DIRECT NUMERICAL SIMULATION OF A SPATIALLY EVOLVING SUPERSONICTRANSITIONAL/TURBULENT BOUNDARY LAYER

Hiroshi MaekawaGraduate School of Engineering,

Hiroshima University1-4-1 Kagamiyama, Higashi-Hiroshima-shi, 739-8527 Hiroshima, Japan

maekawa@mec.hiroshima-u.ac.jp

Daisuke WatanabeGraduate School of Engineering,

Hiroshima University1-4-1 Kagamiyama, Higashi-Hiroshima-shi, 739-8527 Hiroshima, Japan

watanabe@hiroshima-u.ac.jp

Kougen OzakiGraduate School of Engineering,

Hiroshima University1-4-1 Kagamiyama, Higashi-Hiroshima-shi, 739-8527 Hiroshima, Japan

mkgnoz@hiroshima-u.ac.jp

Hajime TakamiEnvironmental Engineering Division,Railway Technical Research Institute

2-8-38 Hikari-cho, Kokubunji-shi, 185-8540 Tokyo, Japantakami@rtri.or.jp

ABSTRACT

Transition mechanism in supersonic °at plate bound-ary layers with isothermal walls up to M = 3 is studiedusing spatially developing DNS. The compressible Navier-Stokes equations are numerically solved using high-orderupwind-biased compact schemes for spatial derivatives (seeDeng, Maekawa, Shen,1996). Navier-Stokes characteristicboundary conditions are used in the streamwise and ver-tical directions, and periodic boundary conditions in thespanwise direction. Random disturbances of isotropic homo-geneous turbulence with the intensity of 4% relative to thefree-stream velocity are introduced at the inlet of a compu-tational box for M = 2:5. The transition scenario consists ofthe following four stages: ¯rst, generation of staggered vorti-cal structures upstream giving rise to streaks, which is ratheroblique and sometimes bifurcated, secondly, elongation ofstreaks in the streamwise direction, thirdly, oscillation ofstreaks with a couple of oblique streamwise vortices andan incipient spot, and ¯nally breakdown of the °ow due togrowth of the spot. The three latter phases in this scenariohave been observed in the low-speed boundary layer experi-ments subjected to high levels of free-stream turbulence byMastubara and Alfredsson (2001). The longitudinal growthof the streaks is closely related to transient growth theory.The distribution of the streamwise velocity °uctuation showsto be similar to the experimental result of the incompress-ible transitional boundary layer, when it is scaled by thedisplacement thickness and normalized with their respectivemaximum. At higher amplitude case of 5% for M = 3, stag-

gered structures are also observed but spanwise large-scalespots with rather complicated vortical structures are gen-erated downstream soon. The shape factor starts around2.6 and decreases to a value of 1.42 in the turbulent region.Reynolds number based on the momentum thickness reaches1400 in the turbulent region. The streamwise velocity °uctu-ations in the turbulent regime show a good agreement withthe experimental result by Konrad (1993) for M = 2:9 tur-bulent boundary layers. Large-scale motions observed in theturbulent regions exhibit similarities to those that have beenfound in incompressible boundary layers. Turbulent statis-tics show that the spanwise scale of the large-scale structureis about 440 wall units, which is consistent with recent PIVmeasurements for the M = 2 turbulent boundary layer byGanapathisubramani et al. (2006). As Adrian (2000) andothers hypothesized that the large-scale coherence in theincompressible turbulent boundary layers is a result of indi-vidual hairpin vortices convecting as groups, the numericalresults show that a few hairpin vortices generated on thelifted-up low-speed streak convect in the supersonic turbu-lent boundary layer. As stated for the turbulent spots in thetransitional region, several hairpin vortices are generated allat once when the large spots grow and merge each other inthe transitional boundary layer. This fact is consistent withthe experimental observations of the large-scale coherencein the supersonic/incompressible turbulent boundary layers.In the simulation the skin friction coe±cient for M = 3 isfound to show a good agreement with experimental resultsfor 2:2 < M < 2:8 compiled by Coles (1954). The simula-

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tion results for M = 2:2, 2:5 and 3 indicate that transitionis delayed as Mach number increases due to the near walldensity gradient.

INTRODUCTION

Over the last two decades, considerable progress hasbeen made in studying incompressible boundary layer °owsthrough experiments, theoretical analyses and quasi-periodicor spatial numerical simulations leading to a fairly compre-hensive picture of the primary stage of transition scenariosand a characterization of structures of inhomogeneous tur-bulence and its properties. Numerical simulations have pro-vided reliable databases necessary for the development ofsuitable turbulence models.

In contrast, the progress in the compressible regime hasnot been as large due to intrinsic di±culties both in the ex-periments and in the numerical simulations. The experimen-tal measurements are limited to basic turbulence quantitiesand by the spatial resolution near the wall. The simulationshave been hampered by low Reynolds number. The un-derstanding of the primary stage of transition scenarios forsupersonic boundary layers is even less pronounced due tothe experimental and numerical di±culties. For transitionalsimulations, a spatial evolving simulation is essential. How-ever, careful treatment of boundary conditions is required sothat acoustic waves are not re°ected back into the domainproducing spurious results. Recently, a few direct numer-ical simulations of compressible turbulent boundary layershave been performed. Rai et al. and Gatski and Erlebackerperformed a direct numerical simulation of a spatial evolv-ing adiabatic °at plate boundary layer °ow at supersonicMach number (M1 = 2:25) that used a fourth-order ac-curate in space upwind-biased ¯nite di®erence technique.More recently, Pirozzoli et al. simulated spatial evolvingboundary layer at supersonic Mach number using weighted-ENO reconstruction of the characteristics inviscid °uxes.They introduced a region of blowing and suction to inducelaminar-turbulent transition. The emphasis of their study isto assess the validity of Morkovin's hypothesis and Reynoldsanalogy, and the controlling mechanism for turbulence. Thenumerical results at Mach number of 2.25 show that the es-sential dynamics of the adiabatic turbulent boundary layer°ow resembles closely the incompressible pattern.

In this study, a spatial DNS of a supersonic, isothermal°at plate boundary layer °ow at 2:2 < M1 < 3 is analyzed.E®ects of both transition and boundary layer growth requirethe application of fully spatial formulation in a direct numer-ical simulation without appealing to either slow growth orextended temporal simplifying assumptions. The emphasisof this study is to assess the primary stage of transitionalscenarios for supersonic boundary layer and the later stageof streak breakdown, and ¯nally turbulent boundary layerfor supersonic °ow up to M1 = 3. Issues associated withthe e®ects of freestream Mach number on the boundary layer°ow with isothermal walls are addressed in this study, suchas: (i) the streamwise vortices and streak formation; (ii) thestreak breakdown and the turbulent spot; (iii) the turbulentspot amalgamation; (iv) the scaling laws of the turbulentboundary layer °uctuation statistics.

COMPUTATIONAL METHOD

To simulate and analyze in detail full con¯gurations isnot currently feasible, su±cient progress has been made toanalyze the dynamics of simpler building block °ows that

provide useful insights into the underlying dynamics of fullcon¯guration compressible °ows. One such °ow is the spa-tially evolving °at plate supersonic boundary layer. In thepresent simulations, NSCBC (Navier Stokes characteristicboundary conditions) are used in the streamwise (x) andvertical (y) directions, whereas the periodic boundary con-ditions are employed in the spanwise (z) directions. ANo-slip isothermal wall and non-re°ecting boundary condi-tions are imposed at y=0 and upper far ¯eld. The lengthof the computational domain in the spanwise directions isdetermined from the previous turbulent boundary layer sim-ulations (Guarini et al. 2001) and linear stability results.Linear stability analysis indicates that wall cooling stabi-lize the ¯rst-mode waves for 2:2 < M1 < 3 (see Mack1984), where no dominant second mode destabilized by cool-ing boundary layer. The three-dimensional ¯rst-modes aredominant in the supersonic boundary layer 2:2 < M1 < 3.In the DNS of the spatially developing boundary layer,the nondimensional equations governing the conservation ofmass, momentum, and energy for a compressible Newtonian°uid are solved. Note that displace thickness at the inletboundary layer is chosen to the representative length scale.Therefore, as shown in Fig.1, the spanwise size is shown as33±¤in, which is large enough for oblique disturbances. theReynolds number based on the displacement thickness, thefree stream velocity and wall viscosity is 1000 at the in-let. The free stream Mach number is ranging between 2.2and 3. To obtain spatially accurate numerical solutions tothe governing equations, high-order (¯fth-order) upwind bi-ased compact schemes (Deng, Maekawa & Shen, 1996) areemployed to discretize the hyperbolic (Euler) terms, andsixth-order compact scheme (Lele, 1992) for the molecularterms. A forth-order Runge-Kutta scheme is used for timeadvancement. Computational details, such as °ux splittingtechnique, grid stretching and minimum grid sizes in thex; y; z directions, are presented in Maekawa et al (2004).

Figure 1: Sketch of the computational domain and Cartesiancoordinates: Rex indicates running Reynolds number basedon the length from the plate inlet.

RESULTS

Streamwise Vortices and Streak FormationThe temporal approximation is only a crude approxima-

tion of a boundary layer spatially developing, where oneworks in a frame traveling with some velocity such as thegroup velocity or the averaged velocity in the boundarylayer, in which the °ow is homogeneous in the streamwise di-rection. We present DNS results of a spatial boundary layer,initiated upstream by a laminar velocity pro¯le with a cor-responding temperature pro¯le superposed on the average°ow, plus a random forcing generated with isotropic homoge-neous compressible turbulence. Figure 2 shows top views ofthe boundary layer structures. The second invariant Q of the

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velocity gradient tensor is employed to represent the vorticalstructure and the surface plot of instantaneous streamwisevelocities minus the laminar pro¯le for the low-speed streak.Figure 3 shows successive snapshots of top view near theinlet. The oblique (not straight) Q structures are apparent¯rst in the close-up view of (a), where low-speed streaks areobserved on the periphery of Q structures. Further down-stream, as seen in (b) and (c), due to the development of apair of oblique Q structures, hairpin structures are generatedabove the lifted-up low-speed streak. These hairpin headsare successively created on the low-speed streak. As canbe seen in the same ¯gure, a packet of hairpins becomes asigni¯cant structure in the transitional boundary layer. Theside view shows that the hairpin packet convects quickly dueto the high-speed free stream velocity in the boundary layer.

Figure 2: Streamwise vortices and streaks for M = 2:5 andRe = 1000: °ow is forced with 4% disturbances. Windowarea corresponds to ¯gure 3 (c).

Figure 3: Downstream development of streamwise vorticesand streaks: u0 = ¡0:1 (dark gray) and Q = 0:001 (lightgray).

Fourier Analysis of Velocity Fluctuation. Figure 4 showsthe results of Fourier analysis for streamwise velocity °uc-tuations at various downstream locations, which may cor-respond to low or high-speed streaks near the wall. Atx = 66±¤in, a °uctuation peak is observed at wave number¯ of 1.8 and at about displacement thickness height. Thewavenumber of 1.8 corresponds to the width of the spanwisedirection, such as 2¼=¯ £ ±¤in in the simulation. This factindicates that the spanwise scale of the streaks approachesthe boundary layer thickness, which shows a good agreementwith the experiments in the incompressible boundary layersubjected to free stream turbulence. Further downstreamlocations at x = 380±¤in, a peak is seen at wave numberof 1:1 » 1:3 which is the same as or somewhat less thanthe observation at x = 66±¤in, because boundary displace-ment thickness at x = 380±¤in is 1.5 times large as that atx = 66±¤in. Note that turbulent spots are developing atx = 380±¤in

(a)

(b)

Figure 4: Wavenumber spectra at di®erent downstream lo-cations for M = 2:5 and Re = 1000: °ow is forced with 4%disturbances.

Figure 5: Comparison of the computational RMS streamwisevelocity pro¯le with experimental data (symbols) of Matsub-ara and Alfredsson (2001).

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(a)

(b)

(c)

Figure 6: Beginning of turbulent spot due to a pair of obliquevortices at the same normal location: (a) Top view of stream-wise vortices and streaks; (b) Low- (minus) and high-speed(plus) streaks and velocity vector in the y-z plane at the lo-cation shown in (a); (c) Q structures (plus), strain dominantstructure (minus) and velocity vector in the y-z plane shownat the location shown in (a).

Streamwise Velocity Fluctuation. Streamwise velocity°uctuations are measured at various downstream locations.Figure 5 shows the distribution of streamwise velocity °uctu-ation scaled by the local displacement thickness and normal-ized with their respective maximum. Figure 5 also presentsthe experimental results of the transitional incompressibleboundary layer measured by Matsubara and Alfredssson,(2001). The numerical pro¯les are similar to the experimen-tal data, but not self-similar as the experimental pro¯les.The most pro¯le peak locations slightly move to the wallcompared to the experimental results. Scaling by the lo-cal displacement thickness would be modi¯ed for supersonicboundary layers.

Turbulent SpotAs shown in ¯gures 3 (c),(d),(e) and (f) and ¯gure 6, the

primary low-speed streak deforms due to the development ofhairpin vortices and the secondary streak appears near bythe primary one. The lifted-up primary streak with hairpinpackets leads to strongly retarded streamwise momentum.Therefore, this con¯guration leads to streak breakdown, asshown in ¯gure 7. This fact indicates that the local ve-locity pro¯le plays an important role to make a turbulentspot. The large-scale motion observed in a turbulent spotoriginates partly from the streak con¯guration with hairpinvortices developing in the streamwise direction.

(a)

(b)

(c)

Figure 7: Structure of turbulent spot: Q = 0:001 andu0 = ¡0:1: (a) Top view of streamwise vortices and streaks;(b) Low- (minus) and high-speed (plus)streaks and veloc-ity vector in the y-z plane at the location shown in (a); (c)Q structures (plus), strain dominant structure (minus) andvelocity vector in the y-z plane at the location shown in (a).

Successive snap shots indicate that hairpin packets ob-served downstream are developing in the streamwise direc-tion ¯rst. Then a cluster of hairpin packets grows in thespanwise direction. As can be seen in the top views of theincompressible boundary layer °ow visualization, structuresdevelops less dependently on the other packet structure ¯rstleading to the large-scale structure in the streamwise x, ver-tical y and spanwise z directions. This mechanism leadsto the inclined hierarchic structure in turbulent spots inthe supersonic boundary, hairpin packets in the outer re-gion and inner mushroom-like density visualized structureswhich is still similar to the experimental observations in highReynolds number turbulent boundary layers (see Matsubaraand Alfredsson 2001). In the experiment of a supersonic tur-bulent boundary layer by Ganapathisubramani et al. (2006),two point correlations show large-scale structures increasingtrend in the streamwise length scale with normal location.The spanwise scale of the uniform-velocity strips increaseswith increasing wall-normal distance. The essential featuresfound in the turbulent spots and the marginal process witheach other are very close to the experimental ¯nding of thesupersonic boundary layer.

Downstream Turbulence Statistics

Mean Flow Evolution. For M = 3, the °ow is forcedwith 5% intensity disturbances. Turbulent spots are devel-

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oping, merging each other downstream, and the turbulentboundary layer are observed in ¯gure 8. Figures 9 (a) and(b) show evolutions of the shape factor of the mean veloc-ity pro¯le and the boundary layer displacement thickness,respectively. This ¯gure indicates that transition is delayedas Mach number increases with the same level of inlet ran-dom disturbances. A comparison of the numerical result forM = 2:5 with the disturbance intensity of 4% (shown indashed line) and 4.5% (dotted line) shows that there is athreshold value to induce transition to turbulence. Further-more, a comparison of the numerical results for M = 2:2(solid line) and those for M = 2:5 (dashed line) with thesame disturbance intensity of 4% indicates that transition isdelayed as Mach number increases.

Figure 8: Bird-eye view of boundary layer for M = 3: °owis forced with 5% disturbances; Q is used to visualize thetransitional/turbulent °ow.

(a)

(b)

Figure 9: Evolution of mean velocity pro¯le:(a) Shape fac-tor;(b) Boundary layer displacement thickness.

The skin friction coe±cient is de¯ned as

Cf = 2³u¿U1

´2 ¹½w¹½1

: (1)

Figure 10 depicts the skin friction coe±cient for M = 2:5and 3:0 °ows as a function of the Reynolds number basedon the boundary layer momentum thickness. Also includedin ¯gure 10 are the experimental data of Coles (1954) andthe skin friction correlation given in Bardina (1997), as wellas the laminar °ow skin friction coe±cient for an isother-mal wall condition. There are very few experimental studiesat the low Reynolds number of the simulation. However,the turbulent simulations compare favorably with the ex-perimental results.

Figure 10: Comparison of the computational skin frictioncoe±cient with experimental data of Coles (1954).

RMS Velocity. Figure 11 shows RMS data of the stream-wise velocity °uctuation forM = 3 at a downstream locationwith the experimental results measured by Konrad (1993)for the M = 2:9 turbulent boundary layer. He used a nor-mal hot-wire close to the wall instead of crossed wire probes,where the e®ect of low Mach number is serious for crossedwires. Therefore, in the experiments good collapse is ob-tained for the streamwise component but the experimentsare inconclusive with respect to the collapse of the othercomponents. There is an additional point which need tobe mentioned in connection with ¯gure 11. Unlike adia-batic wall boundary layers, Reynolds number based on localviscosity in the boundary layer simulation with an isother-mal wall does not vary much across the boundary layer (lessthan 25% increase). The experimental Reynolds number ofthe turbulent boundary layer with an adiabatic wall is muchhigher than the simulation result, so that the viscosity e®ecton the velocity measurement for the probe point near thewall shown in ¯gure 11 may be not so serious as the lowReynolds number experiment.

CONCLUSIONS

Results from three-dimensional spatial direct numericalsimulations of an isothermal supersonic boundary layer pro-vide new physical insights into three-dimensional evolutionsof the streamwise vortices and the streak, the turbulent spot,and the characteristic features of the supersonic turbulentboundary layer. Forcing with isotropic homogeneous turbu-lence at the laminar inlet °ow yields large-scale motions thatincrease in the streamwise length scale with normal loca-

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Figure 11: Comparison of the computational RMS stream-wise velocity pro¯le with experimental data of Konrad(1993).

tion in the transitional and turbulent boundary layer. Thisfeature largely originates from the transitional structure ofstreak breakdown, in which hairpin packets and lifted-uplow speed streak play an important role. A comparison ofthe numerical results for di®erent Mach numbers indicatesthat transition is delayed as Mach number increases mainlydue to the near wall density gradient. The numerical sta-tistical results, such as the skin friction coe±cient and RMSvelocity, show a good agreement with the experiments.

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