numerical investigation of laminar-turbulentnumerical investigation of laminar-turbulent transition...
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NUMERICAL INVESTIGATION OF LAMINAR-TURBULENT
TRANSITION IN A FLAT PLATE WAKE
by
David I. Dratler and Hermann F. Fasel D T I CE ECTEMAY 2 5 1990
g Prepared from work done under Grant .
ONR-N00014-85-K-0412
t Appzoved ioi pu.zkc telease |Disui.w-qU0 Unlimited
Department of Aerospace and Mechanical Engineering
University of Arizona
Tucson, Arizona 85721
February 1990
0 05 22 12,
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Department of Aerospace & Mechanical Engrg.I Tucson, Arizona 85721
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L Taminar-turbulent transition of high-deficit flat ~plate wakes is investigated byI direct numerical simulations using the complete Navier. Stokes equations. The simulationsare based on a spatial model so that both the base flow and the disturbance flow candevelop in the downstream direction. The Navier4 Stokes equations are used in a vorticity-velocity form and are solved using a combination of finite.9difference and spectralI approximations. Fourier series are used in the spanwise direction. Second-order finite-differences are used to approximate the spatial derivatives in the streamwise and trans-verse directions. For the temporal discretion, a combination of ADI, Crank-Nicolson,Iand Adams-Rashforth methods is employed. The discretized velocity equations are solvedusing fast Helmholtz solvers. Code validation is accomplished by comparison of thenumerical results to both linear stability theory and to experiments.-u .. <
3 (over)20L OISTRIBUTION/AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATIONI UNCLASSIFIED/UNLIMITEDZ SAME AS RPT. 0 OTIC USERS CN/A22s. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE NUMBER 22t. OFFICE SYMBOL
(include Area Code)IDr. Michael M. Reischmann Code:N00014 1(202) 696-4406 N
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-Calculations of two- and/or three-dimensional sinuous mode disturbances in the
wake of a flat plate are undertaken. For calculations of two-dimensional disturbances,the wake is forced at an amplitude level so that nonlinear disturbance development maybe observed. In addition, the forcing amplitude is varied in order to determine itseffect on the disturbance behavior. To investigate the onset of three-dimensionality,the wake is forced with a small-amplitude three-dimensional disturbance and a largeramplitude two-dimensional disturbance. The two-dimensional forcing amplitude is variedin order to determine its influence on the three-dimensional flow field. /
Two-dimensional disturbances are observed to grow exponentially at small amplitudelevels. At higher amplitude levels, nonlinear effects become important and the distur-bances saturate. The saturation of the fundamental disturbance appears to be relatedto the stability characteristics of the mean flow. Large forcing amplitudes result inthe earlier onset of nonlinear effects and saturation. At large amplitude levels, aKarman vortex street pattern develops.
When the wake is forced with both two- and three-dimensional disturbances stronginteraction between these disturbances is observed. The saturation of the two-dimen-sional disturbance causes the three-dimensional disturbance to saturate. However, thisis followed by a resumption of strong three-dimensional growth that may be due to asecondary instability mechanism. Larger two-dimensional forcing amplitudes acceleratethe saturation of the two-dimensional and three-dimensional disturbances as well asaccelerate the resumption of strong three-dimensional growth. These interactionsalso result in complicated distributions of vorticity and in a significant increasein the wake width.
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I ACKNOWLEDGEMENTS
I This work was carried out in the transition research group in the Department
of Aerospace and Mechanical Engineering at the University of Arizona. Support
was also provided by the NASA Graduate Student Researchers Program (NASA
Grant Number NGT-50074) and by the Office of Naval Research (ONR Grant
Number N00014-85-K-0412).
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TABLE OF CONTENTSUPage
LIST OF ILLUSTRATIONS ............................... 7
3 ABSTRACT ..................................... .. 16
1. INTRODUCTION ................................ 18
1.1 The Linear Region of Transition ... ............... .... 191.2 The Nonlinear Region of Transition ..... .............. 21
1.3 The Three-Dimensional Region of Transition .... ......... 25
3 2. PROBLEM STATEMENT ..... .................... .. 26
3. GOVERNING EQUATIONS ....... ................... 29
3.1 The Navier-Stokes Equations ....................... 293 3.2 Nondimensionalization ..... ................... .. 31
4. BOUNDARY AND INITIAL CONDITIONS .............. .. 34
1 4.1 Spatial Domain ....... ...................... .. 344.2 Temporal Domain ............................. 354.3 Boundary and Inital Conditions for the Undisturbed Flow . . . . 354.4 Boundary and Inital Conditions for the Disturbed Flow ..... .. 41
U 5. NUMERICAL METHOD ...... .................... .. 50
3 5.1 Spanwise Spectral Approximation ... .............. ... 505.1.1 Fourier Representation of the Boundary and Initial
Conditions for the Undisturbed Flow .... ......... 53
5.1.2 Fourier Representation of the Bounda-y and InitialConditions for the Disturbed Flow .... ........... 55
5.2 Computational Domain ..... .. .................. 60
5.3 Discretization of Spatial Derivatives ................ .. 613 5.4 Discretized Vorticity Equations ..... ............ . .. 61
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5.4.1 Discretized Boundary and Initial Conditions 3for the Undisturbed Vorticity ............ 65
5.4.2 Discretized Boundary and Initial Conditions 3for the Disturbed Vorticity ... ............. ... 66
5.4.3 Equation Systems for the Calculation of the Vorticity . 67
5.5 Discretized Velocity Equations ....... ............... 71
5.5.1 Discretised Boundary and Initial Conditionsfor the Undisturbed Velocity ...... ............ 72 i
5.5.2 Discretized Boundary and Initial Conditions
for the Disturbed Velocity .... .............. .. 74 35.5.3 Equation Systems for the Calculation of the Velocity . 77
5.6 Evaluation of the Nonlinear Terms ................... 87 35.7 Consistency, Stability, and Convergence of the
Numerical Method ..... ................. .. 89 36. RESULTS ................................... .. 92
6.1 Investigations of Two-Dimensional Disturbance Development 95 16.2 Investigations of Three-Dimensional Disturbance Development . 115
7. CONCLUSIONS ............................... .. 128
FIGURES ...................................... .131 3APPENDIX A: SOLUTION OF THE THREE-DIMENSIONAL 3
ORR-SOMMERFELD EQUATION .... .......... 264
APPENDIX B: INFLUENCE OF THE GRID INCREMENT 3ON THE NUMERICAL SOLUTION .... ......... 273
REFERENCES ........ ........................... .. 277 13U3U
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U LIST OF ILLUSTRATIONS
I Figure Page
3 2.1 Spatial domain ....... ......................... 131
5.1 Discretization of the spatial computational domain ... ........ 132
6.1 Base flow streamwise velocity U° for Case-1 ............. .. 133
I 6.2 Sinuous mode eigenfunctions of the two-dimensional Orr-Sommerfeld
equation corresponding to / = .317 and Re 9 = 2&9=Uc = 594.a) amplitudes of U0 , V 0 , and nO; ..................... 134b) phases of U0 , V 0 , and 110, ... ................. ... 135
6.3 Comparison of amplification curves based on the kinetic energyE(z, F) to linear stability theory. Case-i: A2 = .001, A3 = 0,and/ =.317 ....... ........................ . 136
6.4 Comparison of the amplitude distributions of the fundamentaldisturbance (F - 1) to linear stability theory. Case-i: A 2 = .001,
SA 3 =0, and / .317.a) streamwise velocity, U;. 137b) transverse velocity, V'; . . . . . . . . . . . . . . . . . .. . . . . . . 138c) spanwise vorticity, no ..... ................... .. 139
6.5 Comparison of the phase distributions of the fundamental
disturbance (F = 1) to linear stability theory. Case-i: A2 = .001,A 3 = 0, and / = .317.a) streamwise velocity, U0 ; .................. 140b) transverse velocity, V°; . . . . . . . . . . . . . . . . . . . . . . . . 141c) spanwise vorticity, fl ....................... ... 142
1 6.6 Comparison of the streamwise variation of a,., aj, and c. for thefundamental disturbance (F = 1) to linear stability theory.3 Case-i: A 2 = .001, A 3 = 0, and P9= .317.a) streamwise wavenumber, a,.; ...... ................ 143b) amplification rate, a,; ...... .. .................. 144c) phase velocity, c ...... ..................... .. 145
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6.7 Comparison of a Gaussian streamwise velocity distribution to Iexperimental data and to the similarity solution of Goldstein (1929),reproduced from Sato and Kuriki (1961) .............. .. 146 3
6.8 Base flow streamwise velocity U° for Case-2 ............. .147
6.9 Comparison of the base flow centerline streamwise velocity U0 Ifor Case-2 to the similarity solution of Goldstein (1929) ....... 148
6.10 Orr-Sommerfeld eigenvalues for the sinuous mode (mode-I) and the Uvaricose mode (mode-2), corresponding to the streamwise velocitydistribution at the inflow boundary (Case-2) and 3Re= =594.a) streamwise wavenumber, a,.; ... ..... ........... . 149b) amplification rate, aj; ...... .................. .150 Ic) phase velocity, cp ...... ..................... .. 151
6.11 Sinuous mode eigenfunctions of the two-dimensional Orr-Sommerfeld Iequation corresponding to / = .51 and Re@ = I = 594.
a) amplitudes of U', V 0 , and 11'; ... ............... .152b) phases of U', V* , and 11 ....... ................. 153
6.12 Comparison of streamwise velocity amplitude distributions 3of the fundamental disturbance (F = 1) to experimental data.a) Case-2: A2 = .000667, A3 = 0, and j = .51; .......... .. 154b) experimental, reproduced from Sato (1970) ........... .154
6.13 Comparison of the streamwise variation of the mean streamwisecenterline velocity U° and the mean wake half-width to Iexperimental data.a) Case-2: A 2 = .000667, As = 0, and 3 = .51; .......... .. 155b) experimental, reproduced from Sato (1970) ........... .155
6.14 Contours of U0 , V ° , and fl for t = 2176A. Case-3: A 2 = .001,As = 0, and P = .51. Twenty contour intervals between theminimum and maximum values. Solid lines denote positivevorticity and dashed lines denoted negative vorticity. 3a) streamwise velocity, U0 ; .................. 156b) transverse velocity, V'; . . . . . . . . . . . . . . . . . . . . . . . . 157
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1 6.15 Contours of spanwise vorticity f Ol for t = 2176At. Case-3:A 2 = .001, A3 = 0, and / = .51. Thirty contour intervals betweenthe minimum and maximum values. Solid lines denote positivevorticity and dashed lines denote negative vorticity.a) 0 < z/Az < 512 ...... ..................... .. 1593 b)0< z/Az 4..............................6b0 < / x< 340 . . . . . . . . . . . . . . . . . . . . .160
6.16 Contours of spanwise vorticity Wo for t = 2176At. Case-3:A2 = .001, A3 = 0, and / = .51. Thirty contour intervals betweenthe minimum and maximum values. Solid lines denote positive
*vorticity and dashed lines denote negative vorticity.3 a) 0 < z/A < 128 ...... ..................... .. 161
b) 128 < z/Az < 256 ...... .................... . 162c) 256 < z/Az < 384 ......... .................... 163d) 384 < z/Ax < 512 ...... .................... . 164
6.17 Amplification curves of the kinetic energy E(z,F) obtained fromthe time interval 13TF < t < 15TF. Case-3: A2 = .001, A 3 = 0,and,3 = .51 .................................. 165
1 6.18 Temporal variation of the centerline transverse velocity V0 in the
time interval 13TF < t < 15TF. Case-3: A 2 = .001, As = 0,and 8 = .51............................... . 166
6.19 Streamwise variation of the centerline streamwise velocity U° for themean flow and the base flow. Case-3: A2 = .001, A3 = 0, and0 = .51 .................................... .. 167
6.20 Streamwise variation of the wake half-width b for the mean flowand the base flow. Case-3: A2 = .001, A3 = 0, and 0 = .51 . . . . 168
6.21 Streamwise variation of a,. and cp for the fundamental disturbancecomponent (F = 1). Case-3: A2 = .001, A3 = 0, and # = .51.a) streamwise wavenumber, a,.; .... ................ . 169b) phase velocity, cp .......................... 170
3 6.22 Amplification curves of the kinetic energy tA (z, F)for Case-3 (A2 = .001) and Case-4 (A2 = -. 01).a) F=0 ....... ......................... .. 171b) F=19 ................................ . 172c) F =2 ................................ .. 173I
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6.23 Comparison of the mean centerline streamwise velocity U 0 10
for Case-3 (A 2 = .001) and Case-4 (A 2 = -. 01) ........... .. 174
6.24 Comparison of the streamwise variation of the mean half-width bfor Case-3 (A 2 = .001) and Case-4 (A2 = .01) ........... .. 175 1
6.25 Comparison of the base flow centerline streamwise velocity U°
for Case-3 (N = 1024) and Case-5 (N = 1536) ........... .176 36.26 Contours of spanwise vorticity 1Z1 for Case-5. A 2 = .001, A 3 = 0,
- .51. Solid lines denote positive vorticity and Idashed lines denote negative vorticity.a) t = 2176At and 0 < z/AZ < 1024, twenty contour intervals; . 177b) t = 3200At and 0 < z/Az < 1536, ten contour intervals . . . . 178
6.27 Comparison of amplification curves of the kinetic energy E°(x, F)
for Case-3 (N = 1024) and Case-5 (N = 1536). Ia) F = 0 ......... ......................... 179b) F = 1. ....... .. ......................... 180c) F =2 .... ... ... ......................... 181
6.28 Amplification curves of the kinetic energy E°(x, F) obtained fromthe time interval 23TF < t < 25TF. Case-5: A 2 = .001, A 3 = 0,and/ = .51 ..... ... ........................ .182
6.29 Temporal variation of the centerline transverse velocity V0 in the mtime interval 23TF < t < 25TF. Case-5: A 2 = .001, A3 = 0,and/3 = .51 ..... ... ........................ .183 3
6.30 Sinuous mode eigenfunctions of the three-dimensional Orr-Sommerfeld equation corresponding to / - .28, y = .5, and
Ree = 1 = 594.a) amplitudes of Ul, V 1 , and W 1 ; .... .............. .184b) amplitudes of", fl , and fl; ............... 185c) phases of U1 , V1 , and W1 ;. ..................... .. 186d) phases of fW, fW , and . . .... . ......... ..... ... 187
6.31 Comparison of amplification curves of the kinetic energy E 1 (z, F)to linear stability theory. Case-6: A 2 = 0, A3 = .001, 3 = .28, mand 7 = .5 ........ ......................... .188
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3 6.32 Comparison of the amplitude distributions of the fundamentaldisturbance (F = 1, k = 1) to linear stability theory. Case-6:A2 = 0, A 3 = .001, 3- .28, and - = .5.a) streamwise velocity, U1 ; . . . . . . . . . . . . . . . . . . . . . . . . 189b) transverse velocity, VI; ..... .................. . 190c) spanwise velocity, W'..... ..................... 191
6.33 Comparison of the phase distributions of the fundamentaldisturbance (F = 1, k - 1) to linear stability theory. Case-6:A 2 = 0, A 3 = .001, 3 - .28, and 7 = .5.a) streamwise velocity, U; . . . . . . . . . . . . . . . . . . . . . . . . 192b) transverse velocity, V'; . . .. .. . . . . . . . . . . . . 193c) spanwise velocity, W' ...... ................... .. 194
I 6.34 Sinuous mode eigenfunctions of the three-dimensional Orr-Sommerfeld equation corresponding to3 = .51, -f = .5,and Re0 - _ - 594.a) amplitudes of U', V 1, and W1; .... .............. .. 195b) amplitudes of I, f, and il.... ............... 196
c) phases of U1 , V', and W1; 197d) phases of fl, 1, and fl' ..... ................. .. 198
3 6.35 Amplification curves of the kinetic energy tl (z, F). Case-7:A 2 = 0, A3 = .001, 3 = .51, and 7 = .5.a) 6TF < t < 8TF ........ .................... .. 199b) 8TF < t < OTF ...... ..................... .. 200
6.36 Contours of streamwise vorticity w. in the yz-plane for t = 4096At.Case-7: A 2 = 0, A3 = .001,13 = .51, and 7 = .5. Twentycontour intervals between the minimum and maximum values.3 Solid lines denote positive vorticity and dashed lines denotenegative vorticity.a) z = .030 ........ ........................ .201b) z =.146 ............................... .. 202
6.3' Contours of streamwise vorticity w. in the yz-plane, reproducedfrom Meiburg and Lasheras (1988) .... .............. .. 203
6.38 Amplification curves of the kinetic energy Ek(2 , 1). Case-8:A2 = .01, As = .001,/3 =.51, and 7 .5 ...... ........... 204
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6.39 Amplification curves of the kinetic energy tkk(z, 1). Case-9: 3A 2 = .05, A 3 = .001, P = .51, and -y = .5 ............ 205
6.40 Auiplification curves of the kinetic energy 1 (z, F). Case-9: IA 2 = .05, A 3 = .001,)3 = .51, and -y = .5.a) N = 256 ........... ........................ 206b) N =512 ........ ......................... .207
6.41 Temporal variation of two-dimensional and three-dimensionaldisturbance kinetic energy in a free shear layer, reproducedfrom Metcalfe et. al (1987) ....... ................. 208
6.42 Contours of streamwise vorticity w. in the yz-plane for t = 4096At.z = .146. Twenty contour intervals between the minimum andmaximunm values. Solid lines denote positive vorticity and dashed 3lin, £ denote negative vorticity.a) Case-8, A 2 = 0.01 and A3 = 0.001; ... ............. .. 209b) Case-9, A 2 = 0.05 and A3 = 0.001 ... ............. .. 210
6.43 Contours of spanwise vorticity w. in the yz-plane for t = 4096At.x = .146. Twenty contour intervals between the minimum and Imaximum values. Solid lines denote positive vorticity and dashedlines denote negative vorticity.a) Case-7, A 2 = 0.00 and A3 = 0.001; ... ....... ...... .. 211 Ib) Case-8, A 2 = 0.01 and A3 = 0.001; ............. 212c) Case-9, A 2 = 0.05 and A 3 = 0.001 ..... ............. 213 3
6.44 Contours of streamwise vorticity w. in the zy-plane for t = 4096At.z = Ir/2. Twenty contour intervals between the minimum and
maximum values. Solid lines denote positive vorticity and dashedlines denote negative vorticity.a) Case-7, A 2 = 0.00 and A3 = 0.001; ... ............. .. 214 3b) Case-8, A 2 = 0.01 and A3 = 0.001; ... ............. .. 215c) Case-9, A 2 = 0.05 and A3 = 0.001 ..... ............. 216
6.45 Contours of spanwise vorticity w. in the zy-plane for t = 4096At.z = 0. Twenty contour intervals between the minimum and
maximum values. Solid lines denote positive vorticity and dashed Ilines denote negative vortic;ty.a) Case-7, A 2 = 0.00 and A3 = 0.001; ... ............. .. 217b) Case-8, A 2 = 0.01 and A 3 = 0.001; ... ............. .. 218c) Case-9, A 2 = 0.05 and A 3 = 0.001 ..... ............. 219 I
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1 136.46 Contours of spanwise vorticity w, in the zy-plane for t - 4096At
and z = 0. Case-8: A 2 = .01, A 3 = .001, 0 = .51, and = .5.Thirty contour intervals between the minimum and maximumvalues. Solid lines denote positive vorticity anddashed lines denote negative vorticity ..... ............. 220
1 6.47 Contours of spanwise vorticity w, in the zz-plane for t = 4096At.Case-8: A2 = .01, A3 = .001, 6 = .51, and -y = .5.Thirty contour intervals between the minimum and maximum3 values. Solid lines denote positive vorticity and dashed linesdenote negative vorticity.a) y = 2 (y/Ay = 72) ......... .................... 2213 b) y =-2 (y/Ay = 56) ...... .................... 222
6.48 Contours of streamwise vorticity w. in the zz-plane for t = 4096At.Case-8: A 2 = .01, A 3 = .001, 03 = .51, and - = .5.Thirty contour intervals between the minimum and maximumvalues. Solid lines denote positive vorticity and dashed linesdenote negative vorticity.a) y = 2 (y/Ay= 72) ......... .................... 223b) y =-2 (y/Ay =56) ...... ................... .224
6.49 Contours of spanwise vorticity w. in the yz-plane for t = 4096At.Case-8: A2 = .01, A 3 = .001, / = .51, and - = .5.Thirty contour intervals between the minimum and maximumvalues. Solid lines denote positive vorticity and dashed linesdenote negative vorticity.a) z = .172 (x/A = 110) ...... .................. .225b) z = .160 (z/Az = 101) ...... .................. .226
1 6.50 Contours of streamwise vorticity w in the yz-plane for t = 4096At.Case-8: A 2 = .01, A3 = .001, # = .51, and 7 = .5.Thirty contour intervals between the minimum and maximumvalues. Solid lines denote positive vorticity and dashed linesdenote negative vorticity.a) z = .172 (z/Az = 110) ...... .................. .227b) z = .160 (z/Az = 101) ...... .................. .228
6.51 Vorticity amplitude distributions for F = 1. Case-7: A2 = 0,A 3 = .001, 0 = .51, and 7 = .5.a) spanwise vorticity, fl.; ....................... . 229b) spanwise vorticity, 0,; ......... .... .............. 230c) streamwise vorticity, fl_ ..... .................. . 231U
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6.52 Vorticity amplitude distributions for F = 1. Case-8: A2 = .01, 3A 3 = .001,)3 = .51, and -y = .5.a) spanwise vorticity, S1o,; ..... ................... .232 3b) spanwise vorticity, f1.; ..... ................... .233c) streamwise vorticity, IVl ..... .................. .234
6.53 Vorticity amplitude distributions for F = 1. Case-9: A2 = .05,A 3 = .001, P8 = .51, and -y = .5.a) spanwise vorticity, no; ...... ................... .235 Ib) spanwise vorticity, fl,; ...... ................... 236c) streamwise vorticity, f) ...... .................. .237 g
6.54 Streamwise variation of the wake half-width b for Case-4, A2 = 0.01,A 3 = 0.00; Case-7, A 2 = 0.00, A3 = 0.001; and Case-8, A 2 = 0.01,A 3 = 0.001 ....... ......................... .. 238
6.55 Streamwise variation of the wake half-width b for Case-8, A 2 = 0.01,
A 3 = 0.001; and Case-9, A 2 = 0.05, A3 = 0.001 ... ......... 239
B.1 Influence of the grid increment on the amplification of the kineticenergy E°(z, F = 1) for Case-1.a) M = 64 with N = 64, N = 128, and N 256; .......... .240b)IV = 128 with M = 32, M = 64, and M =128 .......... .241
B.2 Influence of the grid increment on the amplitude distributions of thefundamental disturbance (F = 1) of the streamwise velocity Ifor Case-1.a) M = 64 with N = 64, N =128, and N = 256; .......... .242b) N= 128 withM=32, M=64, and M=128 .......... .243
B.3 Influence of the grid increment on the phase distributions of thefundamental disturbance (F = 1) of the streamwise velocityfor Case-1.a) M = 64 with N = 64, N =128, and N = 256; ... ........ 244
b) N = 128 with M = 32, M = 64, and M = 128 .......... .245
B.4 Influence of the streamwise grid increment on the amplificationof the kinetic energy k°(z, F). Case-2: M = 256 with
N = 512, N = 768, and N = 1024.a) the mean component, F = 0; ..... ................ .2463b) the fundamental component, F = 1; ............... .. 247c) the second harmonic, F = 2 .... ................ .. 248
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3 B.5 Influence of the transverse grid increment on the amplificationof the kinetic energy 2°(z,F). Case-2: N = 1024 withM = 128, M = 192, and M 256.a) the mean component, F 0; ... ................ .. 249b) the fundamental component, F = 1; .... ............. 250c) the second harmonic, F = 2 .... ................. 251
B.6 Influence of the streamwise grid increment on the amplitudedistributions of the streamwise velocity. Case-2: M = 256with N = 512, N = 768, and N = 1024.a) the mean component, F = 0; ...... ................ 2523 b) the fundamental component, F = 1; .......... ... 253c) the second harmonic, F = 2 ... ................ .. 254
3 B.7 Influence of the transverse grid increment on the amplitudedistributions of the streamwise velocity. Case-2: N = 1024with M = 128, M = 192, and M = 256.a) the mean component, F = 0; ...... ................ 255b) the fundamental component, F = 1; .... ............. 256
I c) the second harmonic, F = 2 .... ................ . 257
B.8 Influence of the streamwise grid increment on the phase distributionsof the streamwise velocity. Case-2: M = 256 withN = 512, N = 768, and N 1024.a) the mean component, F 0; ...... ................ 258b) the fundamental component, F = 1; .... ............. 259c) the second harmonic, F = 2 ... ................ .. 260
3 B.9 Influence of the transverse grid increment on the phase distributions
of the streamwise velocity. Case-2: N = 1024 withM = 128, M = 192, and M = 256.a) the mean component, F = 0; ... ................ .. 261b) the fundamental component, F = 1; .... ............. 262
I c) the second harmonic, F = 2 .... ................ . 263
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U ABSTRACT
I Laminar-turbulent transition of high-deficit flat plate wakes is investigated
3 by direct numerical simulations using the complete Naver-Stokes equations. The
simulations are based on a spatial model so that both the base flow and the
disturbance flow can develop in the downstream direction. The Navier-Stokes
equations are used in a vorticity-velocity form and are solved using a combination
3 of finite-difference and spectral approximations. Fourier series are used in the
spanwise direction. Second-order finite-differences are used to approximate the
spatial derivatives in the streamwise and transverse directions. For the tempo-
3 ral discretization, a combination of ADI, Crank-Nicolson, and Adams-Bashforth
methods is employed. The discretized velocity equations are solved using fast
I Helmholtz solvers. Code validation is accomplished by comparison of the numeri-
cal results to both linear stability theory and to experiments.
I Calculations of two- and/or three-dimensional sinuous mode disturbances in
the wake of a flat plate are undertaken. For calculations of two-dimensional dis-
turbances, the wake is forced at an amplitude level so that nonlinear disturbance
I development may be observed. In addition, the forcing amplitude is varied in
order to determine its effect on the disturbance behavior. To investigate the
Ionset of three-dimensionality, the wake is forced with a small-amplitude three-
dimensional disturbance and a larger amplitude two-dimensional disturbance. The
two-dimensional forcing amplitude is varied in order to determine its influence on
the three-dimensional flow field.
Two-dimensional disturbances are observed to grow exponentially at small
3 amplitude levels. At higher amplitude levels, nonlinear effects become important
and the disturbances saturate. The saturation of the fundamental disturbance
I
II
17
appears to be related to the stability characteristics of the mean flow. Larger Iforcing amplitudes result in the earlier onset of nonlinear effects and saturation. 3At large amplitude levels, a Kiirmiin vortex street pattern develops.
When the wake is forced with both two- and three-dimensional disturbances, 3strong interactions between these disturbances is observed. The saturation of the
two-dimensional disturbance causes the three-dimensional disturbance to saturate. IHowever, this is followed by a resumption of strong three-dimensional growth gthat may be due to a secondary instability mechanism. Larger two-dimensional
forcing amplitudes accelerate the saturation of the two-dimensional and three- 3dimensional disturbances as well as accelerate the resumption of strong three-
dimensional growth. These interactions also result in complicated distributions of 3vorticity and in a significant increase in the wake width.
IIIIIIIII
II
I CHAPTER 1
18
I INTRODUCTION
IAccording to Sato and Kuriki (1961), in the laminar-turbulent transition of a
flat plate wake three distinct regions can be identified: a linear region, a nonlinear
region, and a three-dimensional region. In the linear region, small disturbances
I in the background flow trigger two-dimensional wave-like fluctuations. Sato and
g Kuriki's measurements indicate that the frequency of the dominant fluctuation in
the wake corresponds to the frequency of the most amplified disturbance predicted
5 by linear stability theory. Once present in the wake, these fluctuations grow ex-
ponentially in the downstream direction and are well described by linear stability
U theory.
Due to the large amplification rates associated with wakes, the amplitudes of
the disturbances rapidly become quite large and nonlinear interactions between the
disturbances become important. The behavior of the disturbances deviate from
linear stability theory predictions but the wake remains two-dimensional (Sato
I and Kuriki, 1961). In this region, the disturbances grow at rates significantly less
g than those predicted by linear stability theory and harmonics of the fundamental
disturbances are generated. Also due to nonlinear effects, the mean flow is altered.
3 As verified by several researchers (Zabusky and Deem, 1971; Aref and Siggia, 1981;
and Meiburg and Lasheras, 1988), this sequence of events leads to the well known
3 Krmjn vortex streets that are observed in flat plate wakes.
Finally, three dimensionality becomes important. The work of Robinson and
Saffman (1982) indicates that the two-dimensional KiLrma.n vortex street is un-
3 stable with respect to three-dimensional disturbances. As a consequence of this
3
I
19 1instability, three-dimensional lambda vortex patterns are formed in the region be- 3tween consecutive Kirmin vortices of opposite sign (Meiburg and Lasheras, 1988).
1.1 The Linear Region of Transition 3Linear stability theory describes the behavior of small amplitude disturbances
in par llel shear flows. In nonparallel shear flows, linear stability theory is not Istrictly applicable but can still be a useful model of small amplitude disturbance 3development. The streamfunction tk of a disturbance described by linear stability
theory has the form I
0(0, Y, z,t) = Real(O(y)ei(a + Yz- ' t)) (1.1) 3where the variables x = (z, y, z) and t are the spatial position vector and time Irespectively. The amplitude 0(y) is an eigenfunction of the Orr-Sommerfeld equa-
tion I
(U-c)(4"-(a2 +- )o)- v"o+ -(""-2( a2 +,y 2)0+(a2 +7 2 )2 0) = 0 (1.2) 3where primes denotes differentiation with respect to y. In equation (1.2), U = 3U(y) is the streamwise velocity distribution of the undisturbed flow, a is the
streamwise wave number of the disturbance, -y is the spanwise wave number of
the disturbance, and 1 is the temporal frequency of the disturbance. The phase 3velocity of the disturbance is c = #/a and Re is the Reynolds number of the
undisturbed flow. In general, a and 8 are complex and -f is real. However, 3interest is usually confined to two cases, namely, to the case of spatially growing
disturbances for which a = (a,, a1 ) is complex and 8 is real, and to the case of Itemporally growing disturbances for which # = (1,.,1) is complex and a is real. 3A complete derivation of equation (1.2) is given by White (1974). '3
I
I
20
I Wakes are inviscidly unstable due to the inflection points in their velocity
I profile. As is characteristic of flows with inflectional profiles, the disturbances ob-
served in wakes have high amplification rates. Also, because the streamwise veloc-
Sity distribution of wakes has two inflection points, two different instability modes
exist. The sinuous mode (mode-i) is highly amplified with an anti-symmetric
I streamwise velocity distribution. The varicose mode (mode-2) is less amplified (at
most frequencies) than the sinuous mode and has a symmetric streamwise velocity
distribution. Because of the higher amplification rates associated with the sinuous
3 mode, it is this mode that is usually observed in experiments (Sato and Kuriki,
1961; Sato, 1970; Miksad et al., 1982).
5 The validity of linear stability theory in describing the initial stages of transi-
tion in near wakes has been verified experimentally. Sato and Kuriki (1961) studied
1 the transition of a flat plate wake subject to acoustical excitation. The results of
these experiments were compared to linear stability calculations of temporally-
amplifying, sinuous mode disturbances. The temporal amplification rates fi, ob-
5tained from linear stability theory, were converted into spatial amplification rates
-£- using a phase velocity transformation. The theoretical eigenfunctions com-
3 pared very well with those obtained experimentally. However, the experimental
amplification rates did not compare as well with those predicted by linear stabil-
ity theory because the phase velocity transformation is not accurate enough for
I dispersive waves.
Mattingly and Criminale (1972) experimentally investigated the behavior of
3 small amplitude disturbances in the near wake of a thin airfoil (NACA 0003) and
compared their results to both spatial and temporal linear stability theory. For
I comparison of temporal amplification rates to experimentally obtained amplifi-
cation rates, the Gaster transformation (Gaster, 1962) was employed. The ex-
UI
21
perimentally obtained amplitude distributions and amplification rates compared Uwell to spatial linear stability theory predictions. The assumption of temporally- Uamplifying disturbances, and using the Gaster transformation, did not produce as
good an agreement with the experimental results. The lack of agreement can be jexplained by the fact that the Gaster transformation is based on the assumption
of small amplification rates jai << 1, which is not satisfied for high-deficit wakes. IFrom their results, Mattingly and Criminale concluded that spatial stability the- gory is more accurate than temporal stability theory in predicting the behavior of
small amplitude disturbances in the near wake of a thin airfoil. 3Mattingly and Criminale also found that the stability characteristics of a
spatially varying wake could be computed by the use of a quasi-uniform (quasi- 5parallel) assumption, i.e. the wake is assumed to be locally parallel. The stability
characteristics of the local mean velocity profile are then computed using the Orr- ISommerfeld equation. Then, the streamwise wave number a is obtained as a 3function of streamwise distance.
Miksad et al. (1982) have shown that linear stability theory is also valid when 3there is more than one dominant disturbance in the wake. They experimentally
studied the early development of two small amplitude sinuous mode disturbances, Iof different frequencies, in the wake of a thin airfoil. Their results indicated that,
at small amplitudes, both disturbances grew exponentially at rates predicted by
linear stability theory. 31.2 The Nonlinear Region of Transition
Due to the large amplification rates associated with high-deficit wakes, the 3linear region of transition in these flows is usually quite small. With the appearance
of large amplitude disturbances, nonlinear effects become important and linear Istability theory is no longer applicable. In contrast to the linear region where 3
I
22
I the disturbances grew quite rapidly, the disturbances evolve more gradually in
the nonlinear region. This is also in contrast with the development in boundary
layer flows where rapid transition to turbulence occurs once nonlinear effects and
3 three-dimensionality become important.
Sato and Kuriki (1961) found that the early (upstream) part of the nonlinear
I region is similar in some respects to the linear region of transition. Although the
amplification rates of the disturbances are significantly lower than those observed
in the linear region, their experimental results indicate that the amplitude dis-
tribution of the fundamental disturbance does not change significantly until the
nonlinearity has persisted for some distance downstream. However, harmonics
I of the fundamental disturbance are generated and the second harmonic becomes
the dominant disturbance component near the wake centerline (Sato and Kuriki,
1 1961). Nonlinear effects also change the mean flow, leading to rapid increases in
the mean centerline velocity and the mean wake half-width (Sato, 1970). The
resulting mean streamwise velocity distribution is much fuller than that of the
3 undisturbed flow, indicating a greater degree of fluid mixing.
As the disturbances travel farther downstream in the nonlinear region, the
3 amplitude distribution of the fundamental disturbance deviates from the shape
predicted by linear theory. The measurements of Sato (1970) indicate that the
S peaks in the streamwise velocity amplitude distribution of the fundamental dis-
I turbance shift towards the outer edge of the wake and also decrease in magnitude
with increasing downstream distance. The peak value in the streamwise velocity
amplitude distribution of the second harmonic component, which initially grew
rapidly, also begins to decrease. Sato speculates that the energy lost by the fluctu-
3 ations might be transferred to the mean flow. Sato's measurements also show that
the mean half-width and the mean centerline velocity, which initially increased in
U
mI
23
the nonlinear region, begin to decrease when the fundamental streamwise velocity Ufluctuation begins to decrease. Sato showed that the production of fluctuation 3energy
I (1.3)
where u' and v' are the time fluctuating components of the streamwise and trans-
verse velocities, V is the Reynolds stress associated with the disturbances, and U
is the mean streamwise velocity, has the same sign as the mean centerline velocity 5gradient. Therefore, when the mean centerline velocity increases the fluctuation
energy should increase, and when the mean centerline velocity decreases the fluctu- m
ation energy should decrease. Also in this region, Sato and Kuriki (1961) observed
velocity "over-shoots" near the outer edge of the wake so that the mean streamwise
velocity attained values greater than its freestream value. IMotivated by the appearance of vortices in wakes, Sato and Kuriki proposed
a "double row vortex model" as a way of explaining the nonlinear behavior of 3wake disturbances. In this model, the vortices alternate in sign and are arranged
in a staggered fashion on each side of the wake centerline. The velocity induced
by this vortex pattern does indeed capture some of the features observed in wakes 3such as the second harmonic near the wake centerline, the velocity over-shoots
near the outer edge of the wake, and the decrease of the mean centerline velocity 3far downstream.
Several different numerical investigations support the double row vortex model. IZabusky and Deem (1971) performed Navier-Stokes simulations of the temporal 3evolution of an unstable disturbance in a parallel wake. Their results indicate
that the saturation of the fundamental disturbance in plane wakes leads to the 3development of a vortex street pattern. Phenomena similar to those observed by
I
5 24Sato and Kuriki, such as the velocity over-shoots, were observed in the numerical
results. Aref and Siggia (1981) and Meiburg and Lasheras (1988) studied the tem-
poral development of vortex rows using inviscid vortex dynamics. They modeled
a plane wake as two vortex sheets of opposite sign. When the vortex sheets were
g perturbed, they evolved into a two-dimensional vortex street.
Several researchers including Sato and Onda (1970), Sato (1970), Motohashi
1(1979) and Miksad et al. (1982) have experimentally investigated the behavior of
g wakes that were disturbed at two different frequencies. By perturbing the wake
with disturbances at two different frequencies, these researchers hoped to gain
3 insight into the complex nonlinear interactions that take place in natural transi-
tion. Sato and Saito (1975) performed experiments to study the interaction of a
3discrete frequency disturbance with the broad band background fluctuations that
were naturally present in the flow. In their experiments, Sato (1970) and Sato and
Onda (1970) found that the nonlinear interactions between the two disturbances
3generated the expected higher harmonics as well as a difference frequency. This
difference frequency is similar to the low frequency fluctuations observed in natu-
I ral transition. In natural transition, the low frequency component is not a single
g frequency but a narrow band of frequencies. The conjecture is that these low fre-
quencies are generated by interactions between the higher frequency fluctuations
I that result from the linear instability.
Miksad et al. (1982) found that interactions between the difference frequency
I and the two fundamental frequencies produced sidebands in the spectrum. These
I sidebands modulate the amplitude of the fundamental fluctuations. Amplitude
modulation in combination with the dispersion relation of the fluctuations pro-
I duces phase modulation of the fluctuation,. Miksad et al. indicate that this mod-
1
S
25
ulation has an important role in spectral broadening, that is in the redistribution 3of energy from a small number of frequencies to a larger number of frequencies.
1.3 The Three-Dimensional Reion of Transition
Investigations of laminar-turbulent transition in plane wakes have dealt mainly
with two-dimensional instability mechanisms. Notable exceptions are the numeri-
cal work of Meiburg and Lasheras (1987 and 1988), the experimental investigations 5of Breidenthal (1980), and the theoretical work of Robinson and Saffman (1982).
Robinson and Saffman (1982) theoretically investigated the stability of vortex 3arrays to small amplitude two- and three-dimensional disturbances. They found
that the staggered pattern of the Krmain vortex street is most unstable with Irespect to three-dimensional, spanwise-periodic disturbances. This implies that 3the breakdown of the vortex streets observed in flat plate wakes should be of a
three-dimensional nature. 5Breidenthal (1980) experimentally investigated the three-dimensional behav-
ior of a turbulent flat plate wake. The plate trailing edge varied periodically in 5the spanwise direction in order to induce three-dimensional disturbances in the
wake. Under the influence of this three-dimensional perturbation, the wake was
observed to form a spanwise periodic array of interconnecting vortex loops. IMeiburg and Lasheras (1987 and 1988) numerically investigated the interac-
tion of a large amplitude, three-dimensional disturbance with a smaller amplitude, 5two-dimensional disturbance. They found that the two-dimensional disturbance
led to the formation of a vortex street. The three-dimensional disturbance, through Ian interaction with the two-dimensional vortex street, formed lambda vortices sim-
ilar to those found in boundary layers. Further interaction between the lambda
vortices and the two-dimensional structures led to the development of closed vortex 3loops similar to those observed by Breidenthal. I
I
I26
I JBAPTER 2
3 PROBLEM STATEMENT
VIn this work, laminar-turbulent transition of an incompressible high-deficit
I flat plate wake is investigated by direct numerical simulations using the com-
plete Navier-Stokes equations. Research efforts are initially focused on the early
I stages of transition where Two-dimensional disturbances are dominant. Of pri-
mary interest is the role of nonlinear effects in the development of disturbances.
3 Subsequently, the focus is on the early stages of three-dimensional breakdown
where both two- and three-dimensional disturbances play an important role. The
onset of this three-dimensionality is investigated by simulating the interactiun of
3two-dimensional and three-dimensional disturbances. Investigations of these in-
teractions might explain how three-dimensional disturbances, which according to
3 linear stability theory are more stable than two-dimensional disturbances, eventu-
ally dominate the wake development.
The numerical simulations are designed to model the physical experiments of
3 Sato and Kuriki (1961) and Sato (1970). The results of these simulations may then
be compared to the results of the physical experiments. The wake that is studied
3is shown in Figure 2.1. The wake is generated by a thin flat plate aligned parallel
to a uniform stream. The boundary layers on the plate are assumed to be laminar
and steady over the entire length of the plate, and hence form a laminar wake
3 when they merge at the plate trailing edge. The wake evolves rapidly downstream
of the plate.
3 The spatial domain in which the numerical simulations take place lies down-
stream of the flat plate trailing edge. The domain is placed near the plate so that
U
I
27
it will contain the high-deficit region of the wake. However, the domain does not 3include the trailing edge of the plate. The plate trailing edge is of primary interest
in studies of receptivity and in studies of absolute instabilities. Although, these
are both important aspects of wake transition, they are not the main focus of this
research.
The calculations are undertaken using a numerical method that is designed
to solve both the two-dimensional and three-dimensional Navier-Stokes equations.
Because of the spanwise periodicity of disturbances that was observed in exper-
iments, the spanwise variation of the wake is represented by finite Fourier se-
ries. Second-order finite-differences are used to approximate the spatial deriva- Itives in the streamwise and transverse directions. Time integration of the Navier- 3Stokes equations is accomplished using a hybrid scheme that is second-order time-
accurate. 3The results of many experimental investigations (Sato and Kuriki, 1961; Sato,
1970; Mattingly and Criminale, 1972) indicate that transition in wakes is the result Iof a spatial instability rather than a temporal instability. Furthermore, Mattingly
and Criminale (1972) showed that the use of the Gaster transformation (1962)
to convert temporal stability results into spatial stability results is invalid for
high-deficit wakes. Therefore, in this work spatial amplification and temporal
periodicity of wake disturbances is assumed. To correctly model disturbances of 5this type, the boundary conditions employed in these simulations must be of the
inflow-outflow type. Spatially growing disturbances are induced in the wake by
time-periodic excitation at the inflow boundary of the spatial domain. 3The remaining chapters describe in more detail other important aspects of
this work. In Chapter 3, the governing equations of these simulations are dis- -cussed. The length, time, and velocity scales that are used to nondimensionalize I
I
I
* 28
5the governing equations are also discussed. The boundary and initial conditions
that are required to complete the mathematical formulation of the physical prob-
1 lem are outlined in Chapter 4. The method used to obtain numerical solutions of
the governing equations is discussed in Chapter 5. Chapter 6 contains a discus-
sion of the results of the numerical simulations. Finally, in Chapter 7 significant
conclusions are presented.
IIIIiI
III
III
I
* 29
3 ClHAPTER 3
3 GOVERNING EQUATIONS
I The Navier-Stokes equations govern momentum transfer in a viscous fluid.
These equations are derived by applying Newton's second law of motion to a fluid
I particle. The assumptions required to obtain these equations are that the fluid is
a continuum and that its stress-strain relationship is Newtonian.
1 3.1 The Navier-Stokes Equations
3 The numerical method is based on the Navier-Stokes equations for incom-
pressible flow. In vector form they can be written as6ru- 1-- 2
t 5V-U = -Vv U (3.1)
3The continuity equation for incompressible flow is
V.11=0 . (3.2)
I The independent variables in equations (3.1) and (3.2) are time 7 and the
spatial position vector K = (2,V,T). The dependent variables are the Eulerian
3 velocity vector field u = (U,U,W) and the fluid pressure 5. These variables depend
on 9 and i. The three coordinate directions 7, Y, and 7 and the corresponding
I fluid velocity components U, V, and W are shown in Figure 2.1. The fluid density
;5 and the kinematic viscosity of the fluid F are assumed to be constant. Variables
with over-bars denote dimensional quantities, while bold-face variables represent
I vector quantities.
The operator V is a vector which in Cartesian coordinates is of the form
I . (3.3)I
3
I30
The Laplacian operator V2 is
-2 82 8 2 83a + + Z (3.4a)
and the operator U . V is
x z (3.4b) I
Equations (3.1) together with (3.2) lead to four scalar equations which in 3principle can be numerically solved for the four dependent variables (U, F, W) and T
as functions of (7, V, i) and ?. However, one difficulty with solving these equations 3is formulating boundary conditions for the pressure. This difficulty is avoided
by solving an alternate formulation of the Navier-Stokes equations in which the mpressure does not appear. This formulation of the Navier-Stokes equations is 3obtained by taking the curl of equation (3.1) and simplifying the result with the
appropriate vector identities. The resulting vector equation (Batchelor, 1967) is 3=_ (3.5)
The Eulerian vorticity vector field U = ( is defined by the relation m
w=-Vxu (3.6)
and the operator U. V is
=,8= 8 8 (3.7)
It is noted that the vorticity vector as defined by equation (3.6) is the negative of
the usual definition of vorticity, U = V x U. However, throughout this work the
term vorticity will be used to denote the flow Variable defined by equation (3.6). gI
31
I Furthermore, because of the negative sign in equation (3.6), the direction of fluid
rotation associated with the vorticity vector U = -V x U is given by a left-hand
rule instead of the usual right-hand rule.
Equation (3.5) is the Navier-Stokes equation in vorticity-velocity formulation,
or alternatively the vorticity transport equation. Coupled to equation (3.5) is the
I Poisson equation3 i=Vx~ (3.8)
which is derived from the definition of vorticity (W = -V x U) using the continuity
I equation (3.2). The vorticity & is also a function of the independent variables Y
and 7.
Equations (3.5) and (3.8) are a system of six scalar equations which can be
3 solved subject to appropriate boundary and initial conditions. As desired, equa-
tions (3.5) and (3.8) also do not contain the fluid pressure 5 as an unknown vari-
able. However, in contrast to equations (3.1) and (3.2), now six scalar equations
must be solved instead of four.
3.2 Nondimensionalization
Equations (3.5) through (3.8) are nondimensionalized using appropriate length,
time, and velocity scales. The independent variables are nondimensionalized using
3 the following formulas:
= = y = Z 7 , and t (3.9)
3 where the variables without overbars are dimensionless. The streamwise length
scale ? is the length of the flat plate. In the transverse direction the length scale
is 3t, which is the momentum thickness of the wake at the trailing edge of the flat
plate. The length scale that is used in the spanwise direction is 1/7 where 7 is the
I
I32
spanwise wavenumber of the three-dimensional disturbance that results from the
excitation at the inflow boundary. The time scale is the ratio of the streanwise Ilength scale 7 to the freestream velocity Uo...
The dependent variables U - (=,VW) and , = are nondimen-
sionalized as follows:
U= , == 1 w= (3.10a)U. 0 V Urs '
andu'9 6 'e _ C8t.jZ =-(3.10b)
The parameter r 2 is the ratio of the transverse length scale 9te to the streamwise
length scale e: r 2 = 0 te/. Similarly, the parameter r3 is the ratio of the spanwise
length scale 1/7 to the streamwise length scale 7: r3 = 1/(7).I
With this nondimensionalization, equations (3.5) and (3.8) become
O5w IVW f(3.11 a)IOt Re
and
V 2 u = g (3.11b)
which are solved for the six unknown variables u = (u, v, w) and w = (w".w,.wI) Ias functions of the dimensionless spatial position vector x = (XI y, z) and dimen- 3sionless time t. The right hand side of equation (3.11a), the vector f = (f., fy, f ),
is the dimensionless version of w • Vu - u. Vw which appears on the right hand
side of equation (3.5). The three components of f areau OU LOu 0 Ow. aw. ow
= W T + W T - B 2- - -- -&- 8w2z , (3.12a)
By Bi, Ow Ow Ow, I-W-- (3.12b)
axw ~ w,+ z X OzB9wi Ow Ow w2 Ow. Z OW.2)
and h=w.++(zT - -ay + (3.12c)
I
I
I
33
I The right hand side equation (3.11b), the vector g - (gz,gy,g,), is the dimension-
less version of V x 9 which appears in equation (3.8). The three components of g
are
r3 1(3.13a)g'=r O r OW '
I1 o&Lff rs & zgy -2O
= 7 Oz r2 (3.13b)
and 1 W 1 .w, (3.13c)g r3 O r3r22
The dimensionless Laplacian has the form
02 1 028:2±22±2,. 8z2+,. (3.14)
and the parameter Re is the Reynolds number Re = V.
Equations (3.11) serve as the basic equations for the simulations described in
this work. Use of such a vorticity-velocity formulation for transition simulations
was suggested by Fasel (1976) and used successfully for both two-dimensional
I (1976) and three-dimensional simulations of transition in boundary layers. The
same formulation was also successfully employed by Pruett (1986) for two- and
three-dimensional transition studies in free shear layers.
IIIIII
I
I 343 CHAPTER 4
BOUNDARY AND INITIAL CONDITIONS
I The governing equations (3.11) represent a set of partial differential equa-
tions. This equation system is parabolic with respect to the time variable t and
is elliptic with respect to the spatial variables z, y, and z. Solution of equations
(3.11) requires specification of suitable spatial and temporal domains. Further-
more, the unknown variables w and u must be specified at some initial time.
3 Finally, boundary conditions must be specified along the entire boundary of the
spatial domain.
3 4.1 Spatial Domain
The governing equations are solved in a three-dimensional rectangular region
U in space that is downstream of the flat plate. The domain extends downstream
from near the trailing edge of the plate and is assumed to extend in the streamwise
direction for several disturbance wavelengths. The transverse thickness of the
domain is assumed to be much wider than the vorticity disturbances that are
present in the wake. The domain extends into the irrotational region of the flow
I field above and below the wake. The spanwise extent of the domain is equal to
one nondimensional spanwise disturbance wavelength X, = 27r.
The spatial domain and coordinate system are shown in Figure 2.1. The z di-
3 rection corresponds to the streamwise direction which is parallel to the freestream
velocity U . The transverse direction is denoted by the independent variable
I y. The third coordinate direction z corresponds to the spanwise direction and is
parallel to the trailing edge of 'the flat plate. For convenience, the origin of the
coordinate system is placed at the trailing edge of the flat plate . The spatial
II
35I
domain extends in the streamwise direction from zo to ZN, in the transverse di-
rection from yo to YM, and in the spanwise direction from z = 0 to z = 27r. The
spatial domain is then given by
ZO :_ Z :_ ZN , (4.1a)
YO :5Y :YM , (4.b)I
and 0<z_<2v" (4.1c)
4.2 Temporal Domain IThe time integration of the governing equations is divided into two separate
procedures or steps. In the first step, the governing equations are solved to obtain
the steady laminar wake that appears behind the flat plate. This is done by
integrating the governing equations (3.11) with respect to time, subject to time-
independent boundary conditions (no forcing), until a steady state solution is
obtained. This calculation begins at an initial time t = tL1 < 0for which initial 3values of the dependent variables U(X,y,z,tL,) and W(Xy,z,tL,) are specified.
The calculation ends at time t = 0, at which time the steady solution is obtained. IThis steady wake is referred to as the base flow.
In the second step of the calculation, the response of the wake to time-
dependent excitations that are introduced at the inflow boundary is investigated. 3The governing equations are integrated in time from t = 0 to t = t L, . The initial
values of the dependent variables at time t = 0 are given by the base flow. I4.3 Boundary and Initial Conditions for the Undisturbed Flow 3
The undisturbed wake is assumed to be two-dimensional, so that
w(z,y,z,) = 2 z(z,y,z,t) = WY(zy,z,t) = 0 (4.2)
II
U
1 36I The remaining flow variables u(z,y,z,t), v(z,y,z,t), and w,(z,yj,z,t) are solu-
tions of the governing equations (3.11). Restricted to two-dimensional flows, the
governing equations become3 &O &z &&, O 1 (f 2W5 1 82W ,.)
& + +,--- +Vi = - 2 8y (4.3a)&: C9 y Re 8 2 r 2 ,
i92u 1 2u r30w (4.3b')
+7- 2 = 72, '
and 2v 1 2v - _ (4.3c)a " r282 r2 8Z
3 Because the base flow is two-dimensional, the spatial domain reduces from a three-
dimensional region to a two-dimensional plane that is perpendicular to the z axis.
I Boundary conditions for the unknown variables u, v, and w. need to be specified
along the boundaries of this plane. These boundary conditions are independent of
time and are chosen such that the solution of equations (4.3) represents a laminar,
3 high-deficit, flat plate wake.
Inflow Boundary (z - zo)
I The inflow boundary of the spatial domain is located downstream of the flat
plate trailing edge. Therefore, the velocity and vorticity distributions at this
boundary should correspond to those of a spatially-developing, high-deficit wake.
3 For most of the calculations in this work, a Gaussian distribution is used to rep-
resent the streamwise velocity at the inflow boundary. The Gaussian distribution
I compares reasonably well to experimentally obtained streamwise velocity profiles
3 (Sato and Kuriki, 1961). In a few other calculations, a hyperbolic secant function
is used to represent the inflow streamwise velocity. The hyperbolic secant dis-
3 tribution also compares reasonable well with experimentally obtained streamwise
velocity profiles (Mattingly and Criminale, 1972)UI
37
With the use of either a Gaussian distribution or a hyperbolic secant dis- 3tribution, the streamwise velocity is then known as a function of y at the inflow
boundary. However, since the z dependence of the streamwise velocity is unknown,I
being given by the solution of the Navier-Stokes equations, the transverse velocity
cannot be related to the streamwise velocity using the continuity equation (3.2).
Instead, v must be given an arbitrary value. In this work, the Reynolds number of 3the wake Re = 2 is large. Therefore, except for very near the flat plate trailing
edge, the transverse velocity is much smaller than the streamwise velocity and can H
be set to zero at the inflow boundary with reasonable accuracy.
For the base flow calculation, the inflow boundary condition is
u(zo,y,z,) = u'(y) , (4.4a)
v(zo,y,z,t)=O , (4.4b) 3and w'(zo,y,z,t) = wI(y) (4.4c)
where the superscript ( )' refers to the fact that the functions u'(y) and wI(y) are
the streaxnwise velocity and spanwise vorticity distributions at the inflow bound-
ary. If the streamwise velocity is specified to have a Gaussian distribution, then 3u'(y) is given by
u'(y) -1- (1 - Uc)e- "' (4.5a) Iwhere
ln(2) (4.5b)
Alternatively, u'(y) can be given by
2='y)- 1 - (1 - Uc)sech2 (o7y) (4.6a) ,
II
38
3 wherearccosh(V,2)U = b2(4.6b)
The spanwise vorticity is given by
= (Y) (4.7)
In equations (4.5) and (4.6), U, is the centerline velocity of the wake at the inflow
3 boundary and b is the wake half-width at the inflow boundary. The parameters
U, and b are chosen so that the resulting base flow closely models the high-deficit
U wakes observed in experiments.
Outflow Boundary (X = zN)
For spatially-developing wakes, a proper outflow boundary condition is not
3 easily found since the flow condition at this location would be obtained by the
solution of the governing equations and is therefore not known a priori. However, it
is possible to determine an outflow boundary condition without requiring advance
3 knowledge of the solution. Suppose the two-dimensional version of equations (3.5)
and (3.8) are nondimensionalized using the boundary layer scaling
Y --X = =~Rei , V = -Rel , and w.= (4.8)1 1 UM0 U00 tT0 0Ret
The parameters 1, U.., and Re are defined in section 3.2. The resulting dimen-
3 sionless equations, for steady flow, are
+ R-,e 82 8 2 (4.9a)1 O u W2u 8w83 Re82+ L92 ' o (4.9b)
1 8 2 iV O _
and + - . (4.9c)Re8: 2 + 2
U n ~~a i '-
39 UFor high Reynolds number flows Re > 1, as in these investigations, the stream-
wise diffusion term is the smallest term in each of the three equations (4.9a)
through (4.9c). Therefore, the streamwise diffusion terms are set equal to zero, Uresulting in the outflow boundary conditions
02uz2=(Nyzt) -- 0 , (4.10a)
&2vX2(ZN, y,z,t) = 0 , (4.lOb)
and -Z2 (zN, y,z,t) _0 (4.10c)
These boundary conditions were first proposed by Fasel (1976). 3Freestream Boundaries (y = yo) and (y = yM) 3
As stated previously, the base flow is assumed to be a high Reynolds number
flow. Therefore, the rotational part of the wake is confined to a region very near 3to its centerline and the streamwise velocity rapidly approachs its freestream value
as y - ±-o. The boundary conditions that satisfy these assumptions are, for the Ulower freestream boundary, 3
u(z,yo,z,t) = 1 , (4.11a) 3O;(zIyo9zt)=0 , (4.11b)
and WZ(z,yo,z,t)=O . (4.11c)
Similarly, for the upper freestream boundary: Uu(z, YM, z,I= , (4.12a) 3
- (Z,/MZ,t)= I0 ,(4.12b) 3and W.(Z,yM,Z,t) = 0. (4.12c) U
U
II
40
IEquations (4.11b) and (4.12b) follow directly from the conditions (4.11a) and
(4.12a) using the continuity equation.
Auxiliary Condition (y = 0)
3 The boundary conditions described up to this point are sufficient to ensure
the uniqueness of the solution of the vorticity equation (4.3a) and the Poisson
I equation for the streamwise velocity, equation (4.3b). However, these boundary
conditions are not sufficient to ensure the uniqueness of solutions of the transverse
velocity equation (4.3c). This was shown by Pruett (1986) who notes that if a
3function V(z, y) is a solution of equation (4.3c), subject to the boundary conditions
described previously in this section, then the functionIV(z,y) + c(z - z0 ) (4.13)
where c is a constant, is also a solution of equation (4.3c). Therefore, an infinite
number of solutions satisfy equation (4.3c) and the boundary conditions described
* in this section.
Therefore, an additional condition has to be specified that will ensure that
3 the solution of equation (4.3c) is unique. Experimental evidence indicates that
the steady streamwise velocity is symmetric with respect to the wake centerline
(y = 0). Therefore, the transverse velocity v is antisymmetric with respect to the
wake centerline (y = 0) and is equal to zero at the centerline. The transverse
velocity v is therefore required to satisfy the conditionUv(z,0, z,t) = 0 (4.14)
3at the wake centerline.I
I
I
411
Initial Conditions (t = tL) 3This calculation starts at time t = tL, for which initial valueJ of the unknown
variables u, v, and w, are specified. In this work, the initial conditions are chosen 3such that the flow profile is independent of z and is identical to that of the inflow
boundary. These conditions are
(z,y,Z,,,) = ul(y) (4.15a) IV(ZX,YZ, tL,) = 0 , (4.15b) 3
and W,(X,Y,Z,tL,) = W(Y) (4.15c)
4.4 Boundary and Initial Conditions for the Disturbed Flow NFor calculations of the disturbed flow, which may be three-dimensional, bound- 3
ary conditions have to be specified for all boundaries of the three-dimensional
domain. IInflow Boundary (z = z0 )
In a physical experiment, disturbances in wakes may originate from a variety
of sources. Wake disturbances may arise from freestream turbulence, roughness 3on the flat plate, sound, etc. Disturbances can also be artificially introduced into
wakes through loudspeakers, flaps at the plate trailing edge, or through heater 3strips on the plate. In this work, the disturbances are introduced at the inflow
boundary. For this purpose, solutions of the Orr-Sommerfeld equation are used. IThus, the velocity and vorticity components at the inflow boundary are a 3
combination of those of the steady base flow discussed in the previous section and
of time varying components that then excite the disturbances in the wake. The 3inflow boundary conditions are of the form
3u,(z0,y, z,t) = uss~z0,y) + P (z0, y, t) + Pd(z0,y,t0cos(z) , (4.16a)
UU
42
IV(20) YJz It) 0 + ,1(ZO, Y,t) + p.31(ZO, y, t)COS(Z),
3W(z, 17YzIt) =0 +Pd(zo,y,t)cos(Z) , (4.16c)
w (ZoIY1 Z~I) t 0 + P.3 (Zo,Y, t)COS(Z) I (4.16d)
IW 1 (zo , ZIt) =0 + pw3d(Xo,Y,it)COS(Z) I (4.16c)
3 ~(.o and 1 .(Z 0 , t) = XS ws(ZO, Y) + p(,2 ,,t +Pd(ZO, Y,COS(Z)
3 The functions uss(xo,y) and wAzss(xo,y) are the streamnwise velocity and
spanwise vorticity components of the base flow at the inflow bounr y. The func-
3tions pu~d(Xo,Y, t), p.2d(Xo, yt), and P,2f(zo, y, t) are two-dimensional disturbances
obtained from the Orr-Sommerfeld equation. These functions are
p2d(X0 , Y, t) =r(t)A2 U~(Y)CoS(a~dzO - /3t + 02d(y)) , (4.17a)
I p~d(~ 0 ~j) =r(t)A2VAdycs~~z r /3 + V() (4.17b)
3 and P -To,Y,it) =r(t)A2 &(yc(dz A ~3 + Wq_ (4.17c
3The functions pld(ZO, Y,t), p~ld(Z 0 , Y,t), p3d(. 0 , ,Jt), p,3 To,Y, t), p.3d( o,y,i)
and P,,d(x0 , y, t) are three- di mensional disturbances obtained from the three-dimen-
I sional Orr-Sommerfeld equation. These functions are
3 ~ ~pd(Z0 , Y, t) U3?(t)s~Y)CoS(a~dzo _ f8t + 03,d(y)), (48)
5 p~d~z0 ~j, t) r(t)Asvdyco3 ~ z A r 3 + (4.18a)
p3d(ZO Yt) =r(i)As 3d (Y)cos(a 3dzO _ pt + 03,d(y)) , (4.18b)
Upw~( Zo, Y, 0) =r(i)Aw 3d(Y)COS(a~dzo _ O3t + 03d(y)) (41c
3. ~ZO, Y,i0 =r(t)A3 W.3d(Y)COS(a4dZo _-#t + 031d(y)) ,(4.18d)
and Pdz0 ,Y,t) =r(t)A3 w(cocdz A3 '() (4.18f)
I43 3
For the three-dimensional disturbances, a superposition of two oblique waves of
equal and opposite spanwise wavenumbers is used.
In the preceding paragraph, the parameter ar was the real part of the eigen- 3value of the two-dimensional Orr-Sommerfeld equation corresponding to the fre-
quency 3. The parameter a3d was the real part of the eigenvalue of the three- 3dimensional Orr-Sommerfeld equation corresponding to the frequency /3 and the
nondimensional spanwise wavenumber -' = 1. The functions (uAd(y), 0'(y)), 3(v2Ad(y), 4 Ud(y)), and (w 2 Ad(y), 4d(y)) are the amplitudes and phases of the eigen- 3function of the two-dimensional Orr-Sommerfeld equation corresponding to the
frequency /3. The functions (uAd(y), Sd(y)), (v d(y), 3 d(y)), (WA d(y),¢ d(y)),(w 3d(y),,3d(y)), (WY 3d(y), 3d (y)), and (w.2d(y), 02d(y)) are the amplitudes and
phases of the eigenfunction of the three-dimensional Orr-Sommerfeld equation 3corresponding to the frequency /3 and the nondimensional spanwise wavenumber
-y = 1. Both the two-dimensional and the three-dimensional eigensolutions cor-
respond to the streamwise location zo at which the steady streamwise velocity 3distribution is uss(zo, y). The amplitude of the disturbances at the inflow bound-
ary is controlled by the parameters A2 and A3 . The solution procedure used to 3obtain eigensolutions of the Orr-Sommerfeld equation is discussed in Appendix A.
The function r(i) that appears in equations (4.17) and (4.18) is chosen to min-
imize any transient disturbance that might arise when the excitation is initiated.
The function r(t) has the values
r(t) = { 2, if0 <t < w/ ; (4.19) 3Outflow Boundary (z = ZN) 3
For numerical simulations of spatially-amplifying disturbances in shear flows,
the specification of appropriate outflow boundary conditions represents a major IIU
I!
* 44
3 difficulty. Numerical simulations of spatially-periodic, temporally amplifying dis-
turbances avoid this problem because then periodic inflow-outflow boundary con-
3 ditions can be employed.
For the numerical simulation of two-dimensional, spatially-amplifying, small-
amplitude disturbances in a flat plate boundary layer, Fasel (1976) employed the
condition
X 2 = (4.20)
at the outflow boundary. In equation (4.20), u' = u - uss is the streamwise per-
3 turbation velocity and a,. is the real part of the disturbance wavenumber. Similar
conditions were used for the other flow variables. In Fasel's numerical simulations,
3 the boundary condition (4.20) allowed small-amplitude disturbances to pass unre-
flected through the outflow boundary. However, equation (4.20) is formally exact
only for small amplitude neutrally stable disturbances. Therefore, while equa-
3 tion (4.20) is a suitable approximation for the slowly amplifying disturbances that
are observed in flat plate boundary layers, it is definitely not appropriate when
3 disturbances are highly amplified, as in high-deficit wakes.
Pruett (1986) attempted to use outflow boundary conditions that were sim-
ilar to equation (4.20) for the simulation of spatially-amplifying disturbances in
3 free shear layers. However, as Pruett notes, amplification rates are quite large in
free shear layers, and therefore the previous condition is not valid. Pruett found
3 that boundary conditions similar to equation (4.20) allowed the fundamental dis-
turbance component to pass through the boundary unreflected, but reflected any
nonlinearly generated harmonics of the fundamental fluctuation.
3 As an alternative to boundary conditions such as equation (4.20), Pruett
employed a moving outflow boundary for his calculations. The moving outflow33
U
45 3boundary is implemented in the following manner. First, the base flow is com- -puted in the entire spatial domain for which z0 < z < N. Then, the disturbed
flow is calculated in a smaller subset of the spatial domain bounded by z = zo and IZ = ZMB(t) < zN where z = ZMB(t) is the instantaneous location of the moving
boundary. The moving boundary is initially located at a specified distance down-
stream of the inflow boundary. As the calculation progresses and the disturbances 3propagate downstream, the moving boundary propagates downstream ahead of
the disturbances so that the flow remains undisturbed at this boundary. As a 3consequence, Dirichlet boundary conditions that enforce an undisturbed flow can
be employed at the location of the moving boundary. When the moving boundary Ireaches the fixed outflow boundary at z = ZN, the calculation is stopped.
For the present simulations, boundary conditions similar to those used by
Pruett are employed. However, instead of using a moving outflow boundary, the 3undisturbed flow condition is enforced at the outflow boundary located at x = ZN.
The disturbed flow is then calculated in the entire spatial domain. Furthermore, 3the domain is specified to be large enough so that the disturbances do not reach
the outflow boundary before the calculation is stopped. The resulting boundary Iconditions are
U(ZN,y,Z,t) = uSS(ZN,y) , (4.21a)
V(ZN,y,Z,t) = VSS(XN,y) , (4.21b) IW(ZN,y,z,t)=O , (4.21c) 3
=(ZNYZit)= 0 , (4.21d)
w,(ZN,y,z,t)=O , (4.21e) 3and W(ZN,y,Z,t) =WzSS(ZN,Y) (4.21f) 3
As before, the subscript ( )ss denotes the steady state wake or base flow. II
I
* 46
I Freestream Boundaries (y = Yo) and (y = yM)
The freestrearn boundary conditions are based on the assumption that distur-
I bances at these locations decay exponentially at rates predicted by linear stability
theory. These boundary conditions are now derived.
An equation for the streamwise perturbation velocity u' = u - uss can be
3 obtained by subtracting the base flow streamwise velocity equation (4.3b) from
the streamwise component of equation (3.11b). The resulting equation is
a 0 U' 1 2U , 1 2 u' r3 Wz' 1 aw(Tx _ + r2 2 + _ r3 6z 2 r 22y 7 Oz 42
1 where the spanwise perturbation vorticity wz - wZss is denoted by w,'. Assum-
3 ing that the velocity and vorticity fluctuations at the freestream boundaries are
described by linear stability theory, then
u'(x,y,z,t) = Real(fi(y)ei(Qz+z - t)) (4.23)
U and analogous formulas are valid for the other velocity and vorticity components.
3 Then, from Squire's equation
I - (a 2 + 1 + iaRe (U - c)) "i, = -iReU'i (4.24)
it can be shown that for Re >> 1, 4j3 - 0 for Jyj >> 1. Furthermore, from
the continuity equation and the definition of u;, and 4, it can also be shown
3 that for velocity fluctuations that decay exponentially when IyI >> 1, then a.
0 and u4 = 0 for IyI >> 1. Combining equation (4.22) with equation (4.23),
3 and assuming that the vorticity fluctuations are negligible for large IyI, then the
equation
y2 (y) = r (c4 + - .)fi(y) (4.25)
!2 3
U
47 3is valid if the disturbance is assumed to neutrally stable (ai = 0). In reality 3however, wake disturbances are usually highly amplified and therefore equation
(4.25) is not strictly applicable. However, for the calculations presented in this 3work, the assumption of neutrally stable disturbances at the freestream boundaries
did not prove detrimental. Equation (4.25) has the solution Ul(y) = Ae Be(4.26)
Assuming that fl(y) is finite for all y, then for y -4 -oo, B = 0 and fi(y) satisfies 3the differential equation
di-(y) =r2 1 (4.27)
dy -+ 73 (.7
Combining equation (4.27) with equation (4.23) results in the following boundary
condition for the streamwise perturbation velocity u'(z, y, z, t) at y = yo: 3
09' + 73 U '(0,yolzit) (4.28) 3Similarly, the boundary condition at y = YM is 3
8u' 1 C
")-r2 + - u '(z,yM,z,t) (4.29)
In a similar manner, the following boundary conditions for the spanwise ve- 3locity w are obtained:
z z,t) - F2 a2 + "jw(ZYo,Z) (4.30a)
andSC1(Z,yM,Z,i) = -2 a + 0-w(zyMZi) (4.30b)
ey3
II
48
I Boundary conditions for the transverse velocity v are obtained from the con-
3 tinuity equation. These boundary conditions are
e(z,' yz,t) = - 0 (XYo, z, t) (4.31a)
andZYM, Z, 0 (1,Y%,,,, - 8W(,' z, 0 (4.31b)
a( ' z
3 The vorticity at both freestream boundaries is assumed to be zero. At the
lower boundary (y = yo)Iw(z,y0,z,t)=0 (4.32a)
I (z ,Yo,z,t) = 0 (4.32b)
I I(z, Yo,ZI) = 0 (4.32c)
I and at the upper boundary (y = yM)
I W(Z,YM,Z,It)=0 (4.33a)
3 ~W(z,yM,Z,t) = 0, (4.33b)
WZ(z,YM,z,t)=0 (4.33c)
One further consideration is the form of the coefficient r2 a + 9 in equa-
tions (4.28), (4.29), and (4.30). These boundary conditions were derived based
on the assumption of three-dimensional disturbances for which a, = a 3d and forwhich the coeffcient V/a becomes r2 /a' d2 + J. For two-dimensional
3 disturbances, the coefficient r2 / '4+ _r becomes simply a2dr2. When both two-
dimensional and three-dimensional disturbances are present in the wake it is not
I
!
49 3difficulty is resolved by decomposing the boundary conditions (4.28), (4.29), and
(4.30) into two-dimensional and three-dimensional components. Then, for the two- idimensional component of equations (4.28), (4.29), and (4.30), r2 ca + 2 is seti
ual 2 For the three-dimensional component of equations (4.28), (4.29), Iisseqeulao 2c to+ 4 Tisdcopoito wllband (4.30), rb set + io
discussed further in the next chapter.
Lateral Boundaries (z = 0) and (z - 27r) IAs discussed in section 4.1, the spanwise width of the spatial domain is equal
to one spanwise wavelength Az = 2ir. Therefore, periodic lateral boundary condi- Itions are employed. These boundary conditions for the ( component are n
u(z,y,O,i)=--u(z,y,27r, t) , (4.34)with analogous conditions for all other variables. I
Initial Conditions (t = 0) 3This calculation starts at time t = 0. At this time an undisturbed flow
is assumed and therefore the flow variables are set equal to the values of the 3previously computed base flow. Thus
Iu(z,y,z,O) =uss(z,y) , (4.35a) 3v(z,y,z,O) = vss(z,y) , (4.35b)
tv(:,y,z,O) =0 , (4.35c)I
w=(z,y,z,O) =0 , (4.35d) 3w(z,y,z,O) =B0 , (4.35e)
and w(z,y,z,O) =w 3,ss(z,y) (4.354)1
II
II
50
I CHAPTER 5
I NUMERICAL METHOD
I The governing equations (3.11) together with the boundary and initial condi-
tions described in the previous chapter are solved using a combination of spectral
and finite-difference methods. Fourier series are used to represent the spanwise
3 variation of the dependent variables. Spatial derivatives in the streamwise and
transverse direction are approximated using second-order finite-differences. The
3 vorticity equation (3.11a) is numerically integrated in time using a combination of
an Alternating-Direction-Implicit (ADI) method, the Crank-Nicolson method, and
3 the second-order Adams-Bashforth method. The nonlinear terms are computed
spectrally at each time step. The discretized Helmholtz equations are solved using
fast Helmholtz Solvers. The numerical method discussed here is based on the work
3 of Pruett (1986).
5.1 Spanwise Spectral Approximation
3 As discussed in Chapter 4, the disturbance flow is assumed to be periodic
in the spanwise direction with wavelength A. = 27r. Because of this spanwise
I periodicity, the streamwise velocity can be represented by a truncated Fourier
series of the form
u(zy,z,t) L U(zy,t)e (5.1)k=-fI
with analogous Fourier representations for the other flow variables. The Fourier
3 coefficients Uk ', Vk, W" , fl , flh, and flk are functions of the two spatial variables
z and y and the time variable t. The Fourier coefficients can be rewritten as the
vectors
g UA = (Uh, V ' W k) (5.2a)
3
3
51
and k (5.2a)
which are the Fourier representations of the velocity vector u = (u, v, w) and the 3vorticity vector w = (w=,w 1,we). Furthermore, because the physical variables
u = (u, v, w) and w = (w., wyIw) are real numbers, the Fourier coefficients satisfy 3the relations
Uk('cY,t) = O-k(, t) (5.3a)
and flk(zy,t)=A_ k (z, y, t) (5.3b)
where - denotes the conjugate of a complex number. Equations (5.3) are valid for
all values of k.
In the governing equations (3.11), the physical variables are replaced by their 3Fourier representations given by equation (5.1). Orthogonality of the exponential
functions e k' in the spanwise interval 0 < z < 27r decouples the three-dimensional 3governing equations into K + 1 sets of two-dimensional equations of the form
012 1 ~ 1''a 10 2n ,, \ ,19X 2e _2 + 2) = F:, (5.4a)1
tRe ( 2 (4
an~ 1(a2nk +1 a2n k~ ~~ FV Re + - - I) =F ,(5.4b)- -- 2 2 - ) cy 20 Y
1(82 1 8 2nU (I k k) k =F.I
2 +2 2XJ~~+ 2 r 3,
0-2VU 1 82V k _ 2 = k (5.4d
82Vk I 82V _vk G, (5.4e)
82 1 82 Wk 2 W" = aW . (5.4f)
Equations (5.4) are solved for 0 < k < K to obtain the complete set of Fourier 3coefficients. The Fourier coefficients for which k < 0 are obtained from equations I
I
I
I 521 (5.3). Once the Fourier coefficients are obtained from equations (5.4), the physical
variables u and w can be reconstructed using equations (5.1).
The functions F , Fk, and Fk are Fourier representations of the nonlinear
l terms f., fv, and f-. These functions are related by the expressions
4f.(Z'Y'z't)= F'(z' yt)e ikz ) (5.5a)
3 f,(zyzt)= Fl(z,y,t)eii'z , (5.5b)
k1=-4fK
and f(x,y,z,t) = F.'(z,'y,t)ei'kz (5.5c)k=- K
The functions f , fy, and fz are defined by equations (3.12).
The functions G;, G ,and G, are Fourier representations of g., g., and g..
U These functions are related by the expressions
'1=fg(-',Y9,, ? t) G'(z, Y, e'' , (56,a)
9,(XY~z~t) = G, 'C,y,tOe'" (5.6b)
and g,(z,y,zt) z G*(xy,t)e'1z (5.6c)
The functions g, g., and g are defined by equations (3.13). The functions G.,
5 G , and Gk have the simple form
k = L ik (5.7a)
II = ilk n k r3 (5.7b)
i an; 1 ankand G! . (5.7c)3r 8z r &y
I
I
53 3Equations (5.4) are solved for the Fourier coefficients, subject to the Fourier 3
repr-zentations of the boundary and initial conditions from Chapter 4. For the
undisturbed flow calculation, Fourier representations of the boundary and initial Uconditions from Section 4.3 are employed. For the disturbed flow calculation,
Fourier representations of the boundary and initial conditions from Section 4.4
are employed. 35.1.1 Fourier Representation of the Boundary and Initial Conditions for the
Undisturbed Flow 3The Fourier representation of the boundary and initial conditions for the
undisturbed flow are discussed in this section. The undisturbed flow is two- 3dimensional, so that 3
Uk = Wk a = 1 k = 0 for .k (5.8)
Furthermore, for k = 0
1(Z, t) = DY(z ,y,t) = W°(X,y,t) = o . (5.9)
The remaining Fourier coefficients U°(z,y,t), V°(z,y,t), and S°z,y,t) satisfy 3th e e q u a tio n s 1 1 2 0
[0 2l 1 82 fl,0.8o - , 1 (zq + 2n - ) = rF , (5.10a)
,' U° 1 a2 U° r3 flo I+ - - (5.10b)
f 2 V ° 1 a 2Vf (5.10c) Iand - Oy; = - (1
r2 ;2
which are the Fourier analogs of equations (4.3) in Chapter 4. The nonlinear term 3F.(z,y,t) in equation (5.1Oa) is
F.(Z,8t) (U O : VIln: +(5.11)ex \ i+
I
UI
54
I Solution of equations (5.10) requires the Fourier representation of the boundary
and initial conditions from Section 4.3.
Inflow Boundary (z = zo)
3 The Fourier representation of the inflow boundary conditions (4.4) are
U°(zo,y,t) = u'(y) , (5.12a)
V°(zO,y,t)=0 , (5.12b)
and fl0(zo, y,t) = w4(y) (5.12c)
3 Outflow Boundary (M = XN)
The Fourier representation of the outflow boundary conditions (4.10) are
9 - (zN,y,t) = O , (5.13a)
82
82VO- -2 (zN y )--O , (5.13b)
and -- (XN,y,t) = 0 (5.13c)
Freestream Boundaries (y = Yo) and (y = YM)
3 The Fourier representation of the lower freestream boundary conditions are
3 U0 (z,yo,t) = 1 , (5.14a)
OIVo
-=-(Xyott) = 0 , (5.14b)
Sand fl0(z, yo,t) = 0 . (5.14c)
5 Similarly, the Fourier representation of the upper freestream boundary conditions
are
S U°(z,YM,t) = 1 , (5.15a)-- (Zymt) = 0 , (5.15b)
and IO(XYMI)=0 (5.15c)
!55 3
Auxiliary Condition (y = 0) 3The auxiliary condition is
V°(x,0,t) = 0 (5.16) 3Initial Conditions (t = tL,) I
The initial conditions for the base flow calculation are
U(,y,tiL)=uI(y) , (5.17a)
V 0 (X, Y,tL,) = 0 , (5.17b)
and nfl(Z,y,t,)=W(y) (5.17c)
5.1.2 Fourier Representation of the Boundary and Initial Conditions for the
Disturbed Flow 3The Fourier representation of the boundary and initial conditions that are
used for the calculation of the disturbed wake are discussed in this section. In gen- 3eral, the disturbed wake is three-dimensional. This requires solving the governing
equations for the three velocity components and the three vorticity components. 3Boundary and initial coiditions are required for all of these components.
Inflow Boundary (z = z 0 )
The Fourier representation of the inflow boundary conditions are shown below. 3For k = 0, the boundary conditions are
U°(zo,y,i) = uss(zo,y) + Pd(zo,y,) (5.18a)
V°(ZO,y,i) = 0 + pV2d(Z 0 ,y,,) I (5.18b)
w°(o,y,) = o , (5.18c)
fC0,v,,Tt) = 0 , (5.1sd)
no"(o,1yt) = 0 , (5.18e)
) p1d (Z ) (5.18t)and fl°(zo,y,t) = WS 5ss(zo,y) + P(zo,y,5.18 I
I
II
56
I For k = 1, the inflow boundary conditions are
VU(zo,,t) = -Pd( 0,o,,t) (5.19)
3 with similar conditions for the other flow variables. For k > 1, the Fourier com-
ponents of the velocity and vorticity at the inflow boundary are
Uk(zo,y,t) = 0 etc. (5.20)
Outflow Boundary (z = ZN)
3 At the outflow boundary, the wake is assumed to be undisturbed. For k = 0,
the boundary conditions (4.21) become
UO(ZN,y,t) = USS(ZN,y) , (5.21a)
V°(ZN,Y,t) = VSS(XN,Y) ,(5.21b)
W°(N,,t)=O , (5.21c)
1.(XN,y,t) = O , (5.21d)
I n1(ZN,y,t) = 0(5.21e)
flZ(ZN,y,t) = W. SS(ZN,Y) (5.21f)
I and for k > 0, become
Uk(ZN,y,t) = 0 etc. (5.22)
I Freestream Boundaries (y = yo) and (y = yM)
IIn Chapter 4, it was shown that a disturbance described by linear stability
theory behaves like
-Oy (z tY)- -=±r2 r4 + -7'u(z,y,z,i) (5.23)
I o
I
II
57
as y -+ ±oo. For two-dimensional disturbances, equation (5.23) simplifies to I
y'z't= ±r2a,.u'(z,y,z,t) (5.24)
However, as noted in Chapter 4, equations (5.23) and (5.24) are not easily em-
ployed in their present form. The freestream boundary conditions are dealt with 3more easily in terms of the Fourier transformed variables.
If equation (4.28) is Fourier decomposed with respect to the spanwise direction 3z, then U' satisfies boundary conditions similar to equation (5.23) and U ° satisfies
boundary conditions similar to equation (5.24). One other question concerns the
value of a,. in these equations. For k = 0, a,. is set equal to a 2 d which is associated
with the two-dimensional excitation at the inflow boundary (see equations 4.17).
For k = 1, a,. is set equal to a3d which is associated with the three-dimensional Iexcitation at the inflow boundary (see equations 4.18). For k > 1, the wake is 3assumed to be undisturbed at the freestream boundaries.
The Fourier representation of the lower freestream boundary conditions (4.28), 3(4.30a), (4.31a), and (4.32) are shown below. For k = 0, the boundary conditions
are I
"-(zYOt) = r 2 dU"°'y t) , (5.25a) 3-9-(X)y0,t)= -8- (ZYot) , (5.25b)
W =0 , (5.25c)
f°( ZYO,t) = 0, (5.25d)
110(Z'y0, 0 = 0 , (5.25e) 3and flC(z,Y0,t) = 0 (5.25f) 3
I
!
58
As mentioned previously, primes denote the perturbation component of a variable.
3 For k = 1, the boundary conditions are
. =,yout) = a2 "d2 + - (5.26a)
8v1 7 O0= 2 r2Ul(110I) "Y(xyolt) = - a---(Zyo,t)-iwl(Yo,t) I (5.26b)
Zl 0= r2 + 1wl(zyo,?t) (5.26c)
f_1(.,yOt) = 0 , (5.26d)
3 ,--(x,yo,t) --= 0 , (5.26e)
and £l2(x,yo,t) = 0 (5.26f)
For k > 1, the boundary conditions are
I Uk(z,yo,t) = 0 etc. (5.27)
I The upper freestream boundary conditions are similar to those at the lower
freestream boundary. For k = 0, these conditions are
-(",YMt) = 2 (,yM,) (5.28a)
ay (='YM't) = 2d a.(z0~t
-- (z, I/,t)= =y,), (5.28b)
w 0 =0 , (5.28c)
fl° (Z, yM, 0) = 0 , (5.28d)
I fl°(ZyM, t)=0 , (5.28e)
3 and fl°.(z, yM, )-0 . (5.28f)
g For k 1, the boundary conditions are
aUl---== ad 2 + UI(zyM,t) , (5.29a)
3
!
59I
-- (ZlyMt) = -- .(ZIYMI -iW1 (ZYM,it) , (5.29b)
fl. (z YM, 0) = 0 (5.29d) 31Zy(Z,yM,t) = 0 , (5.29e)
and fl°(z,yM,t)=0 . (5.29f) I
For k > 1, the upper freestream boundary conditions are I
Uk(X,YM,t) = 0 etc. (5.30) 3Equations (5.25a) and (5.28a) enforce transverse exponential decay of the two- I
dimensional velocity disturbances. Equations (5.26a), (5.26c), (5.29a), and (5.29c)
enforce transverse exponential decay of the three-dimensional velocity disturbances Ithat correspond to the first spanwise mode k = 1. £Initial Conditions (t = 0)
As initial conditions for the calculation of the disturbed wake, the velocity 3and vorticity distributions of the two-dimensional base flow are prescribed in the
entire integration domain. Thus, for k = 0 the initial conditions are £U°(z,yO)=uss(zY) , (5.31a)
V°(z,y,0) =vss(z,y) , (5.31b) 3W°(zIyO) =0 I (5.31c)
fI°(z'y,0) =0 , (5.31d) Ifl°(z,y, 0) =0 , (5.31e) 3fl°z,y, O) = wss(z,y) (5.31!)
I11' (' Y 0 =W. S Z'Y) 5.1I
I
1 60
and for k > 0, the initial conditions are
3 Uk(z,Y,O) = 0 etc. (5.32)
3 5.2 Computational Domain
As a result of the Fourier decomposition of the dependent variables, the spatial
1 domain reduces to a two-dimensional plane in space (z, y) that is perpendicular
I to the z-axis. This domain is discretized into a finite number of uniformly spaced
grid points. These grid points have coordinates (x,,y,) given by the relations
Z,=xo+nAx for n=0,...,N (5.33a)
I and y,=yo+mAy for m=0,...,M (5.33b)
I The number of grid increments in the streamwise and transverse directions are
denoted by N and M respectively. The constants Ax and Ay are the grid sizes
I in the streamwise and transverse directions respectively. The discretized spatial
g domain is shown in Figure 5.1.
The temporal domain is divided into discrete uniformly spaced time levels ti,
I such that
i1 =IAt for I=L 1 ,...,0,...,L 2 (5.34)
where At is the time step between each time level. The base flow calculation is
3 performed in the time interval
5 tL, < t _< 0 (5.35)
3 which corresponds to
L, _ 1I5 0 (5.36)
I
61
The disturbed flow is calculated in the time interval I
0 < ti _< tL, (5.37) 3
which corresponds to 3O<l<L2 • (5.38)
Approximate numerical solutions of the governing equations (5.4) are obtained
at the spatial and temporal locations (Zn,ym,ti). The approximate solutions and 1their correspondence to the spatial and temporal grid points are denoted by
U.,,., = Uk (z,ym,,tI) etc. (5.39)
5.3 Discretization of Spatial Derivatives
All spatial derivatives in equations (5.4) are approximated using the second- Iorder finite-difference formulas shown below. Consider a complex function O(C, 77)
with known values at the equally spaced points, (0,17), ( i,10),.. • , (CI -l,17),
(CI, 77j). Then the first parial derivative ±±". can be approximated by 3i _ 4j) ' i+l,j - oi-ij + O(A 2) (5.40) 1
2A
and the second partial derivative o4I) can be approximated by 32 0( 1j) = -+ O(A ) (5.41)
a2 A 2 +I
where A is the grid increment in the direction. 35.4 Discretized Vorticity Equations
A combination of different methods are used to solve the vorticity equations 3(5.4a) through (5.4c). The methods have been chosen because of their numerical
I
I
I 62I sstability, their accuracy, and their computational efficiency. The Alternating-
Direction-Implicit (ADI) method is used to discretize the streamwise and trans-
3 verse diffusion terms. This method was selected because it is second-order time-
accurate and is unconditionally stable when used for solving linear diffusion equa-
tions. Use of the Crank-Nicolson method for the spanwise diffusion term re-
j tains both second-order time-accuracy and good stability characteristics. Fi-
nally, the second-order Adams-Bashforth method is used for the nonlinear terms.
I This method possesses favorable stability characteristics and is second-order time-
accurate. The spatial derivatives are discretized using equations (5.40) and (5.41).
I The resulting discretized vorticity equations can be solved using noniterative meth-
odThis choice of numerical techniques results in a two-step method for inte-
I grating the vorticity equations. In the first integration step, the streamwise dif-
fusion term is explicit and the transverse diffusion term is implicit. In this step,
3 the vorticity, which is known at t = lAt, is obtained at the intermediate time
tj+ I= (1 + !)At. For the first integration step, the discretized streamwise vortic-
ity equation is
I ADI-Explicitk'k k. +
______________ 1 / l m fi n,M,i
I At/2 Re AX2
ADI-Implicit Crank-Nicolson
+1l - 2+k + 0 k 2 fl k + , )J+ z,n,.+1,1+1 z. , 1,,+j .,.,m -1,1+j k " ( , , ,+i + n,, ,)
3 Adams-Bashforth% Fh3Fk - F "
2 , (5.42a)
I3
633
the discretized transverse vorticity equation is3ADI-Explicit
n- i (k -21
,n,l,+i y,n',m.I1 (y,n iSM11 -' 2flkAt/2ReA
ADI-Iraplicit Crank -Ni colson
+ 2AY2 2Adams-Eashforth
3Fk -F= y,n,m,l 2 y~rt~mL,-), (5.42b)3
and the discretized spanwise vorticity equation isADI -Explicit
l ,n~n,+i- ~nrnl ~ nI,~i- 2 Z,n,m,i + -IM~
ADI-Implicit Crank-Nicolson
+ nk2
____ +____
+ z,n~m+111+1 z ,n,k zz-11. ,,1,+1 ~~,
Adams -Bashforth
3F k - Fk= ( in~ 2 .~~~- (5.42c)
In the second integration step, the streamwise diffusion term is implicit andI
the transverse diffusion term is explicit. In this step, the vorticity is obtained at
t = (I + l)At. For this step, the discretized streamwise vorticity equation isADI -Implicit
________________ ( z+1m,1+1 n,m, 1~ ,,+i + f1:n-l,m,l+1
At/2 Re Az A 2
ADI -Explicit Crank -Nikolson
2nk, + n~~k, k2 n k nm ,L+l + ":nm~~x,n,m+1,1+f z n,m,l+i zn,m-1L*__ ~
+ r&y 2 2Adams-Bashforth
3F k - Fuknmi 53= x,fl,m7,1+i z~~)(54a
2I
1 64
g the discretized transverse vorticity equation is
ADI-Implicit
V n,m,L+l -~~~~ ~~~~~ nmil+"ynIi,+
At/2 Re Ax 2
ADI-Explicit Crank-Nicolson
+ y,n,m+1,1+4 V 'n,in,L- y,n,mn-1,1+.I __________+__________
I Ad&rn:-Baahforth2
3,m,i+. - F,n,m 1(36
= ( 2
5and the discretized spanwise vorticity equation is
ADI-Implicit
____________________ z,n+l,m,4- 2fl&~,m,l+1 z n&-1,M,Il
At/2 AeK A 2
IADI-Explicit Crank-Nicolson
+ ,n,m+1,1 ~~~~ ~~-,+I k zc nZmfl.?l + " fzm+ 4
Adams-Bashforth
3k-Fz,n,m,1+2 xn (5.43c)
I Equations (5.42) and (5.43) are valid in the interior of the spatial domain at
the points (OnYm) where
g1 < n< N -1 (5.44a)
and 1 <m <M -1 .(5.44b)
Equations (5.42) and (5.43) are also valid for times t1 where
Lj,. ,. L2(5.45)
65 £At the domain boundary, the boundary conditions from Sections 5.1.1 and 5.1.2 1are used to obtain the vorticity components.
5.4.1 Discretized Boundary and Initial Conditions for the Undisturbed Vorticity 3For the base flow calculation, the vorticity boundary and initial conditions
from Section 5.1.1 are discretized. These boundary conditions are discussed here. 5At the inflow boundary, the spanwise vorticity is
k, ,J = WRYm) (5.46)
At the outflow boundary, equation (5.13c) combined with equation (5.42c)
results in IN~ml~ ZIM I n z,N,vn,1+i ,~~~ zN,m-1,1+~
z ,N,,i+ -- z,N,m,1 1 zf/,N,.m+1,L+j- 2fl0 ,N.,+ +fl~,~-l,0
At/2 Re r2Ay2
= 3F0 N~, - F0 , ,= Fzml 2 z)N~m-i
(5.47) £Equation (5.13c) combined with equation (5.43c) results in 3
- o 1 n o - 21nm +z,N,m,l+l Z,N,m,l+I _ __ _ _,N,+I,+ ,N,_,_+_ +_ ,N,_-_,_+_
At/2 Re r)3FO F,
= Z,N,m,L+l z,N,MS 548l2 )(5.48)
At the freesteam boundaries, the spanwise vorticity is 3Oz°,n,o,l 0 (5.49a)
and ' = 0 (5.49b)z,n,M,1 54b
The initial condition for the base flow vorticity is
zn,M,L1, = Wz,(Y) (5.50)
I
I
!
I 663 5.4.2 Discretized Boundary and Initial Conditions for the Disturbed Vorticity
The boundary and initial conditions from Section 5.1.2 are used for the cal-
culation of the disturbed vorticity. The discretized versions of these boundary
5 conditions are discussed here.
At the inflow boundary, the vorticity boundary conditions for which k = 0
3 are
-o , (5.51a)
and = 0 , (5.51b)
n Z,,-,l =Wzss(Xoym) PZ,(zo,y,,) (5.51c)
For k 1 1, the boundary conditions are
Pw.(0 ,I YM, ti) I (5.52a)Y Ot,0MJ---= pw ,t0,ymt,) I(5.52b)
and 2 p.o,y,,t) (5.52c)
For k > 1, the vorticity components at the inflow boundary are
n, o,,,, = 0 etc. (5.53)
are At the outflow boundary, the vorticity boundary conditions for which k = 0
Ia,N,m = 0 , (5.54a)
NyNml = 0 , (5.54b)
nzN,m,l = WzSS(ZN,yn) (5.54c)
U
IU
67
and for k > 0, are I
nlk,N,ml = 0 etc. (5.55) 1At the freestrearn boundaries, the wake is assumed to be irrotational. For 3
k > 0, the lower freestream boundary conditions are In, = 0 etc. (5.56)
For k > 0, the upper freestream boundary conditions are
=0 etc. (5.57)
Finally, the discretized initial conditions for the disturbed vorticity calculation £are presented. For k = 0, the initial conditions are I
n°,n,..o = 0 , (5.5 8 a)no, = 0 , (5.58b)
n°,n,m,o = WZSS(ntYm) (5.58c) 3and for k > 0, are 5
fI~k,,,,o = 0 etc. (5.59) 55.4.3 Equation Systems for the Calculation of the Vorticit, 3
The discretized vorticity equations, (5.42) and (5.43), combined with the dis-
cretized vorticity boundary conditions form systems of linear algebraic equations 3that are solved for the unknown vorticity. To obtain the vorticity at the interme-
I
U
68
i diate time tj+ , the tridiagonal system
bi a, kn,1,1+'
-(5.60)
al bi a,a, b1 •
z,n,M-1,1+1 k' ( k 'JI+ fk k'I
Cl nft,l,l + d(z,+l,,l - - ,1,1,1)( - N,n) + Hn,1, l
I•
I, Ik SIk nk, k,
ln ,MMl + dn(Z,n+l,M-1,1 + z f,._1,M-1,1)(1 - SN,n) + n M_1,
is solved. For the base flow calculation, equation (5.60) is solved for
3 k=0 1<n<N , and L 1<l<0 (5.61)
For the disturbed flow calculation, equation (5.60) is solved for
0<k< K 1<n<N-1 , and 0<l<L2 (5.62)
3The constants in equation (5.60) areAt
a, = At (5.63a)2Re(r2 Ay) 2
At At (k 2 (5.63b)g = 1 + Re(r 2Ay) 2 +Re r ;.6b_ t A, _L 2 ATN
1 4W'( s 2W WRSz, if # N; (5.63c)1 At , ifn N;
I 4Re ~r3 J'fn N
and d,, -eAzy? if n # N; (3.63d)I0 ifn=N.
I5
69
The function H, ,MI is
k At (5.64F)
n,,1= . (3F;,,,,- , ,,,,,_) (5.64) 3and 6 N,n, the Kronecker delta, is 3
6N,=O n N. (5.65)
To obtain the unknown vorticity at time tje 1 , the tridiagonal system
b2 a 2 z, ,r,L+1
a 2 b2 a 2
(5.66)
a2 b2 a 2 3+a(2+ b2' f
z + (N 'm , + 1
C~~fl~~N~~lml+* + + flN'll+I +,'-_1,-,+)+Hji .3
C2 mL,,+ "d2( l)+m+lL+ -z,1,m1,+ HknL+ a2 ornl+1
CN , r +k 1 < m,-, + a n d d 2 < +,- , + ()+5 .6 7
Sk WQkk , r nk,:Ca2 ,,N,,,.,,,+.,+ d2 z, , ,,,,,,+1,,,+-+ a ,.,-,,, . + ) + .H N ,,,., - a ,,,,,,,+,
is solved. For the base flow calculation, equation (5.66) is solved for 3N' = N , k =0 , 1 < m < M - 1 , and L1:5<0 (5.67)t
For the disturbed flow calculation, equation (5.66) is solved for
N'= N-1 , 0<k<- - 1<m<M-1 , and O<1<L2 . (5.68)
2I-- - 2 ' - ---- *
II
70
The function H,m is t0
H- F, .,) (5.69)
I The constants in equation (5.66) are
I = At (5.70a)a2=2ReAZ2,
b2 = + At + A k (5.70b)
At (k\ 2 At
4Re r 3 Re(r2 Ay)2 (5.70c)
and d 2 = At (5.70d)2Re(r2Ay)2
The constant a2 is
[0, for the base flow calculation; (5.71)a2=a 2 , otherwise.
I The constant
b' is
I 1+{~for the base flow calculation; (5.72)b2 = b2, _4e 3 1 otherwise. (.2
Equations (5.60) and (5.66) are solved for the spanwise vorticity. Equation
systems for the streamwise and transverse vorticity components are identical to
I equations (5.60) and (5.66) and are not shown.
3 For the base flow calculation, equations (5.60) and (5.66) are solved beginning
with the initial data at time t = tL,. For the disturbed flow calculation, equations
3 (5.60) and (5.66) are solved beginning with the initial data at time t = 0. For the
initial integration step of both the base flow and the disturbed flow calculations,
Ithe nonlinear terms are evaluated using the first-order Euler method instead of
I the second-order Adams-Bashforth method.
U
U
71
5.5 Discretized Velocity Equations
The Helmholtz equations, (5.4d) through (5.4f), are discretized using the
second-order finite-difference formulas derived in Section 5.3. The discretized
Helmholtz equations are, for the streamwise velocity component 3U+Im,, - 2Umi, + Uk_1,m, I,
U , 2 U k
U,,k+~ 2Uk ,, + Uk,_~+ nrmA,1 I~, ~-,
Gknm " (5.73a)
for the transverse velocity component r
Vn+l,m,l - 2Vn,,i + Vi1mj
Ax 2
Vk _ 2Vk, +Vk
+ n,m+l,l n,m-1,L
ray 2
SV' , nMi (5.73b) aand for the spanwise velocity component 3
w+ ,+ , - 2W:,m,, + Wni ',mI
W,m+,,, - 2Wn,, + n.M-l
+ r2 2 U=
.nk (5.73c)
The right hand sides of equations (5.73) are
k 3 11 z,n,m+l, - oz,n,m-l,l ik
Z' 7, 2Ay - -Vn,m, , (5.74a)
r-k ik , 3 z n,+l,m,l - 1Z.,n-I,M,I 57 b
Ly,n,m,l r nknM~ _2 ( fk+i,!2k m,2z (5.74b)ar.,r, r2l -
I
UI
72
---- G 1 y,n--,M , -
nG,,,= r3 2Ax
The Helmholtz equations are solved for each integration step, subject to the ap-
5propriate boundary conditions from Sections 5.1.1 and 5.1.2.
Equations (5.73) are valid in the interior of the spatial domain. At the domain
3 boundary, equations (5.73) combined with the discretized boundary conditions are
used to obtain the velocity components.
5.5.1 Discretized Boundary and Initial Conditions for the Undisturbed Velocity
For the base flow calculation, the velocity boundary and initial conditions
from Section 5.1.1 are discretized as discussed below.
At the inflow boundary, the streamwise and transverse velocity components
are
U0 - (Y.) (5.75a)an ,O,m,1 Yn
an , = 0 (5.75b)
I The discretized version of the outflow boundary conditions, (5.13a) and
(5.13b), combined with equations (5.73a) and (5.73b) result in
N - m1, t + U N ,m ,=, O (5.76)
(r 2 Ay) 2 GN,M'I
for the streamwise velocity and
VNO,m+1,l - 2VNOm,. + VN o (577)
(r 2 Ay) 2 = ,N,.,,
for the transverse velocity.
3
UI
73
At the freestream boundaries, the streamwise velocity is SUk = 1 (5.78a)
and U,M, 1 -- 1 (5.78b)
Equations (5.14b) and (5.15b) combined with equation (5.73b) result inV.01,01 -2V.,0, + .LIO'I2Vno,, - 2Vn,o,lV+,o0 1 - 2Vo 0 + ,, + 2 = 0 (5.79)
Az2 r2 Ay 2
at the lower freestream boundary and IVn+iM, - + V 0 2V,_I - 2VOM,j1/0 M~ -- 2VM,-+ -1M, " - 0 (5.80)
+2 r2Ay2 0)
at the upper freestream boundary.
The auxiliary condition
V, =0 (5.81) gimposes antisymmetry on the transverse velocity field. Because of this antisym-
metry, the transverse velocity V °omi is solved only in the lower half of the spatial
domain for which
1< n < N (5.82a)
and 0<m<--M 1 (5.82b)
The transverse velocity in the upper half of the spatial domain is found from the
relation
VO = for 1<m<- and 1<n<N (5.83)2 2 2
1
Finally, the initial conditions for the streamwise and transverse velocities are
Uno,m,Lt U1 (Ymn) (5.84a)
and V.° = 0 (5.84b) 3U
I
I1 5.5.2 Discretized Boundary and Initial Conditions for the Disturbed Velocity
Boundary and initial conditions from Section 5.1.2 are used for the calculation
of the disturbed velocity. The discretized versions of these boundary and initial
conditions are discussed in this section.
At the inflow boundary, the velocity components for k = 0 areIo = USS(ZoYm) + P, d(Zo,ym'tI) (5.85a)
I E02dXjmtIo,,Ui = 0 + P, (Zo,ym,ii) , (5.85b)
and W = (5.85c)
For k = 1, the velocity components are
Sn , -iP:d(T(Imt,y,,) I (5.86a)
o0,,,',- 1p, (.Tojy,,jtI) I(5.86b)
an o,,,,, p .dP ( Z o, Y,,,, ) (5.6c)
For k > 1, the velocity components are
Ukom,, = 0 etc. (5.87)
At the outflow boundary, the velocity components for k = 0 are
N,,MI = USS(XN,Ym) , (5.88a)
VN,mt - Vss(ZN,Ym) , (5.88b)
UN,,n, = 0 (5.88c)
I and for k > 0, are
UV,,n, = 0 etc. (5.89)
U
75
At the lower freestream boundary, equation (5.25a) combined with equationU
(5.73a) results in the following equation for the k = 0 streamnwise velocity:
U0101- 2UIO,0,1 + U%..1,0,1' + 2U.0,1,1 - (2 + 2C.2d)U.,,
- (r ty) 2 (2uss(z., yi) - (2 + 2Cdusz,)) (.0
where C.2d a ~dr2 Ay. Equation (5.25b) combined with equation (5.73b) results
inI
V0 1,1 - 2VI0,0,1 + V 0L'' 2V.',, - 2V 0,
Az 2 vAy2
for the k =0 transverse velocity. For k = 0, the spanwise velocity is
WOO = 0 (5.92)
at the lower boundary.
For k = 1, equation (5.26a) combined with (5.73a) results in the following
equation for the streamwise velocity:
-nOl- 2U.,0,1 + U~.1 ,0 1
Ax 2
+2U,,,, - (2 + 2C.3d)U,O0 1 2
+r 2Ay 2 U,O01 = 0 (5.93)
where Cd-Ayr 2 Var +r()3 Simiflarly, equation (5.26c) combined with3
(5.73c) results in3
n+1,,l - 2Wn',0 ,I + W1 -'I
Ax 2
+ 2W,, 11 - (2 + 2C~d)Wn',0 ,1 _1- Wn 2 = 0 (5.94)It2 AY2 kr3)
r2I
II
76
I for the k = 1 spanwiie velocity. Equation (5.26b) combined with equation (5.73b)
i results in +
I1 __V1
.+,o,- v.,o, .,,,, 2v, ,, - 2V1', _ ( ,O,l
Ay)2 (U.+1,0,- U. 1,0, 1) + iAYW,o,) (5.95)
which is valid for the k = 1 transverse velocity. For k > 1, the velocity components
at the lower freestrearm boundary are
Uo = 0 etc. (5.96)
At the upper freestream boundary, the following equations are valid. For
k 0 0, the velocity boundary conditions are3--+, 2 Ul n _, + 2U,M-1, - (2 + 2C2d)Un,M,l
A: 2 rAy2
(r2 Ay) 2 (2uSs(Xn,yM.-1) - (2 + 2C.2d)uSS(:n,.yM)) , (5.97a)
rt+1,M1 2Vn,, n+ -1,Mj 2v',M-I, - 2Vn0,M,lA:2 + r2Ay 2
1 o - ,-I,M,j) , (5.97b)I. andA:Ayr4 (U.+l,M,1 U ~r
and
A~r
Wn,M'L = 0 (5.97c)
For = 1, the velocity boundary conditions are
- 2U,,,MJ + UI.,M,j
A: 2IU 1 (ld\
2U',M-I,l - (2 + 2C,3d)U -M,M,I 2,+ r 2AY 2 r3 U ,M,I 0 , (5.98a,)2
I77m
2W 1 WI l
Wn+,M, i W ,Ml + n-1,MlAx
2
+ 2W',M_1,1 - (2 + 2C0a )W,M, l )2r Ay2 - Wn,M = 0 , (.98b)
and 3Vn-+,M,l 2Vn,o,1 +"-1,M 2Vn,M-1,L - n M
&Z2 4- ,&y2 r3 V1, M, 1A:2
1r2Ay)2 (E (U 1+,M,1 + UnI,M,I) + iAYWM,I) (5.98c)
For k > 1, the velocity components at the upper freestream boundary are
UkMj = 0 etc. (5.99)
Finally, initial conditions must be specified for the velocity components. For
k = 0, the initial conditions are IU.mo = USS(Xn, ,n) (5.1ooa I
V n,mn,O = VSS(Zn Ym (5.100b)
Wn,m,O = 0 (5.100c)
and for k > 0, the initial conditions are
UOnMo =o0 etc. (5.101)
5.5.3 Eguation Systems for the Calculation of the Velocity
The discretized velocity equations (5.73) combined with the discretized veloc- Iity boundary conditions form systems of linear algebraic equations that are solved
to obtain the velocity components. 3II
mm
78
i Base flow Streamwise Velocity
Equation (5.73a) combined with boundary conditions (5.75a), (5.76), and
(5.78) form a system of linear algebraic equations that is solved for the base flow
3 streamwise velocity. To derive this system of equations, let
I ,
U , ,, : G.,n,l,l
I On= gn = .,(5.102a)
GM1, (M-1)xlx
U0 ,M-z,,,M-1, (M-)xl-2 11 -2 1
3A,(5.102b)
1 -2 11 -2 (M-1)x(M-I)
U0
I and hn (5.102c)I
0
Un,M,1 (M-1)xl
III
I
79
Then, the resulting system of equations 3(A' I 3
I A' II •I
I A' I0 AA NxN #N Nxl 1
(Az 2g, - hjA- O0 )Az2 - h2A
(5.103)
AZ 2 gN- hNA J NxI
is solved for the base flow streamwise velocity. The constant A is IA- AX 2(5.104)
the matrix A' is 3A'=AA-21 , (5.105)
and I is the identity matrix.
Base flow Transverse Velocity
Equation (5.73b) combined with boundary conditions (5.75b), (5.77), (5.79), 3and (5.80) form a system of linear algebraic equations that is solved for the base I
3
II
80
i flow transverse velocity. To derive this system of equations, letV0 Go
-2 1
and A= , (5.106b)
I -- 2 1
I Then, the system of equations
(A' I I1 Az 2gi
I A' I
." .. .. (5.107)
I A; I1 0 AA NxN #N Nxl Nxl
is solved for the base flow transverse velocity in the lower half of the spatial domain,
I for which
1 _<n<N , (5.108a)
Mand 0 m< -- 1 (5.108b)3 -2
I
81
The base flow transverse velocity in the upper half of the spatial domain is obtained 3from the relation
V=- V.. 1 , for 1<m<- and I<n<N . (5.109)?,-M ~ " - n,L'
As previously,
A r2Ay (5.10a)
and
A'=AA-21 (5.110b) UDisturbed Velocity Components for k. 3
Equation (5.73a) combined with the boundary conditions (5.85a), (5.88a),
(5.90), and (5.97a) form a system of equations that is solved for the zeroth spanwise i
mode of the disturbed streamwise velocity UO. To derive this system of equations,
let I1,m,i G(' m~
1I
GoU'O-lI'M'l (N-1)xl G2,N- i,m, (N- 1) xl
-2 11 -2 1 3
.",(5.111b)3
1 -2 11 -2 (N-2)x(N-l)
I
U
0
andhm= (5.111 c)I0
N,,m,l (N-1)
Then, the system of equations
A'-B' 2AI #03 AI A' AI
At A' AI
2AI A'+ B" M+1)x(M+i #M
Az 2go - ho + 2AyAf'
Az 2 g - hi
AZxgM-1 - hm-1AZ 2 gM - hM - 2AyAf" (M+l)xl
3 is solved for the zeroth spanwise mode (k = 0) of the disturbed streamwise velocity.
In equation (5.112), A' is
3 A'= (5.113a)
and A is
A = A 2 (5.113b)
I
I83 3
The matrix A' is
A =A-(2A+A')I , (5.114)
the matrix B' is 3
and the matrix B" is B1=2~ C 515
B" = 2AAyc"I (5.116) 3The vectors f' = [f' and f" = [f,] and the scalars c' and c" result from
the freestream boundary conditions. For the zeroth spanwise mode (k = 0) of the Istreamwise velocity U', 3
, 2d us ~ ss(Xn,Yi) - uss(xn,yo)-a. r2uss(-n,Yo) + , (5.117a)
2d ( USS(3n,YM) - USS(Xn,YM- 1)
adf 'l r. r2USS(XniYM) + A (5.117b)Ay
and N
c a2d (5.117c) USac -dr 2 (5.117d) 3
Equation systems for the velocity components V0, U', V 1, and W 1 are almost
identical to equation (5.112). The only difference between equatic,, (5.112) and Isystems of equations for the other velocity components V °, U 1, V 1, and W 1 are 3the vectors f' = [f,',] and f" = [fn] and the constants c' and c". Therefore, for
these other velocity components, only f,,, f,, c' and c" will be shown. 5For the zeroth spanwise mode (k = 0) of the transverse velocity V °,
= .+1,0,1- Un -1,0,, (5.118a)2Az
U 0 -U 0
f" = U (5.118b)
II
U
1 843 and
c =0 , c =0 (5.118c)
For the first spanwise mode (k = 1) of both the streamwise and spanwise
velocities, U' and W 1,
A" =0 (5.119a)| " = 0 (5.119b)
U and
I2c~ r2 Cr3d 2 +,(5.119C)
*1/ 2
c = r2 a3d2 + (5.119d)
Finally, for the first spanwise mode (k = 1) of the transverse velocity V1,
I 2A-x+ 0- ,01- iW, ,0, , (5.120a)
fU= - 1n+l,M, - U -I 1
2A- W ,M,I (5.120b)Iand
I c' =0 ,c (5.120c)
I Disturbed Velocity Components for k > 1
3 Equation (5.73a) combined with the boundary conditions (5.87), (5.89), (5.96),
and (5.99) form a system of equations that is solved for the disturbed streamwiseUI
3
85
velocity Uk for k > 1. To derive this system of equations, let 3
O =, g,1 , (5.121a)
•• I 3N-,m, (N-1)xlN (N-I)xl
-2 1
and A= (5.121b)
1 -2 1 31 -2 (N-1) x(N-1) I
UIU3I3IU
UI
I Then, the system of equations
86
A' AI #1
Al A' Al
Al A' AI
A1 A' (M-I)x(M-1) OM-1 (M-I)xl
I . (5.122)
A 2gM- 1 (M-1)xl
is solved for the streamwise velocity component Uk. As before,
A 2 (5.123a)
and A= X 2 (5.123b)
I The matrix A' is
A' = A- (2A + A')I (5.124)
Equation systems for W' and for Vk and Wk with k > 1, are identical to equation
3 (5.122) and will not be shown.
Equations (5.103), (5.107), (5.112), and (5.122) are systems of linear alge-
braic equations that are solved for the velocity components. For each system,
3 the coefficient matrix is large and sparse. However, because of the structure of
I
3
87
these matrices, fast Helmholtz solvers can be used to solve these equation systems. 3Details of these techniques are given by Swarztrauber (1977).
5.6 Evaluation of the Nonlinear Terms !
Up to this point in the development of the numerical method, the evaluation 3of the nonlinear terms Fk Fk and Fm at each time step has been
ignored. However, in order to calculate f 1 , fmit and lzk,, MI from equa-
tions (5.42) and (5.43), the nonlinear terms must be computed at times ti-1 , ti,
and t L+j . Generally, computing the nonlinear terms is the most time consuming
aspect of using spectral methods to solve the Navier-Stokes equations. The diffi- -culties arise from the fact that the nonlinear terms do not have a simple Fourier
representation. IThe nonlinear terms can be computed either pseudo-spectrally or spectrally.
The pseudo-spectral method works in the following manner. The physical repre-
sentation of the velocity and vorticity are obtained from the Fourier coefficients 3using equations (5.1). Then, the nonlinear terms f , f,, , and fk,,, are
computed from the formulae I
tk k k k
= k Un+1,m,1 - Un-l,m,l k Un, +l, 1 - Un,+m-l I,,= W,L,, 2Az + Wy'rm'4 2Ay
+ WIz,n,m,1 k k z,n+l,m,l Az ,n-Iml
8z ,m, - n,m,
n'M'I My ntMFIe57z-IP W!n,~~ , (5.125a)k- k I
f n-m~- -n-I~ + k n,. h ,m (sl~ sv"nM~ Z~~'"' 2A2y - yn,~ A
k 8V W k' -
k, _ - VV-- l ,,n,m,l yn-1,ml+ W:,nIMW iftl m4 2 ,A 2A
k kh I,n n,mll y ,n mI k --1,3-Vn,m,i 2A!/1'I In~, (.2b
My 8Z
UI
88
and W-+l,m,l - W--,mL k n,m ILnlm - Wn,m--,
-- k'nz = +,nml rn, zm,l'k k+ Whnl aw I", n" azln+l,m,l - Wz,n-l,m,l
,z Un 2Az
zn,m+l,l z,n,m- l, k z k (5.125c)-n,M, 2Ay nMl 8 InM'l
5 which are the finite-difference versions of equations (3.12).
The z derivatives in equations (5.125) are computed using the formula
- (z)= L ikHkeik (5.126)
whereK
h(z) k 5 He%' c . (5.127)
The nonlinear terms fae,, j and transformed back to
Fourier space using equations (5.5). This results in the nonlinear terms F,,,,
,,,,, ,andFz,,, for all n, m, 1, and k. The pseudo-spectral method, together
I with the use of fast Fourier transforms, has an asymptotic operation count of
I O(NMKLog2K) . (5.128)
U An alternative to the pseudo-spectral method is the spectral method. For the
3 representative nonlinear term W(w= tt (5.129)
I the spectral method proceeds as follows: The Fourier series representation of
3 the dependent variables U(z,, Ym, Zk, ti) and wz(zn, yY, z&, ti) are substituted into
I
I
89
equation (5.129) to get 3
to obtain the Fourier series representation of equation (5.129). This procedureIThe multiplication of the two series in equation (5.130) is performed in order3
is repeated for the other terms that make up the nonlinear part of the govern-
ing equations. In this way, the Fourier representation of the nonlinear terms is 3obtained. The asymptotic operation count for the spectral method is I
O(NMK 2 ) (5.131) IFor large values of K, the pseudo-spectral method requires fewer operations
than does the spectral method. However, for small values of K the spectral method 3is faster. In this work, which is limited to small values of K, the spectral method
is used. I5.7 Consistency, Stability, and Convergence of the Numerical Method 3
Finally, consistency, numerical stability, and convergence of the numerical
method are discussed. A consistent numerical method is one whose truncation 3error approachs zero as the spatial and temporal grid increments approach zero.
A numerical method is stable if the round-off error contained in the numerical Isolution does not grow with time. A convergent numerical method is one in which 3the numerical solution of the discretized partial differential equation approachs
the exact solution of the partial differential equation as the grid sizes approach 3zero.
I
II
90
The relationship between consistency, stability, and convergence of a dis-
5 cretization scheme is given by the Lax Equivalence Theorem (Smith, 1985):Given a properly posed linear initial-value problem and a linearfinite-difference approximation to it that satisfies the consis-tency condition, stability is the necessary and sufficient condi-tion for convergence.
5 In this work, the governing equations are not linear. However, it is still useful to
examine the consistency and stability of the numerical method.
3 The truncation error of the numerical method described in this chapter
is now discussed. Let
L,,, A~)= 0 (5.132)
I represent the difference equations (5.42) and (5.43) and let ! be the solution ofu those difference equations. Furthermore, let fl be the exact solution of the partial
differential equations (5.4). Then, the truncation error ET A,,a,At associated
with the discrete operator LA2 ,AvA± is
3 ET Az, ,Aat(O) (5.133)
3 It can be shown that the hybrid ADI, Crank-Nicolson, Adams-Bashforth dis-
cretization scheme, coupled with second-order finite-difference approximations of
3 the spatial derivatives, has the local truncation error
3 ET,,,,,At = O(Az 2 , Ay 2 , At 2 ) (5.134)
3 This truncation error approachs zero as Az, Ay, and At approach zero and there-
fore the scheme is consistent. Similarly, the discretization scheme used to solve
the Helmholtz equations is consistent and exhibits second-order spatial accuracy.
3 The ntmerical method has spectral accuracy in the spanwise direction.
U
!I
91
Evaluating the numerical stability of the discretization scheme used in
this work is very difficult due to the nonlinearity of the governing equations. How- 3ever, Pruett (1986), who used the same discretization scheme as discussed here,
found that for simulations of instability waves in free shear layers, the condition 3At At AtIlma+. : I-CLZ. + IWl,,z- -< 1 (5.135)
was sufficient to ensure the numerical stability of the hybrid ADI, Crank-Nicolson, IAdams-Bashforth discretization scheme.
For the simulation of instability waves in high-deficit wakes, it is noted Ithat, because of numerical stability considerations, simulations of three-dimensional
disturbances required a smaller time step At than did simulations of two-dimen-
sional disturbances. Furthermore, although a stability criterion has not been de- 3rived, for these simulations the numerical method was stable for the spatial and
temporal resolution that was employed. IIIIIIIII
U
92
I CHAPTER 6
I RESULTS
I As discussed in Chapter 3, the laminar-turbulent transition of wakes is influ-
enced by many nondimensional parameters. A thorough study of wake transition
would require investigating the role of many of these parameters. However, due
1 to the large amount of computer time and memory that is required to solve the
Navier-Stokes equations, a detailed study of the effect of each parameter on wake
I transition is not possible.
In this work, investigations are limited to two areas. First, the influence of
different levels of excitation on the behavior of two-dimensional disturbances is
3 investigated. The results of these investigations are discussed in Section 6.1. Sec-
ondly, the interaction of two- and three-dimensional disturbances is investigated.
* This latter topic will be discussed in Section 6.2.
Before proceeding, the techniques used to analyze the data that result from
I the numerical simulations are discussed. The harmonic content of the flow vari-
ables is obtained by Fourier time series analysis. With t (z, y, t) representing any
one of the flow quantities, the Fourier analyzed variables are obtained from
I +1 _.Lt e i p ' t
3L2 -+with F- 0 1 2 __.. ___1- ' (6.1)
The parameters L, and L2 are the first and last time steps respectively of the
3 time interval in which the data is Fourier analyzed. The parameter Ip,,. denotes
the number of oscillation periods that are analyzed. If one period of oscillation is
I
I
93
analyzed then Ipe, = 1, and if two periods of oscillation are analyzed then Ipe, = 2. 3The parameter c assumes different values depending on the flow variable that is
analyzed. If 4A represents Uk or Olk then c = 1. If ikk represents Vk or a' then
c = r2. Finally, if Ok represents Wk or f]k then c = r3. The parameter i is the 3imaginary number V/'71.
The transformed variable k(z,y, F) is the Fth time harmonic component of 3the flow variable lkk(z, y, t). The Fth harmonic oscillates in time with the frequency
F/3, where f3 is the frequency of the wake excitation.
The amplitude of the Fth harmonic 1 k(z, y, F), for k = 0, is 3= y I, if F = 0; (6.2a)A 21 k(Zy,F)j, if F > 0
and fork >0 is 3,y' if F = 0; (6.2b) I
4(x, y, F) = {4 1¢(z,y,F)1, if F > 0.
The phase of the Fth harmonic k(z, y, F) is
Oh(xy,F) = arctan (IM((Z' y, F))) (6.3a)
if Ok represents Uk, Vk ', or fSl. The phase of the Fth harmonic Ck(z, y, F) is 3= arctan (Re( ik(z Y'F)) (6.3b)
if ?k represents Wk, fk, or fil. Im( ) and Re( ) denote the real and imaginary 3parts of 4k(z, F) respectively.
Additional quantities that are calculated are the wavenumber a., the amplifi- 3cation rate a, and the phase velocity c; of the various harmonic components. For I
I
1I
94
I linear stability theory, in which the base flow is assumed to be parallel, these quan-
3tities are uniquely defined for a given disturbance frequency, spanwise wavenumber,
and flow Reynolds number. Furthermore, for a disturbance that is governed by
3 linear stability theory, ak, , and ck can be computed fromk 9
(6.4al rLST ( I Y) (64a8
i LST - A
and ck P3 (6.4c)PLST - LST3 The values of arLST, aiLST, and CPLST are independent of z and y and are
independent of which flow variable is represented by Ok. However, if expressions
Ianalogous to equations (6.4) are used to obtain similar quantities for a nonparallel
flow, then the coordinates z and y as well as the flow variable that is represented
by Ok have considerable influence on the values of a k a , and c. Gaster (1974)
discusses some of the various ways to calculate a, and C for nonparallel flows.fk a an d kae-acltdi hFor this work, the disturbance quantities a, a1, a c are calculated in the
3 following manner. The streamwise wavenumber ak of the Fth harmonic k(z, y, F)
is calculated from
cfx, F) z y,F) (6.5a)
where the transverse coordinate y is taken to be constant in equation (6.5a). The
Iamplification rate ct of the Fth harmonic k(Z, y, F) is calculated from
a(z,F) = -O' (ln(ikA(zy.".(z),F)) (6.5b)
In formula (6.5b), ym,.(z) is the transverse coordinate, as a function of z, at
which the amplitude 0 kA(z, y, F) attains its maximum value. The phase velocity
3 of the Fth harmonic k(z, y, F) is computed from
, F (6.5c)I
I
I
95 1Finally, the disturbance kinetic energy is calculated from 3
'(zF)= ' (U (z, V, F)2 + v'(z,, F)2 + w' (z, y, F) 2 )dy (6.6) 3The kinetic energy Ek(z, F) is a measure of the disturbance amplitude. Also, 3because the kinetic energy is an integral quantity, it does not depend on the
transverse coordinate y. Amplification rates of the harmonic components can be 3caculated from the kinetic energy using
a F ) (6.7)at~z') = 2 8 \ IUse of equation (6.7) to calculate a is advantageous because, unlike equation
(6.5b), it does not depend on the transverse coordinate y. 36.1 Investigations of Two-Dimensional Disturbance Development
In this section, investigations of two-dimensional disturbance development in 3a high-deficit fiat plate wake are discussed. Several different calculations are under- -taken. With the first two calculations, termed Case-1 and Case-2, the suitability of
the numerical method for the calculation of wake disturbances is tested. In Case-i, Ithe method is tested by comparing the results for small amplitude disturbances to
linear stability theory. In Case-2, the results of the calculation are compared to 1the experimental measurements of Sato (1970). In Case-2, the numerical method 3is tested for larger amplitude disturbances than in Case-1. In Case-3 and Case-4,
the large amplitude behavior of wake disturbances is investigated. Case-4 is iden- -tical to Case-3 except that the amplitude level of the excitation, denoted by A2, is
larger than in Case-3. By calculating disturbances for different excitation levels, 1the influence of the initial amplitude on the disturbance development is observed. 3Finally, in Case-5 the effect of the outflow boundary conditions on the results is
33
iI
96
i investigated. This is done by repeating Case-3 in a longer spatial domain. In all
i of these calculations, the wake is excited with a two-dimensional sinuous mode
disturbance of frequency 3. Finally, because the calculated disturbances are two-
3 dimensional, only U', V ° , and f o. are calculated.
Case-1
U In this case, the behavicr of a small amplitude disturbance in a nonparallel flat
gplate wake is investigated. Because the base flow is nonparallel, there is no proper
way to compare these results to linear stability theory. In fact, Gaster (1974)
3 states that the agreement between linear stability theory, for which the flow is
assumed to be parallel, and experiments (numerical or physical), for which the
flow is nonparallel, can never be better than O(Re- ). Therefore, comparisons
of the numerical results to linear stability theory are qualitative in nature and
can not be used to test the accuracy of the numerical method. However, this
3 calculation can be used to obtain some measure of the suitability of the numerical
method for calculations of small amplitude disturbances.
3 This calculation was performed in a spatial domain bounded by
Ixo= .3 , TN = .7 , Yo = - 10 , and YM = 1 0 (6.8)
3 The transverse extent of the spatial domain was specified so that the vorticity
disturbances would be approximately zero at the freestream boundaries. For the
streamwise extent of the domain, it was required that the domain be long enough
3 to contain several wavelengths of the fundamental disturbance. The domain is
4hown in Figures 2.1 and 5.1.
3 The base flow was calculated subject to the inflow streamwise velocity distri-
bution u'(y) given by equation (4.6a). At the inflow boundary, the wake centerline
I
I
97 1velocity U, and the wake half-width b were I
UC=.5 and b=1.3 (6.9) 3The streamwise velocity component of the computed base flow is shown in Figure 36.1.
As discussed in Section 4.4, the wake is excited using the time-dependent 3boundary conditions (4.16). For this calculation, the disturbance amplitude was
A2 = .001. In terms of the streamwise velocity disturbance u'(z 0 , y) at the inflow
boundary, A2 is given by the relation A2 = max(Iu'(zo,y)) where the function 3max(lu'(xo,y)l) denotes the maximum value of Iu'(o,y)1 with respect to the y
direction. The disturbance frequency was 6 = .317. For this frequency, the Iamplification rate is less than the maximum amplification rate that is predicted
by linear stability theory. Furthermore, the disturbance that corresponds to this
frequency experiences amplification throughout the spatial domain. The Orr- I
Sommerfeld eigenfunctions that correspond to this frequency are shown in Figures
6.2. 1The influence of Az and Ay on the numerical results was investigated to 3
determine the spatial discretization that results in solutions that are reasonable
independent of the grid sizes. A detailed discussion of these investigations is given 3in Appendix B. For this calculation, the grid
N=128 and M=64 (6.10)
was sufficient for this purpose. For the time step At, numerical stability rather
than temporal accuracy was the most severe constraint. Therefore, the influence of
3U
1 98
5 the time step on the numerical results was not investigated. To maintain numerical
stability, the time step was specified to be
At = T with Lnm = 64 (6.11)
The parameter Lm. denotes the number of time steps per fundamental distur-
1 bance period. The response of the wake to the excitation at the inflow boundary
was calculated for five periods TF of the fundamental disturbance (L 2 = 320)
where TF = 2,
3 For this calculation, the spatial domain contained approximately eight wave-
lengths of the fundamental disturbance. Each disturbance wavelength was dis-
3 cretized by approximately sixteen grid points. In the transverse direction, the
spatial domain contained approximately twenty momentum thicknesses (based on
the inflow velocity distribution) with approximately 3.2 grid points per momen-
3 tum thickness. As mentioned previously, investigations of the influence of the grid
sizes on the results of the numerical calculation (see Appendix B) have established
3 the suitability of the spatial discretization.
IIIIUII
99 1The relevant parameters in this calculation are summarized below:
Case-l:
the streamwise location of the inflow boundary, C - .3;
the streamwise location of the outflow boundary, ZN = .7; 3the transverse location of the lower freestream boundary, yo - -10; 1the transverse location of the upper freestream boundary, YM = 10;
the Reynolds number Re - umA, Re - 200000;
the amplitude level of the wake excitation, A2 - .001;
the frequency of the wake excitation, )3 .317; 1the number of streamwise grid increments, N - 128; 3the number of transverse grid increments, M 64;
the number of time steps calculated, L2 320; 3and the time steps per fundamental disturbance period, L,, - 64.
Results of this calculation are shown in Figures 6.3, 6.4, 6.5, and 6.6. Am-
plification curves based on the disturbance kinetic energy &(z, F) are shown in 3Figure 6.3. These amplification curves are compared with analogous curves from
linear stability theory that are based on the amplification rate ac. However, the
amplification of the kinetic energy that results from the Navier-Stokes calculation
arises from two different sources. First, the kinetic energy amplifies due to the 3linear instability. Secondly, the kinetic energy changes due to the slow divergence
of the base flow. However, despite the differences between linear stability theory
and the calculation, the kinetic energy of the fundamental disturbance (F = 1) 3still compares closely to the linear stability theory prediction of the fundamen-
tal disturbance kinetic energy. The mean disturbance component (F = 0) and 3the second harmonic (F = 2) are also present, but are much smaller than the
fundamental disturbance. IIU
1 100
I For the streamwise location z = .35, the amplitude and phase distributions of
the fundamental disturbance component (F = 1) are compared to the amplitude
and phase distributions of the Orr-Sommerfeld eigenfunctions. These compar-
3 isons are shown in Figures 6.4 and 6.5. The amplitudes of the Orr-Sommerfeld
eigenfunctions are multiplied by the constant3m0 L °(z = .3 5 , ymax,1)C T ALST(Y ) (6.12)
so that at y = y,,., they will be exactly equal to the amplitude distribution
I Ak of the calculated fundamental disturbance. The phase distributions of the
Orr-Sommerfeld eigenfunctions are shifted by the constantIdLST = =0( .3 5 ,Ym =_5Y ) - 016LST(Yma) (6.13)I
so that at y = y,,a:, they will be equal to the phase distributions of the calculated
fundamental disturbance. The constant y,,L was defined previously. The function
'OALST(Y) is the amplitude of the Orr-Sommerfeld eigenfunction and OOLST(Y) is
I the phase of the Orr-Sommerfeld eigenfunction. As before, 1,k can represent any
3 one of the flow variables.
As can be observed in Figures 6.4, the amplitude distributions of the fun-
3 damental disturbance are very similar tc. d amplitude distributions of the Orr-
Sommerfeld eigenfunctions. In Figure 6.4c, -. vorticity disturbances are observed
U to be quite small (0(10-8)) at the freestream boundaries. This confirms that the
3 transverse domain was wide enough for this calculation.
The comparison of the phase distributions of the fundamental disturbance
I to those of the Orr-Sommerfeld eigenfunctions is shown in Figures 6.5. In these
figures, reasonably good agreement between the numerical and theoretical phase
I
101 3distributions is observed. The differences between the numerical and theoretical 3phase distributions that are observed at the transverse locations V = ±10 are due
to the freestream boundary conditions. For the streamwise and transverse velocity icomponents, the observed differences are due to the exponential decay boundary
conditions, (5.25a) and (5.28a). For the spanwise vorticity, the differences arise 3because the theoretical spanwise vorticity satisfies exponential decay freestream Iboundary conditions while the calculated spanwise vorticity is set to zero at the
freestream boundaries. 3Finally, in Figures 6.6 the streamwise wavenumber ac, the amplification rate
, and the phase velocity co of the fundamental disturbance are compared to the 3analogous quantities from linear stability theory. As mentioned previously, the
agreement between linear stability theory and the numerical experiments can not I0 0be better than O(R A) Gaster (1974). Furthermore, the values of a?, cit, and cp
depend on the flow variable and transverse location that are used for calculation
of these quantities. The streamwise wavenumber ac and the phase velocity co 3are computed from the flow variables U0 , V0 , and fl0 for the transverse location
y = 5. The amplification rate a9 is computed from all three flow variables as well Ias from the kinetic energy E(z, 1). Equations (6.5) and (6.7) are used to calculate
these values. In Figures 6.6, the observed differences between the theoretical and
calculated values are approximately of the order O(Re-i). Therefore, based on the
results of Figures 6.6, the agreement between the calculation and linear stability
theory is reasonable. Additionally, comparable results were obtained when other 3transverse locations were used to calculate a,. and cp.
The agreement between the calculated results and linear stability theory is
within the limitations imposed by comparisons of parallel linear stability the-
ory to nonparallel numerical simulations. The disturbance amplification rate and
I3
I
102
I wavenumber, ai and a., as well as the disturbance amplitude and phase distribu-
3 tions, all exhibit reasonable agreement with linear stability theory. It is concluded
that the numerical method, with the spatial and temporal discretization that was
I employed, is suitable for the calculation of small amplitude disturbances in flat
plate wakes.
I Case-2
As a further test of the numerical method, the results of this calculation are
compared to the experiments of Sato (1970). In contrast to Case-i, in this calcula-
3 tion the disturbances attain large amplitude levels. To facilitate the comparison of
the numerical results to the experimental data, the parameters for this calculation
are selected so that the calculated wake models the wake from the experiments of
Sato.
ItThis calculation was performed in a spatial domain with boundaries at
3 zo=. 0 3 , ZN=. 75 , yo=-16 , and yM= 1 6 (6.14)
3As in Case-i, the transverse extent of the domain was designed so that the vorticity
disturbances would be approximately zero at the freestream boundaries. In the
3 discussion of Case-5, it will be shown that the streamwise extent of the domain is
sufficiently long for the present case.
The base flow was computed subject to the Gaussian inflow streamwise veloc-
3 ity distribution given by equation (4.5a). This distribution was employed because,
as shown in Figure 6.7, it compares well to experimentally obtained streamwise
3 velocity distributions of high-deficit flat plate wakes (Sato and Kuriki, 1961). At
the inflow boundary, the wake centerline velocity U, and the wake half-width b
I were
3 =.234826 and b = 1.15 . (6.15)
I
103 1The values of U, b, and zo were chosen so that the computed base flow would 3closely model the wake in the experiments of Sato (1970).
The streamwise velocity U° of the computed base flow is shown in Figure
6.8. As observed in Figure 6.9, the streamwise variation of the centerline (y = 0)
velocity of the calculated base flow compares closely to the similarity solution of
Goldstein (1929). The similarity solution was calculated using the series represen-
tation for the inner wake given by Goldstein (1929). The large variation of the
centerline velocity indicates that nonparallel effects may play an irnportant role in ithe initial development of disturbances in high-deficit wakes.
In his experimental work, Sato (1970) found that the frequency of the pre- Udominant small amplitude disturbance in flat plate wakes corresponded almost Iexactly to the frequency of maximum amplification as predicted by linear stabil-
ity theory. Therefore, in this calculation the undisturbed wake was excited at its 3most unstable frequency. This frequency was determined from a linear stability
analysis of the inflow streamwise velocity distribution. The eigenvalues (stream- 3wise wavenumber and amplification rate) of the Orr-Sommerfeld equation were
obtained for a range of frequencies in order to determine the frequency of greatest
instability. The streamwise wavenumber a,, amplification rate aj, and phase ve- -locity % obtained from this analysis are shown in Figures 6.10. As seen in Figure
6.10b, the amplification rate -at of the sinuous mode (mode-i) achieves its max- 3imum value at the frequency /3 = .51. The eigenfunctions of the Orr-Sommerfeld
equation, corresponding to the frequency /3 = .51, are shown in Figures 6.11. 1When this calculation was undertaken, it was thought that Sato (1970) had I
not indicated the initial disturbance levels in his experiments. Therefore, it was
necessary to estimate the amplitude level for the disturbance excitation in the 3calculation. The amplitude of the wake excitation was chosen to be A 2 = .000667 I
I
U104
I as this was thought to be a good estimate of the disturbance level in Sato's (1970)
5 experiments. However, after these calculations were completed, it was learned
that the forcing level in Sato's experiments corresponded to A 2 -1 .001.
5 As in Case-i, the influence of the grid sizes Az and Ay on the numerical results
was investigated in order to determine the spatial discretization that resulted in
I solutions that were reasonable independent of the grid sizes. A discussion of these
g investigations is given in Appendix B. For this calculation, the grid
N = 1024 and M = 256 (6.16)
was sufficient for this purpose. Again, the time step was determined by stability
considerations rather than considerations of temporal accuracy. Therefore, the
3 influence of the time step on the numerical results was not investigated. For this
calculation, the time step
At with La, = 128 (6.17)
was sufficiently small to ensure numerical stability. The response of the wake
3 to the excitation at the inflow boundary was calculated for seventeen oscillation
periods TF of the fundamental disturbance (L 2 = 2176).
3 For this calculation, greater spatial resolution than in Case-1 was required
because of the presence of large amplitude harmonics of the fundamental distur-
bance. The spatial domain contained approximately thirty wavelengths of the
3 fundamental disturbance (based on the wavelength at the inflow boundary), with
approximately 34 grid points per wavelength. In the transverse direction, the
I spatial domain contained approximately 32 momentum thicknesses (based on the
inflow boundary velocity distribution) with approximately eight grid points per
I
105
momentum thickness. As mentioned, investigations of the influence of the grid Isizes on the results of the numerical calculation (see Appendix B) have established Ithe suitability of the spatial discretization.
The relevant parameters in this calculation are summarized below: 5Case-2:
the streamwise location of the inflow boundary, z = .03; 3the streamwise location of the outflow boundary, ZN - .75;
the transverse location of the lower freestream boundary, yo - -16; 5the transverse location of the upper freestream boundary, yM = 16;
the Reynolds number Re = _ Re = 200000;
the amplitude level of the wake excitation, A2 = .000667; 3the frequency of the wake excitation, 0 = .51;
the number of streamwise grid increments, N = 1024; 1the number of transverse grid increments, M = 256;
the number of time steps calculated, L 2 = 2176;
and the time steps per fundamental disturbance period, L,,z - 128. 5Amplitude distributions of the fundamental disturbance component of the
streamwise velocity were obtained by Fourier time series analysis in the time in- Uterval 15TF < t < 17TF. Comparisons of these amplitude distributions to those Ifrom the experiments of Sato (1970) are shown in Figures 6.12. The experimen-
tal results are plotted on an arbitrary scale, but the relative amplitude level at 3each streamwise location is accurately represented in the figure. The streamwise
locations z for which the amplitudes in Figure 6.12a are plotted relate to the Ustreamwise locations X in Figure 6.12b according to
X (6.18)
I
UI
106
I The flat plate length was defined in Chapter 3 and was equal to 300mm in
gSato's experiments. There is good qualitative agreement between the numeri-
cal and experimental amplitude distributions at the streamwise locations z =
3 .067 (X = 20mm), Z = .1 (X = 30mm), and z = .133 (X = 40rmm). At
x = .200 (X = 60mm), the shapes of the amplitude distributions for both the nu-
I merical simulation and the experiment are similar, but the amplitude level relative
to the level at the previous location (z = .133) is smaller for the numerical calcu-
lation than for the experiment. This is due to differences between the excitation
3amplitude in the calculation and in the experiment. Additionally, it is noted that in
Figure 6.12b, the experimentally obtained amplitude distribution for X = 20mm
I is larger than the amplitude distribution for X = 30mm. As this is somewhat
inconsistent with expected behavior, it is possible that the curves are mislabeled
so that the curve labeled X = 30mm actually corresponds to X = 20mm and
3 vice-versa. The comparison of the numerical results to the experimental results
that was discussed in this paragraph was based on the assumption that the curves
* were mislabeled.
g The streamwise variation of the mean centerline velocity and the mean wake
half-width for both the calculation and the experiments of Sato (1970) are shown
3 in Figures 6.13. The horizontal scale 0 <z < 2.67 in Figure 6.13a corresponds to
the horizontal scale 0 < X < 800mm in Figure 6.13b. The mean flow is the zero
Ifrequency (F = 0) component of the disturbed wake and it therefore contains any
nonlinearly generated 0 th harmonic of the fundamental disturbance.
In Figures 6.13, the calculated mean centerline velocity compares reasonably
3well to the experimentally obtained mean centerline velocity (open circles in Figure
6.13b). Both the numerical and experimental centerline velocities increase rapidly.
I
II
107
This rapid rise is due to nonlinear interactions. As observed in Figures 6.13, the Iwake half-width for the calculation is similar to the half-width for the experiment. 3
The results of this calculation exhibit good qualitative comparison witb the
experimental results of Sato (1970). Good agreement between the numerical and
experimental results was obtained for large amplitude levels where nonlinear inter-
actions were important. Therefore, it is concluded that the numerical method is Isuitable for the calculation of large amplitude disturbances in high-deficit wakes.
Case-3
The purpose of this calculation was to investigate the effect of larger ampli- 3tudes on the disturbance development. Therefore, for this calculation all parame-
ters except for the excitation amplitude were kept the same as in Case-2. For this
calculation (Case-3), the excitation amplitude was A 2 = .001. The base flow that
was used as the initial condition was identical to the base flow from Case-2 (see 3Figures 6.8 and 6.9). 3
Figures 6.14 display the flow variables U', V', and 11o at the final time
of the calculation, t = 2176At. Due to the rapid growth of the disturbances,
the wake changes significantly from its undisturbed state. Very near the inflow
boundary, the wake appears undisturbed because the disturbances are still quite 1small in this region of the spatial domain. Beyond z/Az ; 128 (z .12), the
disturbances become large enough so that they are the dominant feature in the
wake. Observing the spar.wise vorticity l o in Figure 6.14c, a pattern develops 3that resembles that )f a K.i-man vortex street. This pattern develops as the
disturbances rewa'! large amplitude levels. As mentioned in Chapter 3, the vorticity 1is defined as w = -V x u.
As seen in Figures 6.14, at this time the disturbances are still quite far up- !
stream of the outflow boundary. Based on the wavelength of the vortex street 3U
108
I observed in Figure 6.14c, the leading edge of the disturbance wave is approxi-
mateiy seven wavelengths upstream of the outflow boundary. Therefore, at this
time the assumption that the wake is undisturbed near the outflow boundary is
clearly satisfied. Furthermore, the vorticity disturbances are confined to a region
near the wake centerline and appear to be quite small at the freestream boundaries.
I In order to see more details of the vorticity field, vorticity plots that corre-
spond to smaller regions of the spatial domain are shown in Figures 6.15 and 6.16.
Figure 6.15a shows the spanwise vorticity 11' in the upstream half of the spatial
domain (.03 < z < .39, 0 < z/Az < 512) and Figure 6.15b shows the spanwise
vorticity 10 for the streamwise interval .03 < z < .27 (0 < z/Az < 340). In
both of these figures, rapid disturbance development is observed. In Figures 6.16,
the spanwise vorticity 110 is shown for the streamwise intervals .03 < z < .12
1 (0 < z/Az < 128), .12 < z < .21 (128 < z/Az < 256), .21 < z < .30
(256 < z/Ax < 384), and .30 < z < .39 (384 < z/Az < 512). These figures
show in great detail the disturbances development that results from the wake
excitation. In Figure 6.16d, a very distinct vortex street pattern is visible.
Amplification curves based on the kinetic energy E°(z, F) are shown in Fig-
I ure 6.17. The kinetic energy is calculated using equation (6.6). The harmonic
content of the flow variables was obtained by Fourier time series analysis of the
velocity components in the time interval 13TF < t < 15Tp. For z < .11, the
streamwise variation of the kinetic energy of the fundamental disturbance (F = 1)
agrees closely with that obtained from linear stability theory calculations. In
the linear stability theory calculations, the streamwise variation of the base flow
was accounted for by the use of a quasi-uniform assumption. The mean distur-
I bance component (F = 0), the second harmonic (F = 2), and the third harmonic
(F = 3) also grow rapidly in the region z < .11. In particular, the second and
I
109
third harmonics grow more rapidly than the fundamental disturbance. Sato and
Kuriki (1961) and Sato (1970) have also observed large mean and second harmonic
components in flat plate wakes.
The saturation of the fundamental disturbance occurs at z ; .11. There, the 3amplitude of the streamwise velocity component of the fundamental disturbance
is approximately twenty percent of the freestream streamwise velocity. Beyond 3X - .11, the fundamental disturbance varies little in the streamwise direction.
The mean disturbance component and the second harmonic also saturate and ireach a state in which they vary little in the streamwise direction. 3
The saturation of the fundamental disturbance, which does not occur for the
small amplitude, linear development of disturbances, may nevertheless be indi-
rectly explained by the changing stability characteristics of the mean flow. As a
simple way of determining the stability of the mean flow, the spatial amplification Irate -ac(z) of the fundamental disturbance is computed from a linear stability
theory analysis of the mean flow. Then, the streamwise variation of the kinetic
energy of the fundamental disturbance is computed from 3to(z,1) = to (zo,1)e- f. ° 2criaz (6.19) I
using the amplification rates of linear stability theory. The streamwise variation of 3the kinetic energy of the fundamental disturbance, as predicted by a linear stabil-
ity analysis of the mean flow, is denoted in Figure 6.17 by the label 'LST MEAN IFLOW'. This simplified model also exhibits saturation of the fundamental distur- -bance. However, the amplitude of the fundamental disturbance after saturation
is over-predicted. These results indicate that the saturation of the fundamental 3disturbance is at least partly due to the changing stability characteristics of the 3
I
II
110
I nonlinearly generated mean flow. Alternatively, the variation of the mean flow
can be thought of as being due to the growth and eventual saturation of the
fundamental disturbance.
3 The presence of the harmonic components F = 0.5, F = 1.5, and F = 2.5
in Figure 6.17 is due to the fact that the wake has not yet achieved a truly time
Speriodic state in the timw' interval 13TF < t -< 15TF. Evidence of the nonperiodic
nature of the wake is given in Figure 6.18 which shows the temporal variation of
the transverse velocity V0 in the time interval 13TF < t < 15TF. It is apparent
3 from Figure 6.18 that, in the time interval shown, the transverse velocity V0 is
not temporally periodic at : = .27. Instead, the amplitude of V° that corresponds
to the streamwise location : = .27 appears to be increasing with time. Fourier
analysis of this data results in the F = 0.5, F = 1.5, and F = 2.5 harmonics
I that appear in Figure 6.17. Later, in the discussion of Case-5 in which a larger
streamwise domain was employed, it will be shown that for the same streamwise
locations the flow variables eventually become temporally periodic after more time
has elapsed. When periodicity is reached, the harmonic components F = 0.5,
F = 1.5, and F = 2.5 decrease to negligible levels. However, as will also be shown
in Case-5, the F = 0.5, F = 1.5, and F = 2.5 harmonics do not influence the other
harmonic components.
The streamwise variation of the mean centerline velocity and the mean wake
half-width are shown in Figures 6.19 and 6.20 respectively. From these figures, it
is apparent that the mean flow is quite different from the base flow and may have
* very different stability characteristics.
The streamwise wavenumber a, and the phase velocity cp of the fundamental
I disturbance are shown in Figures 6.21. They are compared with the corresponding
quantities that are obtained from a linear stability analysis of the calculated mean
I
111 Iflow. The wavenumber and phase velocity that are obtained from this analysis
compare well to the wavenumber and phase velocity of the calculated fundamental
disturbance.
Case-4
The results of Case-3 have shown that a larger excitation amplitude has signif-
icant effects on the disturbance development. For this calculation, the amplitude
was increased by a factor of ten over that in Case-3, resulting in A2 = .01. The
base flow was identical to that of Case-2 and Case-3 (see Figures 6.8 and 6.9). The Iresponse of the wake to the excitation at the inflow boundary, was calculated for
fifteen fundamental disturbance periods (L 2 = 1920). All other parameters were
identical to Case-3.
Amplification curves based on the disturbance kinetic energy of the mean
component F = 0, the fundamental disturbance F = 1, and the second harmonic IF = 2 are shown in Figures 6.22. For comparison, the analogous curves of Case-3
for which A2 = .001 are also displayed in these figures. The amplification curves
for the larger excitation level, A2 = .01, saturate further upstream and at higher
amplitude levels than those for Case-3. However, qualitatively the amplification
curves are quite similar for the two excitation levels. IThe streamwise variation of the mean centerline velocity and the mean wake
half-width are shown in Figures 6.23 and 6.24 and are compared with the corre-
sponding curves from Case-3. For both excitation levels, the centerline velocity
increases rapidly due to nonlinear interactions. However, for the higher excitation
amplitude, the rapid increase in the centerline velocity begins further upstream. iThe wake half-width, shown in Figure 6.24, behaves in a manner similar to the
centerline velocity. The increase in the wake half-width, which is also due to
MII
I
I 112
nonlinear interactions, begins further upstream when the excitation amplitude is
larger.
I The qualitative behavior of the disturbed wake, as observed in Figures 6.22,
6.23, and 6.24, is similar regardless of the excitation amplitude. The main influence
of the larger excitation amplitude is to accelerate the onset of nonlinear interactions
and the saturation of the various disturbance components.
Case-5
With this calculation, the influence of the outflow boundary conditions on the
numerical results is investigated. For this, the calculation of Case-3 is repeated
with a streamwise domain that is 1.5 times longer than the one in Case-3. All
3 other parameters for this calculation were identical to those of Case-3. Thus, the
influence of the outflow boundary conditions can be assessed.
3 This calculation was performed in a spatial domain bounded by
zo = .0 3 , zN = 1.11 , yo = -16 , and yM = 16 . (6.20)
3 The spatial domain was discretized into N streamwise grid increments and M
transverse grid increments withIN=1536 and M=256 . (6.21)
The spatial resolution that resulted from this discretization was identical to that
3 of Case-3. The response of the wake to the excitation at the inflow boundary was
calculated for 25 periods TF of the fundamental disturbance (L 2 = 3200) using
I the same time step as in Case-3. The base flow was computed subject to the same
inflow boundary conditions as discussed in Case-2.
IU
I
113 1The relevant parameters in this calculation are summarized below: 3
Case-5:
the streamwise location of the inflow boundary, Z = .03; Ithe streamwise location of the outflow boundary, N = 1.11; i
the transverse location of the lower freestream boundary, y0 = -16;
the transverse location of the upper freestream boundary, YM = 16;
the Reynolds number Re - 2-I, Re = 200000;
the amplitude level of the wake excitation, A 2 = .001; Uthe frequency of the wake excitation, /0 = .51;
the number of streamwise grid increments, N = 1536;
the number of transverse grid increments, M = 256;
the number of time steps calculated, L2 3200;
and the time steps per fundamental disturbance period, LMZ = 128. 3The streamwise variation of the base flow centerline velocity for this calcula-
tion (N = 1536) is shown in Figure 6.25 together with the corresponding curve of
Case-3 (N = 1024). It is obvious that the centerline velocity is practically identi- Ical for both cases. This is an indication that the outflow boundary had negligible ieffect on the base flow for Case-3.
Contours of instantaneous spanwise vorticity for t = 2176At and t = 3200Ai 3are shown in Figures 6.26. For t = 2176At, the spanwise vorticity contours are sim-
ilar to those in Figure 6.14c (Case-3). The disturbances experience rapid stream- Iwise amplification downstream of the inflow boundary, attain relatively large am-
plitude levels, and eventually dominate the flow field. The wake develops a Kirmin
vortex street pattern similar to what was observed in Figure 6.14c. Figure 6.26b 3shows the spanwise vorticity at t = 3200At. A vortex street pattern is also visible I
I
II
114
I for this time. However, the vortex street has propagated much farther downstream
3 than in Figure 6.26a.
In Figures 6.27, amplification curves f3r the disturbance kinetic energy
3 P°(z,F) of the mean component F = 0, the fundamental disturbance F = 1,
and the second harmonic F = 2 are compared to the corresponding curves from
U Case-3. The disturbance kinetic energy for Case-3 (N = 1024) is computed using
3 equation (6.6) and the Fourier analyzed velocity components from the time interval
13TF < t < 15TF. Similarly, the disturbance kinetic energy for Case-5 (N = 1536)
is computed using the Fourier analyzed velocity components from the time inter-
val 23TF < t < 25TF. From Figures 6.27, it is obvious that the amplification
I curves for the various harmonic components corresponding to Case-5 (N = 1536)
3 are almost identical to the corresponding curves from Case-3 (N = 1024). The
only differences are observed in the region z > .20. These differences are due
to the different time intervals used to Fourier analyze the velocity components
for each case. The otherwise good agreement between the results of Case-3 and
I Case-5 indicates that the influence of the outflow boundary conditions on Case-3
3 is minimal.
Figure 6.28 shows the amplification curves for the kinetic energy E(z, F)
3 that results from the Fourier time series analysis of the velocity components in
the time interval 2 3 tF < t < 25TF. It is observed that the harmonic components
F = 0.5, F = 1.5, and F = 2.5 are smaller by several decades when compared to
3 the results of Case-3 as displayed in Figure 6.17. This reduction is due to the fact
that the wake disturbances are periodic in the time interval 23Tp < t < 25Ty.
3 This periodic behavior is apparent from Figure 6.29 which shows the temporal
behavior of the transverse velocity V0 in the time interval 23TF < t : 25TF.
I
115 I
Clearly, V 0 is approximately periodic for all the streamwise locations that are
shown in Figure 6.29.
6.2 Investigations of Three-Dimensional Disturbance Development IIn this section, investigations of three-dimensional disturbance development
in a high-deficit flat plate are discussed. Several different calculations are under-
taken. With the first calculation (Case-6), the ability of the numerical method
to accurately simulate three-dimensional disturbances is demonstrated. This cal-
culation is the three-dimensional analog of Case-1. In Case-7, the response of Ia wake when subject to a three-dimensional excitation is investigated. In con-
trast to Case-6, in this calculation the base flow exhibits considerable streamwise
variation. In Case-8 and Case-9, the response of the wake to a combination of 3two-dimensional and three-dimensional excitations is investigated.
Case-6 IFor this case, the development of a small amplitude, three-dimensional dis- 3
turbance in a flat plate wake is calculated. The results of this calculation are
compared to linear stability theory in order to verify the ability of the numerical
method to accurately simulate three-dimensional disturbances. For this calcula-
tion, the spatial and temporal domains, and the base flow were identical to Case-1. IThe wake was excited with a three-dimensional sinuous mode disturbance of
amplitude A3 = .001, spanwise wave number 7 = .5, and frequency 3 = .28. The
two-dimensional component of the excitation was set to zero, so that A 2 = 0. IIn terms of the three-dimensional streamwise velocity disturbance u1,d(Zo, y) at
the inflow boundary, A 3 is given by the relation A 3 = max(ud(zo,y)l). The
amplitude and phase of the Orr-Sommerfeld eigenfunctions that correspond to the
frequency 8 = .28 and spanwise wavenumber y = .5 are shown in Figures 6.30. UI
mI
116
I The spatial discretization was identical to that used in Case-l. Three spanwise
modes (K = 6) were calculated. Restrictions on the allowable time step, due
to numerical stability considerations, are more severe for calculations of three-
dimensional disturbances than for the calculation of two-dimensional disturbances.
For this calculation, the time step
At with Ln =- 256 (6.22)
was sufficiently small to ensure numerical stability. Five periods TF of the funda-
3 mental disturbance component were calculated (L 2 = 1280).
The relevant parameters in this calculation are summarized below:
ICase-6:
3 the streamwise location of the inflow boundary, Z = .3;
the streamwise location of the outflow boundary, ZN = .7;
3 the transverse location of the lower freestream boundary, Yo - -10;
the transverse location of the upper freestream boundaryyM - 10;
the Reynolds number Re - UA Re 200000;
the amplitude level of the two-dimensional excitation, A 2 0;
the amplitude level of the three-dimensional excitation, A3 - 001;
the frequency of the wake excitation, - .28;
the spanwise wave number of the wake excitation, t - .5;
the number of streamwise grid increments, N - 128;
3 the number of transverse grid increments, M - 64;
the number of spanwise modes computed, K/2 = 3;
3 the number of time steps calculated, L 2 = 1280;
and the time steps per fundamental disturbance period, LG.=- 256.II
117 1The amplification curves for the disturbance kinetic energy El(z, F) of the
first spanwise mode (k = 1) are shown in Figure 6.31 and are compared to the
corresponding curves from linear stability theory. The amplification curve for the Ifundamental disturbance (F = 1) compares well to the linear stability theory
prediction of the kinetic energy. In the Navier-Stokes calculation, the second
harmonic (F -- 2) is present but is much smaller than the fundamental disturbance. 3For the streamwise location z = .35, the amplitude and phase distributions
of the fundamental disturbance component (k = 1, F = 1) are compared to the 3amplitude and phase distributions obtained from a spatial linear stability theory
analysis of the base flow. This comparison is shown in Figures 6.32 and 6.33. The UOrr-Sommerfeld amplitude and phase distributions are normalized in the same
manner as discussed in connection with Case-1. The amplitude distributions of the
fundamental disturbance are virtually identical to the Orr-Sommerfeld amplitude 3distributions. The phase distributions of the fundamental disturbance also exhibit
good agreement with the Orr-Sommerfeld phase distributions. IThe results of this calculation, as represented by the disturbance kinetic en-
ergy as well as the disturbance amplitude and phase distributions, compared rea-
sonably well to linear stability theory. It is concluded that the numerical method 3is suitable for calculations of three-dimensional disturbances.
Case-7 3With this calculation, the development of a three-dimensional disturbance in
a high-deficit flat plate wake is investigated. In contrast to Case-6, the base flow Ichanges significantly in the streamwise direction and was approximately the same 3as the base flow that was used for Case-2, Case-3, and Case-4. For this case, the
frequency of the excitation was chosen to be identical to the forcing frequency for 3Case-2 through Case-5 so that the results of this calculation can be compared to U
U
I
118
I those earlier cases. The results of this calculation win serve as a reference for other
calculations for which both two-dimensional and three-dimensional disturbances
were introduced.
This calculation was performed in a spatial domain that is bounded by
3 = .03 , ZN =.36 , Yo=- 1 6 , and YM--1 6 (6.23)
As in Case-2, the undisturbed wake was computed subject to a Gaussian
Iinflow streamwise velocity distribution. The resulting base flow modeled the wake
3from the experiments of Sato (1970).
The wake was excited with a three-dimensional sinuous mode disturbance of
3 amplitude A3 = .001, spanwise wavenumber 7 = .5, and frequency = .51. The
amplitude of the two-dimensional component of the excitation was A2 = 0. The
I amplitude and phase of the Orr-Sommerfeld eigenfunctions that correspond to the
frequency j and spanwise wavenumber - are shown in Figures 6.34.
The spatial domain was discretized into N streamwise grid intervals and M
5transverse grid intervals where
3-N=256 and M=128 (6.24)
Three spanwise modes (K = 6) were computed. For this calculation, the time step.
At= 2 with Lma = 512 (6.25)
was sufficient to ensure numerical stability. The time-dependent response of the
3wake was calculated for eight fundamental disturbance periods TF which is equiv-
alent to t = 4096At.
I For this calculation, the spatial domain contained approxlmately eight stream-
wise wavelengths of the fundamental disturbance (based on the wavelength at the
I
119
inflow boundary) with approximately sixteen grid points per streamwise wave- 3length. In the transverse direction, the domain contained approximately 32 mo-
mentum thicknesses (based on the inflow boundary velocity distribution) with Iapproximately four grid points per momentum thickness. Based on the results of 3the two-dimensional calculations, it is felt that the spatial discretization discussed
here results in solutions that are sufficiently independent of the grid sizes. 3The relevant parameters in this calculation are summarized below:
Case-7: Ithe streamwise location of the inflow boundary, ZO .03;
the streamwise location of the outflow boundary, ZN .36;
the transverse location of the lower freestream boundary, yo -16; 3the transverse location of the upper freestream boundary,yM 16;
the Reynolds number Re = L-' , Re 200000;
the amplitude level of the two-dimensional excitation, A2 0;
the amplitude level of the three-dimensional excitation, A3 .001;
the frequency of the wake excitation, - .51; 1
the spanwise wave number of the wake excitation, It .5;
the number of streamwise grid increments, N - 256; 3the number of transverse grid increments, M - 128;
the number of spanwise modes computed, K/2 3; 3the number of time steps calculated, L2 4096; 1and the time steps per fundamental disturbance period, Lma 512.
Amplification curves of the kinetic energy P1 (z,F) are shown in Figure 6.35a.
The disturbance kinetic energy is calculated using the Fourier analyzed velocity 3components that correspond to the time interval 6TF < t < STF. The variation of U
I
I120
I the kinetic energy of the fundamental disturbance (F = 1) agrees closely with that
obtained from linear stability theory calculations. As was the case for calculations
of two-dimensional disturbances, the large amplitude level of the F = 0.5 and
F = 1.5 harmonics is due to the fact that the wake has not achieved a time
periodic state in the time interval 6TF < t < 8TF. At z ; .12, the fundmental
I disturbance appears to saturate. However, this saturation may be due to the
3 nonperiodic component of the wake disturbances.
For comparison purposes, amplification curves of the disturbance kinetic en-
3 ergy that correspond to the time interval 8TF < t < 1OTF are shown in Figure
6.35b. In this figure, the fundamental disturbance does not saturate. In addition,
I the other harmonic components, particularly the F = 0.5 and F = 1.5 harmonics,
3 have decreased relative to their values in Figure 6.35a. This indicates that the
disturbances are not periodic in the time interval 6TF < t < 8TF. The saturation
3 of the fundamental disturbance that was observed in Figure 6.35a is apparently
due to this nonperiodicity.
Figures 6.36 show contours of streamwise vorticity w,, in the yz-plane for
3 t = 4096At. The horizontal scale of the plots in Figures 6.36, 0 < z/Az < 32,
is equivalent to two spanwise wavelengths of the fundamental disturbance. The
3 vorticity shown in Figure 6.36a corresponds to the inflow boundary z = .03 and
is due solely to the three-dimensional excitation at this boundary. Figure 6.36b
shows the streamwise vorticity corresponding to x = .146. For this location, the
3 vorticity concentrations have rotated relative to the vorticity concentrations at the
inflow boundary. The numerical results of Meiburg and Lasheras (1988), which are
3 shown in Figure 6.37, display a similar behavior. Meiburg and Lasheras attributed
the rotation of the vorticity concentrations to the velocity field induced by these
I
II
121
concentrations. In Figures 6.36, the solid contours denote positive streamnwise Ivorticity. Because the vorticity is defined as 3
W=-Vxu , (6.26) Ipositive vorticity induces a counterclockwise azimuthal velocity component and
a counterclockwise rotation of the positive streamwise vorticity concentrations.
For similar reasons, the negative streamwise vorticity concentrations rotate in the
clockwise direction. The vorticity in Figures 6.36 is also distributed in a more corn- Iplicated pattern than the vorticity observed in Figure 6.37 (Meiburg and Lasheras,
1988). It is believed that the simpler vorticity distribution in Figure 6.37, as com-
pared to Figure 6.36b, is due to the simpler initial perturbation that was employed Iby Meiburg and Lasheras (they used inviscid vortex dynamics and disturbed the
wake by perturbing the vortex filaments sinusoidally in the streamwise and span- 3wise directions).
Case-8 and Case-9 IWith these calculations, an attempt is made to investigate certain aspects of
the secondary instability of wakes. These studies are undertaken by calculating the
interaction of a large amplitude two-dimensional disturbance with a smaller ampli- 3tude three-dimensional disturbance. The results of these calculations are compared
to the results of Case-7 and Case-4 in order to observe how the two-dimensional 3disturbance influences the development of the three-dimensional disturbance and
vice-versa. IThe base flow and most relevant parameters were identical to those in Case-7.
For both Case-S and Case-9, the amplitude, frequency, and spanwise wavenumber
of the three-dimensional excitation had the same values as in Case-7, so that 3A 3 =.001 , f3=.51 , and "=.5 . (6.27)
I
II
122
I For Case-8, the amplitude of the two-dimensi:nal excitation had the same value
as in Case-4, so that
A2 = .01 (6.28)
I For Case-9, the amplitude and frequency of the two-dimensional excitation was
I A 2 = .05 . (6.29)
U Amplification curves of the kinetic energy Pk(z, F =- 1) for Case-8 are shown
in Figure 6.38. The curve corresponding to k = 0 represents the kinetic energy
of the two-dimensional disturbance. The curve corresponding to k = 1 is for
3 the kinetic energy of the three-dimensional disturbance. The curve labeled 'LST
MEAN FLOW' is for the kinetic energy E1 (zF - 1) of the three-dimensional
3 disturbance component, as predicted by a linear stability theory analysis of the
mean flow. This last curve is computed in the same manner as discussed previously
I(Case-3). The two-dimensional disturbance (k = 0) grows very rapidly at first and
3 then saturates at an amplitude of (z, F = 1) .20. This behavior is very similar
to that observed for the two-dimensional disturbance development in Case-4 where
3 three-dimensional disturbances were not present.
The three-dimensional disturbance (k = 1) initially grows in a manner similar
I to the three-dimensional disturbance from Case-7. However, the saturation of the
3 two-dimensional disturbance causes a temporary reduction in the amplification
rate of the three-dimensional disturbance. After a brief period of lower ampli-
3 fication, the three-dimensional disturbance resumes stronger growth. However,
the amplification rate is smaller than the amplification rate that was observed
I before the saturation of the two-dimensional disturbance. The reduction in the
three-dimensional growth rate, that is probably caused by the saturation of the
U
I
123
two-dimensional disturbance, may be due to the changing stability characteristics Iof the mean flow. This proposition is supported by the close comparison in Fig- -ure 6.38 of the three-dimensional disturbance and the curve labeled 'LST MEAN
FLOW'. The resumption of rapid three-dimensional disturbance growth, follow- 3ing the saturation of the two-dimensional disturbance, could be explained by a
secondary instability mechanism. UAmplification curves of the kinetic energy tk(z, F = 1) for Case-9 are shown 3
in Figure 6.39. Due to the now larger two-dimensional excitation amplitude, the
two-dimensional disturbance saturates further upstream than in Case-8. As in 3the previous case, the saturation of the two-A;nensional disturbance causes a
strong reduction in the three-dimers~onai amplification rate. After a short period
of reduced three-dimensional growth, the three-dimensional disturbance continues
its amplification. Qualitatively, this behavior is the same as in Case-8. However, as Ia result of the larger two-dimensional excitation amplitude, the reduction in the 3three-dimensional amplification rate and the subsequent resumption of stronger
growth occur further upstream than in Case-8. 3In Figure 6.39, the resumption of strong three-dimensional growth is short-
lived: the three-dimensional disturbance tends to saturate for a second time. How- Iever, this second saturation is in contrast to the observed behavior in Figure 6.38
(Case-8). To further check the validity of the results of Case-9, this calculation
is repeated in a spatial domain that is twice as long (ZN = .69, N = 512) as 3the spatial domain of the original calculation but which in all other respects is
identical to the original calculation. The amplification curves of the disturbance 3kinetic energy E'(z, F) for both the original calculation (ZN = .36, N = 256) and
the long domain calculation (ZN = .69, N = 512) are shown in Figures 6.40. For Ithe long domain calculation (Figure 6.40b), the three-dimensional fundamental 5
U
II
124
disturbance (F = 1) initially grows more slowly after its initial saturation than
3 was the case for the original calculation (Figure 6.40a). However, the fundamental
disturbance does not saturate for a second time as it did in the original calcula-
I tion. Therefore, it appears that the second saturation of the three-dimensional
fundamental disturbance that is observed in Figures 6.39 and 6.40a is due to the
influence of the outflow boundary.
The three-dimensional disturbance behavior observed in Figures 6.38 and 6.39
is similar to that observed by Metcalfe et al. (1987) in their investigations of sec-
I ondary instability in free shear layers (see Figure 6.41). They numerically calcu-
lated the temporal development of a large amplitude two-dimensional disturbance
as it interacted with a smaller amplitude three-dimensional disturbance. They
3 found that the saturation of the two-dimensional disturbance temporarily inhib-
ited the amplification of the three-dimensional disturbance. After a brief period of
I reduced growth, the three-dimensional disturbance continued growing and even-
tually surpassed the two-dimensional disturbance.
For Case-8 and Case-9, the presence of the two-dimensional disturbance ap-
3pears to alter the distribution of vorticity as compared to Case-7. The streamwise
vorticity in the yz-plane for t = 4096At is shown in Figures 6.42. Figure 6.42a
3 shows the streamwise vorticity for Case-8 which corresponds to the streamwise
location z = .146. Figure 6.42b shows the streamwise vorticity for Case-9 for the
same streamwise location. Due to the presence of the two-dimensional disturbance,
3 the streamwise vorticity is distributed quite differently than in Figure 6.36b (Case-
7). In Figure 6.36b, the streamwise vorticity is distributed in a pattern of discrete
I streamwise vortices. For Case-8 and Case-9, in which both two-dimensional and
three-dimensional disturbances are present, the streamwise vorticity is distributed
I
I
125 1in a large number of small vortices. Furthermore, in contrast to Case-7, these 3vortices do not appear to rotate.
Additional displays of the interactions of the two- and three-dimensional dis- 3turbances are shown in Figures 6.43 through 6.45. In Figures 6.43, the spanwise
vorticity w, in the yz-plane for t = 4096W is shown for Case-7, Case-8, and ICase-9. The interaction of the two- and three-dimensional disturbances (Case-8 3and Case-9) significantly alters the vorticity distribution as compared to Case-7
for which only a three-dimensional disturbance was introduced. 3In Figures 6.44, the streamwise vorticity w. in the zy-plane for t = 4096At
is shown for Case-7, Case-8, and Case-9. The two-dimensional disturbance, which 3is present in Case-8 and Case-9, causes a much broader transverse distribution of
vorticity as compared to Case-7 for which the two-dimensional disturbance was Iabsent. In Figures 6.45, similar plots of the spanwise vorticity w. are shown. For 3Case-7 (Figure 6.45a) the spanwise vorticity disturbance is much smaller thar for
Case-8 and Case-9 (Figure 6.45b and 6.45c). For Case-8 and Case-9, the vorticity 3develops in a manner similar to Case-3 (see Figure 6.14c).
Figures 6.46, 6,47, 6.48, 6.49, and 6.50 (Case-8) give detailed views of the Ivorticity fields in the braid regions between the large concentrations of spanwise
vorticity. It is in these regions that Meiburg and Lasheras (1988) observed the for-
mation of lambda vortices. However, from the results of these calculations, no con- 3clusive evidence of these vortices was observed. This might be due to the fact that
the forcing amplitudes that were used for Case-8 were different from those used for 3the simulations of Meiburg and Lasheras (1988). For Case-8, the wake was forced
with a large two-dimensional disturbance and a small three-dimensional distur- Ibance while Meiburg and Lasheras employed a large three-dimensional disturbance 3and a smaller two-dimensional disturbance. Furthermore, there is a considerable
3I
UI
126
difference between the Reynolds number for this work, Ree = U = 594 and
the Reynolds number for Meiburg and Lasheras's calculations, Re9 = 74. Finally,
it was not possible to duplicate the amplitude levels used by Meiburg and Lasheras
3 because a larger three-dimensional forcing amplitude would have required the use
of more Fourier modes in these calculations. This was not feasible due to the
I limitations of the available computer resources.
In Figures 6.51 through 6.53, the amplitude distributions of the spanwise
vorticity for (k = 0, F = 1) and (k = 1, F = 1) and the amplitude distributions of
I the streamwise vorticity for (k = 1, F = 1) are shown. Case-7, Case-8, and Case-
9 are represented in these figures. Due to the influence of the two-dimensional
disturbance in Case-8 and Case-9, the amplitude distributions of the streamwise
and spanwise vorticity are significantly different from those for Case-7 (Figures
* 6.51).
Additional consequences of the interaction of the two-dimensional and three-
dimensional disturbances axe obvious from observing the behavior of the wake
3 half-width b. In Figure 6.54, the wake half-width b corresponding to three different
calculations: Case-4 (A2 = .01,A 3 = 0), Case-7 (A 2 = O,A3 = .001), and Case-
3 8 (A 2 = .01, A 3 = .001) is displayed. For reference the wake half-width of the
base flow is also shown. When only two-dimensional disturbances are present
(A 2 = .01, A3 = 0), the wake as characterized by its half-width b becomes much
broader. However, when forced with only a three-dimensional disturbance (A 2 =
0,A 3 = .001), the wake width does not differ significantly from the width of
3 the base flow. The strongest effect of broadening is observed when the wake is
excited with both two-dimensional and three-dimensional disturbances (Case-8).
3 For Case-8 (A 2 = .01,A 3 = .001), the wake half-width development is initially
very similar to that of Case-4 (A 2 = .01, A3 = 0). However, while for Case-4 the
I1
UI
127
half-width stops increasing when the two-dimensional fundamental disturbance Usaturates, for Case-8 the half-width continues increasing beyond this point. This
additional increase in the wake half-width is due to the presence of the three-
dimensional disturbance. 3In Figure 6.55, the wake half-width for both Case-8 and Case-9 is shown.
For Case-9, in which the two-dimensional disturbance is larger than in Case-8, the Uinitial increase of the half-width begins further upstream. However, the variation of
the wake half-width is similar for both cases once the two-dimensional disturbance
has saturated. 3UUUIIIIIII.I
II
128
* CHAPTER 7
I CONCLUSIONS
I A numerical method has been developed for studying the evolution of two- and
three-dimensional disturbances in high-deficit wakes. Comparison of the results
of this method were made to both linear stability theory and experiments. The
5numerical method was found to be capable of simulating both small and large
amplitude disturbances.
I Simulations of two-dimensional sinuous mode disturbances in a high-deficit
flat plate wake were undertaken. At small amplitude levels, the disturbances grew
exponentially at rates predicted by linear stability theory. At higher amplitude
3 levels, nonlinear effects became important and the disturbances saturated. The
saturation of the fundamental disturbance was found to be related to the stabil-
3 ity characteristics of the mean flow. The main influence of a larger excitation
amplitude on the resulting disturbances was to accelerate saturation. For large
disturbance amplitudes, the wake developed a Kirmin vortex street pattern. Fur-
3 thermore, the influence of the outflow boundary conditions on the results of the
numerical simulations was found to be negligible. The subharmonic component
3 that appeared in the wake was found to be due to a nonperiodic wake disturbance.
For a fixed spatial location, the subharmonic component decreased with time.
Investigations of three-dimensional disturbances were also undertaken. Asso-
3 ciated with three-dimensional disturbances were pairs of counter-rotating stream-
wise vortices that appeared to rotate as a result of the velocity field induced by
3 these vortices. When both two- and three-dimensional disturbances were present,
the saturation of the two-dimensional disturbance caused the three-dimensionalII
129 1disturbance to saturate. Shortly after saturation, the three-dimensional distur-
bance resumed stronger growth, possible because of a secondary instability mech-
anism. Larger two-dimensional forcing amplitudes accelerated the saturation of 3the two-dimensional and three-dimensional disturbances and also accelerated the
resumption of strong three-dimensional growth. IThe interaction of two-dimensional and three-dimensional disturbances re-
salted in complicated distributions of vorticity. Instead of a small number of dis-
crete streamwise vortices, as when the wake was excited with only three-dimensional 3disturbances, a larger number of vortices spread over a much wider transverse re-
gion were present. Furthermore, these interactions resulted in a much broader 3wake distribution than that observed when only two-dimensional or three-dimen-
sional disturbances were excited. I
Future simulations of three-dimensional disturbances in wakes will undoubt- 3edly require larger computational grids in order to better resolve the flow field.
For these simulations to be practical, the numerical method should be modified. It iis felt that greater efficiency could be obtained by solving the Navier-Stokes equa-
tions in velocity-pressure formulation. This would result in both reduced memory Irequirements because a smaller number of variables would have to be stored, and
in reduced computation times because of the smaller number of nonlinear terms.
Furthermore, for simulations of three-dimensional disturbances, the stability 3of the current numerical method was found to be quite restraining. This had
an adverse effect on the computational efficiency. In the future, this inefficiency 3should be avoided by employing numerical methods with more favorable stability
characteristics. 3In this work, interactions of two-dimensional and three-dimensional distur- 3
bances appeared to be an important factor in the three-dimensional breakdown
!UI
II
130
I of the wake. In these simulations, only the initial stage of these interactions was
observed. In the future, more complete simulations of these interactions should
be attempted as these interactions most likely play an important role in the de-
3 velopment of the three-dimensionality in transitional wakes. Of particular interest
is how these interactions lead to the formation of the dominant three-dimensional
3structures that are observed in experiments.
!III!III!IIII
1313
FIGURES
U00
Wake
Flat Plat
Figur 2.1Spatal dmain
II3 132
I3Flat Plate
II
I
II
'1y
Figure 5.1 Discretization of the spatial computational domain.UI
133
JI0I
0U
162432Z 40 46 5664
T- 0 o1I
Figure 6.1 Base flow streamwise velocity U for Case-i. I
II
I 134
I a)
I9rTEEO
U
I I
Lz* LLa
I "8a 0
-I~i
I t
-I I I "
I I
I Figure 6.2 Sinuous mode eigenfunctions of the two-dimensional Orr-Sommer-
feldi equation corresponding to 8 -- .317 andl Reg 594.i a) amplitudes of U° , V ° , and 0'°; b) phases of U ° , V ° , and 11'.
II
I135
_ ___ __I
r4 "'LEGEND
'UIv
LiJ
I S -
I " I -
" II
CDi_LiO ILi
IIr 3l
".,.
I Ii: I
-17 -9.5 O 8.5 17
Ny
Figure 6.2 Continued.
iI
IU
136
LEGENOIF -0.0
. - ..................
F -2.0-- C- LIER THL0RY
III*o
II
0.30 0.33 0.36 0.39 0.42 0.45I x
Figure 6.3 Comparison of amplification curves based on the kinetic energy2 0(x, F) to linear stability theory. Case-i: A2 = .001, A3 = 0,and =.31T.
II I
II
137
a)
LEGENDNA VIER-STOKESL'INEM.. THEORY.. I
0-
I°1?
C:) X " I
dU* I
- _ __ __ __-_
| I I I
-10 -6 -2 2 6 10Y
Figure 6.4 Comparison of the amplitude distributions of the fundamental dis-turbance (F = 1) to linear stability theory. Case-i: A 2 = .001,A 3 = 0, and 6 = .317. a) streamwise velocity, UO; b) transversevelocity, V'; c) spanwise vorticity, Wl. I
I
IU
138
* b)
I LEGENDNAV I NER-STOKESULNEA.HORY.
O_
X
II
- 1 0 -6 -2 '6
Figure 6.4 Continued.
**
II
139
i IC)
LEGENO. NAVIER-STOKES
______-_____I
7
- I
N,
,o II
CD* I
. III
10 -6 -2 26 10I
Figure 6.4 Continued. 3U
3 140
* ~ a)
UU
IILuU Un
CL
* C-
U7LEEN
NAV IER-STOKES*LINA .THEORY..
1 -10 -6 -2 2610
Figure 6.5 Comparison of the phase distributions of the fundamental distur-bance (F = 1) to linear stability theory. Case-i: A2 = .001,A3 = 0, and P8 = .317. a) streamwise velocity, U'; b) transversevelocity, V'; c) spanwise vorticity, no~
141 i
b)
rn
I
i_Ln
I× I
° II I
U, I
>Z
U, 7I
-10- -6 -22 6 l
Figure 6.5 Continued.I
I
142
3 C)
LnIr,x.F
I
Ul0* .... . .. ... . . ,°
* Xe
xL(o
I °
co
0
7 0NI'
LEGENDNRVIER-STOKES..TIER "THEORY""
S-10 - -2 10IY
Figure 6.5 Continued.IU
I-
143 3
Sa)30
LEGENONAVIER-STOKES U .
....i. RiT: k'" QS .NAVIER-STOKES RZ3
"" _ _ _ ____ _~ 1 t' ... IWr- ---- ---- -------
II
C--jC.
III
1 Ii
0.300 0.325 0.350 0.375 0.400x
Figure 6.6 Comparison of the streamwise variation of a,, a,, and cp for thefundamental disturbance (F = 1) to linear stability theory. Case-1: A; = .001, A3 = 0, and 0 = .317. a) streamwise wavenumber, 3a,.; b) amplification rate, a,; c) phase velocity, cp.
!I
UU1 144
] b)
LEGENDNAVIER-STOKES8- ....NNVIM .. ........ .......
NAVI~-TOKES UNqVIER-STOKES V
Ln INTH TORT -
LN
| sr-
0
I*
I
0.300 0.325 0.350 0.375 0.100
IxI Figure 6.6 Continued.
U
II
145 3
C)5
LEGENDNAVIER-STOKES U I
NAVIER-STOKE5 RZ
LOI
IU'U
UrJ I
10. 303.3S0'5 .1350 0
I
Vx
Fu 3I
0
I I
0. 300 O. 325 0.:350 0. 375 0. 1003X I
Figure 6.6 Continued.
[I
I
II146
aX-3oMM L3AX- 40iMMZ7I ___ __ , _, _____i
2 -10 1 2
y = Yjb
FxoIuaz 13. Non-dimenaionai mean.veocity distribution in linear region.
Figure 6.7 Comparison of a Gaussian streamwise velocity distribution to ex-perimental data and to the similarity solution of Goldstein (1929),reproduced from Sato and Kuriki (1961).
IIIIIII
147
.5 4U0 _IccIAI
32 64 120 92 U 25IY/aU
T-0
Figre6.8Bae fowstr~mis veocty * or as-U
1U
1 148
I0I-
LEGENO
aNAVIER-STOKES1 .0 'OLOSTEIN..KE....
1 0-
I0I ,0
Z C-
LI
I
0.03 0.12 0.21 0.30 0.39 0.48 0.57 0.66 0.75IFigure 6.9 Comparison of the base flow centerline streamwise velocity U' for
Case-2 to the similarity solution of Goldstein (1929).
I
II
149
Ia)
I""MODE ... 2" I-
- N-J ~I
U, I0 o
UII
a o.2 0.4 0.8 0.8 1BETR
Figure 6.10 Orr-Sommerfeld eigenvalues for the sinuous mode (mode-i) andthe varicose mode (mode-2), corresponding to the streamwise ve-
locity distribution at the inflow boundary (Case-2) and Rep. 594. a) streamwise wavenumber, a,.; b) amplification
rate, a,; c) phase velocity, cp. 3I
II3 150
* b)
I0 MODE I
* 0Ic cc
z
0 0.2 0.4 0.6 o.e 1I ETA
Figure 6.10 Continued.
I
I
UI
151
cm
I
U,
,,c) I
L IkLODo. I
~I.
0~I
.%,
U, III
I, ' I I I0 02 0.4 0.5 0.8
BETA
IFigure 6.10 Continued. 3
3
!I
152
S..a)
LEGENOIi I RIU
0v
I Li.
I -J I II ... I...
II- I
II
zR'
-IzLi II
Li
c-n
U-
-17 -8.5 0 .51'7*yFigure 6.11 Sinuous mode eigenfunctions of the two-dimensional Orr-Sommer-
feld equation corresponding to ,3 = .51 and Re9 e 594.a) amplitudes of U', VO, and fl' ; b) phases of U', VO, and flo,
II
153 3
I /""I
LEGENO.. 3- -u.v
cn I!
I 'i --i
I 3
_I°: . .. Iz C
Li
L l-
r- i
• .
"" I
-17 -85851'7
yt
F "n
Figure 6.11 Continued.I
I
UI
154
a) -I °'o ,, LEGENO
A3 X -. 0673 ~ X**- . 100
I .f . x - "a"// i.
C-I
LI I
-is -12 -8 -4 0 8 12 16Y
k'! j=630 Hz), /
-10 0 5 10Y (r__)1 Guzz 9. Disributions of fundamental (60 Hz) component. Natural transition.
The ordinate Isle w arbitrary. X (ramn): 7, 20; V, 30; 0, 40; L, 60; A. 130.
Figure 6.12 Comparison of streamwise velocity amplitude distributions of theI fundamental disturbance (F = 1) to experimental data. a) Case-2:
A2 = .000667, A3 = 0, and -. 51; b) experimental, reproducedi from Sato (1970).
IN~
155
a)
0 LEGENOC I WAKE HALF-WIDTH, b
'Q.. ENTERLINE ''-0.
zu
-J0IM
C-
o0.8675 1.3350 2.6025 2.6700XIb)
10 I
0s I
0 200 400 600 S0o
X (MM)3
FIGURE 3. Streamwise variations of velocity on the centre-line U,, 0and half-value-breadth b, 0. Natural transition.3
Figure 6.13 Comparison of the streamwise variation of the mean streamwisecenterline velocity U0 and the mean wake half-width to experi-mental data. a) Case-2: A2 = .000667, A3 = 0, and i3=.51; b)Iexperimental, reproduced from Sato (1970).
I 156
! a)
Ln
C=,
II
II
z- Z-o C 0
F 6 .a f.__Ise._ A
I'
I z9S t, 6 [ 09! 9NI 96 f,9 0£
Figure 6.14 Contours of U0° , V0, and f'l for t - 2176 t. Case-3: A2 =
.001, A3 = 0, and 6 = .51. Twenty contour intervals betweenthe minimum and maximum values. Solid lines denote positivevorticity and dashed lines denote negative vorticity. a) streamwisevelocity, U*; b) transverse velocity, V*; c) spanwise vorticity, fl ° .
U m mll l nimlillllllllllll ~ il l l i
1571
b)I
w.
_________ __-_-_-__-_-_-_-__-_-_-__-_-_-_-
-~~ -I
- - - --I xI
zI9s? zz M1 09 sit i6 i zi
Figur 6.4Cniud
3 158
1 c)
N Xu
li I I
l I Ni I I
Figur 6.1 Cotnud
1593
a)I
'Off
COI
Vr - --UIXI
o I
%, I4so "
I- Il"f
IO/
Figure 6.15 Contours of spanwise vorticity W, for t = 2176At. Case-3: A2 =I
.001, A3 = 0, and 83 = .51. Thirty contour intervals between theminimum and maximum values.. Solid lines denote positive vortic-
ity and dashed lines denote negative vorticity. a) 0 < z/Az < 512,b) 0 < z/Az < 340.
160
N
1 1%x
IR c,0
lo*$W
CD I Ifl*4l I I
of IlIIfNI I folio
IO/Fiue61ICniud
161 3
a)%
i ll ll I I%tl ill t
,l a a
%Ilill I l
,lo :11 1
*aa aile I I
i/ mllmllgl I I I
III IIIIIII I I
t ttttti I I
aaaatttlll a aa atlaatal a I
Z61~~ ~ ~ ~ IL 9 V a ll 96i08 VI
.01 A3=O n 5.Tirtyill cotu inevl ewe h
IIIaaaalIII I a
adv luesl inIItII and i
I IlaaaaaaasI I IIIaaMg lllla
*I aINlISII a
bla)a llll a512.IIIIIII I
aaaaaa a I a '-I
IllilUa l l aawaaaa li~ Ia~a~aa a
a l l a Iauaaa a-aaauaaIIa
aanuuaaalaIIaaIaaaaaaaII aatnItI ulil a aaaaIaaaaaaI a
a IIaaaala a a
n- nmu n a~u v aalues. aa aoi ie eoepstv otcit n dse Iie deot neativeaaa aotcta <zA _1
b) 28_<z/z _ 56 c 256 _<ai~a~ a/ a 8,d 8 <zA
512.inaaa'sNinugg aIaaaaaa
3 162
3 b)
IJa a WE 4
3 03
N xUeoJ
* - % %
IIts
IO/
Figure 6.16 Continued.
1631
%. It%* %
0 rn
5"" -~C31'?
co 01 x,,
w
Atli
.IL
II-/
Figure,6.16 Continued.
1 164
* %
IV) k il , #: t
Is I- -
% . I
% %1 #~ 19 1
I //
Figure~ 6.1 Cotnud
II
165 3C- __ _ _ _ _ I
LEGENOF - 0.0r- . ........ 0 ... . ............ .
F 1 1.0- ... rT-" yI .... I
-1~~~~~ ~ .. .. r....2.S.5...............~ F - 3.0IU LINEAR THEORY
. -----------.
.. .. .........
C:
/''Ii'
0.03 0.11 0.19 0.27 0.35 0.43x I
Figure 6.17 Amplification curves of the kinetic energy E(z, F) obtained fromthe time interval 13TF < t < 15Tp. Case-3: A2 = .001, A3 = 0,and 3 = .51. U
U
II
3 166
[LEGENOI UN ".
K , ..... 9 ..... .....
0
UI'-5 ," ',
I * I I
i#
I
. .. .......... . ............ ..... ... ...... .....
I %! ,.
C.o/
I%
I - - -
0.000 0.015 0.030 0.045 0.060 0.075
T
i Figure 6.18 Temporal variJation of the centerline transverse velocity V ° in the
time interval 13Tp < t < 15T ,. Case-Z: A42 = .001, A3 = 0, and1 0 =.
IU
II
167 I
LEGEND
T- 0.0I
InIoI
CI5 1
Liiz
U""'"I
0. 03 0.12 0.21 0:30 0.39 0:48 0:57 0.66 0.75IxI
IFigure 6.19 Streamwise variation of the centerline streamwise velocity U° forthe mean flow and the base flow. Case-3: A2 = .001, A3 = 0, ardI
II
168
I _ LEGEND
T 0.0
I -
I* '-9
I
--,NI -, ... °
I
0.03 0.12 0.21 0.30 0.39 0.48 0.57 0.66 0.75
x
Figure 6.20 Streamwise variation of the wake half-width b for the mean flowand the base flow. Case-3: A 2 = .001, A 3 = 0, and/ = .51.
I
II
169
a)I
LEGENONAVIER-STOKES U.J Y -
NAVIER-STOKES RZO, _- 0
_---fl -TR'g ---- I
I
I0~
IU,
o III
0.03 0.08 0.13 0.10 0.23 0.28 0.33
x
Figure 6.21 Streamwise variation of a, and c. for the fundamental disturbancecomponent (F = 1). Case-3: A 2 = .001, A3 = 0, and 3 = .51. a)streamwise wavenumber, a,; b) phase velocity, ci.
I
170
NAVI LEGEND.NAVIER -STOKES UO, Y - 4NAVIER-STOKE5- VO,_ --4
* -f L~THUO ---- __------
I C-U.
*.0 0.801 01.3 .8 03
IxFiue62ICniud
II
171
a)
-I°, ...
2" ""J I8 ."
C3 _ ......C
0° I- I
* 3
- I0.03 O11 0.19 0.27 0.35 0.43XI
Figure 6.22 Amplification curves of the kinetic energy At(z, F) for Case-3 I(A2 = .001) and Case-4 (A2 = .01). a) F = 0, b) F = 1, c) F = 2. I
I
1 172
| b)
A2 01
.....................
I r4I.=i.0 .
O-cI
* ,°
LJ .*I
o
U , ,
0.03 0.11 0.19 0.27 0.35 0.43
* x
Figure 6.22 Continued.
I
173
C) 3
2 01I
_..
II
U,
2- II0.03 0.11 0.19 0.27 0.35 0.43
x
Figure 6.22 Continued.
I3.
U3 174
U
LEGEND*i* I.A2 - .01......... .° .....
* 0
I LJc-
I -J
LiI-
zLi*W
I
3
0.03 0.12 0.21 0.30 0.39 0.48 0.57 0.66 0.75I
I Figure 6.23 Comparison of the mean centerline streamwise velocity U0 for
Case-3 (A2 = .001) and Case-4 (A2 = .01).
I
II
175 I
A2 - .01
I
"I-,.
. . 0 . I
I
~I0.03 0.12 0.21 0.30 0.39 0.48 0.57 0.66 0.75
xU
Figure 6.24 Comparison of the streamwise variation of the mean half-width 6
for Case-3 (A2 = .001) and Case-4 (A2 = .0l).
176
* 0 LEGEND
IO N- 1024I
..... N . s0
I LM
* LiI-zw
UU
.030.15 0.27 0.39 0.51 0.683 0.75 0.87 0.99 1.11
I xFigure 6.25 Comparison of the base flow centerline streamwise velocity UO for3 Case-3 (N = 1024) and Case-5 (N = 1536).
1773
a)3
go
miii I
I In
~N
CO
,%a IoIIIIN
=!!! 1 " r , I !
9Sz Z Z6 091 QZ1 6 V9 ZE3Figue 626 Cntors f spnwie vrticty la.for ase5. 2 .01, 3 0
10 ~ ~ ~ ~ ~ ~ ~ , 3 5.Sldlnsdnt oiievriiyaddse ie
denteneatvevotiit. ) t 216A ad z,& : 124twetycotou iteval; ) 30O~ ad z/z 136
ten cntou intevals
II
178
* b)
Co
I I U
I7
I t, ,%
II * ---- ,. -
-. ,.- 0
i - '&.. , .
I ,%,, - -
--o.Q UP. NI N
'Ln
95I *, Z61[ 091 gzi 96 i, o
I Figure 6.26 Continued.
- - i m
II
179
LEGENIN- 1536
I-C I
a 3
-: I
2 I
I I I I
0.03 0. 0.19 0.27 0.35 0.43x 3
Figure 6.27 Comparison of amplification curves of the kinetic energy E°(z, F)for Case-3 (N = 1024) and Case-5 (N = 1536). a) F = 0, b) 3F=1,c) F=2.
II
1U
180
* b)
LEGENO
I
I -.
N 53
II °
0.03 0.11 O.19 0.27 0.35 0.43
x
I Figure 6.27 Continued.
IU
- II181i
c)3
IN 15361
2, 1
II
r
- i
I,
0.03 0.11 0.19 0.27 0.35 0.43
xFigure 6.27 Continued. i
33
II
183
LEGENOIn X -. 03N ....... .. . .. . .
. X - .15
in/
____.__ I/
C . . .. . ........ . . . . . . . . . °. .
C.0
•
0
I " I
N , ' , II
I
I
? " I
0.000 0.015 0.030 0.045 0.060 0.075
-' 0
Figure 6.29 Temporal variation of the centerline transverse velocity V ° in thetime interval 23TF < t < 25TF . Ce-5:A2 = .001, As = 0, andI
/ I
II3 182
ULEGENDF 0...............
F- 1.0F 2...... : .'. Z ........
SI NEAR -FEoRY
------------
'.... ...". ...............
I~~~~ ~ it ..,..1 "_.*. TH_............. .......... .... .
,, .................. .....
ft ,,, ... --
0 °"
*3 0 . 1
o.." /I I
U , I )
"- :' ' J :..
*:3 0.03 0.11 0.19 0.27 0.35 0.43
xFigure 6.28 Amplification curves of the kinetic energy to (z, F) obtained from
the time interval 23TF < t < 25Tp. Case-5: A2 = .001, A3 = 0,and 8 = .51.I
I
1 184
I a)
LEGEND3; X COMPI , .. ... C O R P ....
z COME~a:
a::
zI0* z
L.>Ln
I :L* I-
0
-17 -45 i.5 17y
I Figure 6.30 Sinuous mode eigenfunctions of the three-dimensional Orr-Som-
Ierteld equation corresponding to 0 = .28, -y =f .5, and Rea =~594. ,a) amplitudes of Us, Vs, and WI; b) amplitudes ofNfl, and ill; c) phases of" Us, VI, and WI; d) phases of Ill,
I n ,and fl,.
!-
I
185 3
b) 3
LI LEGEND0I X COMP 3
It I II
z ,WI
- 66 I ' I
Fig 666 Cotine6
CM!-Li
C L6 r 6
C6
0 6
z 6
-17 -8.5 a . 17
Figure 6.30 Continued.
3 186
,....~ ... LEGEND
SX COMP
3r z OM
z
C
3 Li
L- 7
0 1
* >4
-'1 4. 17
LJ~ a.
Fiur 6.0onined
187 3
d)
I LEGEND iiC ... COP....
co
-0 .
I I | I, It
.Itt '
Li
zC :J ,
i- t I
C
I I
1745a '51I I I
Figur 6.3 Contnued
II
I
3 .188
* -
LEGENDI.F - 1 ...................
C LINEAR THEORY
I -
I
I sfl ' ' "
... .
I -"
0.30 0.33 0.36 0.39 0.42 0.45
* x
Figure 6.31 Comparison of amplification curves of the kinetic energy E1(z, F)to linear stability theory. Case-6: A2 = 0, As = .001, 3 = .28,andy=.5.I
!
189
- a)
I LEGENDNAV IER-STOKES 3
.. ITNEARTHE ORY .
ftI .II
! ,
I
10-
I-
LiJ
-0 -6 -2 2
o/D I"
YFigure 6.32 Comparison of the amplitude distributions of the fundamental dis-
turbance (F = 1, k = 1) to linear stability theory. Case-6: A 2 = 0,A 3 = .001, = = .28, and -y = .5 a) streamwise velocity, U1 ; b)
transverse velocity, V 1; c) spanwise velocity, W 1 .
!3
I
190
* b)
LEGENDNAVIER-STOKES
..INERTHEORY.
II
II-
I
II --,I
-2Y/Oy2 0
I Figure 6.32 Continued.
II
191
c)
2- LEGENDI
NRV IER-STOKES
2NR 1TER..
LnIOin
XIM9CL
10 6 2 1'Y/D3
Figur 6.3 Contnued
3192
3 a)
LEGEND3 NRVIER-STOKES.... INERR T OR"..
I
I U:
* :
I -,j fj! I
Cn
! .
II I I I
-10 -6 -2 2 6 toYI Figure 6.33 Comparison of the phase distributions of the fundamental distur-
bance (F = 1,k = 1) to linear stability theory. Case-6: A2 = 0,A3 = .001, fl = .28, and -y = .5. a) streamwise velocity, U'; b)transverse velocity, V1; c) spanwise velocity, W 1 .
Ii
I
L I NIV _ _ _ _ST
I
III
I
-- 12
Fi 7 3"
q.
3
-10 -6 -2 2 6 I
YI
Figure 6.33 Continued. UI
II
194IC)
ILEGEND
NAV I ER-STOKE5.L INEAR THEOR'.
3
III
LO,*Mx
I LCL
*7I
* 40
-10 -6 -2 2 60
I Figure 6.33 Continued.
II
I
195 1a° ) I
n- LEGENDX COMP
0 ... C O P ....
z 3CCU~
,-,-
0 U!
zII0
z
> U3L&::.
tt 1 , I
CD3z
Figure 6.34 Sinuous mode eigenfunctions of the three-dimensional Orr-Som-I
merfeld equation corresponding to j6 = .51, -y = .5, and Res =L m= 594. a) amplitudes of U1, V I , and WI; b) amplitudes ofI
A' , f),, and W.; c) phases of U' , VI, and W1; d) phases of (11I' ,and n,.!
I-I
, ln llnI
I,196
* b)
0
I. LEGENOIo X COMP
Io
1,1 2 : •
., :{ .,
1C 0
I ~,
Cf
z*0
LD0
LiJ
Ln
1 l
Li
17 - . 0 .51'
I
Fi 6.34 Contiu
I
197 3
c) I'I
o 1'
Lj~
> £* a U
Li I
I- I*I, IW!>U}"I I
o * I *
C * I,
Cf) "', I.-0 0iI l Il -
-17 8.5 17
Figure 6.34 Continued.
I
I
198
I ° d)
5 LEGENDx coMP
cr.2" p:. . . . . .
- o:It. ",
Li... A'
* .A,
.oI 4i
I t .4i
0 i
i-. . .,I.0
L...I I I I
Li
-17 4.5 I 8.5 17I
F .
S... minmin~ n ll i i I~ l lli I
I
IIW 3
IIa )
o r -0.0~~. . .... . 0 . . .
~I
/ I+ ; I
Inw/ ,* IoS-I / IC
I I
0.03 0.09 0.15 0.21 0.27 0.33xI
Figure 6.35 Amplification curves of the kinetic energy P(z, F). Case-7:A 2 =
, A 3 = .O01, 6 = .51, and =.5. a) 6TP < t 5 <8T,b)8TF < t 5 IOTF.
I
I3
200
b)
I LEGENDa -. O
2F - 0.5F- 1.0
I 'F 2.5' ... 1..- . ..............
/ -. LINEAR THEORY
! /
-i/
1 1
I .':o
3 0.03 0.09 0.15 0.21 0.27 0.33
I Figure 6.35 Continued.
II
2013
0~01W
96 as 0 g 9S 9 1a
S"" b)
x 3 IIll 5
llq I N
S.I SI CI
-. li S[" 5'I
9Z 0 , I99. T
Figure 6.36 Contours of streamwise vorticity win the lyz-plane for t = 4096At. 3iCase-7: A2 = 0, A3 = .001, Y - .51, and , - .5. Twenty con-tour intervals between the minimum and maximum values. Solid
lines denote positive vorticity and dashed lines denote negativevorticity. a) z - .030, b) z - .146.
I
I
I 202
3 b)
! li
oc
* I-.
I *~
I I
CO "
* IiI I I
ai0
III4 , I
3 V
90 1
1
Figure 6.36 Continued.
I
1 'I I
'6
+3 -3
:- **' r 1i
Fiov 'z . Top and side view s of all v'ortex filamnents at time 1 , 90 Th filamnents of the upper I& ' erare - awn a solid lines. thoae of the lower le~ver aft represented by dashe lines. In addition linesof" onsunt streamwise, vortiitv ame plotted for a verticl (Y. Z)-plan at z - 4.6.i
Figure 6.37 Contours of streamwise vorticity w. in the yz-plane, reproduced I i
from Meiburg and Lasheras (1988).
I. . . . .. . i I I I I I I I I I I II i;
I
204
ICLEGEND
...... ~ 0..- ... .....................LST MEAN FLOW
* C..... -------.. .. £ ? ,- -..- -I 1° /-
,,
II
C)
I -I
0.03 0.09 0.15 0.21 0.27 0.33
I Figure 6.38 Amplification curves of the kinetic energy tk(z, 1). Case-8: A2=
.01, A3 = .001,/,0 = .51, and I'= .5.
II-
C. l I I I
II
205 I
LEGENDa K 0o%k ... - ........................
L5T MEAN FLOW---- --- ------
I I
-A / II
°I I
- I0.30. O,15 0.21 0.27 0.33xI
Figure 6.39 Amplification.5 As- 01curves of51°I the kinetic 5energy A k(z, 1). Caue-9: A2 =I
II
I ,* 206
* a)
I LEGENOo F -0.0
IF- .0_ "'" '-C.E....
I --- --- 'K-
iI /_ , I kl[R[ER
IIL.
II0.03 0.09 0.15 0.21 0.27 0.33X
I Figure 6.40 Amplification curves of the kinetic energy P(z,F). Case-9: A2 =
.05, A 3 = .001, j = .51, and -y = .5. a) N = 256, b) N = 512.II
207
b)
09.III_
_ _ _ _ _
I _o
I
i -=-...... .. . .. ".I
I.. IF 2.. - ~ l.. ....... ...
- p
p ,
Fiue64 ,ote d. o
.- ...-- I
I
I..... , , ,mmm mm m mmm i m
* 208
I
I (
IIN
- \ / .- ~
I\ E.
lop E -4
U -6
0 25 s0 75 100I,
Ficnitz 14. (a) A plot of the hvouon a( the modal a miow as funtuore of timeis .A,~ +'E" + S"- I the thryq lo 4s asaw, ed AWih th. sme Z.W&Vulph " the
fumadal. a lW. aI -04446. A,, a 0.22. AC.oOaa dA,,- I01* M-N- P-$4. (b) Athre.-dimemeiona pmn pee ive plot of the vortiy& ad in figum 10 at f 06. (e) As (a) butAt. - 0.14. (d) As (hi but At a - 0.14
3 Figure 6.41 Temporal variation of two-dimensional and three-dimensional dis-turbance kinetic energy in a free shear layer, reproduced fromMetcalfe et. al (1987).
III
209 3a ) I
N
LO
0It
t I to dI i
I I I 1 it%
I it sI t to
''II.' ti9,
I f I
%II,
I i.It4!
II 9I II
co I I I O9A:
1 I toy I
9, I
96 99 0 S 1 1 Z
r, , ..4 Tw t t ea b n t
mIxmu vu S I l dpsiII
e d t e . A2 = 0 a
0.0;b a9' A 2 = 0.5adA3=001
, ,bi gi i
hI I, l
h", ' ' ,;I \l I UIl,-h ' t
.g 4g, .l~l 4~i~l
II| II
I
II
210
* b)
It IIIit:J
II aa '"L
Io
ItI l11
((1:::
.b.,
3I I i t I I
I t1.
I I ll 10 .I I
1 011 1
.
I I t, 1 1VII I "L 1 N
hI I I I.
I.:: lt:
3 ~' , as ,'I~ *I I L a Q allna ~
aa11 l°
I a,1 l
ll
a II
-III I
-- I- I
8 i a z s si 0," "110"11Faa. Inaa 'II,/
I0/
Figure 6.42 Continued.
II
211
a)__ _ _ _ _ _3
-- ; meaI
0)
as Otto i' 1:
*oa~il% fillI
I ,aaa =
,, *1 1 1m5 I I
, .a , , ,m a a
I ~ IIII I
o l i i .l
ii
aIa It% i I I
LO ID I a.
UI g gIIDil
I l i l 1I4
114 lili
C i,
Iaa
Ia 0:1'0s I I, ItiIaaI I C
6 0II0U H 9S I M,+::~~I I
maximum values. Solid l~in lent posaitiv o t an ase
"a'gIIIIII
Il111i
at l Itma I II
fill
I Iam
ls n n aite I a
_ _ _ _ _ _ _
Ial I
II
0.001; b) Case-8, A 2 = 0.01 and A = 0.001; :) Case-9, A: = 0.05I
and As = 0.001.
II I1%11I
• lllll llIlllll II
3 212
* b)
No ,, m I :f"g 1mg, Ia.:I Will .~ e .
t o t o I . .
~~~~~4 to* H ~ i ,
Omi 8
i s gml l g,
I I in Mim as$.:g ol
INI
to m - ;, : " gl .'.
Oig al I
Iu II f gg1:,g 5 amg g I
to 'm , aCD mg
to~ of 'b'gggggurn'IN ,::
6 dg oo U )a gg 1 is
Figure~ 6.4 Continggd.
II
213
C)
ii 9 I| S Illl
III ::: IliiSI S I I
I...I lugI
.1 I I I
II I IifI: I I IIS, j Il ii~ U
SI, I iiI iii II
IS Hi ll::oi* IIS ilI
I III II I
SII l
II ] fillI I I I
Qp
II I I Ill
I £oill 55 i
' 1l1 IIll
e e, IeI II I
gIlI III
9 ,as Ua, , , nS Il lilii
* 9II I I
-l I1 III I 51S SI
S III I I I II
I III *IIII Il
I IIt' I IglI I 0,II I
* I I III
_I:!! : :
Figure 6.43 Continued.I
S I Ii
II Inel S /ll I
I214I
a)
I CO f.Li
I - ', i
I °?
xX
..I I II
~ 1.0
96 96 do ii 1,9 9 , 0i
10i l,I
Figure 6.44 Contours of streamwise vorticity w. in the zy-plane for t =f 4096&t.z =f T/2. Twenty contour intervals between the minimum and
I maximum values. Solid lines denote positive vorticity and dashed
lines denote negative vorticity, a) Case-7, A2 =f 0.00 and A3 =
0.001; b) Case-8, A2=ff 0.01 and As =f 0.001; c) Case-9, A2 = 0.053 and A3 = 0.001
II
I II" III III CI
2153
b)5
- N
0~0
zz Ig F
Figure 6.4Cniud
3 216
* C)
.I.......
IXXI
co, * ~~ g
IzI6 9 9 U I s P 0
Fiue64*Cniud
217 1
a)
! '~ -
IgIf I I
a
SI3
CDl
I
S tilI I I
jI lllltl I3
6 o' G U R
~lIII Ia,,, - Tsa1I
mum v u . ollllslldno e a I a a
del a ega, A2 = 0
i.I Itll,.'l
I lie alll am
pII ll 1111 IIIII I i
eeIa IItIII3IeI mI e I I
(\J It 1IIIIII
b)Cae-, 2 001aneA = 0.0c aseme , e=00 and
Ai I l0.00mmaue.Sldlnsdnte poslitiv iotct n dseie
b) Cae8 2 =00 n , = 0.01 ) ae-, 2 = .5 nA, =0.001.II
IIIIII
218
3 b)
* wO
O~3 - -2
~- ~ *. - - CC)
it , L
%1
* Is
0~0
I sit I I
Fiur 6.5Cotnud
2195
c)5
Qr'4
z~fI
% 3
I II
96 go 09 9S 0f 0 a
Figure 6.45 Continued.
3 220
NN
N C
10/
3 Figure 6.46 Contours of spanwise vorticity w. in the xy-plane for t = 4096,&tand z=O0. Case-8: A 2 =.01, A .001,6=.51, andy=.5.Thirty contour intervals between the minimum and maimum val-I ues. Solid lines denote positive vorticity and dashed lines denotenegative vorticity.
2213
-------- - ------
------ ------- --_ _ _ _ _ _ _ _ _ _
INI_ ----- -- 3-- -------
DEAN*,. .. aa~a..~.a Dn.. .agae.23LW.
- ------ z
ZE 9 E Oz 91 ZI 90I
Figure 6.47 Contours of spanwise vorticity w, in the zz-plane for t = 4096W.ICase-8: A2 = .01, A 3 = .001, j6 = .51, and -f = .5. Thirty con-tour intervals between the minimum and maximum values. Solidlines denote positive vorticity and dashed lines denote negativevorticity. a) y = 2 (y/4y = 72), b) y = -2 (y/Ay = 56).
3 222
3 b)
U-)
3 NY
a ILI(.\jUX
CC
Figure 6.47 Continued.
223£
a)3
M3to *-
11 ,
N - *
>%
A- X
X :-
I .3 oi - 3t~N
Figure 6.48 Contours of streamwise vorticity w. in the zz-plane for t = 4096&t.3Case-8: A2 = .01, A 3 .-= .001, 0 = .51, and -f = .5. Thirty con-tour intervals between the minimum and maximum values. Solidlines denote positive vorticity and dashed lines denote negative3
vorticity. a) y =2 (y/Ay = 72), b) y = -2 (y/Ay = 56).
3 224
* b)
I - - -, %LO
% ~ ~ % I ' I
-------- --- ------- -- -
I I-
Z6-m
W) It-- -*"'lI . c
* /--------------
3iur 6.48 Cotnud ZO
225
a)
it Iam , li m a
a a m I . , Im a m $ I s m a a 14 aa 1 aa m
m A 11111l _a Ia I, :a maginm ~ am a mmaaa1 1 a, a~ mm40a.
ama soaam aa a a
am "al mm illi : I, Hill a
acm a a amma m t maaIa~mama ~ ~ ~ ~ ~ ~ It Im~ naamm amm
l IN Im aI ~ i ma~ aa o a .aa a ma omin aima m m . amia
III a o Wm aa m INam: a 46Z amm~mamof a 1 , jmm m a mama , Intoa 3am ~ I as a. a1cmammam m.'aa m a mm
to IOma a aa aallmainA inaa a looma aIa ~ a mnm
amI ao a" Iumamm an tom 10,4~ amaa' ~ ~ "= wo mamm mamm 1 .Iam.4 .aa
a ~ ~ ~ I ma tw amaoam,. mm 1 ta o am.na*laammc~Imaa::Leauainm.mm~a' $arnm mmm
a f at to = 'mm m .' % a m flo ammmaam ~ ~ ~ ~ ~ ~ ~ ~ ~ lu maaImamnmn m aaa I mmam
a ma,, Winm maof mam m muama aaaaa
amaa m a~ ~~ns m m ,:a s.mma _0aO ma%' a11 aml ma aI '
am aama a * mam a mamto
.. am0m .to~ I fmlow='a.a t 1mamma Mm mum ama fmmma~~: ama mam m1111afn
Il mam aml a mun ammmJunm
aiue649Cnor of a nws mat t w, inmm mh yza lan fora a 06 t
C ase a Aa =m .01m, a3 a .01 5 , nd -y ma a 5 hrycn
touineral aewe ah minmam mand maimumalus.Soiamne deoepstvIotc n dse ie eoengtv
votct.a) ma .172~ mmmi= 1)b)z=.60(/z 0)
I1 226
3 b)
~~IIIVNN
.... ll. I
m
I t oil$
X ii ~ t 6
I I I I II-I I III
ma I I gg * II tIllmaI I I IgI I
I
lu ll I aI
ii
ill " *l 1, *I
I 11 1 IIl ' mat , , ', gl
f Ullt In I I I
lll ON
I lI I ill tllI I .__ I I u
-Ill1111 I~ili i tilli
eIIIII I 1 1 l o lj] l~ I NI N |liiil * ii
too! 1 I i s
IIII : I I i I
tu lollil I
• ii llli i I •i ll
i ll tmtmI.,- , , ' ',
I, , i I
*,II I I 1III
ill illl
Ii//1 Yll ,,, I ""1I| | I~l*1l I ll
lti i l h I llll
Il.; II a; liltil *l3 l E : I=, '"I1111 i , gIll ll
llltigl
96 so as UL 9 9S ff 01 '
3 ItoiIt
Figure 6.49 Continued.
II "i'l ,,*IiI1 /;'i= , ,,II
227
M)
LO
0~N3
fC
X _fI
ItI
%se It
10/1 1
Figure 6.50 Contours of streamwise vorticity w. in the yz-plane for t = 4096WtCase-8: A2 = .01, A3 = .001, #3 = .51, and -y = .5. Thirty con-Itour intervals between the minimum and maximum values. Solidlines denote positive vorticity and dashed lines denote negative
vorticity. a) z = .172 (z/Ax = 110),b) z =.160 (:14: = 101).
228
b)
I o
II
x0
I No "96 GO 9 zz g 01 Z
Figur 6.5 Contnued
I,
229
a)
i-LEGENO 3IX - .03...... ... 07
0Xo £---: , --- - I
i
L
MII=ti
IDt
-1 -1 -8 - 0 4 2 1
I i11
Ig II
I !o II I
SI 1 I - I 12 16
-16 -12 -O - 2 1
y IFigure 6.51 Vorticity amplitude distributions for F = 1. Case-?: A2 = 0,
A 3 = .001, 0 = .51, and -t = .5. a) spanwise vorticity, fIo; b)
spanwise vorticity, W2.; c) streamwise vorticity, fl .
II = ==-==,.====i i , , , , , I
II1 230
5 b)
LEGENOI IX -. 031
1..... -07
I I-
CLi
I: I
-o
It I
I I
::6:: ' I '
3iur 6.5 Conined
I| iII = II I
IIi
-1 -I 8 - 2 1oIU
I Y III
231
C)
x - .03j.m ' 07..
LEGENO"" .... ... !Q .x - .15
6- 4
Ca,
I'a m -
II6I. II ItI
III
IIIo= a :
i l ,I° i
, ,I0.
*.. " , ,-16 -12 -8 - 4 8 12 15TI
Figure 6.51 Continued.I
II
II3 232
I a)
LEGENDx -. 03 1
I _lI
I I
L4..
1-4 0
yI Figure 6.52 Vorticity amplitude distributions for F = . Case-8: A2 01,
A3 -- .001, 8 = .51, and -f = .5. a) spanwise vorticity, W; b)spanwise vorticity, fI; c) streamwise vorticity, n-
= I
I0*
I
233
b)5I ,
LEGENDX -. 03 5iI ... .... . Y0 7 ..T .x .10=--- ---- -----
I
tit
I IIItI~Ii I
CJ) II II
-16l -12 11 -4 12
FII 6.5 I t3333 Ii i 3%33
,, ii 3,, ie ' i3
I 333 II,
33* I 3 I
o 33 ' ../ .; ":....
1 " 3
, , 3 .' 3
-16 -12 -8 -4 0 4 8 2 16Y
Figure 6.52IContinued.
UI
234
I C)
LEGENOX - .03x..... ..-. 7
I
1 -,
CD
I iLi
"0 'I I iI
I
> 1e -12 Continued.
y~
FigureI 6.5 Cotined
235
LEGEND 3x- .03 I
| : I
IoI I
C
CL 0 C3
fl,
,, , IS" I I
•f I I
0 I ?i: I/ I I: I
/ I I:
-16 -12 -O -4 0 4 8 12 16 IFigure 6.53 Vorticity amplitude distributions for F = 1. Case-9: A2 = .05,
A3 = .001, P3 = .51, and -t = .5. a) spanwise vorticity, 0' ; b)spanwise vorticity, fNl; c) streamwise vorticity, 11.
I
II3 236
I nU,,
LEGEND3 X - .03..... ......
X 0
I
Lj
IC
*A=
,, . .. ,
(I iIl1i I
-16 -12 -8 -4 0 4 a 12 16! 'a
Figure 6.53 Continued.II
IU
237 IC).I
L LEENO 0
0. I
x00
X -J .0519
o £L U
-1 12 -8 -4 0 4 a 1I1
Fiu I Connue,
ii'a, ;1,Ij , I0 ' h'. I "| l I
III/ CI, ; , , , II
Fiue65 Cniud I
I0.i
Figure. 6.5 CInud IIIlil
I
I
238
LEGENDItA2-O.O1 , R3-0.000)(i2-0.01 .R3-0.00 .
II -~ ------------.
I UN'
0.03 0.12 0.2 .0 03 .8 057 0.56 0.75
IxFigure 6.54 Strearmwise variation of the wake half-width b for Case-4, A2 =
0.01, A3 = 0.00; Case-7, A2 = 0.00, A3 = 0.001; and Case-8,A2 = 0.01, A3 = 0.001.
I
I!
239
LEGEND 3IEGN(A2-O.O1,.A3-O.OOI) I
IA2-O.05 A3-0.001)
I f ,LO
Li
_ _ _ __ _ _ _I
I. I....I
0.03 0.12 0.21 0.30 0.39 0.48 0.57 0.66 0.75
x IFigure 6.55 Streamwise variation of the wake half-width b for Case-8, A2 =
0.01, A 3 = 0.001; and Case-9, A 2 = 0.05, A 3 = 0.001.
II
I240
3 _ a)
LEGENDN-64 M-64
N N-256 M-64
I
I =-"
0.30 0.33 0.36 0.39 0.42 0.45
x
I Figure B.1 Influence of the grid increment on the amplification of the kineticenergy tO(z,F = 1) for Case-i. a) M = 64 with N - 64, N =
128, and N = 256; b) N = 128 with M = 32, M = 64, andM = 128.
II
241
b) 3
-LEGENO
M- 32 N- 128"!~~ ~~~ ... -64,-i8......10-, M- 128 N-128
Li
0.30 0.33 0.36 0.39 0.42 0.15xUFigure B.1 Continued.
| I
I
242U* a)
LEGENON-64 M-64
N-256 M-64
* N
I • E -
0*,
I=
-10 -6 -2 2 6 10Y
Figure B.2 Influence of the grid increment on the amplitude distributions ofthe fundamental disturbance (F = 1) of the streamwise velocity.a) M = 64 with N = 64, N = 128, and N = 256; b) N = 128with M = 32, M = 64, and M = 128.
IU
II
243
b)
LEGENDM-32 N-128
M-12.N-'128
' I
Fiue2 Cotnud
I0,×-[ IdI
- I
o I
" I
3 244
; a)
U I -
.. ...............
Ln'
N ,"
! ,
S........ . .
U.-,
D
U'I ' .
!j I
"LEGEND
N-64 M-64N-2.M-64N -2 56 M-64UI I. ""_ _ _ _ _ _ _
I|I I-10 -6 -2 2 6 10
Figure B.3 Influence of the grid increment on the phase distributions of thefundamental disturbance (F = 1) of the streamwise velocity, a)M = 64 with N = 64, N = 128, and N = 256; b) N = 128 withM = 32, M = 64, and M = 128.
IU
I
245
b)
CD
o iii IYi : I
X !
c00 1
LEGENO
M-32.,N-128 3M = 8 N-128
C-. 3-10 -6 -2 2 610
Figure B.3 Continued. III
II
246
3 N a)
LEGENDN-512,M-256
I ..N-i 4.50 -. -...
I -- ---' --- - -- - -- --0
ICI
....... .. .......
I C
0
C
IIO
IT
C
* C
C
I xFigure B.4 Influence of the streamwise grid increment on the .vnplification of
the kinetic energy E(z, F). Case-2: M = 256 with N = 512,3 N = 768, and N = 1024. a) the mean component, F = 0; b) thefundamental component, F = 1; c) the second harmonic, F = 2.I
U
247
b)
LEGEND 3-~~ N-S 12 ,M-256.
N- 102 2
.. . . . . . .
0.03 0.110.190 270.350 .4C I
Fiur BACniud
3 248
3 C)
U - LEGENDN512,M-256| ...... .- , .........------- 2.- .- .6
0
I
I C-
IC
I
0.03 0.11 0.19 0.27 0.35 0.43
XUFigure B.4 Continued.
IU
I249 I
N a)
LEGENDM-128,N-1024-_. .. i92 N-1024 ...
• ' IM- I
--- I
- ICD -I !
0.03 0.11 0.19 0.27 0.35 0.43x
Figure B.5 Influence of the transvere grid increment on the amplification of Ithe kinetic energy E(z,F). Case-2: N = 1024 with M = 128,M = 192, and M = 256. a) the mean component, F = 0; b) thefundamental component, F = 1; c) the second harmonic, F = 2.
3I
IU
250
* b)
U LEGEND-I .M-128,N-1024
..M:.2.,N- I .24
I -
I C
! U-'C)
CDU0I-
0.03 0.11 0.19 0.27 0.35 0.43X
U Figure B.5 Continued.
IU
II
251 3
C)3
LEGENDM-12B,N-1024
- M- eb2.N-1024
0.0 0 . ..M.2 . .,..- 0.5 .43
- I
Nx
F r BII..i
C I: I
- IFigure B.5 Continued. i
I
1 252
I3LEGEND
N-512,M-2563...N'-7 8;=5 ......
* ' 0Lfl
I 0*
I
,MI IU.8
0-
-16 -12 -4 0 12 16Y
I Figure B.6 Influence of the streamwise grid increment on the amplitude dis-
tributions of the streamwise velocity. Case-2: M -- 256 withN = 512, N = 768, and N = 1024. a) the mean component,IF = 0; b) the fundamental component, F -- ; c) the second
harmonic, F = 2.
I0. ___i__miimurI
I
253
b)3
C3 - LEGENDN-512,M-256
o-
I
)I
x
CD
:
3
4- £
-16 - 2 -8 -4 O 4 8 12 16
-J II
Figure B.6 Continued.IIIII1
254
* c)
U LEGENON-512,M-256I ..... ~-768,iM-:S6....
I
IxII
L It I '
U I, ' ,.
IIII .
-16 -i2 -B -4 0 4 8 12 16y
Figure B.6 Continued.*I
I255
a)0
LEGEND3. . ..M1 28' N.024
H0 3
N"'I
I
I
M" I |~
-12 - -4 0 4 8 12 16
Figure B.7 Influence of the transverse gridc increment on the amplitude dis-Itributions of the streamwise velocity. Case-2: N = 1024 withM = 128, M = 192, and M = 256. a) the mean component,
IN,
harmonic, F - 2.
I
I, . aroi c F = I 2.II
I
256I.
LEGENDM-128,N-1024
U C
.... -1i2 -8 --4 0 4 2 16!y
I B! .-
* i
-C
C"-
* 0
II-I.m Fi mmmm=reB7Cn inu e I
II
257 Ic)
0
I' " LEGENO II1-128,N-1024
M-;eb.i2o" I
Li
' I
C- I
-16 -2 "' 4 0 8' 1'2 I'S
Cy
Figure B.7 Continued.I
II
i ! ni |
I
3 258
Sa)
LEGENDN-512,M-256
I
x4
-
I U,
* N,
Ul
UU
I I I I I I Ii
-16 -12 -8 -4 0 4 8 12 16Y
Figure B.8 Influence of the streamwise grid increment on the phase distribu-
tions of the streamwise velocity. Case-2: M = 256 with N = 512,N = 768, and N = 1024. a) the mean component, F = 0; b) thefundamental component, F = 1; c) the second harmonic, F = 2.
U,U
I
259 3
b) 3
LEGENON512M-256
NC o I'
L,
0 '
- I
I IIII U
o ml
U U
0 -
16 -i2 -'e -4 0 4 a 12 16yFigure B.8 Continued.
N 5UD
*nilm an n S I
I3 260
|C)
,,. if ,'S"ILEGENO3,N-512,M-256 .
16, - - 2F B -ontinued
I °I-:,
I :1 I
I : I:
I ,: -
I I.
I'
II i i l I I
II
261
SIa)
LEGENDM"128,N-1O24
.;T ..M.. ..SN f 02A.MY2 N 10 24
x
(-r-f I
Co _ _ _ _ _ _ _ _ _ I
16 -12 -8 -4 0 4 8 12 16
0Y' I
Figure B.9 Influence of the transverse grid increment on the phase distribu-tions of the streamwise velocity. Case-2: N -- 1024 with M = 128,M = 192, and M = 256. a) the mean component, F = 0; b) theI
fundamental component, F = 1; c) the second harmonic, F = 2.
3I
C
I
I
262
3 b)
LEGEND..... MM-". 2' 0~ 02 4 ..3 .- 2 N-1O 4M-126N-102
CDI -,
CD
W
II
I ' 0
C=
"-c
S2- 1
3yF , t
I 0 I
I I I I
2631
C)3
I LEGEND
M-128,N-I024M12N'-10 ION
M-756M N-102
U
C.-'
C37
1 6 -1 8 - 2 1
yIFue B. otiud
264
I APPENDIX A
5 SOLUTION OF THE THREE-DIMENSIONAL
UORR-SOMMERFELD EQUATION
I In this appendix, the finite-difference method used to obtain the eigenvalues
and eigenfunctions of the three-dimensional Orr-Sommerfeld equation for the spa-
3 tial stability problem is discussed. Both the Orr-Sommerfeld equation and the
related Squire's equation are solved using fourth-order finite-differences.
3 For a small amplitude disturbance with velocity and vorticity components
3u(,yzt) = Real(fL(y)ei(az+iYZ- Pt)) (A.la)
v(x,y,z,t) = Real((y)e i ( z+-z - j t)) (A.lb)
w(z, y, z, t) = Real(7b(y)ei(0 + Yz - 6t)) (A.1c)
3 w2 (zz,t) = Real(az(y)e i(*2+'yZ/t )) I (A.ld)
I wS(x,y,z,t) = Real( , (y)ei(0z + iz - ji)) (A.le)
and wz(z,y,z,t)= Real( ,(y)ei(az+'yz-- t)) (A.1f)
the amplitude -b(y) of the transverse velocity is an eigenfunction of the Orr-
3 Sommerfeld equation
bv+...De " -2(a 2 +'y2)f +(a 2 +_ 2 )2 D) = 0 . (A.2)I ctRe
3In equation (A.2), U = U(y) is the streamwise velocity of the undisturbed flow
and is assumed to be independent of z and z. Furthermore, a is the streamwise
3 wavenumber of the disturbance, -y is the spanwise wavenumber of the disturbance,
c = 0/a is the phase velocity of the disturbance where , is the temporal frequency,
I
I
265
Re is the Reynolds number, and i = VW7 is the imaginary number. The notation 3( )' denotes differentiation with respect to y.
Solutions of equation (A.2) are sought in the transverse interval Yr,i, y 3Yma. For wakes, equation (A.2) is subject to boundary conditions at y = Ymin 3and Y = Ymax of the form
=(y,,,,) = a (yni,) , (A.3a) I=(yni) aOf(ymin) , (A.3b)
and V'(Yaz) = -a(y,,,,) (A.3d) IEquation (A.2), coupled with the boundary conditions (A.3), is an eigenvalue I
problem of the form
Lli - cL 2 f (A.4a) 3where 3
LI=U(d 2 _ .2 _ )|1L1 =U ( y 2 - 72) -"
- (d 4 -2 (a 2 + ' )dy-+(a 2 + -2)2) (A.4b)
a n d + R (
/ 4 2
L2 = -a t72 (A.4c)
For specified parameters a, 7, Re, and the base flow velocity U, equation (A.4a)
can be solved for the eigenvalue c and the eigenfunction v. IFor spatially amplifying disturbances, a is obtained as a function of 3, Re, 3
and -y by iteratively solving equation (A.4a) for the eigenvalue c, subject to the
constraint, F(a)I= 0 (A.5)
a= - = 0(A.I
'I266
I "Subsequent values of a are obtained by applying the secant method to equation
5 (A.5) to get
a3+l = aj - F(a') (Fa J+1 a ) (A.6)(F~aj) - F(ai-1))CA6
3where j denotes the iteration level. The iteration scheme is repeated until
a(j3,Rew) is obtained for the desired accuracy. For each iteration level j, the
I Orr-Sommerfeld equation (A.4a) is solved using a finite-difference method.
3 The eigenfunction 0(y) of equation (A.4a) is sought at the discrete points
y,, = yo + may for m = 0,..., M (A.7)
I where yo = Ymin, YM = Ym.., Ay is the spacing between grid points, and M + 1
I is the total number of points in the domain. The function (y) and its derivatives
can be approximated to fourth-order accuracy (Kurtz and Crandall, 1962) usingI1 7 41 7 135g- + 5 -+g-i+ + .9m -+ +-6-, , (A.8Z)
f) l(in-1 + jim+) (A.8b)
1
Vt, 1 1 -g.-2 + 2 g.- 3 2gm +1+m2 (.M T 1 2 3 m+3 2and ' T" (gn-2 - 4gm.- 1 + 6gn - 4g,_ + g_, 2) (A.8d)
where 1, =(y,).
1 Using equations (A.8) to approximate the derivatives in the Orr-Sommer-
feld equation (A.4a) and the boundary conditions (A.3), the following matrix
3
I
267
eigenvalue problem Id, d2 d3 d4 dL -2ds d6 d7 d6 d IA1 B1 C, D1 E1 go
A M*°°* 3I° Io
A 1m B cm Dm Em g
d d. d7 d6 ds gM+ldLi d2 d3 d4 dLi gM+2
o 0 0 0 0 g2 3o 0 0 0 0 9-1
90 IC A.9) I
AM MCM DM EM 10 0 0 0 0 gM+10 0 0 0 0 9M+2 I
is obtained. The coefficients of the matrices in equation (A.9) are 1
A, = U(y.) + aU"(ym) + a2 + bl (A.10a)
Bm = U(y,,) + asU"(ym) + a4 + b2 (A.10b)
Cm = U(ym) + asU"(Ym) + a6 + b3 (A.10c)
Dm B m , (A.10d) 3Em = Am , (A.lOe) 3
and
AM= a7 , (A.11a) I1
268
3 Em = as , (A.11b)
1 C,, = ag , (A.11c)
Dm,, = as , (A.11d)
I Em = aT , (A.lle)
3 and
d I = VY/-a 360- + - 2 (A.12a)
I 4=Ay
d3 = 60 /a +7f (A.12c)
1d = 1 (A.12d)45 2
d5 1-A (a2 + f2 ) (A.12e)d6 = 2 - (a2 +,2) ,(A.12.f)
2 7 5y
d7 = 3 4lAy2 60
3 The constants in equations (A.10) and (A.11) are
a = 1 16 (a 2 + t2) (A.13a)
1
a 2 = - 1 (A.13b)I 360
a3 =2 -T 2 72) (A.13c)
a 4 = - 4 (A.13d)45 '3 41 2 2
as =- 2Ay 2 60 (a2 + 7) (A.13e)
a 4 = -1 4 (A.13f)3 a 6 01 2
a7 = - - +,72) + -I (A.13g)360 12Ay
I
I
I
269
7 2 (A.13h)413(A.13i) 3
aZ = -6-1 (a 2 + - 2) - 32 (A.13i)
an = ,-e (-," 6Ay 2 (a 2 + 7'2) + 1 (a 2 + 2)2) , (A.14a)
7;)+e ( 22 +()a It (A.14b)
b3 = i 6 4 32 (a 2 + -t) + L(a 2+ -t2)2) (A.14c)
The eigenvalues and eigenfunctions of equat:an (A.9) are obtained using the IIMSL routine EIGZC. As equation (A.9) is a M + 5 dimensional matrix eigenvalue
problem, M + 5 eigenvalues and eigenfunctions are obtained. However, only two 1eigenvalues and two eigenfunctions have physical meaning. 3
Of the M + 5 values of the complex eigenvalue c, only those for which the real
part of c, C ,. = R eal( = 3 () a (A.15)
c,.=Reala = + a ?_
satisfies
ct < 1 forfl < 1 (A.16) 3are considered to be physically meaningful. In this work, 3 is always less than
one. Of the eigenvalues that satisfy (A.16), one corresponds to a sinuous mode Idisturbance and one corresponds to a varicose mode disturbance. The sinuous
mode eigenvalue is the one for which
ci = Imag( = --"- (A.17)Ia .a + a?9
attains its maximum value and for which the corresponding centerline transverse 3velocity is nonzero. Alternatively, the varicose mode eigenvalue is the one for ,I
!1
1
270
which ci attains its maximum value and for which the corresponding centerline
5transverse velocity is zero.
Once a is obtained for the given parameters 3, f, and Re, it remains to
5 construct the velocity and vorticity components from the eigenfunction g,,,. For
each transverse location y,, the transverse velocity 6, is obtained from
I 7 41 7 1I, =3i60gm -2 + Tjgm-i + Fogm + Tgm1+ T6.m- (A. 18)
Procedures to obtain the remaining velocity and vorticity components depend on
the whether the disturbance is two-dimensional or three-dimensional.
5For two-dimensional disturbances (-1= 0), the streamnwise velocity is obtained
from the continuity equation
= -,,,m (A.19)
where b' is given by equation (A.8b). The spanwise vorticity component u, is
obtained from
Wzm 7n M - M) (A.20)a
where 6" is given by equation (A.8c). The remaining velocity and vorticity com-
ponents are
Wm, m = , M = U; " = 0 (A.21)
For three-dimensional disturbances (/ 5 0), the velocity and vorticity compo-
nents are obtained in the following manner. The transverse vorticity cj, is obtained
from Squire's equation
"- (a 2 + 72 + iaRe (U - c)) a. = -iReU'Tyi (A.22)
I
271
Squire's equation is solved in the transverse interval yi, y 5 Y,.. subject to 3the Dirichlet boundary conditions g
0I yi.) , (A.23a)
and u.J(ym,.) = 0 (A.23b)
Fourth-order finite-differences are used to discretize the derivatives in equation
(A.22). The resulting system of equations, 5G0 16 -1 wyo
16 G, 16 -1 wy1 I-1 16 G 2 16 -1 w 2
-1 16 GM- 2 16 -1 W'M-2
-1 16 GM-1 16 WYM-1-1 16 GM awIM
f2
fM-2fm-
is solved for the transverse vorticity 4j,. The coefficients of the matrix in equation
(A.24) are 3G,, = -12Ay 2 (a 2 + _72 + iaRe (U(y,.) - c)) - 30 (A.25)
I
U
272
3 and the right hand side of equation (A.24) is composed of the elements
5 f, - -iU'(y,)y,,,'12Ay 2 (A.26)
The remaining velocity and vorticity components can be obtained from the
I transverse velocity 0 and the transverse vorticity Jv. The spanwise vorticity C3, is
3 obtained from the equation
"zm =-s ((a2"" + i m ) (A.27)
Equation (A.27) is derived from the v Poisson equation and the relationship
I +O + i-101 = 0 (A.28)
In equation (A.27), the derivative V' is obtained from equation (A.8c) and C0Jm
is obtained using the fourth-order finite-difference formula
Ym = 2h (Y-2 - 80/- Oml ;m2(.9
i The streamwise vorticity is obtained from equation (A.28), rewritten as
'5 lzM_+ WY"M (A.30)
where O 'is obtained from equation (A.29). The streamwise velocity is obtained
from the relation
fI )a ( Wp + ar(A.31)which is derived from the definition of the transverse vorticity. Finally, the span-
S wise velocity is obtained from the continuity equation
3 = ia?,,+O- , (A.32)i7
3 In both equations (A.31) and (A.32), the velocity derivative 6' is obtained from
equation (A.8b).II
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273
3 APPENDIX B
i INFLUENCE OF THE GRID INCREMENT
ON THE NUMERICAL SOLUTIONIFor the calculations that are discussed in Chapter 6, it is desired to determine
the influence of the grid increment on the numerical solutions. This is done by
If repeating the calculations of Case-I and Case-2 for several different computational
grids. For Case-i, small amplitude disturbances are calculated and therefore, this
3case is used to test the influence of the grid increment when nonlinear effects are
unimportant. For Case-2, larger amplitude disturbances, as compared to Case-i,
are calculated. This case is used to test the influence of the grid increment when
nonlinear effects are significant.
For both Case-1 and Case-2, the streamwise and transverse grid increments
5are varied independently so that the variation of the calculated flow field with the
grid increment may be observed. With the domain size held constant, the grid
increments are varied by changing the number of grid points. For the temporal
discretization, numerical stability considerations require that a much smaller time
step be used than is required based solely on temporal accuracy. Therefore, the
3 time step is not varied for either Case-1 or Case-2.
Case-1 is recalculated using the grids
M=64, N=64, (B.la)
SM= 64, N = 256, (B.lb)
N = 128, M = 32, (B.lc)
and N - 128, M = 128. (B.id)I,3
I
274 1In addition, the results of Case-1 that are described in Chapter 6, for which 3
N = 128 and M = 64, (B.2) £are compared with the calculations described here. All other parameters for these
calculations are identical to those discussed in Chapter 6 with regard to Case-1.
Figures B.1 show amplification curves of the fundamental disturbance for 3various grid increments. In Figure B.la, the amplification curves corresponding to
N = 128 and N = 256 are relatively close together and exhibit exponential growth. 5However, for N = 64 the amplification curve is significantly different from those
corresponding to N = 128 and N = 256 and does not exhibit exponential growth Ias would be expected for a small amplitude disturbance. In Figure B.lb, the
amplification curves corresponding to M = 64 and M = 128 are almost identical
while the amplification curve corresponding to M = 32 exhibits small differences 3with the other two curves. For all values of M, the fundamental disturbance grows
exponentially. 5Figures B.2 display the amplitude distribution of the fundamental disturbance
component of the streamwise velocity for various grid increments. In Figure B.2a, Ithe amplitude distributions for N = 128 and N = 256 are quite dose and compare 3well to linear stability theory. For N 64, the shape of the amplitude distribution
is similar to those for N = 128 and N = 256, but its amplitude level is significantly 3higher than for the other curves. In Figure B.2b, the amplitude distributions for
M = 64 and M = 128 are quite close. For M = 32, the amplitude distribution is 5similar in shape to those for M = 64 and M = 128 but clearly requires a smaller
transverse grid increment.
Figures B.3 display the phase distribution of the fundamental disturbance
component of the streamwise velocity for various grid increments. In Figure B.3a, SI
275
Sthe phase distributions for N = 256 and N = 128 are similar in shape and rea-
isonably close together. For N = 64, a significant phase differcme.-" between this
curve and those for N = 128 and N = 256 is present. In Figure B.3b, the phase
I distributions for all values of M are in reasonable agreement.
From these results it is apparent that the size of the grid increment has
significant influence on the calculated flow field. However, it appears that for the
grid used for Case-1 in Chapter 6, for which
M=64 and N=128 , (B.3)
the calculated flow field is reasonable well resolved and is not significantly altered
5 when the finer grids (N = 256, M = 64 and N = 128, M = L28) are employed.
Case-2 is recalculated using the grids
M = 256, N =512, (B.4a)
I M = 256, N =768, (B.4b)
SN= 1024, M = 128 (B.4c)
and N=1024, M=192. (B.4d)
In addition, the results of Case-2 that are discussed in Chapter 6, for which
M = 256 and N = 1024, (B.5)
are compared with the calculations discussed here. All other parameters for these
3 calculations are identical to those discussed in Chapter 6 with regard to Case-2.
Since nonlinear effects are important for this flow field, the fundamental dis-
5turbance as well as the mean and second harmonics components are displayed for
various grid increments. Figures B.4 and B.5 display the amplification curves for
I
276 3the mean component F = 0, the fundamental disturban ,e F = 1, and the second 5harmonic F = 2- In Figure B.4a (F = 0) and B.4b (F = 1), the amplification
curves corresponding to N = 768 and N = 1024 are virtually identical. Beyond 'the point of saturation, the amplification curves for N = 512 deviate from those
corresponding to N = 768 and N = 1024. In Figure B.4c, the second harmonic I(F = 2) appears to be more sensitive to the streamwise grid increment than the 3other harmonic components and appears to require greater streamwise resolution
than was used in these calculations. In Figures B.5, the amplification curves for 3all three harmonic components, F = 0, F = 1, and F = 2, are virtually identical
regardless of the transverse grid increment. IFigures B.6 through B.9 display the amplitude and phase distributions of the
streamwise velocity for F = 0, F = 1, and F = 2. As was observed for the
amplification curves, the amplitude and phase distributions vary more with the 5streamwise grid increment than with the transverse grid increment. In particular,
the second harmonic component displays the most sensitivity with respect to the 3streamwise grid increment.
From the results presented in Figures B.4 through B.9, it would appear that
for M = 256, the transverse grid increment is sufficiently small so that the mean 5component, the fundamental component, and the second harmonic component are
adequately represented. However, for the streamwise grid increment corresponding 5to N = 1024, the results are not quite as conclusive. It appears that the mean £component and the fundamental component are adequately represented with this
grid increment. However, it is not quite as clear whether the second harmonic is 3adequately resolved as it displays a significant variation with the streamwise grid
increment. £I£
277
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