplifiers, threedifferential amplifiers, a dual phase-lock amplifier, afunction generator, an eight-channel oscilloscope,a single-channel spectrum analyzer, fourth-orderband-pass filters, and two tracking filters. A three-dimensional traverse system is included in the facility.The x traverse guide rods are mounted exterior to thetest section parallel to the tunnel side walls. A slotted,moveable plastic panel permits the insertion of thehot-wire strut through the tunnel side wall. Thetraverse system has total travel limits of 3700 mm,100 mm, and 300 mm in thex, y, and z directions,respectively, wherex is in the free-stream flow direc-tion, y is normal to the wing chord plane, andz spansthe tunnel. The data-acquisition system automaticallymoves the probe within the boundary layer for each setof measurements after an initial manual alignment.Thex traverse is driven by stepping motors through alead screw with a minimum step size of 286µm. Theyandz traverses are operated by precision lead screws(2.54 mm lead, 1.8 percent per step) which give mini-mum steps of 13µm.

Further details of the wind tunnel, data-acquisitionsystem, and operating conditions of the UWT arediscussed by Saric (ref. 123) and Saric, Takagi, andMousseux (ref. 124).

3.2. New Test Section

A new test section was designed and fabricated forthese experiments in the UWT. Figure 4 shows a pho-tograph of the new test section with the liner underconstruction. It is fully interchangeable with the exist-ing test section. The 45° swept-wing model, whichweighs approximately 500 kg, is supported by a thrustbearing mounted to the floor of the new test section.With the model weight supported on the thrust bear-ing, the two-dimensional model angle of attack can beeasily changed from−4° to +4° in steps of 1°. Con-toured end liners must be fabricated and installedinside the test section for each angle of attack. Oncethe system of model and end liners are installed in thenew test section, the entire unit replaces the existingtest section. This unit allows alternate tests of thecrossflow experiment and other experiments in theUWT without disrupting the attachment and alignmentof the model in the test section.

10

4. Model and Liner Design

Section 4 gives the design procedure for theexperiment. The expected pressure distributions on theselected airfoil in free air and on the swept wing in theUWT including wind tunnel wall-interference effectsare shown. Linear stability analyses for stationary andtravelling crossflow waves and Tollmien-Schlichtingwaves at the maximum chord Reynolds number areperformed. The experimental test condition and a test-section liner shape to simulate infinite swept-wingflow are selected.

4.1. Airfoil Selection

In order to investigate crossflow vortex develop-ment and growth in isolation from other boundary-layer instabilities, it is necessary to design or select anexperimental configuration that strongly amplifies thecrossflow vortices while keeping the other instabilitiessubcritical. The NASA NLF(2)-0415 airfoil (refs. 121and 122) is designed as a low-drag wing for commuteraircraft with unswept wings. It has a relatively smallleading-edge radius and no concave regions on itsupper surface. The NLF(2)-0415 airfoil shape and the-oretical pressure distribution for the design angle ofattack of 0° are shown in figures 5 and 6. The mini-mum pressure point on the upper surface at thiscondition is at 0.71 chord. The decreasing pressurefrom the stagnation point to the minimum pressurepoint is intended to maintain laminar flow on theunswept wing by eliminating the Tollmien-Schlichtinginstability.

4.1.1. Pressure Gradient Effects

As discussed earlier in section 2, positive or nega-tive pressure gradients act to generate boundary-layercrossflow on a swept wing. For the present applicationon a 45° swept wing, the NASA NLF(2)-0415 airfoilfunctions as a nearly ideal crossflow generator whenoperated at a small negative angle of attack. Itsrelatively small leading-edge radius eliminates theattachment-line instability mechanism for the range ofReynolds numbers achievable in the UWT. TheGörtler instability is not present because no concaveregions are on the upper surface. The negative pres-sure gradient on the upper surface keeps the Tollmien-Schlichting instability subcritical tox/c = 0.71 for

angles of attack at or below the design angle of attackof 0°.

Figures 7, 8, 9, and 10 show the NASANLF(2)-0415 airfoil pressure distributions predictedwith the Eppler airfoil code (ref. 125) for angles ofattack of −4°, −2°, 2°, and 4°, respectively. Thesecomputations neglect viscous effects and assume thatthe airfoil is operating in free air; that is, no wind tun-nel wall interference is present. Note that forα = −4°,−2°, and 0°, the minimum pressure point on theupper surface is located at aboutx/c = 0.71. Beyondx/c = 0.71 the pressure recovers gradually at first andthen more strongly to a value somewhat greater thanthe free-stream static pressure (Cp > 0) for all anglesof attack shown in figures 6 to 10. For positive anglesof attack, the minimum pressure point shifts far for-ward tox/c < 0.02. Forα = 2°, the pressure recovery isvery gradual tox/c = 0.30 followed by a slight acceler-ation to a second pressure minimum atx/c = 0.71. Forα = 4°, a relatively strong pressure recovery followsthe pressure minimum and a nearly flat pressureregion is observed over the middle portion of theairfoil.

This shift in the pressure distribution with angle ofattack has important implications for the strength ofthe boundary-layer crossflow generated in the leading-edge region. The strength of the crossflow varies withthe magnitude of the pressure gradient, the extent ofthe pressure gradient region, and the local boundary-layer thickness. The leading-edge crossflow is drivenmost strongly by the strong negative pressure gradi-ents for the positive angles of attack, but because theextent of the negative pressure gradient region is quitesmall and the boundary layer is very thin near the lead-ing edge, very little boundary-layer crossflow is actu-ally generated. Furthermore, for the positive angles ofattack, the positive pressure gradient that follows thepressure minimum overcomes the initial leading-edgecrossflow to drive the crossflow in the opposite direc-tion. This positive pressure gradient also acceleratesthe development of Tollmien-Schlichting waves. Fornegative angles of attack, the negative pressure gradi-ent in the leading-edge region is a somewhat weakercrossflow driver, but the negative pressure gradientregion (0≤ x/c ≤ 0.71) is much larger. Thus, as theangle of attack decreases from 4° to −4°, the leading-edge crossflow increases in strength. This indicatesthat the desired crossflow-dominated test condition

11

should be achieved atα = −4°. Interaction betweenTollmien-Schlichting waves and crossflow vorticesgenerated in the pressure recovery region is possiblefor α = 4°. Quantitative computational results to sup-port these statements are presented in section 4.2.

Figures 6 to 10 show that a considerable range ofpressure distributions is achievable by varying themodel angle of attack. To ensure even more flexibilityin the pressure distributions, the model is alsoequipped with a 20-percent-chord trailing-edge flap.Figures 11 to 14 show typical effects of the20-percent-chord flap for the nominal design angle ofattack of 0° and a range of flap-deflection angles from−20° to 20°. Using this flap-deflection range, the air-foil lift is changed from a negative value forδf = −20°to a large positive value atδf = 20° with correspondingupper surface pressure gradients that vary from mildlynegative to strongly positive. However, these calcula-tions neglect viscous effects, which yield some verystrong positive pressure gradients that are probably notphysically achievable in the wind tunnel. But theyindicate that changes in the angle of attack and flapdeflection angles can be used together to achieve alarge range of pressure gradient conditions on theupper surface.

4.1.2. Wind Tunnel Wall Interference Effects

The large model chord of 1.83 m was selected topermit the examination of the crossflow vortex devel-opment in a relatively thick (2 to 4 mm) boundarylayer. However, wind tunnel wall interference effectsare expected when a 1.83-m chord model is installedin a 1.37-m square test section. To eliminate the influ-ence of the walls on the model pressure distribution,the model could be surrounded by a four-wall test sec-tion liner that follows streamline paths in free air flow.At each end of the swept wing, the liner would have tofollow the curved streamlines as shown in figure 1.The liner would have to bulge on the walls oppositethe airfoil surfaces to accommodate the flow over thewing shape. However, contoured top and bottom wallshapes make visual observation of the model very dif-ficult during testing.

For the present experiment, a two-wall linerdesign was selected. In this approach, the wind tunnelwalls opposite the upper and lower wing surfaces werenot contoured to match the free-air streamlines but

were simply left flat. However, the presence of the flatwalls must be accounted for in the design of the end-liner shapes and in the data interpretation. To accom-plish this, a two-dimensional airfoil code (MCARF)that includes wind tunnel wall effects (ref. 126) wasmodified for 45° swept-wing flow. The influence ofthe flat tunnel walls on the pressure distribution isshown in figures 15, 16, 17, 18, and 19 for angles ofattack of−4°, −2°, 0°, 2°, and 4°, respectively. Theinfluence of the flat walls on the airfoil pressure distri-butions is not negligible, but the qualitative features ofthe pressure distributions remain the same when thewall interference is included. Negative angles ofattack still produce gradual accelerations of the flow tothe minimum pressure point atx/c = 0.71, whereaspositive angles of attack give a rapid drop to minimumpressure near the leading edge followed by pressurerecovery to a nearly constant level in the midchordregion. The required end-liner shapes to achievequasi-infinite swept-wing flow are discussed insection 4.5.

4.2. Stability Calculations

Extensive stability calculations were conductedprior to any experiments in order to determine theappropriate parameter range for this study. Twoboundary-layer stability codes—MARIA (ref. 45) andSALLY (ref. 44) are used to predict the performanceof the experimental configuration to assure (to theextent possible) that the experimental parameter rangecovers the physical phenomena of interest. Both codesuse mean laminar boundary-layer profiles computedwith the Cebeci swept and tapered wing boundary-layer code (ref. 126) with pressure boundary condi-tions such as those shown in figures 15 to 19. TheMARIA code analyzes the stationary crossflow insta-bility subject to the constraint of constant crossflowvortex wavelength. It does not actually solve thecrossflow eigenvalue problem discussed earlier in sec-tion 2.4, but estimates the local spatial growth ratesfrom a range of known solutions to the Orr-Sommerfeld equation for crossflow velocity profiles.On the other hand, the SALLY code can analyze eitherthe crossflow or Tollmien-Schlichting instabilitieswith a variety of constraint conditions. In the SALLYcode, the crossflow instability is not limited to station-ary vortices; travelling crossflow modes are alsopermissible.

12

The boundary-layer stability analysis methods arestrictly eigenvalue solvers that give local disturbancegrowth rates. TheeN method of transition predictionemploys the integrated amplification factors(N-factors) as functions of location on the wing asgiven by

(1)

where the local spatial amplification rateαi indicatesamplification wheneverαi < 0. The values ofαi aredetermined by applying the eigenvalue solver atnumerous locations along a streamline for variousinstability Fourier components. Each Fourier compo-nent is specified by its frequency and the eigenvaluesolution must be constrained by some parameters tomake the integral of equation (1) physically meaning-ful. As mentioned in section 2.4, this constraintparameter is often selected in an apparently arbitraryfashion. In this paper, the fixed wavelength constraintis used for crossflow vortices, whereas the maximumamplification constraint is employed for Tollmien-Schlichting wave calculations. At this point, the con-stant wavelength constraint for crossflow is simply anassumption; however, a full justification for this selec-tion based on the experimental observations are pre-sented later in section 5. Because this investigation isaimed at the examination of crossflow vortices in theabsence of primary Tollmien-Schlichting waves, it iscritical that the strength of the Tollmien-Schlichtinginstability not be underestimated. Hence, we make theselection of the maximum amplification constraint forTollmien-Schlichting waves.

The factorA/A0 represents the amplification fromthe neutral point (x/c)0 to an arbitrary location (x/c)and is obtained as

(2)

for each disturbance component. The maximumN-factor (Nmax) for each wavelength is obtained bycontinuing the integration in equation (1) to the end ofthe amplification range as

(3)

where (x/c)e indicates the end of the amplificationregion. The amplification region may end because ofthe occurrence of a second neutral point. The maxi-mum amplification over the entire crossflow zone(0 ≤ x/c ≤ 0.71) is given by

(4)

or, equivalently, the natural logarithm of the amplifi-cation ratio is given by

(5)

which is, of course, still a function of the disturbancecomponent wavelength.

4.2.1. Stationary Crossflow Vortices

Figures 20 to 34 show predicted stationary cross-flow vortex growth rates, local amplification factors(N-factors), and maximum amplification factors(Nmax) computed with the MARIA code (ref. 45) forthe 45° swept wing installed in UWT at angles ofattack ranging from−4° to +4°. The growth rates arenormalized with respect to the chord. Travelling cross-flow vortices, which are more highly amplified thanstationary vortices, are considered in section 4.2.3.Emphasis is placed on the stationary vortices becausethey arise because of surface roughness effects thatseem likely to dominate on practical wing surfacesoperated in low-disturbance wind tunnel or flight envi-ronments. (See Bippes and Mueller, ref. 101.) Thesecomputations set an upper bound on the stationarycrossflow vortex amplification ratios by assuming thatlaminar flow is maintained to the beginning of thestrong pressure recovery region atx/c = 0.71 for thehighest achievable chord Reynolds number of3.81× 106. Of course, the amplification of crossflowvortices may cause boundary-layer transition beforex/c = 0.71 for this or even lower Reynolds numbers.

The local spatial growth rate is shown for eachangle of attack in figures 20 to 24. The data are plottedfor a range of wavelength for each angle of attack.Note that the distribution of local amplification is con-siderably different for the five cases. Short wavelengthdisturbances are amplified over a fairly narrow range

N x/c( ) αi dxc--–

x/c( )0

x/c

∫=

AA0------ e

N x/c( )=

N x/c( ) αi dxc--–

x/c( )0

x/c( )e∫=

AA0------

max

eNmax=

AA0------

max

ln Nmax=

13

near the leading edge, whereas the amplificationregion for the longer wavelengths begins fartherdownstream and continues to the beginning of thestrong pressure recovery atx/c = 0.71. In all cases, themaximum local amplification occurs in the leading-edge region (x/c < 0.10) and is of similar magnitude.Downstream of the leading-edge region (x/c > 0.10)the amplification rates vary considerably from case tocase. Forα = −4°, the growth rates for intermediatewavelengths level off at a plateau slightly greater thanhalf the initial short wavelength amplification peak.As α increases from−4°, this plateau level decreasesuntil it disappears completely atα = 2°. Forα = 2° and4°, the amplification region divides into two crossflowregions. Atα = 2°, both these crossflow regions areassociated with mean flow accelerations, the first inthe leading-edge region and the second in the slightacceleration region fromx/c = 0.20 to 0.71. (Seefig. 18.) For α = 4°, the two crossflow regions areassociated with pressure gradients of opposite sign(fig. 19) with the mean boundary-layer crossflowgoing in opposite directions. The first region corre-sponds to the leading-edge negative pressure gradient(and inboard crossflow), whereas the other is associ-ated with the relatively strong positive pressure gradi-ent following the early pressure minimum atx/c = 0.02(outboard crossflow). Thus, the crossflow instabilitypattern changes progressively as the angle of attack isincreased fromα = −4°, where fairly strong crossflowamplification continues following the initial crossflowsurge, to a complete reversal of the crossflow directionwhenα = 4°.

Figures 25 to 29 showN-factors obtained byapplying equation (1) for the five angles of attack. Thevalues ofN(x/c) are shown as functions of location onthe wing for various ratios of wavelength to chord.Short wavelength disturbances are shown to beginamplification in the thin boundary layer near the lead-ing edge, reach maximum amplification in the range0.10 <x/c < 0.30, then decay back to initial intensitylevels. Mid and long wavelength vortices begin ampli-fication farther downstream from the leading edge andcontinue to grow to the beginning of the strong pres-sure recovery atx/c = 0.71. Values ofNmax obtainedby continuing the integration of equation (3) over theentire crossflow region (0≤ x/c ≤ 0.71) are displayedin figures 30 to 34 as functions of the wavelength foreach angle of attack. The maximum stationary cross-flow amplification decreases progressively as the

angle of attack is increased fromα = −4° to 2°. TheNmax curves peak at 15, 9.5, 4.4, and 0.5 forα = −4°,−2°, 0°, and 2°, respectively. Forα = 4°, the leading-edge crossflow is negligible and the pressure recoverycrossflow is fairly weak (Nmax = 2.3). These resultsindicate that significant stationary crossflow amplifi-cation should occur forα = −4° and −2°, moderatecrossflow atα = 0°, and only minimal amplificationfor

Previous correlations between computed station-ary crossflow amplification factors and experimentaltransition locations in low disturbance wind tunnelsindicate thatNmax at transition is about 7 (ref. 45).Thus, selecting eitherα = −4° or −2° should ensuresufficient crossflow amplification to cause transi-tion on the wing at the highest Reynolds number,Rc = 3.81× 106. In fact, crossflow-generated transitionshould occur well ahead of the pressure minimum atx/c = 0.71 in the more extreme case (α = −4°) andmove progressively back toward the pressure mini-mum as Reynolds number is decreased.

4.2.2. Tollmien-Schlichting Waves

Significant stationary crossflow vortex amplifica-tion is predicted in section 4.2.1 for the selected con-figuration whenα = −4° or −2°. The experimentalgoal is to examine crossflow vortex amplification andbreakdown in the absence of Tollmien-Schlichtingwaves. Figure 35 shows the maximumN-factors forTS amplification predicted by the SALLY code asfunctions of frequency forα = 0°, 2°, and 4°. Themaximum amplification rate constraint (envelopemethod) is employed for these computations. In thismethod, the wave orientation angle is allowed to varywhile the code searches for the maximum amplifica-tion rate at the selected frequency. Examination of thecomputational results indicates that at least two peaksare possible in the local amplification rate solutions,one nearψ = 0° and the other nearψ = 40°. The irreg-ularity of theN-factor curves in figure 35 is probablycaused by the code switching back and forth betweenthese two possible solutions.

Figure 35 shows large TS amplification forα = 4°,much weaker disturbance growth forα = 2°, minimalamplification atα = 0°, and no amplification for nega-tive angles of attack. The large TS amplification for

α 2°.≥

14

α = 4° is easily anticipated from the pressure distribu-tion shown in figure 19. The relatively strong positivepressure gradient in the region 0.02 <x/c < 0.18strongly excites TS waves. Forα = 2°, much less TSamplification results from the weaker positive pres-sure gradient in the region 0.02 <x/c < 0.10. (Seefig. 18). For α = 0°, the flow accelerates (negativepressure gradient) tox/c = 0.71; as a result, figure 35shows minimal TS amplification. Forα = −4° and−2°,figures 15 and 16 show that fairly strong flow acceler-ations continuing tox/c = 0.71 prevent any TS amplifi-cation. Thus,α = −4° and −2° produce the desiredflow conditions—strong crossflow amplification withno Tollmien-Schlichting wave growth.

4.2.3. Travelling Crossflow Vortices

Travelling crossflow vortices are examined theo-retically forα = 4° at the maximum Reynolds number,Rc = 3.81 × 106, with the SALLY stability codesubject to the constraint of fixed vortex wavelength.Table 1 summarizes the predictedNmax values for arange of frequencies and wavelengths where the localamplification rates are integrated using equation (3)over the entire crossflow region (x/c)0 ≤ x/c ≤ 0.71.The local amplification rates, integratedN-factors, andtotal amplification values for these cases vary in amanner similar to the MARIA code results shown infigures 20, 25, and 30. The frequenciesf investigatedrange from−50 to 500 Hz and include stationary vorti-ces (f = 0) as a subset. The negative frequency wavesmay be physically possible and simply correspond towaves that travel in the direction opposite to the direc-tion of the wave-number vector. The orientation of thewave-number vector is shown in appendix C.

Table 1 shows that the most amplified wavelengthvaries slightly with frequency but in all cases lies inthe range 0.004≤ x/c ≤ 0.006. This slight adjustmentof the maximum-amplification wavelength is probablycaused by local pressure gradient effects and is notconsidered to be particularly significant. The station-ary vortex results are very similar to those obtainedwith the MARIA code. The wavelength having maxi-mum total amplification for both codes isλ/c = 0.004,but the maximumN-factor from the SALLY code islower—Nmax = 13.1 compared with 15.0 from theMARIA code. This difference is not surprising sincethe MARIA code does not actually solve theboundary-layer stability eigenvalue problem but only

estimates the amplification rates from known solu-tions. On the other hand, the maximum predictedN-factor for all cases investigated isN = 17.3 fortravelling crossflow waves withf = 200 Hz andλ/c = 0.005. Thus, the travelling crossflow vortices arepredicted to be considerably more amplified (by thefactor e4.2 = e17.3/e13.1 = 66.7) than the stationarywaves. Of course, the actual vortex strength dependsnot only on the amplification factor but, also, on theexternal disturbance input. That is, the receptivity por-tion of the transition process is equally important inthe vortex development, growth, and eventual break-down. The moving vortices are driven by time-varyingsound and vorticity fluctuations in the free stream,whereas local surface roughness and discontinuitiesare most important for stationary vortices. The balancebetween these two types of disturbance input is criticalto developments in the transition process.

4.2.4. Crossflow–Tollmien-Schlichting Interaction

The goal of the present experiment is to examinecrossflow vortex development and growth in theabsence of Tollmien-Schlichting waves. However, theresults of sections 4.2.1 and 4.2.2 indicate two testconditions where the potential interaction betweencrossflow vortices and TS waves may be fruitfullypursued. The most promising of these conditions is atα = 0° where moderate crossflow amplification andweak TS waves are predicted. The other possibleinteraction condition exists atα = 4° where verystrong TS waves and weak pressure-recovery cross-flow should coexist. These instability estimates areindependent of any such interaction effects themselvesbecause they are computed with linear stabilitymethods.

Figure 32 shows that forα = 0° the maximumamplified stationary crossflow isNmax = 4.6, whereasfigure 35 shows that the TS amplification peaks atNmax = 3. The presence of the moderate strengthcrossflow vortices may sufficiently distort the meanflow velocity profiles so as to produce enhanced TSwave amplification and early breakdown to turbu-lence. If, however, these disturbance intensities areinsufficient to generate mode interaction, the distur-bance intensities can be increased by one of two meth-ods. The simplest way to increase the interaction is toincrease the Reynolds number, which will increase thestrength of both fundamental instabilities. However,

15

this way is probably not possible in the UWT becausethe calculations presented are forU∞ = 35 m/s, whichis near the tunnel speed limit. The other alternative isto boost the disturbance intensities by the selective useof two-dimensional or three-dimensional roughnesselements and sound. This alternative is similar to theuse of vibrating ribbons to introduce disturbances intoflat-plate TS instability experiments.

4.3. Selection of Experimental Test Condition

In sections 4.1 and 4.2, we have discussed the air-foil selection process, wind tunnel wall interferenceeffects, and boundary-layer stability analysis. TheNASA NLF(2)-0415 airfoil is selected as a strongcrossflow generator with minimal TS wave amplifica-tion. The interference effects of installing a large wingmodel in the UWT are found nonnegligible. Theseeffects do not change the basic character of the pres-sure distributions and, therefore, do not change theexpected instability characteristics. The stationarycrossflow instability is found to be strong forα = −4°and−2° but to get progressively weaker as the angle ofattack is increased. Forα = 2°, the crossflow instabil-ity essentially disappears and only a fairly weak pres-sure recovery crossflow region is found forα = 4°.The Tollmien-Schlichting instability is determined tobe very strong atα = 4° and to get progressivelyweaker as the angle of attack is reduced. This instabil-ity is predicted to be totally absent for angles of attackless than zero. Travelling crossflow vortices are exam-ined forα = −4°, where it was shown that the travel-ling waves are more amplified than stationary vorticesby a factor of 66.7. Selecting the test point for thecrossflow-dominated transition experiment is nowappropriate.

The selected test point is atα = −4°. This condi-tion has the strongest crossflow instability and noTollmien-Schlichting wave amplification. This selec-tion allows the isolated examination of crossflow vor-tex development and growth. In addition, with thepredicted crossflow being very strong at this angle ofattack, the Reynolds number can be reduced from themaximum to achieve a range of test conditions wherecrossflow-induced transition is likely. The effect ofReynolds number variation on the crossflow instabil-ity is examined in section 4.4; section 4.5 illustratesthe wind tunnel liner shape required to achieve quasi-infinite swept-wing flow.

4.4. Reynolds Number Variation

Figure 36 shows the effect of decreasing Reynoldsnumber on the strength of the stationary crossflowinstability computed with the MARIA code. The peakof the maximumN-factor curve is seen to decreasefrom Nmax = 15 to 8.5 as the Reynolds number isreduced fromRc = 3.81× 106 to 2.0× 106. The peakN-factor is reduced approximately in proportion to theReynolds number reduction; however, this corre-sponds to a nearly 700 fold reduction in the totalamplification. Thus, a very large range for the cross-flow vortex strength can be achieved simply by vary-ing the test Reynolds number for the selected testcondition ofα = −4°.

4.5. Test Section Liner Shape

The pressure distributions and boundary-layer sta-bility predictions in sections 4.1, 4.2, and 4.3 are com-puted with the assumption that the flow could beapproximated as that on an infinite swept wing (i.e.,no spanwise pressure gradients). The infinite sweptwing produces a three-dimensional boundary layercaused by the combined effects of wing sweep andchordwise pressure gradient, but the boundary-layerprofiles and stability parameters are invariant alonglines of constant chord. This ideal situation is notpossible if a swept wing is installed in a wind tunnelwith flat sides on all four walls. With a large chordmodel installed in a flat-walled wind tunnel, pressure-interference effects will produce a highly three-dimensional pressure pattern and, potentially, a highlythree-dimensional boundary-layer instability and tran-sition pattern. To obtain a flow field that is invariantalong lines of constant chord, one must employ con-toured wind tunnel liners. In the most idealized condi-tion, all four walls of the wind tunnel would becontoured to follow stream surface shapes for an infi-nite swept wing in free air. For the present applicationof a large chord model installed in the UWT, the lessrestrictive approach of contouring only the end linersis adopted. For this approach to be successful, theinterference due to the flat side walls adjacent to theupper and lower wing surfaces must be properly takeninto account. These effects are considered by employ-ing a modified version of the MCARF two-dimensional airfoil code (ref. 127) that includes theeffects of wind tunnel side walls by modeling both thewing and tunnel walls by singularity distributions.

16

Figures 37 to 39 show various contour lines on theend liners designed for the NASA NLF(2)-0415 airfoilwhen operated at an angle of attack of −4° in theUWT. The liner coordinates (xL,yL,zL) are parallel tothe streamwise coordinates (xs,ys,zs) defined in appen-dix A with the origin taken at the liner entrance. Fig-ure 40 shows a schematic diagram of the model andliners installed in the UWT. These lines are computedwith a modified version of a code called TRACES thatwas written by H. Morgan of Langley Research Centerto use output from the MCARF code. The TRACEScode is modified to include a constant velocity compo-nent along the span of the 45° swept wing (i.e., theinfinite swept-wing approximation). Twenty-fivestreamline tracks are computed for the end liners, butfor clarity of presentation, only six are shown infigure 37. The lines are projected in figure 37 onto theXL-ZL tunnel-liner coordinate plane. The model lead-ing edge is located atxL/c = 1.00, which is 1 chorddownstream of the liner origin. The trailing edge ofthe model is located atxL/c = 2.00. The streamlinesshown include lines near each flat side wall(zL0/c = 0.306 and−0.417), lines just above and belowthe wing surface (zL0/c = 0 and 0.028), and lines inter-mediate between the model and the tunnel walls(zL0 = 0.139). Note that the streamlines near the wallsare nearly flat as required by the presence of the flattunnel wall. The other streamlines curve and bulge asthey pass the model location. The approximate modelshape is discernible from the separation of the stream-lines around the model. The negative model angle ofattack is indicated by the downward curve of thestreamlines just ahead of the model leading edge.

Figure 38 shows the lateral deflections of the endliner required to follow the curved streamlines overthe swept wing. Again 25 streamline paths arecomputed, but only 6 are shown for clarity. The linesall begin with an initial deflection of zero at the linerorigin and gradually curve as the model leading edgeis approached. In the neighborhood of the model, thestreamlines curve more sharply as they pass throughregions of strong pressure gradient. Note that thestreamlines nearest to the wing surface (zL0/c = 0 and0.028) had zero lateral deflection at the liner originand are separated at the trailing edge by about 0.02c(38 mm). This offset of the streamlines is due to thelift of the wing that causes the upper and lower surfacestreamlines to deflect different amounts as they passover the model. The total thickness of liner material

can be seen from figure 38 to be just under 0.11c(0.2 m). The liner contours on the two ends of theswept-wing model must be complementary so that apositive deflection on one wall corresponds to a nega-tive deflection on the other wall. To accommodatethese contours in the end liners, the initial liner thick-ness is taken to be 0.127 m on each end. This leavesabout 38 mm of excess material on one end of themodel with slightly less than 25 mm minimum thick-ness on the other end.

Figure 39 shows another view of the liner surfaceshape. Here surface lines in theYL-ZL plane are shownfor various longitudinal positions along the liner. Atthe liner origin (xL/c = 0), the contour is flat and thedeflection is taken to be zero. At the model leadingedge (xL/c = 1.00), the liner is deflected to negativeyLvalues over the upper surface side of the model(zL > 0) and a portion of the lower surface side. Theliner lateral deflection is purely negative for the uppersurface and purely positive for the lower surface of themodel at the midchord position (xL/c = 1.50). Note thatthere is an abrupt jump in the liner contour from theupper to lower surfaces of the model at this location.The jump occurs through the model location itself.This jump or discontinuity continues into the wakeregion (xL > 2.00) due to the lift on the model.

A schematic view of the model and end linersinstalled in the UWT is shown in figure 40. The modelis mounted with the wing chord plane vertical and thecontoured liners located on the floor and ceiling of thetest section. The contraction section of the tunnel isequipped with fairings that go from the existing con-traction contours to an initial liner depth of 0.127 m atthe entrance of the test section. The contractionfairings are each cut from a single large slab ofpolystyrene material. The end liners are manufacturedby laminating 51 mm by 152 mm by 1.22 m (2 in. by6 in. by 4 ft) pieces of polystyrene material into blocksto form the required liner thickness. The surface con-tour is then cut into each laminate block with a heated-wire apparatus. This process results in a faceted shapeto the liners when all the laminate blocks are assem-bled into the complete liner. Figure 4 is a photographof the composite liner during installation in the newUWT test section. To complete the liner constructionthe polystyrene block surface is sanded lightly toremove the facets and the surface is covered with athin layer of heat shrink plastic film.

17

5. Experimental Results and Discussion

The experimental results are presented, analyzed,and compared with predictions from the linear stabil-ity theory in section 5. Appendix B outlines the hot-wire signal interpretation procedure. Measured wingpressure distributions are given. The stationary cross-flow vortex pattern and the transition line are visual-ized with sublimating chemical and shear sensitiveliquid crystal surface coatings. Free-stream andboundary-layer velocity spectra are shown. Velocityprofiles and contour plots are given for the extensivehot-wire measurements taken across a single station-ary crossflow vortex track fromx/c = 0.20 to 0.55 atRc = 2.37× 106 andα = −4°. These data include themean velocity, stationary crossflow disturbance veloc-ity, and narrow-band-pass travelling wave velocitycomponents in the streamwise direction. Theoreticalstationary crossflow disturbance velocity data sup-plied by Fuciarelli and Reed (ref. 128) are presentedand transformed to various coordinate systems forcomparison with the experimental results. Theoreticalvelocity-vector plots are shown overlaid on the experi-mental velocity contours plots. Observed stationarycrossflow vortex wavelengths and growth rates arecompared with theoretical predictions.

5.1. Free-Stream Flow Quality

The UWT is designed to operate as either anunsteady wind tunnel or as a conventional low-turbulence tunnel. The tunnel is equipped with analuminum-honeycomb mesh and seven turbulencedamping screens which limit the free-stream turbu-lence level to less than 0.04 percentU∞ in the low tur-bulence mode. For the present experiment the largechord model and associated end liners add distur-bances that increase the background turbulence levelsomewhat, but it generally remains less than 0.09 per-cent U∞, which is still excellent flow quality for thecrossflow experiments. A typical free-stream velocityspectrum measured with a hot wire forRc = 2.66× 106

is shown in figure 41. Most of the free-stream distur-bance energy is concentrated at low frequencies.Above 10 Hz the energy rolls off with increasing fre-quency to about 100 Hz, where the spectrum dropsbelow the electronic noise.

5.2. Pressure Distributions

Figures 42 and 43 show the measured wing-pressure distributions on the upper surface. These dataare measured for three different free-stream velocities,and the three sets of data are almost indistinguishable.The pressure taps are located in streamwise rows withone row near the top end of the model and the otherrow near the bottom end. The data presented in thesetwo figures are the swept-wing pressure coefficients(Cp,3) that differ from the airfoil pressure coefficients(Cp) given in section 4 by the square of the cosine ofthe sweep angle as

(6)

wherep is the local surface pressure andp∞, ρ∞, andU∞ are the free-stream pressure, density, and velocity,respectively. For the top end of the model, the mea-sured pressure distribution is in general agreementwith the predicted curve, but the theoretical pressuredistribution slightly underestimates the measurementsover the whole model (fig. 42). The underestimate islargest in the region 0.05< x/c < 0.40. Examination ofthe top end liner contour indicates that the liner isslightly thinner near the model leading edge thandesigned, which probably accounts for the underpre-diction of the pressure in this region. At the lower endof the model, the experimental pressure distribution iswell predicted to aboutx/c = 0.25, but the pressureminimum nearx/c = 0.70 is underpredicted (fig. 43).This underprediction may occur because the test sec-tion floor is inclined to offset normal tunnel wallboundary-layer growth on all four tunnel walls. Boththe top and bottom rows of pressure taps are locatedwithin 5 to 15 cm of the liner surfaces; this tunnel wallboundary-layer interference probably exaggerates theinfluence of liner-contour errors as compared with theimpact felt in most of the flow field. Because the mea-sured pressure distributions differ only slightly fromthe predicted distributions, a reasonable approxima-tion of infinite swept-wing flow appears to have beenestablished in the central portion of the test region.The free-stream and boundary-layer hot-wire mea-surements confirm this.

Cp,3

p p∞–

0.5ρ∞U∞2

----------------------- Cp cos2Λ= =

18

5.3. Flow Visualizations

A naphthalene-trichlorotrifluoroethane spray isused to place a white sublimating coating over theblack model surface. The naphthalene sublimes fasterin regions of high shear; this allows the visualizationof the stationary crossflow vortices and clearly indi-cates the transition location. Figures 44 to 48 shownaphthalene visualization photographs forα =−4° andchord Reynolds numbers ranging from 1.93× 106 to3.27× 106. The flow is from left to right in the figureswith fractions of chord indicated by the markings at10-percent-chord intervals. In each figure, the naph-thalene coating is absent over approximately the first15 percent of chord because of the high laminar shearstress in this region. From approximatelyx/c = 0.15 tothe jagged transition line, the stationary crossflow vor-tex pattern is clearly evident. The vortex spacing isdetermined by counting the number of light and darkstreak pairs over a length of 10 cm. The wavelength isobserved to remain constant over the model at eachtest condition. This observation is in agreement withthe findings of Saric and Yeates (ref. 112). In contrastwith the results obtained by Arnal and Juillen (ref. 90),no vortex dropouts or other adjustments to the vortexspacing are observed. The laminar region is termi-nated in each case at a jagged transition line producedby overlapping turbulent wedges. Table 2 shows theaverage transition location and measured vortex spac-ing as a function of the chord Reynolds number. Thetransition location is estimated from the photographsas the average of the beginning and ending locationsof the turbulent wedges. Figure 49 shows a closeupphotograph of a heavy coating of naphthalene forRc = 2.65× 106 where the stationary crossflow vortextraces can be seen to continue into the turbulent wedgeregions.

In addition to naphthalene, liquid crystal coatingsare also used to visualize the crossflow vortex streaksand transition pattern. Figure 50 shows an example ofa green liquid crystal flow visualization photograph.The black and white view shown does not adequatelydemonstrate the patterns that are visible in a colorimage. The stationary crossflow vortices are visible asalternating green and black streaks and the transitionlocation is indicated by an abrupt shift to a deep blue.However, this technique proved less satisfactory thanthe naphthalene visualization. The crossflow streaksand the transition location are less obvious in the

liquid crystal photographs than in the naphthalenevisualizations. Perhaps the relatively low shear stressin the present application limits the utility of the liquidcrystal technique. Similar results were obtained whenthese studies were repeated in a cooperative programwith Reda using his technique (ref. 129).

The repeatability of the transition pattern is inves-tigated by marking the jagged transition line on themodel with a felt-tipped pen following a naphthaleneflow visualization run. The naphthalene visualizationsare repeated at the same Reynolds number after sev-eral days, during successive tunnel entries, and evenafter the screens are removed, cleaned, and reinstalledwith virtually the same transition patterns observed.The visualization is also repeated with liquid crystals,and again, essentially the same transition patterns areobserved. This agreement indicates that the stationaryvortex traces and the transition pattern are dominatedby small-scale surface roughness effects that are notsignificantly influenced by the two different flow visu-alization techniques or the facility condition. Indeed,Bippes and Mueller (ref. 101) find that when theymove their flat-plate model laterally in the open-jettest section the vortex streak and transition patternsremain fixed and move with the plate.

5.4. Transition Locations

Boundary-layer transition locations are deter-mined by several methods including interpretations ofhot-film and hot-wire voltage signals and sublimatingchemical flow visualizations. The transition locationsare determined from the flow visualization photo-graphs by the abrupt shift in sublimation rate of thenaphthalene coating due to turbulence-induced shearstress increases. The sharp change from the streakednaphthalene pattern to black background thus marksthe transition location. The rms voltage responses ofthe hot-film gauges are plotted as functions of theReynolds number. The point on the curve where theslope increases abruptly with increasing Reynoldsnumber is taken as the transition point. For theboundary-layer hot-wire probes, the onset of abruptvoltage spikes in the time-dependent voltage signal istaken as the transition indicator. Thus, all these meth-ods indicate the beginning of the transition processwith the hot wires and hot-film gauges providing localtransition measurements and the flow visualizationgiving a global view of the transition pattern.

19

Figure 51 is a summary plot of transition measure-ments on the swept wing versus chord Reynolds num-ber forα = −4°. A transition band is indicated for thenaphthalene flow visualization results. The beginningof the band indicates the origin of the most forwardturbulent wedge and the end of the band is the locationwhere the wedges merge. Points are shown for hot-wire transition measurements atx/c = 0.40 and for hot-film transition measurements at several locations.Thus, the naphthalene flow visualization technique iscalibrated. For chord Reynolds numbers greater than2.3 × 106, the transition location is observed to beahead of the pressure minimum atx/c = 0.71. For theseReynolds numbers the transition process is presumedto be completely crossflow dominated.

5.5. Boundary-Layer Spectra

Figures 52 to 55 show the rms velocity spectrafor a hot wire located within the boundary layer atx/c = 0.40, as the chord Reynolds number is increasedfrom 2.62× 106 to 3.28× 106. This Reynolds numberrange is selected because transition is expected tooccur in the neighborhood ofx/c = 0.4 as seen fromfigure 51. In figure 52, an amplified-response band isnoted near the blade-passing frequencyf bp. As theReynolds number increases in figures 53 and 54, theresponse band nearf bp broadens and a second, higherfrequency amplified band emerges. For this testcondition,f bp is approximately equal to the maximumamplified crossflow frequencyfcf,max. These frequen-cies fall within the lower frequency amplified band.The higher frequency band corresponds to approxi-mately 2fcf,max. In addition, the blade passing plusstators frequency,fbpsand 2fbp, are in the higher fre-quency band. Figure 55 shows the spectrum forRc = 3.28× 106, which is in the turbulent flow region.Here the spectrum is flattened with similar energy lev-els at all frequencies to 500 Hz. A comparisonbetween the predicted crossflow frequency responseand the measured spectrum is shown in figure 56. Thelower frequency response band corresponds to a por-tion of the predicted moving crossflow vortex amplifi-cation range nearfbp. The higher frequency responseband is located at the extreme upper end of the pre-dicted amplified frequency range where the predictedamplitude rapidly decreases with increasing fre-quency. Bippes and Mueller (ref. 101) observe travel-ling crossflow waves that tend to dominate the flowsin relatively high disturbance tunnel environments.

They find that stationary crossflow vortices dominatein low disturbance tunnels. Saric and Yeates (ref. 112)do not observe travelling crossflow vortices. However,they do observe a stationary wavelength spectrumwith a broad peak near the theoretically predictedmaximum amplified wavelength and an additionalsharp peak at half the predicted wavelength. Reed(ref. 7) is able to explain this development as a para-metric resonance between the primary crossflow vorti-ces that develop relatively far downstream andvortices of half this wavelength, which are slightlyamplified in the thin upstream boundary layer.

5.6. Boundary-Layer Hot-Wire Surveys

5.6.1. Streamwise Velocity Measurements

Constant-temperature hot-wire anemometers areused to make detailed mean streamwise velocityprofile measurements across a single stationary cross-flow vortex for α = −4° and Rc = 2.37 × 106. Themeasurements are made at intervals ofx/c of 0.05from x/c = 0.20 to 0.55 with two hot-wire elements. Asingle hot-wire probe is located inside the boundarylayer, and a second single wire probe is located in thefree stream. Both wires are oriented parallel to themodel surface and perpendicular to the free-streamvelocity vector. The ratio of the velocity indicationsfrom the two anemometers yields the streamwiseboundary-layer velocity ratio. The hot-wire calibrationand data reduction procedure is given in appendix B.The experimental error analysis is given in appen-dix C. Figure 20 shows that the stationary crossflowvortices become unstable atx/c = 0.05, whereasfigure 51 shows that the average transition line lies atapproximatelyx/c = 0.58. Thus, the measurementlocations cover a large portion of the unstable cross-flow region from slightly downstream of the first neu-tral point to just ahead of the transition location.

A high-shear vortex track (i.e., dark streak) on themodel is marked with a soft felt-tipped pen followinga sublimating chemical flow visualization study. Thebeginning point of the track is arbitrarily chosen as themidspan location forx/c = 0.20. For most locations,the measurements are made at seven spanwise loca-tions across the vortex along lines parallel to the lead-ing edge of the 45° swept wing. These seven profilesrepresent six steps across the vortex with the first andseventh profiles expected to be essentially the same.

20

The spanwise measurement locations are separated byintervals of∆s = 1.6 mm. The spanwise step size isdictated by the step size in the downstream and lateraldirections and the desire to make an integral numberof steps across the vortex; this can be demonstrated asfollows. Figure 47 shows that the stationary vortextracks lie at an angle of approximatelyθv = 5° withrespect to the free-stream direction. Then, lines paral-lel to the leading edge cut across the vortex tracks atθM = Λ − θv ≈ 40°, whereΛ is the wing sweep angle.Now, the wavelength measured parallel to the leadingedge can be obtained asλ45 = λ/cosθM. Table 2 indi-cates that forRc = 2.37× 106 the ratio of wavelengthto chordλ/c is approximately 0.004 orλ = 7.32 mm.Thus,λ45 = 9.5 mm.

Both the mean and fluctuating velocity compo-nents are measured simultaneously by separating theanemometer output signals into DC and AC compo-nents. The AC component is quite small and can notbe measured accurately in its raw state. It is measuredby blocking the DC component of the signal, amplify-ing the remaining fluctuating signal, narrow-band-passfiltering at f = 100 Hz, and amplifying again before thecomputer analog to digital (A/D) converter measuresthe signal. The amplifier gains are then divided out toobtain the final fluctuating signal values. The selectedcentral frequency off = 100 Hz is chosen because it isnear the maximum amplified frequency for travellingcrossflow vortices as indicated both by computationsand experimental hot-wire spectra.

The motion of the hot-wire probe inside theboundary layer is controlled by the data-acquisitioncomputer after the initial position is set by hand. Thisinitial alignment is accomplished by locating the hot-wire probe above the intersection of the marked vortextrack and the local fractional chord line. The startingpoint for each boundary-layer survey station is set inthis fashion. During the data-acquisition procedure,the experimenter has to actively observe the hot-wireAC signals on an oscilloscope, adjust the amplifiergain settings to assure maximum signal strength withoverranging the instruments, and stop the traversemechanism before the probe collides with the modelsurface. The data-acquisition computer measures theanemometer voltages and moves to the next point onlyafter acceptance of the data by the experimenter.

The velocity data are presented in figures 57 to120 in two forms—velocity profiles at several span-

wise stations across the crossflow vortex and velocitycontours over the 45° spanwise measurement cuts.Figures 57 to 64 show the actual velocity ratiosobtained from the hot-wire data reduction procedureoutlined in appendix B. The height above the wingsurface is determined by extrapolating the velocitydata to a zero value at the surface for each profile. Thepresence of the stationary crossflow vortex is indi-cated in figures 65 to 72 by subtracting the averagevalue of the streamwise velocity from the local profilevalues. The resultant disturbance velocity profilesshow the excess or deficit of velocity produced by thestationary vortex. An alternate representation of thestationary crossflow vortex disturbance intensity isgiven in figures 73 to 80 where reference-computedvelocity profiles are subtracted from the local velocitymeasurements. In both representations, the velocitydifference is made nondimensional by dividing by thelocal boundary-layer edge velocity magnitude. Thus,the plots represent local disturbance intensity values,but because the boundary-layer edge velocityincreases slightly fromx/c = 0.20 to 0.55, the velocityratios are scaled down by a small amount asx/cincreases. The root-mean-square velocity profiles fortravelling waves off = 100 Hz are given in figures 81to 88. The velocity values are again nondimensional-ized by the local boundary-layer edge velocity. Con-tour plots of the mean streamwise velocity across thevortex are shown in figures 89 to 96. Stationary vortexvelocity contours are plotted in figures 97 to 104 forUs,avg removed and in figures 105 to 112 forUs,refremoved. Figures 113 to 120 show rms intensity con-tours forf = 100 Hz.

5.6.2. Spanwise Variation of Streamwise Velocity

Figure 57 shows that the mean velocity pro-files across the vortex are very similar in shapeat x/c = 0.20, but there is already some variation infullness of the profiles due the presence of the station-ary crossflow vortex. As the stationary vortex growsin strength in the downstream direction, the variationin the velocity profiles across the vortex increases. Atx/c = 0.35 (fig. 60), some profiles have developed dis-tinct inflectional shapes, whereas other profiles remainrather full. Figure 64 shows that atx/c = 0.55 (only ashort distance ahead of the breakdown region) all sixvelocity profiles have taken on a distorted inflectionalshape and several profiles are severely distorted intoS-shaped profiles. These highly distorted streamwise

21

velocity profiles are expected to respond very differ-ently to streamwise or secondary instabilities than theundisturbed profiles.

As noted earlier, for each fractional chord loca-tion, the first velocity profile measurement is centeredon the dark (high-shear) vortex track marked with thefelt-tipped pen. In examining figures 57 to 64, it is evi-dent that for the minimum measurement height the ini-tial profile is very near the smallest velocity ratio andthus the highest surface shear at each station except atx/c = 0.20, 0.50, and 0.55. At these stations, the maxi-mum shear location appears to be approximately 2 or3 mm away from the initial profile location. Earlier theflow visualization patterns were noted to be repeatablefor matching Reynolds numbers. However, lateralshifts of the whole vortex pattern by a small fractionof a wavelength are not unexpected. The deviationof the initial profile from the maximum shear locationat x/c = 0.20, 0.50, and 0.55 could be due to a smallshift in the vortex pattern or to a small lateral mis-alignment (≈2 to 3 mm) of the traverse rig at thebeginning of the measurements. The influence of thismisalignment at the beginning of the measurementregion is evident in the results that follow.

5.6.3. Disturbance Profiles

Figures 65 to 72 show the stationary crossflowdisturbance velocity profiles determined by subtract-ing the average streamwise velocity ratio at a givenheight from the measured velocity ratio at each loca-tion. The abscissa scale is chosen to keep the localvelocity scales essentially the same for each chordwisestation. As noted previously, the local boundary-layeredge velocity, which is used as a reference value,increases slightly fromx/c = 0.20 to 0.55 so that thedisturbance velocity ratios are scaled down slightlywith this nondimensionalization asx/c is increased.For the crossflow instability, the disturbance vortexaxes are nearly streamwise and the primary distur-bance vortex components in a streamwise coordinatesystem are the velocity componentsv andw. The per-turbation in the streamwise direction (velocity compo-nent u) is a secondary effect due to the convectionarising from the velocity componentsv andw. How-ever, the streamwise component (component of theboundary-layer velocity has a large gradient in thedirection perpendicular to the wing surface sothat when combined with small convective velocity

componentsv and w, it produces a large secondarystreamwise velocity perturbation. This streamwisevelocity perturbation is the one shown in figures 65to 72 and later in figures 73 to 80. Over a single vortexwavelength, these perturbation velocity profiles areexpected to exhibit either excesses or deficits from themean, depending on the lateral location in the stream-wise vortex. Over that portion of the vortex wherethe velocity componentsy and w convect the highmomentum flow from the outer portion of the bound-ary layer toward the surface, the local profile shouldhave an excess (or bulging) shape. On the other hand,when the vortex velocity components convect lowmomentum flow away from the surface, the local pro-file is expected to have a deficit shape.

Note that in figures 65 and 67 only five velocityprofiles are measured. Nonetheless, it is apparent infigure 65 that some of the local disturbance profilesshow excess velocities, whereas others show deficitvelocities as expected. However, the profiles infigure 65 have two unexpected features. First, the dis-turbance profiles do not approach 0 for large values ofthe heighty above the wing surface; this is due to tem-perature drift effects in the UWT. The UWT has notemperature control; the test temperature is governedby the ambient temperature in the wind tunnel build-ing (which is cooled by an electrical air-conditioningunit) and, more importantly, by the power input to thewind-tunnel fan. To eliminate this effect for the othermeasurement stations, the tunnel was run in a preheatmode for 30 to 45 min before acquiring boundary-layer disturbance data. This preheat time is used eachday to verify instrumentation connections, filter set-tings, and so forth. The second anomalous aspect ofthe profiles in figure 65 is the bulge in excess and def-icit velocities belowy ≈ 0.5 mm. These bulges areunexpected and are almost as large as the maximumdisturbance intensities found fory ≈ 1.2 mm. Thepresence of these velocity perturbations is traced toresidue left behind by cleaning the model with alcoholand supposedly lint-free cloths. This contaminationhad not been noted earlier during the preliminaryvelocity profile measurements probably because sin-gle velocity profile measurements were generallymade following a flow visualization study in whichthe model surface was effectively cleaned by thetrichlorotrifluoroethane solvent used with the naphtha-lene. The model cleaning procedure was modified to atwo-step procedure—cleaning first with alcohol and

u)

du/dy( )

22

then with distilled water. A single velocity profilemeasurement atx/c = 0.20 confirmed that the residueproblem was solved, but the complete set of velocityprofiles atx/c = 0.20 were not measured again due tolack of sufficient time. The data at all subsequent mea-surement stations are taken following the two-stepmodel cleaning process and no further contaminationproblems are encountered.

Examination of figures 65 to 72 shows that themaximum disturbance intensity of the stationarycrossflow vortex grows progressively fromx/c = 0.20to 0.55. Atx/c = 0.20, the profiles show either excessor deficit velocities only. But byx/c = 0.35, some ofthe disturbance profiles have taken on definite cross-over shapes. These crossover profiles have bothexcess and deficit velocity regions. At the last mea-surement station (x/c = 0.55) the maximum distur-bance intensity exceeds 20 percent of the edgevelocity and all the profiles have taken on highlydistorted shapes. The nature and significance ofthese crossover profiles are discussed further insection 5.7.3 where the disturbance velocities are dis-played as contour plots.

Figures 73 to 80 show stationary crossflow distur-bance profiles obtained by a different procedure. Herelocal theoretical velocities are subtracted from themeasured profiles to yield the local disturbance vortexintensities. Note that the angle of attack for the refer-ence case is taken asα = −5° rather than the actualangle of attack,α = −4°. This adjustment in the theo-retical angle of attack is required because the theoreti-cal profiles forα = −4° are obviously fuller than theexperimentally measured profiles. The reason for thisdiscrepancy is uncertain, but it may arise from a slightflow angularity in the UWT test section or a minormisalignment of the model mounting bearing.

The stationary crossflow profiles in figures 73to 80 agree in general trends with those shown infigures 65 to 72. For both sets of figures, themaximum disturbance intensity grows progressivelywith increasingx/c, definite crossover profiles developby x/c = 0.35, and all profiles are highly distorted atx/c = 0.55. However, there are some slight differencesbetween removing the average velocity profile and thetheoretical velocity profile from the measured data.These differences arise because the averaged profilesfrom the experiment include flow history effects

produced by the presence of the stationary crossflowvortices within the boundary layer, whereas the theo-retical profiles completely neglect this effect. Themost notable of the differences in the two sets ofprofiles is observed by comparing figures 65 and 73for x/c = 0.20. In figure 73, the influence of the surfacecontamination discussed earlier produces velocity def-icits in all profiles fory < 0.5 mm. In figure 65, thisdeficit effect is included in the averaged profile, and asa result, the disturbance profiles are not biased towarda deficit condition. Of course, this deficit effect is anexperimental error which would have been removedcompletely by retaking the data atx/c = 0.20 if timehad allowed. Other notable, and experimentally moresignificant, differences are observed forx/c > 0.45where flow history effects become more pronounced.This effect is shown more clearly in the disturbance-velocity contour plots.

Root-mean-square velocity profiles for travellingwaves are shown in figures 81 to 88. As mentionedpreviously, these data are measured simultaneouslywith the mean velocity by splitting the hot-wireanemometer signal into mean and fluctuatingcomponents. The fluctuating component is amplified,narrow-band-pass filtered, and amplified again beforerecording with the UWT A/D converter system. Theselected central frequency for the narrow-band-passfilter is f = 100 Hz, which is near the frequency ofmaximum amplification according to both experimen-tal and theoretical considerations. Again, these dataare plotted with an abscissa scale that is essentiallyunchanged over the range of measurement locations;this allows for easy visual examination of the distur-bance amplification with increasingx/c. For travellingcrossflow waves, rms-averaged profiles of the stream-wise velocity are expected to yield profiles with a sin-gle maximum and, of course, only positive values.Furthermore, in the absence of nonlinear distortionscaused by the stationary crossflow vortices or the pres-ence of some other travelling waves in the same fre-quency range, the rms velocity profiles are expected tobe identical at each spanwise location.

For the first two measurement stations (x/c = 0.20and 0.25, figs. 81 and 82), the disturbance intensitiesare quite small and the velocity profiles have essen-tially the same shape at all spanwise locations acrossthe stationary crossflow vortex. Here the rms stream-wise velocity profiles have a single lobe (or maxi-mum) as expected for travelling crossflow vortices.

23

By x/c = 0.30, the shapes of the travelling wave veloc-ity profiles have begun to distort, but the profiles con-tinue to have single-lobed shapes. This distortion ofthe profile shape may arise from the development ofother travelling wave disturbance modes of the samefrequency but different direction of travel or the non-linear distortion of travelling crossflow waves by thestrong stationary crossflow vortex layer. Forx/c > 0.40(figs. 85 to 88), definite double-lobed travelling wavedisturbance velocity profiles are apparent at somespanwise locations across the stationary crossflowvortex. Betweenx/c = 0.40 and 0.50, the travellingwaves grow considerably in strength. However, fromx/c = 0.50 to 0.55, the largest amplitudes decrease bygreater than a factor of 2. Note that even at their maxi-mum intensity, the travelling waves are quite small ascompared with the strength of the stationary crossflowvortex (e.g., only 3.5 percent as large). Thus, itappears that the travelling waves which initially havevery low amplitude, grow and distort considerably for0.40 <x/c < 0.50, and then decay forx/c > 0.50. Thenature of the distortion of the travelling waves due tothe stationary crossflow vortices is more apparentwhen viewed as contour plots in section 5.6.4.

5.6.4. Streamwise Velocity Contour Plots

Contour plots of the mean velocity, stationarycrossflow disturbance intensities, and the travellingwave disturbances are given in figures 89 to 120.These plots show the various experimentallydetermined quantities plotted on a grid which is 4 mmdeep in they (surface normal) direction and extends9.5 mm along a 45° swept line parallel to the wingleading edge. As mentioned previously, the span-wise coordinate cuts across the stationary crossflowvortex tracks at approximately a 40° angle and thestationary vortex wavelength along a line parallel tothe leading edge isλ45 = 9.5 mm. The abscissa istaken aszλ = −(zm −zm,0)/λ45 and is a local coordinatewith zλ = 0 on the marked stationary vortex track andzλ > 0 in the direction of spanwise motion of the hot-wire probe. With this coordinate selection, theabscissa actually runs in the direction opposite to themodel spanwise coordinatezm. The plots show the sit-uation an observer would see when looking upstreamfrom the hot-wire probe location. These data are plot-ted for 0.20≤ x/c ≤ 0.55. Forx/c = 0.20 and 0.30, thedata are not measured across the full crossflow vortexwavelength; but, for the other stations, these data

are shown for a full stationary crossflow vortexwavelength. For each boundary-layer station, allvelocities are made nondimensional by dividing bythe local streamwise boundary-layer edge velocity.Because the edge velocity increases by about 10 per-cent fromx/c = 0.20 to 0.55, the actual velocities arescaled down by this factor.

Figures 89 to 96 show the mean streamwiseboundary-layer velocity ratio in the tunnelcoordinate frame. In the absence of stationary cross-flow disturbance vortices, the velocity contours areexpected to be flat and parallel to the wing surface.Figure 89 shows that the mean velocity contoursat x/c = 0.20 are nearly flat and parallel to the wingsurface. But some influence of the stationary cross-flow vortex is already present at this forward locationwith the contour levels somewhat wavy and inclinedslightly toward the surface for increasing values ofzλ.The waviness of the contours increases withx/c untilthe contours obviously bulge upward at approximatelythe middle of the wavelength forx/c = 0.35 (fig. 92).This upward bulge of the contours corresponds to low-momentum fluid being swept upward from the wingsurface by the stationary crossflow vortex. The bulgecontinues to grow asx/c increases until the contoursactually begin to roll over like a breaking wavefor x/c = 0.50 (figs. 95 and 96).

This mean velocity contour pattern is consistentwith expectations for boundary-layer flow withembedded stationary crossflow vortices. The flowvisualization photographs (figs. 44 to 48) show thatthe crossflow vortex axes are aligned almost parallelto the free-stream velocity vector. The instability pro-duces a layer of counterrotating disturbance vorticesthat combine with the mean boundary-layer crossflowto yield a layer of crossflow vortices all with the samerotational direction (corotating pattern). This patterndevelops because the flow is most unstable to thecrossflow instability at some small angle to the purecrossflow direction (ref. 45). The mean-velocity pro-file in the most unstable crossflow direction has acrossover shape with flow streaming in the crossflowdirection near the wing surface but in the oppositedirection farther out from the surface (ref. 8). Whenthis crossover velocity profile is combined with thecounterrotating disturbance vortices, it reinforces thestrength of one pair of vortices while cancelling theother. This produces the observed velocity field with

us/us,e

24

flow streaming in the crossflow direction near thewing (zλ direction) while the flow in the outer part ofthe boundary layer flows in the opposite direction.Hence, the breaking-wave pattern seen in figures 95and 96 is caused.

As previously noted, the initial velocity profile ateach measurement station is intended to be on the lineof maximum shear as determined by the flow visual-ization study. But because the measurements are madeover many days of wind tunnel testing, it is not unex-pected that small shifts (≈2 to 3 mm) in the location ofthe stationary vortex pattern occur. Such shifts in thevortex pattern can be deduced from the mean stream-wise velocity contour plots (figs. 89 to 96). The loca-tion of maximum surface shear stress is determinedqualitatively by observing the grouping of streamwisevelocity contours near the wing surface. Figures 95and 96 show that the maximum shear stress pointappears to have shifted by 1.5 to 3 mm in thezλ direc-tion. This shift is even more obvious in the stationaryvortex velocity field.

Stationary crossflow vortex velocity contours areplotted in figures 97 to 104 and 105 to112. The firstset of contours corresponds to disturbance velocitiesdetermined by removing the averaged mean velocityprofile from the local measured velocities. The secondset of disturbance velocity contours are computed bysubtracting the theoretical mean velocity values fromthe measured velocities. For each set, the disturbancevelocity values are nondimensionalized by the localboundary-layer edge velocity that increases slightlyfrom x/c = 0.20 to 0.55. The range of contour levels upto ±0.20 are used for all measurement locationsfor easy data comparisons. The two sets of contoursagree in general shape and levels of the velocity con-tours. Some relatively small differences can be notedfor x/c = 0.50 and 0.55, where the averaged meanvelocity profile is distorted by the presence of the sta-tionary crossflow vortex, but the theoretical profilesignore this effect. Thus, the following discussion ofthe evolution of the stationary crossflow disturbancevelocity contours applies equally well to either set offigures.

The expected stationary crossflow disturbancevelocity pattern has excess velocities at the extremesof the plotting field (zλ = 0 and 1) and deficit velocitiesnear the middle of the field (zλ = 0.5). This is because

the initial velocity profile (zλ = 0) is measured at thesupposed maximum shear point, which should corre-spond to high momentum fluid being swept toward thewing surface. This situation should, of course, recur1 full wavelength away (zλ = 1). Velocity deficitsshould occur where the stationary crossflow vortexsweeps flow away from the surface (zλ = 0.5).Figures 97 and 105 show that atx/c = 0.20 the pres-ence of the stationary crossflow vortices is alreadydetectable with velocity variations exceeding±0.20 . The expected velocity pattern is not evidentfor x/c = 0.20 or 0.25 (figs. 97 and 98 and 105 and106), but it emerges forx/c = 0.30 (figs. 99 and 107).The periodicity of the velocity perturbations is clearlyevident for 0.35≤ x/c ≤ 0.55 (figs. 100 to 104 and 108to 112). Forx/c = 0.35, 0.40, and 0.45 the excessvelocities occur at the ends of the measurementzone and the deficits in the middle as expected. Forx/c = 0.50 and 0.55, the peak excess velocities areshifted in the +zλ direction by approximately 2 mm.This shift seems to correspond to and is consistentwith the observed shift in the maximum shear stresslocation as discussed previously.

The stationary crossflow disturbance velocities arequite small at the beginning of the measurementregion (x/c = 0.20). The disturbances grow progres-sively larger with increasingx/c to x/c = 0.50 untilthey exceed±0.20 Fromx/c = 0.50 to 0.55, thedeficit velocities continue to increase in intensity, butthe velocity excesses drop sharply. This decrease inexcess velocity intensity seems surprising because thelinear stability analysis presented earlier (fig. 20)shows that the stationary crossflow disturbancevortices should be amplified all the way to the pres-sure minimum atx/c = 0.71. For the present case(Rc = 2.37× 106), the average transition line is deter-mined by the flow visualization studies to be atapproximatelyx/c = 0.58. (See fig. 51 and table 2.) Itmay be that betweenx/c = 0.50 and 0.55 energy isbeing extracted from the stationary crossflow vorticesand pumped into some other disturbance mode such asthe secondary instability mode.

Figures 113 to 120 show contour plots of thetemporal rms velocities of travelling waves withf = 100 Hz forx/c = 0.20 to 0.55. These velocities areagain made nondimensional by dividing by the localboundary-layer edge velocity. It was indicated earlierthat f = 100 Hz corresponds to a peak in both the

us,e

us,e

us,e.

25

measured and theoretical boundary-layer velocityspectra. The rms values measured are quite small withthe maximum values being approximately 0.7 percent

The same contour levels are used for all the plotsso that the disturbance levels can be readily compared.In the absence of complicating factors such as nonlin-ear interaction with stationary crossflow vortices orthe presence of other travelling waves in the same fre-quency range, the rms velocity contours are expectedto be flat and parallel to the wing surface.

The travelling wave intensities are essentially 0for 0.20< x/c > 0.30 (figs. 113 to 115). Forx/c = 0.35and 0.40, the peak disturbance amplitudes range fromabout 0.07 to 0.1 percent (figs. 116 and 117) andthe contour lines are very roughly parallel to the wingsurface, as anticipated for undistorted travelling cross-flow waves. The disturbance velocity profiles shownearlier in figures 83 and 84 for these locations alsohave the expected single-lobed shapes. Byx/c = 0.40(fig. 118) definite closed-contour shapes havedeveloped. This corresponds to the development ofdouble-lobed disturbance profiles (fig. 86). Betweenx/c = 0.45 and 0.50 the travelling wave disturbancestrength grows dramatically with the maximum rmsintensity reaching 0.7 percent The disturbanceintensity also departs strongly from the ideal of equaldistribution along the span to peak sharply near thecenter of the vortex wavelength. Perhaps significantly,figure 63 shows that near the middle of the measure-ment zone the streamwise velocity profile takes on adistinctly distorted S-shape. Fromx/c = 0.50 to 0.55,the strength of the travelling waves drops precipi-tously (fig. 120) and the maximum contour levels shiftlocation. The sharp drop in travelling wave intensityoccurs even though the streamwise mean velocity pro-files continue to develop ever more distorted S-shapedprofiles (fig. 64). This development is in the sameregion where the stationary crossflow vortices areobserved to decrease in strength although the decreaseis not as pronounced in the stationary vortex case. Asmentioned previously, in the region fromx/c = 0.50to 0.55 energy may be transferred to other high-frequency modes, which lead to laminar flow break-down in the neighborhood ofx/c = 0.58. The nonuni-form nature of the rms disturbances along the spanseems to indicate that the travelling waves detectedmay not be travelling crossflow vortices but someother travelling waves in the same frequency range(possibly Tollmien-Schlichting waves generated at thelocations of the S-shaped mean velocity profiles).

5.7. Experimental and TheoreticalComparisons

In section 5.6, experimental velocity profiles andcontours are shown along a single vortex track on the45° swept wing forα = −4° at Rc = 2.37× 106. Bothmean and disturbance velocities extracted from themean data are given. In this section, those experimen-tal data and other results obtained from them are com-pared with linear stability theory predictions suppliedby Reed using her theoretical code (ref. 128). Thistheoretical code is used because it gives both growthrates and disturbance eigenfunction profiles, whereasthe MARIA (ref. 45) and SALLY (ref. 44) codesemployed earlier give only growth rates. The meanvelocity profiles supplied to Reed and shown in sec-tion 5.7.1 were computed with the method of Kaupsand Cebeci (ref. 126). As mentioned previously, theo-retical data forα = −5° are used for this comparisonbecause these data seem to yield a better match to theexperimental data which are measured at a nominalangle of attack of−4°. A small flow angularity in theUWT test section or a slight misalignment of themodel could account for this difference.

5.7.1. Theoretical Disturbance Profiles

Figures 121 and 122 show the mean velocity pro-files at the experimental measurement stations com-puted with the method of Kaups and Cebeci (ref. 126).The velocity components are given in a model-oriented coordinate system (xm,ym,zm) with xm perpen-dicular to the wing leading edge,ym normal to thewing chord plane, andzm parallel to the wing leadingedge. (See appendix A.) Note that the spanwise veloc-ities are taken to be negative because aleft-handed coordinate system was used so that thecrossflow wave numbers are both positive. The exper-imental data presented in section 5.6 are shown fromthe perspective of the hot-wire measurement probelooking upstream. This constitutes essentially a con-version of the experimental coordinate frame into aleft-handed system. Thus, the experimental and theo-retical data can be compared directly. Stationarycrossflow instability eigenfunctions are shown infigures 123 to 125 for each of the coordinate direc-tions. The computations are for a fixed wavelengthof λ = 7 mm, which essentially matches the experi-mentally observed wavelength ofλ/c = 0.004 orλ = 7.3 mm. The profiles are scaled to match the

us,e.

us,e

us,e.

wm/ut,e

26

experimentally determined maximum streamwisedisturbance amplitudes at each measurement station.This scaling is permissible, of course, since linear sta-bility theory predicts the actual disturbance intensityonly to within a multiplicative constant. The phaserelationships between the velocity components are notshown in the figures, although they are criticallyimportant to the determination of the spatial velocityfield of the instability waves. Note also that in themodel-oriented coordinate frame, the disturbancevelocities in the chordwise directionum and the span-wise directionwm are of the same order, whereas thenormal velocity componentvm is an order of magni-tude smaller.

For comparison with the experimental data, thevelocity profiles given in figures 121 to 125 must berotated about theY-axis to two other coordinate refer-ence frames. One of these frames, the streamwiseframe (xs,ys,zs) is oriented withxs parallel to the free-stream velocity vector andys perpendicular to thewing chord plane, whereas in the other frame, thewave-oriented frame (xw,yw,zw), xw is along the vortexaxis andzw is parallel to the wave-number vector. Ofcourse,ym, ys, andyw are all parallel. The experimen-tal measurements are made in the streamwise coordi-nate frame (us,ys,zs). The theoretical vortex flowpattern in the wave-oriented frame (xw,yw,zw) issuperposed onto the experimental data plots. The rela-tionships between these coordinate frames are given inappendix A.

The mean flow velocity components in the(xs,ys,zs) frame are shown in figures 126 and 127. Inthis coordinate frame (appendix A), the cross-streammean velocities are considerably smaller than thestreamwise velocities Likewise the cross-streamdisturbance velocities are much smaller than thestreamwise-disturbance velocities (See figs. 128and 129.) Figures 130 to 133 show the mean and dis-turbance velocity components in the wave-orientedcoordinate frame. Here both the mean and disturbancevelocity components in thezw direction are an order ofmagnitude smaller than the respective velocities com-ponents along the vortex axis. In this frame,ww is thesame order of magnitude asvw = vm. Thus, in both thewave-oriented and streamwise-oriented coordinateframes the velocities along theX-axes are much largerthan the velocity components in the other twodirections.

The nature of the disturbance vortex flow is illus-trated in figures 134 to 136. A vector plot of (vw,ww),across a single vortex wavelength is shown infigure 134 in the wave-oriented coordinate frame. Thevectors are the projections of the disturbance velocityvectors onto theyw-zw plane. The disturbance is seento consist of a pair of counterrotating vortices within asingle wavelength. The vortex cells are skewed so thata central counterclockwise rotating vortex is borderedon each side by a portion of the alternate clockwiserotating vortices. In figure 135 the mean plus distur-bance velocity vectors, are plotted overa single wavelength. Here the mean normal velocity

which is quite small has been neglected. Note thatthe mean velocity (fig. 131) completely dominatesthe vector field masking the presence of any distur-bance vorticity. The presence of the disturbance vor-ticity can be illustrated by arbitrarily scaling thevelocities by a factor of 100. This is shown in figure136 where it is apparent that the mean plus disturbanceflows combine to produce a single counterclockwiserotating vortex per wavelength. That is, the total flowconsisting of disturbances superposed on a base flowcontains a layer of corotating vortices.

5.7.2. Disturbance Profile Comparisons

Experimental stationary crossflow disturbanceprofiles (from ) are presented with thelinear-theory eigenfunction magnitudes in figures 137to 144. In each case the streamwise disturbance veloc-ity profiles are shown. Similar results are found byusing the experimental profiles determined from

(figs. 73 to 80) but these are not shown.The experimental profiles are determined by takingthe spatial rms of the individual profiles (figs. 65to 72) across the stationary vortex. This procedure isthe spatial analog for a stationary wave of taking thetemporal rms of a travelling wave. All profiles areplotted on the same abscissa scale (made nondimen-sional by reference to the local boundary-layer edgevelocity) for easy visual comparison of the disturbancegrowth with distance along the wing. Because the lin-ear stability theory gives the disturbance velocitiesonly to within a multiplicative constant, the theoreticaleigenfunctions are scaled to match the maximumexperimental disturbance intensities. Note that the the-oretical eigenfunctions have only a single lobe. (See,for example, fig. 128.)

wsus.ws

us.

vw,ww+ww( )

vwww

vw

us us,avg–

us us,avg–

27

At x/c = 0.20 (fig. 137), the theoretical and experi-mental profiles are of similar shape in the region of themaximum amplitude neary = 1 mm, but the twocurves diverge in the near-surface region and in theouter flow. As mentioned previously, the measuredprofiles at this location are thought to contain experi-mental errors that are rectified for the remainingmeasurements. The near-surface results are affectedby a lint-contaminated surface and the outer flow mea-surements are affected by tunnel-temperature drift. Forx/c = 0.25, 0.30, 0.35, and 0.40 (figs. 138 to 141), theexperimental and theoretical profiles are of similarsingle-lobed shapes. However, the point of maximumdisturbance intensity is slightly higher in the boundarylayer for the theoretical eigenfunctions than for theexperimental profiles.

For x/c = 0.45 and beyond (figs. 142 to 144), theexperimental profiles take on double-lobed shapes thatcontrast with the single-lobed theoretical eigenfunc-tions. The point of maximum disturbance strength forthe theoretical profiles lies between the two maxima ofthe experimental profiles. Recall from the earlier dis-cussion that forx/c = 0.45 and beyond, the local exper-imental disturbance profiles take on crossover shapesthat are not anticipated from the linear theory. This isevident in the local profile plots of figures 70 to 72 and78 to 80 as well as the disturbance velocity contourplots of figures 102 to 104 and 110 to 112. Figures 94to 96 show that the mean streamwise-velocity con-tours for 0.60 < < 0.90 rise sharply from themodel surface and begin to roll over. This rollover isdue to the presence of the stationary crossflow vortexand becomes evident in section 5.7.3.

Thus, there is general agreement between theshapes of the experimental disturbance velocityprofiles and the theoretical eigenfunctions up to aboutx/c = 0.40 where the rms intensity of the stationaryvortices is about 7 percent of But forx/c > 0.45,the presence of the stationary crossflow vorticesdistorts the experimental disturbance profiles intodouble-lobed shapes not predicted by the linear the-ory. This does not necessarily mean that nonlineareffects are present, even though the stationary distur-bance intensities (±20 percent atx/c = 0.50) arewell beyond the small perturbation limits assumed inthe linear theory. In fact, the observed effects maysimply be because of flow history. That is, the strongstationary crossflow vortices continually lift low-speed fluid up from the surface and push high-speed

fluid downward so that the flow wraps around the vor-tex axis. More is shown on this point in section 5.7.3.

5.7.3. Velocity Contour Plots and Vector Plots

Figures 145 to 152 show theoretical velocityvectors superposed on the experimental streamwisevelocity-contour plots. Theoretical velocity vectors aresuperposed on the stationary crossflow disturbancevelocity contour plots in figures 153 to 160. Thevelocity vectors in these figures havevw scaled up by afactor of 100 (as in fig. 136) so as to illustrate the pres-ence of the stationary crossflow vortex.

Recall that the experimental procedure outlined insection 5.6 called for the experimental profile mea-surements to be made at various spanwise locationsacross a single stationary crossflow vortex. To accom-plish this a single dark vortex track is traced on themodel with a felt-tipped pen following a flow visual-ization run. For each fractional chord location, the hot-wire probe is manually centered above this trace. Afterthe initial manual setup, the traverse motion is com-puter controlled in both normal and spanwise steps.The dark vortex track in the flow visualization studycorresponds to high-shear path under the stationarycrossflow vortex pattern. Thus, this procedure shouldassure that the measurement locations move in stepsfrom a high-shear region through minimum shear andback to high shear again. However, the experimentalsituation is not quite this simple. The entire flow visu-alization pattern is found to be highly repeatable evenmonths apart. The pattern repeats in detail down to thejagged transition line and the individual vortex tracks.But, the vortex wavelength is onlyλ = 7.3 mm so thatsmall errors in the manual alignment of the traversesystem or even very small shifts in the location of thevortex track can impact the relative location of themaximum shear. Examination of the streamwise-velocity plots in figures 89 to 96 shows that the maxi-mum shear point (judged by how closely the velocitycontours are bunched) is not always located atzλ = 0.To account for this effect, the maximum shear point inboth the experimental and theoretical flow patterns isdetermined. The phase of the theoretical flow patternis then shifted to align the maximum shear points inthe theoretical and experimental flows.

Examination of figures 145 to 152 shows that thevariations of the mean streamwise-velocity contours

us/us,e

us.

us

28

over the vortex wavelength can be anticipated fromthe vector plots. The streamwise-velocity contoursspread out when the velocity vectors are directed awayfrom the surface and they crowd together wheneverthe velocity vectors point toward the surface. In partic-ular, near the surface, the streamwise-velocity con-tours approach each other to produce the high surfaceshear (i.e., largedU/dy) when the velocity vectorsare directed downward. Low surface shear (i.e., smalldU/dy) results when the velocity vectors are directedupward and the contour lines spread out.

Figures 153 to 160 show that the qualitative fea-tures of the streamwise-disturbance velocity contourplots can also be anticipated from the velocity vectorfield. A plume of low-speed fluid is observed inregions where the velocity vectors are directed sharplyoutward from the model surface. Concentrations ofhigh-speed fluid near the model surface are found inregions where the velocity vectors are directed sharplytoward the surface. Furthermore, both the low- andhigh-speed regions are skewed in a counterclockwisepattern consistent with the theoretical velocity vectorpattern. Note that flow history effects are not expectedto produce qualitative differences between the contourand vector plots as found in the disturbance profile andeigenfunction comparisons.

Thus, when the maximum shear points of theexperimental and theoretical data are matched, thequalitative features of the flow variables are consistentwith expectations gleaned from the velocity vectorfield. In particular, both the streamwise-velocity con-tours and the stationary crossflow disturbance velocitycontours distort in patterns consistent with the pres-ence of a single counterclockwise rotating vortex. Thispattern of qualitative agreement between the theoreti-cal and experimental flow fields persists throughoutthe measurement region fromx/c = 0.20 to 0.55 incontrast to the disturbance profile and eigenfunctioncomparisons that diverge forx/c > 0.45.

5.7.4. Wavelength Comparison

Tables 3 to 5 show the results of crossflowstability calculations performed by using the SALLYcode (ref. 44) subject to the constraint of constant vor-tex wavelength. As mentioned previously, the naph-thalene flow visualization photographs show constantcrossflow vortex wavelengths over the entire regionfor a given Reynolds number. The calculations are

begun at the neutral point and continued to the averagetransition location as indicated in table 2. Tables 3 to 5correspond to chord Reynolds numbers of 2.37× 106,2.73× 106, and 3.73× 106, respectively. For all threetest conditions the most amplified frequency is non-zero. The maximum amplified frequency increaseswith Reynolds number fromfcf,max = 100 Hz atRc = 2.37 × 106 to fcf,max = 300 Hz at the maximumchord Reynolds number. The maximumN-factor attransition is found to be about 9.1 at the lowerReynolds number and about 8.5 for the higherReynolds numbers. These results agree with earliercalibrations of the crossflow stability problem as indi-cated by Dagenhart (ref. 45). Surface and streamlinecurvature effects have not been considered in thisanalysis, but this may not be significant since both thesurface and streamline curvatures are small over mostof the unstable flow region. The wavelength of maxi-mum stationary crossflow vortex amplification isplotted in figure 161 where it is compared with theexperimental observations given in table 2. The exper-imental and theoretical curves have similar trends withwavelength decreasing as chord Reynolds numberincreases, but the theoretically predicted wavelengthsare approximately 25 percent larger than thoseobserved experimentally. This discrepancy may arisebecause the crossflow vortex pattern (having a con-stant wavelength over the entire wing) is establishedwell forward on the wing where the boundary layer isrelatively thin. Swept flat-plate experiments generallyhave shown closer agreement between the theoreti-cally predicted wavelength and the observed wave-length than the predicted disturbance as opposed toswept-wing studies. Perhaps the blunter nose of theswept wing is an important factor in establishing thesmaller wavelength.

5.7.5. Growth-Rate Comparison

The stationary crossflow vortex growth rate isestimated by numerically differentiating the amplitudedata shown in section 5.7.2. There are several possiblechoices for the disturbance amplitude function such as

(7)

(8)

A1 x( ) us,max x( )=

A2 x( ) 1ymax----------- us x,y( ) yd

0

ymax

∫=

29

or

(9)

The simplest choice is given in equation (7) wherethe amplitude function is taken as the maximum of thestreamwise velocity disturbance profiles as shown infigures 137 to 144. The second choice given inequation (8) is to use the average of the streamwisedisturbance velocity over the thickness of the bound-ary layer to represent the disturbance amplitude. Athird possibility is to use the rms value of the distur-bance profile as in equation (9). Then the growth rate(made dimensionless by referring to the chord length)is computed as

(10)

wherei is 1, 2, or 3. If values from the smooth theoret-ical eigenfunctions shown in figures 137 to 144 aresubstituted in equations (7) to (9) the resulting growthrates are essentially the same irrespective of the choiceof the amplitude function.

Figure 162 shows the various growth-rate esti-mates obtained from the experimental disturbance pro-files and theoretical predictions from the MARIA code(ref. 45) and from Reed’s computations (ref. 128). Theexperimental growth rates are computed from both theprofiles shown in figures 137 to 144 and from similardata determined from The theoreticalgrowth rates peak ahead of the first measurement sta-tion atx/c = 0.20 and decrease approximately linearlyover the measurement zone from 0.20≤ x/c ≤ 0.55,with the two codes predicting slightly different values.In contrast, the several experimental growth-ratecurves have a distinct up and down pattern over themeasurement range and the experimental growth ratesare all at or below the level of the theoretical esti-mates. This may be because of nonlinear saturation ofthe stationary crossflow vortices. The several experi-mental growth-rate curves differ considerably at eachmeasurement station. The variations in growth rateestimated with the various amplitude functions appearto be a measure of the roughness of the experimentalprofiles since the smooth theoretical profiles yield

essentially the same growth-rate estimate, no matterwhich amplitude function is employed.

6. Conclusions

An experimental configuration is designed andconstructed to permit the examination of a wholerange of problems associated with the development,growth, and breakdown of crossflow vortices in aswept-wing flow. Careful control of the model andwind tunnel geometries creates a benchmark experi-mental setup for the study of swept-wing flows. Therange of problems that can be addressed with thisexperimental configuration include the investigationof crossflow vortex growth and development in acrossflow-dominated flow, the interaction of cross-flow vortices with Tollmien-Schlichting waves,surface-roughness effects on crossflow disturbancereceptivity, and crossflow vortex breakdownmechanisms.

In the present investigation, we focus largely onthe first of these possible research problems. In partic-ular, a small negative angle of attack is selected so thatthe resulting favorable (i.e., negative) pressure gradi-ent eliminates primary Tollmien-Schlichting waveswhile strongly amplifying the crossflow vortices. Thebulk of the measurements taken at a chord ReynoldsnumberRc of 2.37× 106 consists of extensive hot-wireprobe surveys across a single stationary vortex track.Both steady and narrow-band-pass travelling wavedisturbance velocities are determined in steps acrossthe vortex track at fractional chord locationsx/c rang-ing from just downstream of the neutral stability pointto just ahead of the transition location. The data arepresented as local velocity profile plots and as isolinecontour plots across the stationary vortex. The experi-mental results are compared with theoretical eigen-function shapes, growth rates, and vector velocityplots.

The following conclusions are drawn:

1. Transition locations are determined by usingsurface-mounted hot-film gauges, boundary-layer hot-wire probes, and flow visualization in the range fromx/c = 0.80 at the minimum test chord Reynolds num-ber Rc = 1.932× 106 to x/c = 0.30 at the maximumchord Reynolds numberRc = 3.271× 106. The localReynolds number at transition varies across the range

A3 x( ) 1ymax----------- us x,y( )[ ]2

yd0

ymax

∫=

σi1Ai-----

dAi

d x/c( )----------------=

us us,ref.–

30

from 1.14× 106 to 1.54× 106, which indicates thatsome roughness effects may be important.

2. The maximum theoretical crossflowN-factorsfor travelling crossflow vortices at transition rangefrom 8.5 to 9.1 in agreement with previous calibra-tions of the linear stability method. However, the cor-responding N-factors for the dominant stationarycrossflow vortices are in the range from 6.4 to 6.8.

3. The boundary-layer hot-wire spectra areobserved to contain mostly low-frequency oscillationsat the lower test Reynolds numbers. With increasingReynolds number, two bands of amplified frequenciesare observed. The first of these bands is near theblade-pass frequency and within the range of ampli-fied travelling crossflow waves predicted by the lineartheory. The second amplified-frequency band falls atapproximately twice the blade-pass frequency and atthe upper frequency limit of the band of amplifiedtravelling crossflow waves. The travelling waves inthe first frequency band are thought not to betravelling crossflow waves, but perhaps Tollmien-Schlichting waves generated locally in the highlydistorted mean flow.

4. The measured mean velocity profiles showslight variations across the stationary vortex trackeven at the first measurement station atx/c = 0.20. Thevariations across the vortex grow with downstreamdistance until distinct S-shaped profiles are observednear the middle of the measurement span atx/c = 0.45.By x/c = 0.55, the measured profiles all the way acrossthe stationary vortex have taken on highly distortedS-shapes. The mean streamwise-velocity contours areshown to be approximately flat and parallel to themodel surface atx/c = 0.20, but byx/c = 0.50 to 0.55the velocity contours in the outer portion of the bound-ary layer actually begin to roll over under the continu-ing action of the stationary crossflow vortex.

5. The local stationary vortex disturbance profileshave single-lobed shapes with either purely excess ordeficit velocities at the forward measurement stationsas expected from theoretical considerations. But, for

the local stationary disturbance profilestake on distinct crossover shapes not predicted by lin-ear theory. The maximum stationary vortex distur-bance intensities reach levels of 20 percent of the localboundary-layer edge velocity just before transition.

The stationary crossflow vortex disturbances have lit-tle influence on the velocity contour pattern at the for-ward measurement stations, but byx/c = 0.30 a distinctpattern forms with a plume of low-velocity fluid risingfrom the model surface near the middle of the mea-surement span and concentrations of high-velocityfluid near the wing surface at the ends of the measure-ment span. Forx/c = 0.50 and 0.55, the excess and def-icit velocities reach maximum intensities of 20 percentof the local boundary-layer edge velocity, but theestablished flow pattern is shifted approximately onefourth of the wavelength toward the wing root. Thisshift is thought to be caused by either a slight mis-alignment of the traverse mechanism or a small shift inwhole stationary crossflow vortex pattern.

6. The travelling wave rms profiles at the forwardlocations have single-lobed shapes as expected fromlinear theory, but develop double-lobed shapes for

which are not predicted by the linear the-ory. The travelling wave rms disturbance intensitypeaks at 0.7 percent of the local boundary-layer edgevelocity which is more than an order of magnitudesmaller than the strength of the stationary crossflowvortex. The travelling wave disturbances are found tobe very weak with no significant pattern evident untilx/c = 0.45 where closed-contour isolines appear. Theseclosed-contour isolines differ from the flat contoursexpected from linear stability theory. The travellingwave disturbance intensity peaks strongly near themiddle of the measurement span atx/c = 0.50 and thenabruptly decreases. The travelling wave disturbanceenergy may be transferred to some other instabilitymechanism as the transition location atx/c = 0.58 isapproached.

7. The experimental streamwise disturbancevelocity functions are found to have single-lobedshapes very similar to those predicted by linear stabil-ity theory for 0.20≤ x/c ≤ 0.40. The maxima of thetheoretical eigenfunctions are located slightly higherin the boundary layer than are the experimentalmaxima. For the experimental disturbancefunctions take on double-lobed shapes. The theoreticaleigenfunction maximum is located at a height betweenthe two experimental maxima. The root-mean-squaredisturbance strength at the breakpoint between thesingle- and double-lobed experimental profiles isabout 7 percent of the local boundary-layer edgevelocity.

x/c 0.45,≥

x/c 0.45,≥

x/c 0.45≥

31

8. Qualitative agreement with the experimentallyobserved flow features is obtained throughout themeasurement range when theoretical velocity vectorplots (from linear stability theory) are superposed ontothe experimental contour plots.

9. A fixed wavelength stationary crossflow vortexpattern is observed for all flow visualization condi-tions. No vortex dropouts or other adjustments to thevortex spacing are observed in the flow visualizationregion which extends from approximatelyx/c = 0.15to 0.80.

10. The wavelengths observed in the flow visual-ization studies are found to be approximately 20 per-cent smaller than the wavelengths predicted by lineartheory. This is probably because of the fact that thefixed stationary vortex wavelength is established wellforward on the model where the boundary layer is stillrelatively thin. Perhaps the swept-wing nose radius isan important factor in establishing the smaller vortexwavelengths since swept flat-plate experiments gener-ally have closer agreement between theoretical andobserved wavelengths.

11. Three different measures of the experimentalgrowth rate are found to yield similar trends which dif-

fer from the theoretically predicted growth rate. Non-linear saturation of the vortex strength appears to haveoccurred. The measured growth rates are found to beat or below the values predicted by linear theory. Also,the experimental growth rate alternately increases anddecreases over the measurement range, whereas thelinear theory predicts an approximately linear decreasewith downstream distance over the measurementspace.

The present investigation contributes to animproved understanding of the physics of the cross-flow instability in a swept-wing flow. The stationarycrossflow vortices which are highly sensitive to small-scale surface roughness effects dominate the distur-bance flow field and the transition process eventhough travelling waves are more amplified accordingto the linear stability theory. The features of theobserved flow field evolve from qualitative agreementwith expectations from the linear stability theory forthe forward measurement stations to highly distortedprofiles with marked differences between the observa-tions and the theoretical predictions. A benchmarkexperimental data set for the crossflow instability isgenerated for comparison with results from advancedcomputational codes currently under development.

32

Appendix A

Relationships Between CoordinateSystems

Figure A1 shows a swept wing in a right-handedCartesian coordinate system (xm,ym,zm), wherexm istaken perpendicular to the wing leading edge,ym isperpendicular to the wing chord plane, andzm is paral-lel to the wing leading edge. A positive wing sweepangleΛ is shown and the flow is from left to right. Theboundary-layer edge velocity is given by

(A1)

where and for attached flow.The angle of the boundary-layer edge velocity withrespect to theZm-axis is obtained as

(A2)

and 0≤ ε ≤ π/2. The total wave number is given by

(A3)

whereαr is the wave number in thexm direction andβris the wave number in thezm direction. The waveangle of the disturbance is then

(A4)

whereθ > π/2 for crossflow disturbances. And finally,the wave orientation angle with respect to the localboundary-layer edge velocity is obtained as

(A5)

The model-oriented coordinates just described areobtained by rotation about theYs-axis in the stream-wise coordinate system (xs,ys,zs) by the wing sweepangleΛ. Herexs is parallel to the free-stream velocityvector. The relationship between these two coordinateframes is given as

(A6)

or, inverting, as

(A7)

And, the relationship between the wave-oriented coor-dinate system (xw,yw,zw) and the model coordinates isobtained as a rotation by the angleθ about theYm-axisas

(A8)

Reed’s left-handed coordinate system is shown infigure A2 where theZ-axes are all directed in theopposite directions from those in the right-handed sys-tems used in figure A1. Equations (A1) to (A8) stillapply, but all the rotations are taken in the oppositedirection. In particular, the wing sweep angleΛ is nownegative. Also, as a consequence of this shift

and for attached flow. Theangle of the boundary-layer edge velocity vector withrespect to theZm-axisε is now greater thanπ/2 and thecrossflow wave orientation angleθ is less thanπ/2.

Ut,e um,e2

wm,e2

+=

um y( ) 0≥ wm y( ) 0≥

ε tan1– um,e

wm,e-----------

=

αT αr2 βr

2+=

θ tan1– αr

βr------

=

ψ θ ε–=

zm

xm Λcos Λsin

Λsin– Λcos

zs

xs

=

zs

xs Λcos Λsin–

Λsin Λcos

zm

xm

=

zw

xw Λcos Λsin

Λsin– Λcos

zm

xm

=

wm y( ) 0≤ um y( ) 0≥

33

Figure A1. Coordinate system relationships for swept wing.

U∞

Λ

Xm

ZmZs

Zw

α

Xw

θcf

Xs

Figure A2. Left-handed coordinate systems for swept wing.

U∞

Xm

Zs

Zw, α

Xw

θcf

Xs

–Λ

Zm

34

Appendix B

Hot-Wire Signal InterpretationProcedure

The free-stream and boundary-layer velocitymeasurements are performed by using Dantec 55M01constant-temperature anemometers equipped with55M10 CTA standard bridges with bridge resistanceratios of 1:20. The hot wires are Dantec type 55P15miniature boundary-layer probes having 5-mplatinum-plated tungsten wires which are 1.25 mm inlength. The probe tines are 8 mm long and are offset3 mm from the probe axis. Standard 4-mm-diameterprobe supports are used. The three-dimensionaltraverse system (described in section 3) is used to sup-port and move the probes through the flow field. Thetraverse system is mounted external to the test sectionwith only the probe-support sting extending through asliding opening in the test section wall. The sting con-sists of a composite element and an aluminum strut.The composite element is 5 mm thick, 0.425 m long,and its chord tapers from 64 mm at the base to 50 mmat the tip. The aluminum strut dimensions are 13 mmby 76 mm by 0.324 m. Both the steady state and fluc-tuating hot-wire signals are sampled simultaneouslywith the 16-channel MASSCOMP 12-bit A/D con-verter which can sample at an aggregate rate of up to1 MHz. The fluctuating voltage signal is narrow-band-pass filtered using a Spectral Dynamics SD122equipped with a 4-pole Butterworth tracking filterwith 10-Hz passband.

The voltage response of a constant-temperaturehot-wire anemometer can be assumed to have the form

(B1)

whereρ is the ambient air density,U is the velocity,T0is the total temperature, andE is the anemometer volt-age response. Differentiating equation (B1) gives

(B2)

Thus, a small voltage change is dependent on smallchanges in the density, velocity, and total temperature.

To reduce the complexity of the functional rela-tionship given in equation (B1) can be accomplishedby eliminating or at least minimizing the variations inρ and T0 so that the anemometer response dependssolely on the velocity. The UWT has no heatexchanger system to maintain a desired tunnel temper-ature. The tunnel total temperature increases with testtime until an equilibrium condition is achieved. Forthe present experiment, the tunnel flow is preheated byoperating the tunnel at the expected test condition for45 to 60 min before hot-wire probe calibration. Thisprovides sufficient time for the flow temperature toreach its equilibrium value. The air density depends ontwo factors—atmospheric pressure and flow tempera-ture. To minimize atmospheric pressure effects, thehot-wire calibrations are conducted before each data-acquisition run. These steps ensure thatdρ anddT0 arenearly zero and can be neglected in equation (B2).Then, equation (B1) can be simplified to

(B3)

The hot-wire probes are calibrated in the UWTflow by varying the free-stream velocity in stepsacross the range of velocities expected during theexperiment. Typically 12 velocities are used for eachcalibration. Equation (B3) is not actually used forprobe calibration; instead,

(B4)

is used whereg(E) is a fourth-order least-squarescurve fit to the calibration data. Then,f (U) is deter-mined as

(B5)

Differentiating equation (B3) gives

(B6)

Now we assume that

(B7)

and

(B8)

E F ρ U T0, ,( )=

dEdFdρ------- dρ dF

dU------- dU

dFdT0---------- dT0+ +=

E f U( )=

U g E( )=

E f U( ) g1–

E( )= =

dEdfdU------- dU f ′ dU= =

e E e′+=

u U u′+=

35

wheree is made up of a steady (or DC) voltageE and asmall fluctuating voltage andu consists of a steadyvelocity U and a small fluctuating velocity Substi-tuting equations (B7) and (B8) into equation (B3) andexpanding in a Taylor series while neglecting higherorder terms (since they are assumed to be small) give

(B9)

Subtracting equation (B3) from equation (B9) yields

(B10)

or solving for gives

(B11)

Since and are small deviations from thesteady values ofU andE, we can apply equation (B11)not just at a single point in time but for and

as functions of time while holding con-stant and then take the root-mean-square of these func-tions to get

(B12)

which gives the rms velocity fluctuations as a functionof the measured rms voltage output from the hot-wireanemometer circuit.

For boundary-layer velocity profile measure-ments, we desire the ratio of local velocity to theboundary-layer edge velocity. Two hot-wire probesare used for this measurement—one probe located inthe boundary layer and the other in the external flow.Both probes are mounted on the traverse strut andmoved together as the boundary-layer velocity profileis measured. The probe in the external flow is notlocated at the edge of the boundary layer but is, in fact,located approximately 15 cm from the boundary-layerprobe. During the traverse, the two hot-wire probesmove only about 4 mm. Over this distance the externalflow probe detects only negligible variations in thevelocity, but the boundary-layer probe sees the veloc-ity decrease from the edge value to near zero as thesurface is approached. The boundary-layer velocityratio cannot be obtained directly as the ratioU2/U1becauseU1 is not at the boundary-layer edge. How-ever, the desired velocity ratio is given by

(B13)

whereU/Us,e is the boundary-layer velocity ratio,U1is the external flow velocity, andU2 is the boundary-layer velocity. By scaling the measured velocity ratioU2/U1 in equation (B13) by the velocity ratio observedat the maximum distance from the surface we normal-ize the profile to unity at the boundary-layer edge.This accounts for the fact that the external-flow hot-wire probe is not at the boundary-layer edge.

e′,u′.

E e′+ f U u′+( ) U( ) f ′ U( )u′+= =

e f ′ U( )u′=

u′

u′ e′f ′ U( )---------------=

u′ e′

u′ t( )e′ t( ) f ′ U( )

u′rms

e′rms

f ′ U( )---------------=

U2

Us,e----------

U2/U1

U2/U1( )y,max

----------------------------------=

36

Appendix C

Error Analysis

Kline and McClintock (ref. 130) discuss theeffects of experimental measurement errors oncomputed data in various experimental situations.They discuss both single- and multiple-sample experi-ments, but their primary emphasis is on describinguncertainties in single-sample experiments. Formultiple-sample experiments, statistical methods canbe used to establish both the mean values and varia-tions from the mean. However, in single-sampleexperiments, errors in the results computed fromexperimentally measured quantities can only be esti-mated. Kline and McClintock showed that the uncer-tainty ∆R for the computed result

(C1)

can be obtained as

(C2)

wherevi represents the measured quantities used in thecomputation ofR andwi represents the expected errorranges for the measured quantities.

In the present experiment, the range of measuredquantities is limited to static and dynamic pressures,pressure differentials, flow temperature, and hot-wireanemometer voltages. From these measured quantitiesthe free-stream velocity, the surface pressure coeffi-cients, boundary-layer and edge velocities, and, mostimportantly, the boundary-layer velocity ratios aredetermined. The free-stream velocity can be obtainedfrom the incompressible Bernoulli equation and theperfect-gas equation of state as

(C3)

where is the free-stream velocity and the mea-sured quantities are the dynamic pressure thestatic pressure and the static temperature and

R is the gas constant from the equation of state. Thesurface pressure coefficient is given by

(C4)

where is the pressure coefficient andp is the localsurface pressure. The boundary-layer and edge veloci-ties are obtained from the hot-wire calibrationfunctions

(C5)

whereas

is the inverse of the hot-wire calibration function.

Equation (C2) can be applied to equations (C3) to(C5) to obtain uncertainty estimates for andU as

(C6)

(C7)

(C8)

Equations (C6) and (C7) can be straightforwardlyapplied because estimates of the uncertaintiesinvolved are easily obtained. However, equation (C8)is much more difficult to apply since an estimate of theuncertainty in the hot-wire anemometer voltage ismuch more difficult to ascertain. This difficulty can beovercome by recognizing that in the present experi-ment some of the hot-wire measurements can be con-sidered as multiple-sample measurements, whereasother measurements must be regarded as single-sample measurements.

R R v1 v2 v3 … vn, , , ,( )=

∆RR∂vi∂

------- wi 2

i=1

n

∑=

U∞2q∞RT∞

p∞----------------------=

U∞q∞,

p∞, T∞,

Cp

p p∞–

q∞----------------

pD

q∞-------= =

Cp

U g E( )=

f U( ) g1–

E( )=

U∞, Cp,

∆U∞U∞

------------wq∞

2q∞----------

2 wT∞

2T∞----------

2 wp∞

2p∞----------

2

+ +=

∆Cp

Cp-----------

wpD

2pD----------

2 wq∞

2q∞----------

2

+=

∆UU

--------wE

U df /dU( )--------------------------

2

=

37

The most important hot-wire measurementsinvolve determining the boundary-layer velocity ratioas

(C9)

where is the velocity indicated by the hot-wireprobe inside the boundary layer and is the velocityindicated in the outer flow. The quantities and

must be regarded as single-sample measure-ments even though and are evaluated as timeaverages of repeated measurements taken at a fre-quencyfs of 1 kHz over a 30-sec interval. On the otherhand, and can be regarded asmultiple-sample measurements and analyzed statisti-cally since these two variables are measured repeat-edly during a hot-wire survey of the boundary layer.

According to the instrument handbook theuncertainty in the measurement ofq∞ andp∞ in equa-tion (C6) is 0.08 percent of reading, butq∞ is observedto oscillate due to a very low-frequency modulation ofthe fan controller at about 1 percent of reading. Thus,the expected uncertainties forq∞ andp∞ are taken as

and

Also, the thermocouple is found to be in error by

Substituting these uncertainties into equation (C6)gives

If the uncertainty inCp is evaluated at the maxi-mum pressure point, then substituting

into equation (C7) yields

The uncertainties in and areevaluated statistically at each fractional chord mea-surement station. The standard deviation for isfound to be between 1 and 3 percent of for allmeasurement locations exceptx/c = 0.55, where itreached 5.56 percent. More importantly, the standarddeviation in the velocity ratio is much smaller rangingfrom 0.15 to 1.38 percent.

An alternate method to estimate the error in theboundary-layer velocity ratio can be derived by usingKing’s law as the calibration function for a constant-temperature hot wire

(C10)

where we taken = 1/2. Or, solving forU gives

(C11)

Strictly speaking, the calibration coefficientsdepend on the temperatures of the hot wire and theflow as

(C12)

and

(C13)

Now, suppose that the flow temperature changesfrom the calibration temperature giving

(C14)

and

(C15)

URU

Us,e----------

U2/U1

U2/U1( )y,max

----------------------------------= =

U2U1

U2U2/U1

U1 U2

U1 U2/U1( )y,max

∆q∞ 0.02 torr= q∞ 2.0 torr=( )

∆ p∞ 0.6 torr= p∞ 720.0 torr=( )

∆T∞ 1.5 K–= T∞ 309.0 K=( )

∆U∞U∞

------------ 0.006=

wpD0.02 torr= pD q∞ 2.0 torr= =( )

∆Cp

Cp----------- 0.014=

U1 U2/U1( )y,max

U1Us,e

E2

A BUn

+=

UE

2A–

B----------------

2

=

TwT f

A A1 Tw T f–( )=

B B1 Tw T f–( )=

AAcab Tw T f–( )

Tw Tcab–------------------------------------=

BBcab Tw T f–( )

Tw Tcab–------------------------------------=

38

where and are the values of and deter-mined at the calibration temperature Substi-tuting equations (C14) and (C15) into equation (C11)for hot wires 1 and 2 and taking the ratio gives

(C16)

where

But, if equation (C16) reduces to

an estimate of the error in is obtained bytaking the ratio of equations (C16) and (C17) andsquaring the result. Doing so for a typical set of hot-wire calibration data with the maximum temperatureshift taken to be the effect of tem-perature drift is found to be negligible at the boundary-layer edge, but it increases as is decreased. Formost of the boundary layer, 0.25≤ ≤ 1.0, theerror does not exceed 2.7 percent. The maximum erroris 5.8 percent at the minimum velocity ratio of

Acab BcabTcab( ).

U2/U1

U2

U1-------

BC1

BC2----------

E22

AC2T–

E12

AC1T–--------------------------=

TTw T f–

Tw Tcab–------------------------=

U2

U1-------

BC1

BC2----------

E22

AC2–

E12

AC1–----------------------=

U2/U1

T f Tcab– 4°C,=

U2U2/U1

U2/U1 0.1.=

39

References

1. Reed, Helen L.; and Saric, William S.: Stability ofThree-Dimensional Boundary Layers.Annual Review ofFluid Mechanics, Volume 21, Ann. Rev., Inc., 1989,pp. 235–284.

2. Saric, William S.: Laminar-Turbulent Transition: Fun-damentals. Special Course on Skin Friction DragReduction, AGARD-R-786, 1992.

3. Morkovin, M. V.: On the Many Faces of Transition.Vis-cous Drag Reduction, C. Sinclair Wells, ed., PlenumPress, 1969, pp. 1–31.

4. Schlichting, Hermann (J. Kestin, transl.):Boundary-Layer Theory. McGraw-Hill Book Co., Inc., 1968.

5. Smith, A. M. O.; and Gamberoni, Nathalie: Transition,Pressure Gradient, and Stability Theory. Rep.No. ES 26388, Douglas Aircraft Co., Inc., 1956.

6. Van Ingen, J. L.: A Suggested Semi-Empirical Methodfor the Calculation of the Boundary Layer Tran-sition Region. Rep. V.T.H.-74, Tech. HogeschoolVliegtuigbouwkunde, 1956.

7. Reed, Helen L.: Wave Interactions in Swept-WingFlows. Phys. Fluids, vol. 30, no. 11, Nov. 1987,pp. 3419–3426.

8. Gray, W. E.:The Effect of Wing Sweep on LaminarFlow. Tech. Memo. Aero 255, British R.A.E., Feb.1952.

9. Owen, P. R.; and Randall, D. G.:Boundary-Layer Tran-sition on a Sweptback Wing. Tech. Memo. Aero 277,British R.A.E., May 1952.

10. Stuart, J. T.: The Basic Theory of the Stability of Three-Dimensional Boundary-Layers. Rep. No. F.M. 1899,British Natl. Phys. Lab. (Rep. No. 15,904, A.R.C.), May1953.

11. Gregory, N.; Stuart, J. T.; and Walker, W. S.: On the Sta-bility of Three-Dimensional Boundary Layers WithApplication to the Flow Due to a Rotating Disk.Philos.Trans. R. Soc. London, ser. A, vol. 248, no. 943, 1955,pp. 155–199.

12. Brown, W. B.: Exact Solution of the Orr-SommerfeldStability Equation for Low Reynolds Numbers. Rep.No. BLC-43, Northrop Aircraft, Inc., 1954.

13. Brown, W. B.:Extension of Exact Solution of the Orr-Sommerfeld Stability Equation to Reynolds Numbers of4000. Rep. No. NAI-55-548 (BLC-78), Northrop Air-craft, Inc., 1955.

14. Brown, W. B.:Numerical Calculation of the Stability ofCross-Flow Profiles in Laminar Boundary Layers on a

Rotating Disc and on a Swept-Back Wing and an ExactCalculation of the Stability of the Blasius Velocity Pro-file. Rep. No. NAI-59-5 (BLC-117), Northrop Aircraft,Inc., 1959.

15. Pfenninger, W.; Gross, Lloyd; and Bacon, John W., Jr.(appendix I by G. S. Raetz):Experiments on a 30°Swept 12%-Thick Symmetrical Laminar Suction Wing inthe 5-Ft by 7-Ft Michigan Tunnel. Rep. No. NAI-57-317(BLC-93), Northorp Aircraft, Inc., 1957.

16. Bacon, J. W., Jr.; Tucker, V. L.; and Pfenninger, W.:Experiments on a 30° Swept, 12% Thick SymmetricalLaminar Suction Wing in the 5- by 7-Foot University ofMichigan Tunnel. Rep. No. NOR-59-328 (BLC-119),Northrop Aircraft, Inc., Aug. 1959.

17. Pfenninger, W.; and Bacon, J. W., Jr.: About the Devel-opment of Swept Laminar Suction Wings With FullChord Laminar Flow. Boundary Layer and Flow Con-trol, Volume 2, G. V. Lachmann, ed., Pergamon Press,1961, pp. 1007–1032.

18. Gault, Donald E.:An Experimental Investigation ofBoundary-Layer Control for Drag Reduction of aSwept-Wing Section at Low Speed and High ReynoldsNumbers. NASA TN D-320, 1960.

19. Boltz, Frederick W.; Kenyon, George C.; and Allen,Clyde Q.: Effects of Sweep Angle on the Boundary-Layer Stability Characteristics of an Untapered Wing atLow Speeds. NASA TN D-338, 1960.

20. Pfenninger, W.: Laminar Flow Control Laminarization.Special Course on Concepts for Drag Reduction.AGARD-R-654, 1977.

21. Jaffe, N. A.; Okamura, T. T.; and Smith A. M. O.: Deter-mination of Spatial Amplification Factors and TheirApplication to Predicting Transition. AIAA J., vol. 8,no. 2, 1970, pp. 301–308.

22. Mack, Leslie M.: Linear Stability Theory and the Prob-lem of Supersonic Boundary-Layer Transition.AIAA J.,vol. 13, no. 3, Mar. 1975, pp. 278–289.

23. Mack, Leslie M.: Transition Prediction and Linear Sta-bility Theory. Laminar-Turbulent Transition, AGARD-CP-224, Oct. 1977, pp. 1-1–1-22.

24. Mack, Leslie M.: Boundary-Layer Linear Stability The-ory. Special Course on Stability and Transition of Lami-nar Flow, AGARD-R-709, June 1984, pp. 3-1–3-81.

25. Hefner, Jerry N.; and Bushnell, Dennis M.: Applicationof Stability Theory to Laminar Flow Control. AIAA-79-1493, July 1979.

26. Bushnell, D. M.; and Malik, M. R.: Application of Sta-bility Theory to Laminar Flow Control—Progress and

40

Requirements.Stability of Time Dependent and Spa-tially Varying Flows, Springer-Verlag, 1987, pp. 1–17.

27. Berry, Scott A.; Dagenhart, J. Ray; Yeaton, Robert B.;and Viken, Jeffrey K.: Boundary-Layer Stability Analy-sis of NLF and LFC Experimental Data at Subsonic andTransonic Speeds. SAE Paper 871859, Oct. 1987.

28. Arnal, D.; Casalis, G.; and Juillen, J. C.: Experimentaland Theoretical Analysis of Natural Transition on‘Infinite’ Swept Wing. Laminar-Turbulent Transition,R. Michel and D. Arnals, eds., Springer-Verlag, 1990,pp. 311–325.

29. Creel, T. R., Jr.; Malik, M. R.; and Beckwith, I. E.:Experimental and Theoretical Investigation ofBoundary-Layer Instability Mechanisms on a SweptLeading Edge at Mach 3.5.Research in Natural Lami-nar Flow and Laminar-Flow Control, Jerry N. Hefnerand Frances E. Sabo, compilers, NASA CP-2487, Part 3,1987, pp. 981–995.

30. Bieler, H.; and Redeker, G.: Development of TransitionCriteria on the Basis of eN for Three Dimensional WingBoundary Layers.Flows With Separation, DGLR, 1988,pp. 103–116.

31. Collier, F. S., Jr.; Bartlett, D. W.; Wagner, R. D.; Tat,V. V.; and Anderson, B. T.: Correlation of BoundaryLayer Stability Analysis With Flight Transition Data.Laminar-Turbulent Transition, R. Michel and D. Arnal,eds., Springer-Verlag, 1990, pp. 337–346.

32. Parikh, P. G.; Sullivan, P. P.; Bermingham, E.; andNagel, A. L.: Stability of 3D Wing Boundary Layer on aSST Configuration. AIAA-89-0036, Jan. 1989.

33. Collier, Fayette S., Jr.; Johnson, Joseph B.; Rose,Ollie J.; and Miller, D. S.: Supersonic Boundary-LayerTransition on the LaRC F-106 and the DFRF F-15 Air-craft. Part 1: Transition Measurements and StabilityAnalysis. Research in Natural Laminar Flow andLaminar-Flow Control, Jerry N. Hefner and Frances E.Sabo, compilers, NASA CP-2487, Part 3, 1987,pp. 997–1014.

34. Obara, Clifford J.; Vijgen, Paul M. H. W.; Lee,Cynthia C.; and Wusk, Michael S.: Boundary-LayerStability Analysis of Flight-Measured Transition Data.SAE Paper 901809, 1990.

35. Lee, Cynthia C.; Wusk, Michael S.; and Obara,Clifford J.: Flight Experiments Studying the Growth ofthe Disturbances in the Laminar Boundary Layer. SAEPaper 901979, 1990.

36. Horstmann, K. H.; Redeker, G.; Quast, A.; Dressler, U.;and Bieler, H.: Flight Tests With a Natural LaminarFlow Glove on a Transport Aircraft. AIAA-90-3044,1990.

37. Waggoner, Ed G.; Campbell, Richard L.; Phillips,Pam S.; and Hallissy, James B.: Design and Test of anNLF Wing Glove for the Variable-Sweep TransitionFlight Experiment. Research in Natural Laminar Flowand Laminar-Flow Control, Jerry N. Hefner andFrances E. Sabo, compilers, NASA CP-2487, Part 3,1987, pp. 753–776.

38. Obara, Clifford J.; Lee, Cynthia C.; and Vijgen,Paul M. H. W.: Analysis of Flight-Measured Boundary-Layer Stability and Transition Data. AIAA-91-3282,Sept. 1991.

39. Berry, Scott; Dagenhart, J. R.; Brooks, C.W.; and Harris,C. D.: Boundary-Layer Stability Analysis of LaRC8-Foot LFC Experimental Data.Research in NaturalLaminar Flow and Laminar-Flow Control, Jerry N.Hefner and Frances E. Sabo, compilers, NASACP-2487, Part 2, 1987, pp. 471–489.

40. Harvey, William D.; Harris, Charles D.; and Brooks,Cuyler, W., Jr.: Experimental Transition and Boundary-Layer Stability Analysis for a Slotted Swept LaminarFlow Control Airfoil. 4th Symposium on Numerical andPhysical Aspects of Aerodynamic Flows, CaliforniaState Univ., 1989.

41. Arnal, D.; Juillen, J. C.; and Casalis, G.: The Effects ofWall Suction on Laminar-Turbulent Transition in Three-Dimensional Flow. Boundary Layer Stability andTransition to Turbulence, FED-vol. 114, ASME , 1991,pp. 155–162.

42. Maddalon, D. V.; Collier, F. S., Jr.; Montoya, L. C.; andPutnam, R. J.: Transition Flight Experiments on a SweptWing With Suction. Laminar-Turbulent Transition,R. Michel and D. Arnal, eds., Springer-Verlag, 1990,pp. 53–64.

43. Runyan, L. J.; Bielak, G. W.; Behbehani, R.; Chen,A. W.; and Rozendaal, R. A.: 757 NLF Glove FlightTest Results.Research in Natural Laminar Flow andLaminar-Flow Control, Jerry N. Hefner and Frances E.Sabo, compilers, NASA CP-2487, Part 3, 1987,pp. 795–818.

44. Srokowski, Andrew J.; and Orszag, Steven A.: MassFlow Requirements for LFC Wing Design—LaminarFlow Control. AIAA-77-1222, 1977.

45. Dagenhart, J. Ray:Amplified Crossflow Disturbances inthe Laminar Boundary Layer on Swept Wings With Suc-tion. NASA TP-1902, 1981.

46. Malik, M. R.; and Orszag, S. A.: Efficient Computationof the Stability of Three-Dimensional CompressibleBoundary Layers. AIAA-81-1277, June 1981.

47. Malik, Mujeeb R.:COSAL—A Black-Box CompressibleStability Analysis Code for Transition Prediction

41

in Three-Dimensional Boundary Layers. NASACR-165925, 1982.

48. Herbert, Thorwald: A Code for Linear Stability Analy-sis. Instability and Transition, Volume II, M. Y.Hussaini and R. G. Voigt, eds., Springer-Verlag, 1990,pp. 121–144.

49. Arnal, D.: Some Transition Problems in Three-Dimensional Flows. Instability and Transition, Vol-ume I, M. Y. Hussaini and R. G. Voigt, eds., Springer-Verlag, 1990, pp. 130–135.

50. Saric, William S.: Low-Speed Experiments: Require-ments for Stability Measurements.Instability and Tran-sition, Volume I, M. Y. Hussaini and R. G. Voigt, eds.,Springer-Verlag, 1990, pp. 162–176.

51. Stetson, K. F.: Hypersonic Boundary-Layer Transition.Second Joint Europe/U.S. Short Course in Hypersonics.U.S. Air Force Academy, 1989.

52. Malik, Mujeeb R.: Group Summary: CompressibleStability and Transition.Instability and Transition,Volume II, M. Y. Hussaini and R. G. Voight, eds.,Springer-Verlag, 1990, pp. 233–234.

53. Poll, D. I. A.: The Effect of Isolated RoughnessElements on Transition in Attachment-Line Flows.Laminar-Turbulent Transition, R. Michel and D. Arnal,eds., Springer-Verlag, 1990, pp. 657–667.

54. Arnal, D.; and Aupoix, B.: Hypersonic Boundary Lay-ers; Transition and Turbulence Effects.Aerothermo-dynamics for Space Vehicles, B. Battrick, ed.,ESA-SP-318, 1991, pp. 25–38.

55. Lees, Lester; and Lin, Chia Chiao:Investigation of theStability of the Laminar Boundary Layer in a Compress-ible Fluid. NACA TN 1115, 1946.

56. Lin, C. C.: The Theory of Hydrodynamic Stability.Cambridge Univ. Press, 1955, pp. 75–82.

57. Dunn, D. W.; and Lin, C. C.: On the Stability of theLaminar Boundary Layer in a Compressible Fluid.J.Aeronaut. Sci., vol. 22, no. 7, 1955, pp. 455–477.

58. Lees, Lester; and Reshotko, Eli: Stability of a Com-pressible Laminar Boundary Layer.J. Fluid Mech.,vol. 12, pt. 4, 1962, pp. 555–590.

59. Mack, Leslie M.: Computation of the Stability of theLaminar Compressible Boundary Layer. Methods inComputational Physics, Volume 4, Berni Alder, SidneyFernbach, and Manuel Rotenberg, eds., Academic Press,Inc., 1965, pp. 247–299.

60. Mack, L. M.: The Stability of the Compressible LaminarBoundary Layer According to a Direct Numerical

Solution. Recent Developments in Boundary LayerResearch, Part I, AGARDograph 97, 1965, pp. 329–362.

61. Mack, L. M.:Boundary Layer Stability Theory. NASACR-131501, 1969.

62. Lekoudis, Spyridon G.: Stability of Three-DimensionalCompressible Boundary Layers Over Wings With Suc-tion. AIAA-79-0265, 1979.

63. Mack, Leslie M.: On the Stability of the BoundaryLayer on a Transonic Swept Wing. AIAA-79-0264,1979.

64. Mack, Leslie M.: Compressible Boundary-Layer Stabil-ity Calculations for Sweptback Wings With Suction.AIAA J., vol. 20, no. 3, Mar. 1982, pp. 363–369.

65. El-Hady, Nabil M.: On the Stability of Three-Dimensional Compressible Nonparallel BoundaryLayers. AIAA-80-1374, 1980.

66. Reed, H. L.; Stuckert, G.; and Balakumar, P.: Stabilityof High-Speed Chemically Reacting and Three-Dimensional Boundary Layers.Laminar-TurbulentTransition, D. Arnal and R. Michel, eds., Springer-Verlag, 1990, pp. 347–358.

67. Balakumar, Ponnampalam; and Reed, Helen L.: Stabil-ity of Three-Dimensional Supersonic Boundary Layers.Phys. Fluids A, vol. 3, no. 4, Apr. 1991, pp. 617–632.

68. Padhye, A. R.; and Nayfeh, A. H.: Nonparallel Stabilityof Three-Dimensional Flows. AIAA-81-1281, 1981.

69. Nayfeh, A.: Stability of Three-Dimensional BoundaryLayers.AIAA J., vol. 18, 1980, pp. 406–416.

70. Nayfeh, A. H.: Three-Dimensional Stability of GrowingBoundary Layers. Laminar-Turbulent Transition,R. Eppler and H. Fasel, eds., Springer-Verlag, 1980,pp. 201–217.

71. Reed, H. L.; and Nayfeh, A. H.: Stability of Compress-ible Three-Dimensional Boundary-Layer Flows. AIAA-82-1009, 1982.

72. Malik, M. R.; and Poll, D. I. A.: Effect of Curvature onThree-Dimensional Boundary Layer Stability. AIAA-84-1672, 1984.

73. Viken, J.; Collier, F. S., Jr.; Wagner, R. D.; and Bartlett,D. W.: On the Stability of Swept Wing Laminar Bound-ary Layers Including Curvature Effects.Laminar-Turbulent Transition, R. Michel and D. Arnal, eds.,Springer-Verlag, 1990, pp. 381–388.

74. Mueller, B.; Bippes, H.; and Collier, F. S., Jr.: The Sta-bility of a Three Dimensional Laminar Boundary LayerOver a Swept Flat Plate.Instability and Transition,Volume II, M. Y. Hussaini and R. G. Yoigt, eds.,Springer-Verlag, 1990, pp. 268–277.

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75. Collier, F. S., Jr.; and Malik, Mujeeb R.: CurvatureEffects on the Stability of Three-Dimensional BoundaryLayers.Fluid Dynamics of Three-Dimensional Turbu-lent Shear Flows and Transition, AGARD CP-438,1990. (Available from DTIC as AD A211 101.)

76. Lin, Ray-Sing; and Reed, Helen L.: Effect of Curvatureon Stationary Crossflow Instability of a Three-Dimensional Boundary Layer. AIAA J., vol. 31, no. 9,1993, pp. 1611–1617.

77. Lekoudis, S. G.: Resonant Wave Interactions on a SweptWing. AIAA J., vol. 18, no. 1, 1980, pp. 122–124.

78. Fischer, T. M.; and Dallmann, U.: Theoretical Investiga-tion of Secondary Instability of Three-DimensionalBoundary-Layer Flows. AIAA-87-1338, 1987.

79. El-Hady, N. M.: Evolution of Resonant Wave Triads inThree-Dimensional Boundary Layers. AIAA-88-0405,1988.

80. Bassom, Andrew P.; and Hall, Philip:On the Inter-action of Stationary Crossflow Vortices and Tollmien-Schlichting Waves in the Boundary Layer on a RotatingDisc. NASA CR-181859, 1989.

81. Bassom, Andrew P.; and Hall, Philip:Concerningthe Interaction of Non-Stationary Cross-Flow Vorticesin a Three-Dimensional Boundary Layer. NASACR-182037, 1990.

82. Bassom, Andrew; and Hall, Philip:Vortex Instabilitiesin 3D Boundary Layers. The Relationship BetweenGoertler and Crossflow Vortices. NASA CR-187456,1990.

83. Bassom, Andrew P.; and Hall, Philip: Vortex Instabili-ties in Three-Dimensional Boundary Layers: The Rela-tionship Between Görtler and Crossflow Vortices. J.Fluid Mech., vol. 232, Nov. 1991, pp. 647–680.

84. Arnal, D.; Coustols, E.; and Juillen, J.C.: Experimentaland Theoretical Study of Transition Phenomena on anInfinite Swept Wing.La Recherche Aerosp., no. 4, 1984,pp. 39–54.

85. Arnal, D.; Habiballah, M.; and Coustols, E.: LaminarInstability Theory and Transition Criteria in Two- andThree-Dimensional Flow.La Recherche Aerosp., no. 2,1984, pp. 45–63.

86. Arnal, D.; and Coustols, E.: Application of Two andThree-Dimensional Criteria for Calculating Transitionsand Boundary Layers Over Swept Wings.Improvementof Aerodynamic Performance Through Boundary LayerControl and High Lift Systems. AGARD CP-365, 1984.(Available from DTIC as AD A147 396.)

87. Michel, R.; Arnal, D.; and Coustols, E.: StabilityCalculations and Transition Criteria in Two- or Three-

Dimensional Flows. Laminar-Turbulent Transition,V. V. Koslov, ed., Springer-Verlag, 1985, pp. 455–462.

88. Arnal, D.; Coustols, E.; and Jelliti, M.: Transition toThree-Dimensional Flow and Laminarization of theBoundary Layer on a Swept-Back Wing.Colloqued’Aerodynamique Appliquee, 22nd (Lille, France),Nov. 13–15, 1985.

89. Michel, R.; Coustols E.; and Arnal, D.:Transition Cal-culations in Three-Dimensional Flows. TP 1985-7,ONERA, 1985.

90. Arnal, D.; and Juillen, J. C.: Three-Dimensional Transi-tion Studies at ONERA/CERT. AIAA-87-1335, 1987.

91. King, Rudolph A.: Mach 3.5 Boundary-Layer Transi-tion on a Cone at Angle of Attack. AIAA-91-1804,1991.

92. Poll, D. I. A.: Some Observations of the Transition Pro-cess on the Windward Face of a Long Yawed Cylinder.J. Fluid Mech., vol. 150, 1985, pp. 329–356.

93. Michel, R.; Arnal, D.; Coustols, E.; and Juillen, J. C.:Experimental and Theoretical Studies of BoundaryLayer Transition on a Swept Infinite Wing.Laminar-Turbulent Transition, V. V. Kozlov, ed., Springer-Verlag,1985, pp. 553–562.

94. Kohama, Y.; Ukaku, M.; and Ohta, F.: Boundary-LayerTransition on a Swept Cylinder.Frontiers of FluidMechanics, Shen Yuan, ed., Pergamon Press, 1988,pp. 151–156.

95. Choudhari, Meelan; and Streett, Craig L.: BoundaryLayer Receptivity Phenomena in Three-Dimensionaland High-Speed Boundary Layers. AIAA-90-5258,1990.

96. Fischer, Thomas M.; and Dallmann, Uwe:TheoreticalInvestigation of Secondary Instability of Three-Dimensional Boundary-Layer Flows With Applicationto the DFVLR-F5 Model Wing. DFVLR-FB-87-44,1987.

97. Fischer, T. M.; and Dallmann, U.: Primary andSecondary Stability Analysis of Three-DimensionalBoundary-Layer Flow.Phys. Fluids A, vol. 3, 1991,pp. 2378–2391.

98. Nitschke-Kowsky, P.; and Bippes, H.: Instability andTransition of a Three-Dimensional Boundary Layeron a Swept Flat Plate.Phys. Fluids, vol. 31, 1988,pp. 786–795.

99. Bippes, H.: Instability Features Appearing on SweptWing Configurations. Laminar-Turbulent Transition,R. Michel and D. Arnal, eds., Springer-Verlag, 1990,pp. 419–430.

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100. Mueller, B.: Experimental Study of the TravellingWaves in a Three-Dimensional Boundary Layer.Laminar-Turbulent Transition. R. Michel andD. Arnal, eds., Springer-Verlag, 1990, pp. 489–498.

101. Bippes, H.; and Mueller, B.: Experiments on theLaminar-Turbulent Transition on Swept Wings. FlowsWith Separation, DGLR Paper 88-05, 1988, pp. 3–16.

102. Meyer, F.; and Kleiser, L.: Numerical Simulation ofthe Nonlinear Evolution of Perturbation in a ThreeDimensional Boundary Layer.Flows With Separation,DGLR Paper 88-05, 1988, pp. 39–40.

103. Meyer, F.; and Kleiser, L.: Numerical Simulationof Transition Due to Crossflow Instability.Laminar-Turbulent Transition, R. Michel and D. Arnal, eds.,Springer-Verlag, 1990, pp. 609–619.

104. Singer, Bart A.; Meyer, F.; and Kleiser, Leonhard:Nonlinear Development of Crossflow Vortices.Insta-bility and Transition, M. Y. Hussaini and R. G. Voigt,eds., Springer-Verlag, 1990, pp. 300–312.

105. Meyer, Friedrich: Numerical Simulation of Transitionin Three-Dimensional Boundary Layers. ESATT-1203, 1991.

106. Fischer, Thomas M.: AMathematical-Physical Modelfor Describing Transitional Boundary-Layer Flows.Volume 1—The Linear and Nonlinear DisturbanceDifferential Equations. DLR-FB-95-06 ESA TT-1242,1994.

107. Spalart, P. R.: Direct Numerical Study of Cross-flow Instability. Laminar-Turbulent Transition,R. Michel and D. Arnal, eds., Springer-Verlag,1990, pp. 621–630.

108. Reed, Helen L.; and Lin, Ray-Sing: Stability of Three-Dimensional Boundary Layers. SAE Paper 871857,1987.

109. Lin, Ray-Sing: Stationary Crossflow Instability on anInfinite Swept Wing. Ph.D. Thesis, Arizona StateUniv., 1992.

110. Malik, M. R.; and Li, F.: Three-DimensionalBoundary-Layer Stability and Transition. SAEPaper 921991, 1992.

111. Herbert, Th.: Boundary-Layer Transition—Analysisand Prediction Revisited. AIAA-91-0737, 1991.

112. Saric, W. S.; and Yeates, L. G.: Experiments on theStability of Crossflow Vortices in Swept-Wing Flows.AIAA-85-0493, 1985.

113. Nitschke-Kowsky, Petra:Experimental Investigationson the Stability and Transition of Three-DimensionalBoundary Layers. ESA-TT-1026, 1986.

114. Mueller, Bernhard: Experimental Investigation ofCross-Flow Instability in the Linear and Non-LinearStage of the Transition Region. ESA-TT-1237, 1990.

115. Bippes, H.; Mueller, B.; and Wagner, M.: Measure-ments and Stability Calculations of the DisturbanceGrowth in an Unstable Three-Dimensional BoundaryLayer.Phys. Fluids A, vol. 3, 1991, pp. 2371–2377.

116. Kachanov, Y. S.; and Tararykin, O. I.: The Experimen-tal Investigation of Stability and Receptivity of aSwept-Wing Flow. Laminar-Turbulent Transition,R. Michel and D. Arnal, eds., Springer-Verlag, 1990,pp. 499–509.

117. Saric, W. S.; Dagenhart, J. Ray; and Mousseux, Marc:Experiments in Swept-Wing Transition. Numericaland Physical Aspects of Aerodynamic Flows IV,Tuncer Cebeci, ed., Springer-Verlag, 1990.

118. Dagenhart, J. Ray; Stack, J. Peter; Saric, William S.;and Mousseux, Marc C.: Crossflow-Vortex Instabilityand Transition on a 45 Degree Swept Wing.AIAA-89-1892, 1989.

119. Dagenhart, J. R.; Saric, William S.; Hoos, Jon A.; andMousseux, Marc: Experiments on Swept-WingBoundary Layers. Laminar-Turbulent Transition,R. Michel and D. Arnal, eds., Springer-Verlag, 1990,pp. 369–380.

120. Radeztsky, Ronald H., Jr.; Reibert, Mark S.; Saric,William S.; and Takagi, Shohei: Role of Micron-Sized Roughness in Swept-Wing Transition. SAEPaper No. 921986, 1992.

121. Somers, Dan M.; and Horstman, Karl-Heinz: Designof a Medium-Speed, Natural-Laminar-Flow Airfoilfor Commuter Aircraft Application. DFVLR IB129-85/26, 1985.

122. Somers, Dan M.: Subsonic Natural-Laminar-FlowAirfoils. Natural Laminar Flow and Laminar FlowControl, R. W. Barnwell and M. Y. Hussaini, eds.,Springer-Verlag, 1992, pp. 143-176.

123. Saric, William S.: The ASU Transition Research Facil-ity. AIAA-92-3910, 1992.

124. Saric, William S.; Takagi, Shohei; and Mousseux,Marc C.: The ASU Unsteady Wind Tunnel and Funda-mental Requirements for Freestream Turbulence Mea-surements. AIAA-88-0053, 1988.

125. Eppler, Richard; and Somers, Dan M.:A ComputerProgram for the Design and Analysis of Low-SpeedAirfoils. NASA TM-80210, 1980.

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126. Kaups, Kalle; and Cebeci, Tuncer: Compressible Lam-inar Boundary Layers With Suction on Sweptand Tapered Wings.J. Aircr., vol. 14, no. 7, 1977,pp. 661–667.

127. Stevens, W. A.; Goradia, S. H.; and Braden, J. A.:Mathematical Model for Two-Dimensional Multi-Component Airfoils in Viscous Flow. NASA CR-1843,1971.

128. Fuciarelli, David A.; and Reed, Helen L.: StationaryCrossflow Vortices.Phys. Fluids A, vol. 4, no. 9, 1992,p. 1880.

129. Reda, Daniel C.: Liquid Crystals for Unsteady SurfaceShear Stress Visualization. AIAA-88-3841, 1988.

130. Kline, S. J.; and McClintock, F. A.: Describing Uncer-tainties in Single-Sample Experiments.Mech. Eng.,vol. 75, no. 1, 1953, pp. 3–8.

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Table 1. Crossflow Stability Analysis With SALLY Code forα = −4° andRc = 3.81× 106

Frequency,fNmax for wavelength,λ/c, of—

0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

−50 4.8 10.1 11.2 10.0 8.7 7.3 6.1 5.10 4.9 10.7 13.1 12.6 11.6 10.4 9.3 8.3

50 5.1 10.9 14.5 14.6 13.8 12.8 11.7 10.7100 5.2 10.6 15.4 16.0 15.5 14.6 13.5 12.4200 5.2 9.4 15.6 17.3 17.2 16.4 15.3 13.8300 5.2 8.2 13.8 16.2 16.6 15.8 14.6 12.9500 4.8 6.2 7.3 7.6 7.2 6.5 5.3 4.4

Table 2. Transition Locations and Wavelengths From Naphthalene Flow Visualization

Reynolds number,Rc

Transition location,(x/c)tr

Wavelength,λ/c

1.92× 106 0.78 0.00502.19 0.732.37 0.58 0.00402.73 0.45 0.00343.27 0.33 0.00293.73 0.30 0.0024

Table 3.N-Factors at Transition Computed With SALLY Code forα = −4°, Rc = 2.37× 106, and (x/c)tr = 0.58

Frequency,fNtr for wavelength,λ/c, of—

0.004 0.005 0.0055 0.006 0.007 0.008 0.009

0 6.2 6.8 6.8 6.7 6.2 5.6 4.550 8.2 8.5 8.2 6.6

100 8.3 9.1 9.1 7.8 7.4150 7.2 8.4 8.6 8.5 7.1200 5.3 6.3 6.9 6.8 5.5300 2.8 2.6 2.4 1.9 1.1

46



0.003 0.004 0.005 0.0055 0.006 0.007 0.008

0 4.7 6.5 6.2 5.6 5.0 4.350 7.3 7.6 7.4 7.2 6.6

100 7.6 8.3 8.2 8.1 7.6150 7.3 8.5 8.5 8.5 7.9200 6.4 8.1 8.1 8.2 7.7300 4.7 5.2 5.3 5.5 5.3400 3.3 3.0 2.9 2.5 2.2



0.0025 0.003 0.004 0.0045 0.005 0.006 0.007

0 6.1 6.4 6.0 5.5 5.0 4.1 3.350 7.0 6.8 6.4 5.9

100 7.4 7.5 7.2 6.7150 7.6 8.1 7.8 7.3200 7.7 8.4 8.2 7.8300 7.2 8.5 8.4 8.0400 6.4 7.8 7.8 7.3500 5.6 6.2 5.9

47

Figure 1. Curved streamlines over swept wing.

Figure 2. Boundary-layer velocity profiles on swept wing.

Inviscid streamline

ut,e(xt)

U∞Crossflowdirection (zt)

Λ

wt

ut

zt

xt

y

Wall shear

Crossflowcomponent

Inflectionpoint

Tangentialcomponent

Tangent toinviscidstreamline

48

Figure 3. Plan view of Arizona State University Unsteady Wind Tunnel. Dimensions are in meters.

Figure 4. New UWT test section with liner under construction.

7.7 5.0 11.1

2.55.0

1.4

5.15.6

Fan

Shutters

Test section

Diffuserextension

Settling chamber

49

Figure 5. NASA NLF(2)-0415 airfoil.

Figure 6. NASA NLF(2)-0415 design point pressure distribution atα = 0° andδf = 0° in free air.

0 .20 .40 .60 .80 1.00–.40

–.20

0

.20

.40

x/c

y/c

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

50

Figure 7. NASA NLF(2)-0415 pressure distribution forα = −4° andδf = 0° in free air.

Figure 8. NASA NLF(2)-0415 pressure distribution forα = −2° andδf = 0° in free air.

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

51

Figure 9. NASA NLF(2)-0415 pressure distribution forα = 2° andδf = 0° in free air.


0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

52

Figure 11. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = −20° in free air.

Figure 12. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = −10° in free air.

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Upper

Lower

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

53



0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

54

Figure 15. NASA NLF(2)-0415 pressure distribution forα = −4° andδf = 0° in UWT.

Figure 16. NASA NLF(2)-0415 pressure distribution forα = −2° andδf = 0° in UWT.

UWTFree air

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Upper

Lower

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Lower

Upper

UWTFree air

55

Figure 17. NASA NLF(2)-0415 pressure distribution forα = 0° andδf = 0° in UWT.


0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Upper

Lower

UWTFree air

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Upper

Lower

UWTFree air

56


Figure 20. Local spatial growth rates for stationary crossflow vortices atα = −4° andδf = 0° in UWT atRc = 3.81× 106.

0 .20 .40 .60 .80 1.00–1

0

1

2

3

x/c

–Cp

Upper

Lower

UWTFree air

1

11

2

2

2

2

2

3

3

33

3

3

3

3

4

4

44

4

44

44

44

44

44

44 4

5

5

5

5 5

55

5 5 5 5 5 5 5 5 55

5

66

6

66

6

6 6 6 6 6 6 6 6 6 6 6

6

7

7

77

7

7 77

7 7 7 7 7 7 7 7

7

8

8

8 8

8 88

88

8 8 8 8 8 8

8

99

99

99

99

99

9 9 9 9 9

9

aa

aa

aa

aa

aa a a a a

a

bb

bb

bb

bb

b b b b b

b

cc

cc

cc

cc c c c c

c

dd

dd d

dd d d d d

d

ee

ee

ee e e e e

e

fff

ff

f f f f f

f

123456789abcdef

0 .20 .40 .60 .80 1.00

10

20

40

30

50

0.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.02000.02250.0250

x/c

λ/c

–αi c

57

Figure 21. Local spatial growth rates for stationary crossflow vortices atα = −2° andδf = 0° in UWT atRc = 3.81× 106.

Figure 22. Local spatial growth rates for stationary crossflow vortices atα = 0° andδf = 0° in UWT atRc = 3.81× 106.

1

11

2

2

2

2

3

3

3

3

3

3

4

44

4

4

4

4

4

4

5

5

5 55

5 5 5 55

5 5 5 55

55

5 5

6

6

66 6 6

66

6 6 6 6 6 6 6 66

6

77

7

7 7 7

77

7 7 7 7 7 7 7 7 7

7

8

8

88

8

8 8

8 8 8 8 8 8 8 8 8

8

99

99

9 9

9 99

9 9 9 9 9 9

9

aaa

a a

a aa

aa a a a a

a

bbb

bb

bb

bb b b b b

b

ccc

cc

cc

c c c c c

c

dd d dd

dd d d d d

d

e ee e

ee e e e e

e

f ff

ff f f f f

f

123456789abcdef

0 .20 .40 .60 .80 1.00

10

20

40

30

50

0.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.02000.0225

x/c

λ/c–α

i c

1

2

2

3

3

3

4

4

4

4

45

5

5

5

5

5

6

66

6 66 6 6

6 6 6 66

66 6

6 67 7

7 7 7 77

77 7 7 7 7 7 7 7

7

8 8 8 88

88

88 8 8 8 8 8

88

9 9 99

99

99 9 9 9 9 9

99

aa

aa

a a a a a aa

a

bb

bb b b b b b

bb

cc

c c c c c cc

c

dd d d d d d

dd

e e e e e e e ef f f f f f f f

123456789abcdef

0 .20 .40 .60 .80 1.00

10

20

40

30

50

0.00060.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.0200

x/c

λ/c

–αi c

58



1

1

2

2

2

3

33

3

4

4

4

5

55

566 77 7 7 7 7 78 8 8 8 8 8 89 9 9 9 9 9 9

aa a a a a abb b b b b

cc c c cd d d

123456789abcd

0 .20 .40 .60 .80 1.00

10

20

40

30

50

0.00060.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.0150

x/c

λ/c–α

i c

1

1

2

2

2

3

3

3

4

4

5

5

5

66

7

7 7

7

77

8

8

88

8

8

88

89

9

9 9

9

9

99

9 9 9 9 9a

a

a aa

aa

aa a a a a ab

b

b bb

bb

b b b b b b bc

cc c

cc c c c c c c c cd

dd d d d d d d d d d dee e e e e e e e ef f f f f

123456789abcdef

0 .20 .40 .60 .80 1.00

10

20

40

30

50

0.00060.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.0200

x/c

λ/c

–αi c

59

Figure 25.N-factors for stationary crossflow vortices atα = −4° andδf = 0° in UWT atRc = 3.81× 106.

Figure 26.N-factors for stationary crossflow vortices atα = −2° andδf = 0° in UWT atRc = 3.81× 106.

111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 122

2

22 2 2

2

2 2 2 2 2 2 2 2 2 2 2333

3

3

33

3 3 33

3

3

33 3 3 3 344

4

4

4

4

4

4

4

44

44

44

44 4

555

5

5

5

5

5

5

5

5

5

5

5

5

5

5

66 66

66

6

6

6

6

6

6

6

6

6

6

66

777

77

7

7

7

7

7

7

7

7

7

7

7

7

88 88

88

8

8

8

8

8

8

8

8

88

99 9 99

99

99

9

9

9

9

9

99

aa a a aa

aa

aa

aa

aa

a

bb b b bb

bb

bb

bb

bb

c c c c c cc

cc

cc

cc

d d d d d dd

dd

dd

d

e e e e ee

ee

ee e

ff f f f ff

ff

f f

g g g g g g gg

g g

123456789abcdefg

0 .20 .40 .60 .80 1.00

15

10

5

0.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.02000.02250.02500.0275

x/c

λ/c

N(x

/c)

1111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1222

2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2333

33

3 3 33

3 3 3 3 3 3 3 3 3 3 3444

44

44 4 4 4

44

44 4 4 4 4 4555

55

55

55

55

55

55 5 5 5 5

66 66

66

66

66

66

66

66

66

77 7 7 77

77

77

77

7

77

77

7

88 8 8 88

88

88

88

88

88

8

99 9 9 99

99

99

99

99

99

aa a a a aa

aa

aa

aa

aa

bb b b b bb

bb

bb

bb

b

cc c c c c cc

cc

cc

c

dd d d d d d dd

dd d

e e e e e e e e e e e

f f f f f f f f f f

g g g g g g g g g

123456789abcdefg

0 .20 .40 .60 .80 1.00

15

10

5

0.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.02000.02250.0250

x/c

λ/c

N(x

/c)

60

Figure 27.N-factors for stationary crossflow vortices atα = 0° andδf = 0° in UWT atRc = 3.81× 106.


1111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12222 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 23333 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3444

4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4555

5 5 5 5 5 5 55

5 5 5 5 5 5 5 5 5666 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

77 7 7 7 7 7 7 77

77

77

77 7 7

88 8 8 8 8 8 88

88

88

88

8 8

9 9 9 9 9 9 9 99

99

99

99 9

a a a a a aa

aa

aa

a a

b b b b b b bb

bb b b

c c c c c c c c c c c

d d d d d d d d d d

e e e e e e e e e

ff f f f f f f fg g g g g g g g

123456789abcdefg

0 .20 .40 .60 .80 1.00

15

10

5

0.00060.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.01500.01750.02000.0225

x/c

λ/c

N(x

/c)

11111 122222 233333

34444 45555 5666 6 77 7 7 7 7 78 8 8 8 8 8 89 9 9 9 9 9 9

aa a a a a abb b b b bcc c c cd d d

123456789abcd

0 .20 .40 .60 .80 1.00

15

10

5

0.00060.00080.00100.00150.00200.00300.00400.00500.00600.00800.01000.01250.0150

x/c

λ/c

N(x

/c)

61


Figure 30. MaximumN-factors for stationary crossflow vortices atα = −4° andδf = 0° in UWT atRc = 3.81× 106.

111122223333444 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5666 7 77

7 7 7 7 7 7 7 7 7 7 7 7888

88

8 8 8 8 8 8 8 8 8 8

999

99

9 9 9 9 9 9 9 9 9 9

aa aa

a a a a a a a a a a a

bb b b b b b b b b b b b b b

c c c c c c c c c c c c c c

dd d d d d d d d d d d d dee e e e e e e e e e e ef f f f f f f f f f f f

123456789abcdef

0 .20 .40 .60 .80 1.00

15

10

5

0.00060.00080.00100.00150.00150.00200.00200.00300.00400.00500.00600.00800.01000.01250.0150

x/c

λ/c

N(x

/c)

0 .01 .02 .03 .04

15

10

5

λ/c

Nm

ax

62

Figure 31. MaximumN-factors for stationary crossflow vortices atα = −2° andδf = 0° in UWT atRc = 3.81× 106.

Figure 32. MaximumN-factors for stationary crossflow vortices atα = 0° andδf = 0° in UWT atRc = 3.81× 106.

0 .01 .02 .03 .04

15

10

5

λ/c

Nm

ax

0 .01 .02 .03 .04

15

10

5

λ/c

Nm

ax

63



0 .01 .02 .03 .04

15

10

5

λ/c

Nm

ax

0 .01 .02 .03 .04

15

10

5

λ/c

Nm

ax

64

Figure 35. MaximumN-factors for Tollmien-Schlichting waves forα = 0°, 2°, and 4° andδf = 0° in UWT atRc = 3.81× 106.

Figure 36. MaximumN-factors for stationary crossflow vortices atα = −4° andδf = 0° in UWT for a range of Reynoldsnumber.

0 1000 2000 3000 4000

4

2

0

15

10

5

f, Hz

Nm

ax

α, deg

3.81 × 106

3.002.502.00

0 .01 .02 .03 .04

15

10

5

λ/c

Rc

Nm

ax

65

Figure 37. Streamline traces of wind tunnel end liner onXL-ZL plane forα = −4°.

Figure 38. Lateral deflections of end-liner surface at various distances from wing chord plane forα = −4°.

–.60

–.40

–.20

0

.20

.40

.60–0.417–0.139 0 0.028 0.139 0.306

zL0

0 .50 1.00 1.50 2.00xL/c

z L/c

2.50 3.00 3.50

–.10

–.05

0

.05

.10

–0.417–0.139 0 0.028 0.139 0.306

zL0

0 .50 1.00 1.50 2.00xL/c

y L/c

2.50 3.00 3.50

66

Figure 39. End-liner contours inYL-ZL plane at various longitudinal positions forα = −4°.

–.50 –.25 0 .25 .50

01.0

2.172.01.5

.100

.050

0

–.050

–.100

xL/c

zL/c

y L/c

67

Figure 40. Wind tunnel test section with swept-wing model and end liners installed.

U∞

68

Figure 41. Free-stream velocity spectrum forRc = 3.27× 106.

Figure 42. Measured and predicted model pressure coefficients at upper end of model forα = −4°.

10–2

10–3

10–4

10–5

10–6

10–7

10–8

101 102 103

u rms, 1

/Hz

f, Hz

'

MeasuredPredicted

.40

–.40

–.600 .20 .40 .60 .80 1.00

.20

–.20

0

–Cp,

3

x/c

69

Figure 43. Measured and predicted model pressure coefficients at lower end of model forα = −4°.

MeasuredPredicted

.40

–.40

–.600 .20 .40 .60 .80 1.00

.20

–.20

0

–Cp,

3

x/c

70

Figure 44. Naphthalene flow visualization atα = −4° andRc = 1.93× 106.

71


72


73


74


75

Figure 49. Naphthalene flow visualization with vortex tracks in turbulent regions shown.

76

Figure 50. Liquid-crystal flow visualization.

Figure 51. Transition location versus Reynolds number atα = −4°.

BlueGreen/black

1.5 2.0 2.5 3.0 3.5 4.0 × 1060

.20

.40

.60

.80

1.00

NaphthaleneHot wiresHot films

(x/c

) tr

Rc

77

Figure 52. Boundary-layer velocity spectrum atα = −4° andRc = 2.62× 106 atx/c = 0.40.


10–2

10–3

10–4

10–5

10–6

10–7

10–8

101 102 103

u rms, 1

/Hz

f, Hz

fbp'

10–2

10–3

10–4

10–5

10–6

10–7

10–8

101 102 103

u rms, 1

/Hz

f, Hz

fbp

fcf,max fbps

'

78



10–2

10–3

10–4

10–5

10–6

10–7

10–8

101 102 103

u rms, 1

/Hz

f, Hz

fbp

fcf, max fbps

2fbp

'

10–2

10–3

10–4

10–5

10–6

10–7

10–8

101 102 103

u rms, 1

/Hz

f, Hz

'

79

Figure 56. Measured and predicted boundary-layer velocity spectra atα = −4° andRc = 2.92× 106.

Figure 57. Streamwise velocity profiles atx/c = 0.20,α = −4°, andRc = 2.62× 106.

10–2

10–3

10–4

10–5

10–6

10–7

10–8

101 102 103

u rms, 1

/Hz

f, Hz

fbp

fbps

2fbp'

4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

80



4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

0.833

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

81



4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

0.833

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

0.833

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

82



4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

0.833

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

0.833

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

83


Figure 65. Stationary crossflow disturbance velocity profiles forf = 0 Hz atx/c = 0.20,α = −4°, andRc = 2.37× 106 obtainedfrom

4

3

2

1

0 .2 .4

0

0.167

0.333

0.500

0.667

0.833

.6us/us,e

.8 1.0 1.2

zλ

y, m

m

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.667

.50

zλ

y, m

m

(us – us,avg)/us,e

us us,avg.–

84



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,avg.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.667

.50

zλ

y, m

m


us us,avg.–

85



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,avg.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,avg.–

86



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,avg.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,avg.–

87



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,avg.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.667

(us – us,ref)/us,e

.50

zλ

y, m

m

us us,ref.–

88



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,ref.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.667

.50

zλ

y, m

m


us us,ref.–

89



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,ref.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,ref.–

90



4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,ref.–

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,ref.–

91


Figure 81. Travelling wave disturbance velocity profiles forf = 100 Hz atx/c = 0.20,α = −4°, andRc = 2.37× 106.

4

3

2

1

0–.25 .250–.50

00.1670.3330.5000.6670.833

.50

zλ

y, m

m


us us,ref.–

4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

zλ

y, m

m

us,rms/us,e

92



4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

0.833

zλ

y, m

m

us,rms/us,e

4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

zλ

y, m

m

us,rms/us,e

93



4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

0.833

zλ

y, m

m

us,rms/us,e

4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

0.833

zλ

y, m

m

us,rms/us,e

94



4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

0.833

zλ

y, m

m

us,rms/us,e

4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

0.833

zλ

y, m

m

us,rms/us,e

95


Figure 89. Mean streamwise velocity contours atx/c = 0.20,α = −4°, andRc = 2.37× 106.

4

3

2

1

0 .002 .004 .006 .008 .010

0

0.167

0.333

0.500

0.667

0.833

zλ

y, m

m

us,rms/us,e

2 345 56 678

9AB CDE FGHI

JJ

K

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

L

L

L

L

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevel

y, m

m

zλ, mm

96



2 234 45

6 67 789 9A AB BC CDEEFG GH

H

IJ J

KK

L

L

L

L L

L

L

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevely,

mm

zλ, mm

D

35

8

F

2 23 34455667

789A

ABC

CDEFG

G

H

I

J

K

L

L

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevel

y, m

m

zλ, mm

97



2 23

4 45 56 67 7

89 9

AB BC

D DE

EF

F

G

G

HI

I

J

J

K

K

LL

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevely,

mm

zλ, mm

C

H

A8

3

23 34 45 56 6778 8

9 9

AB

CCD

DE

EF

FG

GH

H

I

I

J

J

K

L

L L

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevel

y, m

m

zλ, mm

BA

2

K

98



2334 4

5678

9

9AA

BC

C

D

DE

E

F

F

G G

H

H

H

I

IJJ

J

K

K

L

L L

L

L L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevely,

mm

zλ, mm

2568

B

2 2334

45

56

67

78

89 9A

A

B

B

C

C

D

D

E

E

F

F

G

G

G

H

H

H

I

I

IJ

J

J

K

K

L

LL

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevel

y, m

m

zλ, mm

99


Figure 97. Stationary crossflow vortex velocity contours obtained from atx/c = 0.20,α = −4°, andRc = 2.37× 106.

2 234

45

56 67

7

8

8

9

9

AA

B

B

C

C

D D

E

E

F

F

FG

G

G

H

H

H

HI

I

I

I

J

J

K

K

L L

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

1.000.950.900.850.800.750.700.650.600.550.500.450.400.350.300.250.200.150.100.050.00

u/ueLevely,

mm

zλ, mm

K

3

A

B

B

B

B

B

B

BB

C

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevel

y, m

m

zλ, mm

us us,avg–

100



A

A

B

B

B

B

B

B

B B

B

B B

C

C

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevely,

mm

zλ, mm

us us,avg–

8

9A

A

B

B

B

B

B

B

B

C

C

D

E

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevel

y, m

m

zλ, mm

us us,avg–

101



8

9

9

9

A

A

A

A

B

BB

B

B

B

B

B

BB

B

B

C

C

C

D

D

D

E

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevely,

mm

zλ, mm

us us,avg–

6 7

7

8

8

9

9

9

A

A

A

A

B

B

B

B

B

B

B

B

B

BB

B

B

C

C

C

C

D

D

E

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevel

y, m

m

zλ, mm

us us,avg–

102



6

6

7

7

7

8

8

8

9

9

9

9

A

A

A

A

A

B

B

B

B

B

B

B

B

BB

C

C

C

C

C

D

D

D

E

E

EF

F

F

G

G

GH

H

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevely,

mm

zλ, mm

us us,avg–

3

4

4

4

5

5

5

6

6

6

7

7

7

78

88

8

9

9

9

9

9

A

A

A

A

A

B

B

B

B

B

B

B

C

C

C

CD

D

D

DE

E

E

E

F

F

F

G

G

H

H

II J

K

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevel

y, m

m

zλ, mm

us us,avg–

103



1

2 3

3

4

4

4

5

5

5 6

6

6

7

7

7

7

8

8

8

8

8

8

9

9

9

9

9

A

A

A

A

A

A

B

B

B

BB B

B

B

B

B

C

C

C

C

C

C

D

D

D

D

DE

E

E

E

F

F

F

F

G

G

G

H

H

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.200.180.160.140.120.100.080.060.040.020.00

–0.02–0.04–0.06–0.08–0.10–0.12–0.14–0.16–0.18–0.20

(ucf)avgLevely,

mm

zλ, mm

us us,avg–

9

A

AA A

AA

B

B

B

B

B

B

B

C

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevel

y, m

m

zλ, mm

us us,ref–

104



9

A

A

A

A

A A

B

B

B

B

B

B

B

B

B B

B

B

B B

B

C

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevely,

mm

zλ, mm

us us,ref–

8

9

A

A

B

B

BB

B

BB

B

C

C

D

D

E

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevel

y, m

m

zλ, mm

us us,ref–

105



77

8

8

9

9

9

9

A

A

AA

A

A

A

B

B

B

B

B B

B

B

B

C

C

C

D

DE

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevely,

mm

zλ, mm

us us,ref–

78

99

9 A

A

A

A

A

B

B B

B

B

B

B

B

B

B

B B

C

C

C

C

D

D

E

E

F

G

H

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevel

y, m

m

zλ, mm

us us,ref–

106



3445

5

6

6

7

7

7

8

8

8

8

9

9

99

9

9

A

A

A

A

A

A

B

B

B

B

B

B

B

B B

C

C

C

C

D

D

D

D

E

E

E

F

FG G

H

H

I

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevely,

mm

zλ, mm

us us,ref–

2

3

4

4

5

5

6

6

6 7

7

7

78

8

8

8

9

9

9

9

A

A

A

A

AB

B

B

B

B

B

B

C

C

C

C

C

D

D

D

E

E

E

EF

F

G

G

G

H

H I

I

J

K

KL

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevel

y, m

m

zλ, mm

us us,ref–

107


Figure 113. Travelling wave rms velocity contours forf = 100 Hz atx/c = 0.20,α = −4°, andRc = 2.37× 106.

2

23

3

3

3

4

4

4

4

5

5

5

5

6

6

6

6

7

7

7

7

7

7

8

8

88

8

89

9

9

99

9

A

A

A

A

A

A

B

B

B

B

B

BC

C

C

C

CD

D

D

E

E

E

F

F

F

G

G

GH

I

I

J

L

K

J

I

H

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.2000.1800.1600.1400.1200.1000.0800.0600.0400.0200.000

–0.020–0.040–0.060–0.080–0.100–0.120–0.140–0.160–0.180–0.200

(ucf)refLevely,

mm

zλ, mm

us us,ref–

2

2

2

2

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevel

y, m

m

–zm/λ45

108



22

2 2

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevely,

mm

–zm/λ45

2 2

2

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevel

y, m

m

–zm/λ45

109



2

2

22

22

3

3

3

3

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevely,

mm

–zm/λ45

3

2

2

2 2

33

33

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevel

y, m

m

–zm/λ45

110



2

2

2

3

3

3 3

4

4

4

4

5

5

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevely,

mm

–zm/λ45

2

2

2 2

3

3

33

3

4

4

4

4

4

4

5

5

5

5

5

6

6

6

6

6

7

7

7

7

8

8

8

8

9

9

9

A

A

A

BB

C

C

DD EE

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevel

y, m

m

–zm/λ45

8

111


Figure 121. Theoretical mean chordwise velocity profiles forαref = −5° andRc = 2.37× 106. Theory from reference 126.

2

2

2

2 2

33

3

3 3

4

4

4

4

4

4

44

5

55

5

6

66

7

G

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

4

3

2

1

0 2 4 6 8 10

0.00700.00650.00600.00550.00500.00450.00400.00350.00300.00250.00200.00150.00100.00050.00010.0000

urmsLevely,

mm

–zm/λ45

5

4

3

2

1

0 .2 .4

0.200.250.300.350.400.450.500.55

.6um/ut,e

.8 1.0 1.2

x/c

y, m

m

112

Figure 122. Theoretical mean spanwise velocity profiles forαref = −5° andRc = 2.37× 106. Theory from reference 126.

Figure 123. Theoretical stationary crossflow disturbance velocity profiles (chordwise component) forαref = −5°andRc = 2.37× 106. Theory from reference 128.

4

3

2

1

0–1.00–1.20 –.80

0.200.250.300.350.400.450.500.55

–.60wm/ut,e

–.40 –.20 0

x/c

y, m

m

4

3

2

1

0 .05

0.200.250.300.350.400.450.500.55

.10um/ut,e

.15 .20

x/c

y, m

m

113

Figure 124. Theoretical stationary crossflow disturbance velocity profiles (surface normal component) forαref = −5°andRc = 2.37× 106. Theory from reference 128.

4

3

2

1

0 .002

0.200.250.300.350.400.450.500.55

.004vm/ut,e

.006 .008 .010

x/c

y, m

m

114

Figure 125. Theoretical stationary crossflow disturbance velocity profiles (spanwise component) forαref = −5° andRc = 2.37× 106. Theory from reference 128.

Figure 126. Theoretical mean streamwise velocity profiles forαref = −5° andRc = 2.37× 106. Theory from reference 126.

4

3

2

1

0 .05

0.200.250.300.350.400.450.500.55

.10wm/ut,e

.15 .20

x/c

y, m

m

4

3

2

1

0 .20 .40

0.200.250.300.350.400.450.500.55

.60us/ut,e

.80 1.00 1.20

x/c

y, m

m

115

Figure 127. Theoretical mean cross-stream velocity profiles forαref = −5° andRc = 2.37× 106. Theory from reference 126.

Figure 128. Theoretical stationary crossflow disturbance velocity profiles (streamwise component) forαref = −5°andRc = 2.37× 106. Theory from reference 128.

4

3

2

1

0 .20 .40

0.200.250.300.350.400.450.500.55

.60ws/ut,e

.80 1.00 1.20

x/c

y, m

m

4

3

2

1

0 .05

0.200.250.300.350.400.450.500.55

.10us/ut,e

.15 .20

x/c

y, m

m

116

Figure 129. Theoretical stationary crossflow disturbance velocity profiles (cross-stream component) forαref = −5°andRc = 2.37× 106. Theory from reference 128.

Figure 130. Theoretical mean velocity profiles along vortex axis forαref = −5° andRc = 2.37× 106. Theory fromreference 126.

4

3

2

1

0 .05

0.200.250.300.350.400.450.500.55

.10ws/ut,e

.15 .20

x/c

y, m

m

4

3

2

1

0 .2 .4

0.200.250.300.350.400.450.500.55

.6uw/ut,e

.8 1.0 1.2

x/c

y, m

m

117

Figure 131. Theoretical mean velocity profiles perpendicular to vortex axis forαref = −5° andRc = 2.37× 106. Theory fromreference 126.

Figure 132. Theoretical stationary crossflow disturbance velocity profiles along vortex axis forαref = −5° andRc = 2.37× 106.Theory from reference 128.

4

3

2

1

0–.40–.60 –.20

0.200.250.300.350.400.450.500.55

0ww/ut,e

.20 .40 .60

x/c

y, m

m

4

3

2

1

0 .05

0.200.250.300.350.400.450.500.55

.10uw/ut,e

.15 .20

x/c

y, m

m

118

Figure 133. Theoretical stationary crossflow disturbance velocity profiles perpendicular to vortex axis forαref = −5°andRc = 2.37× 106. Theory from reference 128.

Figure 134. Theoretical stationary crossflow disturbance velocity vectors across single vortex wavelength forαref = −5°andRc = 2.37× 106.

4

3

2

1

0 .05

0.200.250.300.350.400.450.500.55

.10ww/ut,e

.15 .20

x/c

y, m

m

0 .2 .4 .6zw/λ

y, m

m

.8 1.0

1

2

3

4

119

Figure 135. Theoretical total velocity vectors (disturbance plus mean flow) across single vortex wavelength forαref = −5°andRc = 2.37× 106.

Figure 136. Theoretical total velocity vectors (disturbance plus mean flow) across single vortex wavelength with normalvelocity components scaled 100 forαref = −5° andRc = 2.37× 106.

0 .2 .4 .6zw/λ

y, m

m

.8 1.0

1

2

3

4

0 .2 .4 .6zw/λ

y, m

m

.8 1.0

1

2

3

4

120

Figure 137. Experimental streamwise disturbance velocity profile determined from and theoretical eigenfunctionfor x/c = 0.20,α = −4°, andRc = 2.37× 106.


4

3

2

1

0 .05 .10us/us,e

.15 .25.20

y, m

mExperiment from rms [(us - us,avg)/us,e]Theory, reference 128

us us,avg–

4

3

2

1

0 .05 .10us/us,e

.15 .25.20

y, m

m

Experiment from rms [(us - us,avg)/us,e]Theory, reference 128

us us,avg–

121



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0 .05 .10us/us,e

.15 .25.20

y, m


us us,avg–

4

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0 .05 .10us/us,e

.15 .25.20

y, m

m


us us,avg–

122



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0 .05 .10us/us,e

.15 .25.20

y, m


us us,avg–

4

3

2

1

0 .05 .10us/us,e

.15 .25.20

y, m

m


us us,avg–

123



4

3

2

1

0 .05 .10us/us,e

.15 .25.20

y, m

m


us us,avg–

4

3

2

1

0 .05


.10us/us,e

.15 .25.20

y, m

m

us us,avg–

124

Figure 145. Experimental mean streamwise velocity contours and theoretical vortex velocity vector field forx/c = 0.20,α = −4°, andRc = 2.37× 106.


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128

Figure 153. Experimental stationary crossflow disturbance velocity contours and theoretical vortex velocity vector field forx/c = 0.20,α = −4°, andRc = 2.37× 106.


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Figure 161. Theoretical and experimental stationary crossflow vortex wavelengths forα = −4° andRc = 2.37× 106.

Figure 162. Theoretical and experimental stationary crossflow vortex growth rates forα = −4° andRc = 2.37× 106.

.006

.004

.005

.002

.003

.001

02.01.5

Theory, reference 45Experiment

2.5Rc

3.0 4.0 × 1063.5

λ/c

25

20

15

10

5

0

–5.200

σ1 from us – us,avg

.40x/c

.60 1.00.80

σ

σ1 from us – us,avgσ1 from us – us,avgσ1 from us – us,refσ1 from us – us,refσ1 from us – us,refTheory, MARIA CodeTheory, reference 128

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July 1999 Technical Publication

Crossflow Stability and Transition Experiments in Swept-Wing FlowWU 522-31-11-03

J. Ray Dagenhart and William S. Saric

L-17658

NASA/TP-1999-209344

Dagenhart: Langley Research Center, Hampton, VA; Saric: Arizona State University, Tempe, AZ.

An experimental examination of crossflow instability and transition on a 45° swept wing was conducted in theArizona State University Unsteady Wind Tunnel. The stationary-vortex pattern and transition location are visual-ized by using both sublimating chemical and liquid-crystal coatings. Extensive hot-wire measurements wereobtained at several measurement stations across a single vortex track. The mean and travelling wave disturbanceswere measured simultaneously. Stationary crossflow disturbance profiles were determined by subtracting either areference or a span-averaged velocity profile from the mean velocity data. Mean, stationary crossflow, and travel-ling wave velocity data were presented as local boundary layer profiles and contour plots across a single stationarycrossflow vortex track. Disturbance mode profiles and growth rates were determined. The experimental data arecompared with predictions from linear stability theory.

Swept-wing transition; Experiments; Crossflow instability; Natural laminar flow 150

A07

NASA Langley Research CenterHampton, VA 23681-2199

National Aeronautics and Space AdministrationWashington, DC 20546-0001

Unclassified–UnlimitedSubject Category 34 Distribution: StandardAvailability: NASA CASI (301) 621-0390

Unclassified Unclassified Unclassified UL

crossﬂow stability and transition experiments in swept-wing...

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