AIAA 03–3726Preliminary DNS Database ofHypersonic Turbulent BoundaryLayersM. Pino MartinDepartment of Mechanical and Aerospace EngineeringPrinceton University, Princeton, NJ

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33rd AIAA Fluid Dynamics Conference and Exhibit23-26 June 2003, Orlando, Florida

AIAA 2003-3726

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Preliminary DNS Database of Hypersonic

Turbulent Boundary Layers

M. Pino Martin

Department of Mechanical and Aerospace Engineering

Princeton University, Princeton, NJ

We are in the process of building a direct numerical simulation database of supersonic

and hypersonic boundary layers that will be available to the scientific community. The

data obtained from direct numerical simulation of Mach 4 turbulent boundary layers

at Reθ up to 9480 are presented, including different wall temperature conditions. The

procedures for initializing the turbulent fields and determining the grid resolution are

described. Under the proper transformation, the turbulent kinetic energy shows that

the isothermal wall reduces the magnitude of the turbulence production and dissipation

mechanisms. The presence of shock waves is illustrated. These data are currently being

used for the calibration of compressible LES1 and RANS2 turbulence models.

Introduction

The study of high-speed boundary layers is impor-tant in advancing supersonic and hypersonic flighttechnology. In a high-speed boundary layer, the ki-netic energy is substantial and the dissipation dueto the presence of the wall leads to large increasesin the temperature. Therefore, a high-speed bound-ary layer differs from an incompressible one in thatthe temperature gradients are significant. Since thepressure remains nearly constant across the bound-ary layer, the density decreases where the temperatureincreases. Thus, to accommodate for an equivalentmass-flux, a supersonic boundary layer must growfaster than a subsonic one. The extra growth modifiesthe freestream, and the interaction between the invis-cid freestream and the viscous boundary layer affectsthe wall-pressure distribution, the skin friction and theheat transfer. Furthermore, the high temperature inthe boundary layer leads to air reactions. To improveour understanding of the flow physics and to calibrateturbulence models, we need accurate experimental andcomputational databases of high-speed flows.

Direct numerical simulations can provide a vastamount of accurate data that can be used to ana-lyze turbulent boundary layers at high Mach num-bers. Based on a better understanding of the realflow physics and using DNS data, accurate turbulencemodels for high-speed flows can be developed, cali-brated and tested. Recent advances show that buildinga detailed DNS database of fundamental flows at su-personic and hypersonic conditions is attainable. Forexample, Guarini et al.3 perform a DNS of a Mach2.5 boundary layer at Reθ=1577; Adams4 performs a

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copyright owner.

M Reθ Twall Status3 3000 isothermal in progress4 9480 adiabatic completed4 7225 isothermal in progress8 3600 isothermal in progress

Table 1 Flow conditions and simulation status forthe turbulent boundary layer database.

DNS of the turbulent boundary layer over a compres-sion ramp at Mach 3 and Reθ=1685; and Martin &Candler5, 6 perform DNS of reacting boundary layersat Mach 4. One of the achievements of this DNS workis the ability to accurately simulate turbulent flows athigh Mach numbers while reproducing complex flowphysics, shock waves and chemical non-equilibriumeffects, permitting the study of turbulence under dif-ferent flow conditions.

Studying physical phenomena via numerical and ex-perimental databases requires controlled inflow condi-tions. This presents a challenge for numerical sim-ulations since turbulent flows are highly non-linearand initialization procedures and simulation transientsmake the final flow conditions difficult to control.

In this paper, we present our progress in building aDNS database of hypersonic turbulent boundary lay-ers. The governing equations and numerical methodare first introduced. Then, we present an initializa-tion procedure to minimize transients while matchingthe desired skin friction and Reynolds number. TheMach number, Reynolds number based on momentumthickness, and the wall temperature condition for theturbulent boundary layers are given in Table 1. Exper-imental data for the Mach 3 case are being gathered atthe Gas Dynamics Laboratory in Princeton University.The experimental data for the Mach 8 boundary layer

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American Institute of Aeronautics and Astronautics Paper 03–3726

M∞ ρδ Uδ θ × 104 µδ × 104 ρw uτ δ × 103 µw × 104

adiabatic 4 0.48 5684 3.66 1.06 0.132 270.25 6.5 2.90isothermal 4 0.48 5798 2.70 1.04 0.507 147.00 3.42 1.01adiabatic 0.3 1 104 1.70 0.18 1.0 4.69 1.51 0.18

Table 2 Dimensional boundary layer edge and wall parameters in SI units for the Mach 4 simulations.

is compiled in Baumgartner7 and at present no exper-imental data is available for the Mach 4 conditions.Here, we describe the Mach 4 DNS data including theevolution from the initial condition to the realistic tur-bulent field and the estimated data accuracy. We thencomment on the status of the more stringent Mach 8DNS data.

Conserved equations for DNS

The equations describing the unsteady motion of aperfect gas flow are given by the mass, momentum,and total energy conservation equations

∂ρ

∂t+

∂

∂xj

(ρuj) = 0,

∂ρui

∂t+

∂

∂xj

(ρuiuj + pδij − σij) = 0,

∂ρe

∂t+

∂

∂xj

((ρe + p

)uj − uiσij + qj

)= 0,

where ρ is the density; uj is the velocity in the j di-rection; p is the pressure; σij is the shear stress tensorgiven by a linear stress-strain relationship

σij = 2µSij −2

3µδijSkk,

where Sij = 12(∂ui/∂xj + ∂uj/∂xi) is the strain rate

tensor, and µ is the temperature dependent viscosityand is computed using a power law; qj is the heat fluxdue to temperature gradients

qj = −κ∂T

∂xj

,

where κ is the temperature dependent thermal conduc-tivity; and e is the total energy per unit mass givenby

e = cvT + 12uiui,

where cv is the assumed constant specific heat at con-stant volume.

Numerical Method

The numerical method combines a weighted essen-tially non-oscillatory (WENO) scheme for the invis-cid fluxes with an implicit time advancement tech-nique. The third-order accurate, high-bandwidth,WENO scheme was designed for low dissipation and

high bandwidth8 and provides shock-capturing, whichis necessary at the Mach numbers that we consider.The time advancement technique is based on the Data-Parallel Lower-Upper (DPLR) relaxation method ofCandler et al.

9 and was extended to second-order ac-curacy by Olejniczak & Candler.10 The derivativesrequired for the viscous terms are evaluated using 4th-order central differences. We use supersonic boundaryconditions in the freestream and periodic boundaryconditions in the streamwise and spanwise directions.Thus, the boundary layer is develops temporally. Thevalidity of periodic boundary conditions is briefly dis-cussed in this paper. A more detailed discussion canbe found in Xu & Martin.11

Flow Conditions

The freestream conditions are M∞ = 4, T∞ = 5000K, and ρ∞ = 0.5 kg/m3. These conditions are rep-resentative of the boundary layer on a 26◦ wedge ata free-stream Mach number of 20 and 20 km altitude.The temperature in these simulations is chosen to behigh so that the same simulations could be used tostudy the effect of chemical reactions.5,6 The perfectgas assumption is used in the present study, which isintended to illustrate our initialization procedure andthe accuracy of our simulations.

The Reynolds numbers based on the momentumthicknesses are 9480 and 7225 for the adiabatic andisothermal simulations, respectively. These Reynoldsnumbers correspond to a freestream velocity of 5708m/s. The corresponding turbulent Reynolds numbersor Karman numbers, Reτ = ρwuτδ/µw where δ is theboundary layer thickness, . The values of Reτ are 800and 2450 for the adiabatic and isothermal simulations.For both cases, uτ is two orders of magnitude largerthan that of typical subsonic boundary layers. For theisothermal simulation the relatively higher density andlower viscosity values at the wall lead to a larger tur-bulent Reynolds number. The dimensional values thatwe use to compute these Reynolds numbers are givenin Table2. For comparison, the values for a Mach 0.3turbulent boundary layer computed in Martin et al.

12

are also provided. The Mach 0.3 DNS data compareswell with experimental data.

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American Institute of Aeronautics and Astronautics Paper 03–3726

Tw δ+ Lx/δ Ly/δ Lz/δ ∆x+ ∆y+ Nx Ny Nz

adiabatic 800 6.5 1.6 11 13.5 5.1 384 256 128isothermal 2450 3.7 0.9 14 24 9.2 384 256 150

Table 3 Grid resolution and domain size for the direct numerical simulations.

Resolution Requirements and

Initialization

The mean velocity, density, and temperature pro-files, and the inner parameters

uτ =

õw

ρw

(∂u

∂z

)

w

, zτ =µw

ρwuτ

,

at the desired Mach number are being obtained froma Baldwin-Lomax Reynolds-averaged Navier-Stokessimulation13 for the Mach 3 and Mach 8 cases. TheMach 4 initial mean profiles12 for the present simula-tions were obtained from a k-ε RANS code and theydid not match the law of the wall exactly. In turn,the simulations led to longer simulation transients toreach a statistically stationary state than those beingobtained for the Mach 3 and Mach 8 cases. Notice thatz is the normal direction. The fluctuating velocityfield is obtained by normalizing the velocity fluctu-ations from the incompressible Mach 0.3 DNS12 bythe ratio of the inner parameters at the high Machnumber to that at M =0.3. We have yet to deter-mine the effect of using a Morkovin scaling for thefluctuating velocity. The turbulent field is mappedonto a computational domain that is also normalizedin wall units. Thus, the initial turbulence structuresand energy spectra resemble those of a realistic turbu-lent boundary layer. Our preliminary experience withthe Mach 8 turbulent boundary layer simulations isthat this initialization procedure may allow for shortsimulation transients provided that the mean turbu-lent profiles are accurate. The initial fluctuations inthe thermodynamic variables are estimated using thestrong Reynolds analogy.14

The DNS computational domain size and structuredgrid resolution required for the simulations is deter-mined based on the characteristic large length scale,δ; the characteristic small, near-wall length scale, zτ ;and grid convergence studies. The computational do-main must be large enough to contain a good sampleof the large scales. On the other hand, the grid res-olution must be fine enough to resolve the near wallstructures. The first requirement gives the size of thecomputational domain, whereas the later one gives anestimate on the grid resolution in wall units. Thus,increasing the ratio of the large to small scale δ/zτ in-creases the required number of grid points. Ultimately,grid convergence studies or comparisons with experi-

mental or semi-empirical data will determine the finalresolution.

Let us consider the computational size and resolu-tion estimates for the present simulations. From Table2, the ratio of the large to small scales i.e. the Karmannumber δ+ = δ/zτ , or turbulent Reynolds numberReτ is 800 and 2450 for the adiabatic and isothermalsimulations, respectively. Thus, the isothermal simula-tion will require a reduced number of grid points. Toalleviate the resolution requirements for the isother-mal simulation we use a shorter computational domainin both streamwise and spanwise directions, yet longenough to prevent correlation of the large scales. Also,if we were to use the same number of grid points perlarge structure for both simulations, the resolutionwould be coarser in wall units for the isothermal case.We apply this criteria, and we use slightly coarser res-olution in wall units for the isothermal case. Theseconstrains are relaxed for the adiabatic case. For bothcases, the wall-normal grid is stretched. The first pointaway from the wall is located at z+ = 0.13, and thereare 21 grid points below z+ = 10. The final do-main size and grid resolution are given in Table 3.These correspond to meshes with (384 × 256 × 128)and (384 × 256 × 150) grid points in the streamwise,spanwise and wall-normal directions for the adiabaticand isothermal simulations, respectively. The consis-tency of the computational size is considered duringthe simulation by ensuring that the large structures inthe inflow are not correlated with the large structuresin the middle of the computational domain. The ade-quacy of the resolution in wall units is assessed via gridresolution studies and comparison with semi-empiricaldata.

By examining the two-point correlations, it wasfound that for the adiabatic simulation the lengths of6.5δ and 1.6δ in the streamwise and spanwise direc-tions, respectively, are adequate. For the isothermalsimulation, 3.7δ and 0.9δ are sufficient. In the wall-normal direction, the height of the domain is set sothat acoustic disturbances originating at the upperboundary do not interact with the boundary layer nearthe wall. The resulting grid spacing and resolution ineach direction are given in Table 3. Note that althoughthe resolution in wall units is coarser for the isother-mal simulation, the resolution in outer units is nearlytwice as large as it is for the adiabatic case.

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American Institute of Aeronautics and Astronautics Paper 03–3726

z+

⟨u+

⟩ VD

10-1 100 101 102 103 1040

5

10

15

20

25

30

35

40

adiabaticisothermal2.44 log z+ + 5.5z+

Fig. 1 Van-Driest transformed velocity profilesscaled on inner variables for the Mach 4 boundarylayers.

Accuracy of the Mach 4 Turbulent

Boundary Layer Data

The van Driest transformed velocity for the adia-batic and isothermal simulations nearly collapse upto the wake region as it is shown in Fig. 1. For theadiabatic simulation the intercept of the logarithmiclayer is C = 5.5 and gives a value of uτ that is un-derpredicted roughly by 8% when compared with theexperimental value,15 C = 5.1. For the isothermalsimulation, the constant for the logarithmic region isC = 6.6. This simulation should be improved by mod-ifying the resolution slightly in the near wall region.

Hopkins and Inouye16 present a survey comparingdifferent theories to predict the turbulent skin frictionin supersonic and hypersonic boundary layers. Theyfind that the van Driest II17 theory gives the best pre-diction. For a given Reθ, we compute Cf from theKarman-Schoenherr equation given by

1

FcCf

= 17.08[log(FθReθ)]2 + 25.11 log(FθReθ) + 6.012

where Fc and Fθ are van Driest II transformation func-tions computed as

Fc =0.2rM2

e

(sin−1 α + sin−1 β)2,

Fθ =µe

µw

,

where r is the recovery factor and α and β are calcu-lated by

α =2A2 − B√4A2 + B2

,

β =B√

4A2 + B2,

z+

⟨ρ⟩u

′2 rms

/ρw

u τ2

100 101 102 103 1040123456789

10

adiabaticisothermal

Fig. 2 Longitudinal turbulence intensity for theMach 4 simulations.

with

A =

√0.2rM2

e

F,

B =1 + 0.2rM2

e − F

F,

F =Tw

Te

.

For the adiabatic simulation, the predictions given bythe DNS and the Van Driest II theory are within 2%.Hopkins et al.

18 find that the Van Driest II theorypredicts the experimental data within about 10% er-ror for Tw/Taw > 0.3. For the isothermal simulationTw/Taw is about 0.26.

Figure 2 shows the longitudinal turbulence intensi-ties using Morkovin scaling. The peak value for theisothermal simulation is slightly overestimated proba-bly due to the underprediction on uτ . The adiabaticsimulation agrees well with the incompressible predic-tions. The turbulent kinetic energy budget for theadiabatic simulation is shown in Fig. 3. We observethat the turbulent kinetic energy transfer mechanismsresemble those of a Mach 2.5 adiabatic DNS.3 Thesefindings allow us to be confident in the quality of thedata.

Figure 4 shows the two-dimensional energy spectrafor the more challenging isothermal simulation. Thespectra have been obtained by averaging in time andin planes parallel to the wall for z+ = 975, 145, and7, respectively. The grid resolution can be assessedusing the value of kmaxη, where kmax is the maximumwave number and η is the Kolmogorov length scale.We find that kmaxη varies from 0.3 near the wall to1.2 in the wake. These values have been found ade-quate in previous numerical simulations.3, 19 Figure 5shows the two-point correlations for the same simula-tion. There correlations converge to small values inthe spanwise direction near the edge of the bound-ary layer. Thus, the domain size is marginally largeenough to enclose an adequate sample of the large

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American Institute of Aeronautics and Astronautics Paper 03–3726

z+

Bud

gett

erm

s

0 10 20 30 40-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Dissipation

Production

Pressurediffusion

Viscousdiffusion

Turbulentdiffusion

Mach 4

Guarini et al

Fig. 3 Turbulent kinetic energy budget for theadiabatic case.

k δ100 101 102 103

10-14

10-12

10-10

10-8

10-6

10-4

10-2

(a)kδ-5/3

k δ100 101 102 103

(b)kδ-5/3

k δ100 101 102 103

(c)kδ-5/3

Fig. 4 Two-dimensional energy spectra at (a) z+=

975, (b) z+= 145, and (c) z+

= 7 for the isothermalsimulation. streamwise, spanwise and

wall-normal velocity component.

turbulent structures in the wake region. In the log-arithmic and near wall regions, the correlations arenegligible. Note that the small correlation that ap-pears near the wall in the streamwise direction is ex-pected. Both computations20 and experiments21 haveshown that the near-wall streaky structures are morecoherent in the streamwise direction for high Machnumbers. The same degree of correlation is found forthe adiabatic simulation, where the domain size is 1.7times longer than that of the isothermal case in thestreamwise direction.

Periodicity and statistical convergence

The amount of energy that is present in a super-sonic boundary layer and the small computational do-mains that are used in the present simulations, makeit possible to use periodic boundary conditions in thestreamwise direction. A time developing boundarylayer simulation is valid provided that (i) the flow canbe considered quasi-steady, i.e. the flow adjusts to itslocal (in time) conditions much faster than the bound-ary layer thickness changes, and (ii) for the purposesof gathering statistics, the time sampling is shorterthan the time scale for boundary layer growth. A flow

y / δ0 0.1 0.2 0.3 0.4

-0.4

0.0

0.4

0.8 (b)

x / δ0 0.5 1 1.5 2

-0.4

0.0

0.4

0.8

u′v′w′

(a)

x+0 1000 2000 3000 4000 5000

-0.4

0.0

0.4

0.8(e)

y+0 500 1000

-0.4

0.0

0.4

0.8 (f)

x+0 1500 3000 4500 6000 7500 9000

-0.4

0.0

0.4

0.8(e)

y+0 1000 2000

-0.4

0.0

0.4

0.8 (f)

Fig. 5 Two-point correlations at (a)-(b) z+= 975,

(c)-(d) z+= 145, and (e)-(f) z+

= 7 for the isother-mal simulation. streamwise, spanwiseand wall-normal velocity component.

that satisfies these conditions evolves slowly and canbe viewed as a good approximation of a static stationof a boundary layer.

The growth time, adjusting time, and sampling timecan be estimated as

tgrowth =

(1

δ

dδ

dt

)−1

, tΛ =δ

Ue

, tsample =δ

uτ

,

respectively. Where δ and uτ are the averaged bound-ary layer thickness and wall friction velocity and Ue isthe velocity at the edge of the boundary layer. Forthe simulations, tΛ is at least two orders of magnitudesmaller than tgrowth, and tsample is less than tgrowth.Thus, the temporal development of the boundary layeris negligible during an appropriate data collectiontime. These premises have been further corroboratedin Xu and Martin11 by comparing temporal and ex-tended temporal simulation results for the adiabaticMach 4 simulation.

It should be noticed that it is necessary to initializethe boundary layer to a nearly equilibrium state for re-alization of these conditions. If the initial flow field isfar from equilibrium, a long temporal transient devel-ops before the flow settles down to a quasi-stationarystate. In this scenario, the final skin friction and the

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American Institute of Aeronautics and Astronautics Paper 03–3726

t uτο / δ

u τ/u

το

0 0.5 1 1.5 20.6

0.8

1.0

1.2

adiabaticisothermal

Fig. 6 Temporal evolution of the friction velocity.

Reynolds number are hard to control.Figure 6 shows the temporal evolution of the friction

velocity uτ normalized by its initial value. The timeis made non-dimensional by uτo/δ. For the adiabaticsimulation, uτ reaches a stationary state in about 1.3periods of time. For the isothermal simulation, uτ

reaches a stationary value after about 0.3tuτo/δ. Allstatistics are computed in time and space during oneuτo/δ period of time after the simulations reach thestatistically stationary state.

The statistics were gathered in space and timein over fifty large-eddy turnover characteristic times,δ/Ue, and included thirty three computational boxes.

Shockwaves and Shocklets

Experimental data21–23 have shown that the turbu-lence structure in a boundary layer is very similar forincompressible and compressible flows. As in the Mach2.5 DNS data of Guarini et al.,3 the Mach 4 turbulentstatistics are very similar to those found in incompress-ible boundary layers. In this section we focus on thecompressibility effects and the differences between theadiabatic and isothermal simulations, for which themaximum fluctuating Mach number (M−〈M〉) is 0.35and 0.30, respectively.

An index of the compressibility is given by

χ =〈 |∇ · u′|2 〉〈 |∇ × u

′|2 〉 ,

which represents the amount of turbulent kinetic en-ergy in the compressible modes of motion relative tothe amount of incompressible turbulent kinetic energy.Figure 7a shows that in the near wall region, the ratioof compressible to incompressible energy is as large as0.13 and 0.15 for the adiabatic and isothermal simu-lations, respectively. Figure 7b shows a large increasein the value of χ near the boundary layer edge. Thisresult indicates the possible presence of shocks in theouter part of the boundary layer. since a jump in χ in-dicates a jump in the compressible dissipation ( 4

3∇·u′).

The predominant mechanism for the generation ofshock waves in these flows occurs when slow moving,

z+

χ

0 100 200 300 400 500 6000.00

0.04

0.08

0.12

0.16

adiabaticisothermal

(a)

z+

χ

101 102 1030.0

0.5

1.0

1.5

2.0

2.5

3.0

adiabaticisothermal

(b)

Fig. 7 Relative compressible to incompressibleenergy ratio (a) near the wall and (b) near theboundary layer edge.

x / δ

z/δ

3.0 3.5 4.0 4.5 5.0 5.5

0.4

0.6

0.8

1.0

1.2

1.4

(a)

y/δ=0.6

y / δ0.0 0.5 1.0 1.5

0.450.420.390.360.330.290.260.230.200.170.14

(b)

x/δ=4.2ρ (kg/m3)

Fig. 8 Density contours on the (a) streamwise-wall-normal and (b) spanwise-wall-normal plane forthe adiabatic direct numerical simulation.

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American Institute of Aeronautics and Astronautics Paper 03–3726

x / δ

z/δ

2 2.5 3 3.5 40

0.5

1

(a)

x / δ

z/δ

2 2.5 3 3.5 40

0.5

1

(b)

Fig. 9 Contours of (a) pressure and (b) divergenceof the velocity on a streamwise-wall-normal planefor the adiabatic simulation.

near-wall gas is lifted into the high-speed outer flow.Such a bursting event is illustrated in Figure 8, whichshows the density in a streamwise/wall-normal planeand a spanwise/wall-normal plane. Note that the low-speed gas has a significantly higher temperature andlower density than the outer region gas, and thus theburst event is easily identified by the low density re-gions.

Detecting the location of shock waves in this flow isvery difficult. Following the approach of Blaisdell et

al.,24, 25 we search for locations of large compressionand streamwise increase in the static pressure. Wethen sample the data along instantaneous streamlinesupstream and downstream of the candidate shock loca-tion and compute the upstream relative Mach number,Mrel

1 , using

u1 − u2 = a1Mrel1 − a2M

rel2

where the downstream relative Mach number, M rel2 ,

is a known function of M rel1 . The variables u1, u2,

a1, and a2 are pre- and post-shock velocity and speedof sound obtained from the simulation data. We findthat there tends to be very little deflection of the flowacross most of the candidate shock waves, so we usenormal shock relations to obtain M rel

2 . Once we havefound the relative Mach number of the shock, we cancompute the pressure rise across the shock and com-pare it to the DNS data.

Figure 9 plots contours of pressure and divergencein a streamwise/wall-normal plane. The highlighted

z / δ

-Ru′

T′

0 0.2 0.4 0.6 0.8 1-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

adiabaticisothermal

Fig. 10 Fluctuating streamwise velocity and tem-perature correlation coefficient.

region shows the location of a candidate shock wave.Sampling the data at the upstream and downstreamconditions, we find that the shock wave has a relativeMach number of about 1.10. This gives a pressure risethat is consistent with the data, and we conclude thatthere is indeed a shock wave in this region. Again,because of the complicated interaction between thenear-wall gas and the outer flow, the identification ofthe pre- and post-shock conditions is difficult. Thus,it is not possible to calculate the exact shock Machnumber.

We have found shock waves in regions closer to thewall than that illustrated above. For the adiabaticboundary layer, the maximum shock Mach number isabout 1.12. From Figure 9, it is clear that the high-lighted shock wave has a small length scale, with awall-normal dimension of only about 0.03δ. This ischaracteristic of the other shock waves that we haveidentified in the simulation data.

Since the amount of compressible energy is signif-icant in the logarithmic and boundary layer edge re-gions, the transport of turbulent kinetic energy due tosound radiation (acoustic waves) must be addressed.An index of such interaction is the correlation co-efficient for the streamwise-velocity and temperaturefluctuations −Ru′T ′ . Figure 10 plots this correlationcoefficient. For the adiabatic simulation, −Ru′T ′ isalways positive since the mean gradients of velocityand temperature are of opposite sign. For the samesimulation, −Ru′T ′ peaks at 0.9 near the wall. For theisothermal simulation, −Ru′T ′ is -1.0 at the wall wherethe mean temperature and velocity gradients are of thesame sign. For this simulation, the turbulent Reynoldsnumber is high and the boundary layer edge is char-acterized by an energized turbulent wake. Thus, nearthe boundary layer edge where most of the turbulentkinetic energy is in the compressible modes, −Ru′T ′

increases. For both simulations, −Ru′T ′ is near 0.6 for0.3 ≤ z/δ ≤ 0.7, as it is for incompressible flows. Thisresult is also consistent with the DNS data of Guariniet al.

3 for a flat plate at a free-stream Mach number

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American Institute of Aeronautics and Astronautics Paper 03–3726

of 2.5.Gaviglio26 postulated that a possible mechanism

that contributes to maintaining the magnitude ofRu′T ′ within the boundary layer is the small-scaleinternal intermittency of turbulence due to compress-ibility. Here we have seen that for the isothermalsimulation, Ru′T ′ is maintained in the wake due to thelarge-scale intermittency of the wake and compress-ibility. However, elsewhere within the boundary layer,Ru′T ′ follows the trend of the incompressible flowdata.26 Thus, no effect of the small-scale intermittencydue to compressibility is observed in the present study.The ratio of compressible to incompressible turbulentkinetic energy away from the boundary layer edge isabout 0.15 for the present simulations. For the condi-tions chosen, the local compressibility effects are notsubstantial enough to counteract the de-localizationprovided by vortical pressure fluctuations. Experimen-tal data shows no effect of the shockwaves or shockletsin the dynamics of compressible boundary layers atMach numbers up to 8,7, 27 except for a reduction inthe length scale.28

Effect of Wall Temperature on Energy

Transfer

After having examined the effects of compressibil-ity, we will consider the turbulent energy mechanisms.There are four energy exchange mechanisms that takeplace in turbulent boundary layers: transport, pro-duction, dissipation and diffusion of turbulence. Thebudget equation for the turbulent kinetic energy is

∂

∂t(ρ k )+w

∂

∂z(ρ k ) = P+T+Πt+Πd+φdif +φdis+ST

where

P = −ρu′′i w′′

∂ui

∂z,

T = −1

2

∂

∂zρu′′

i u′′i w′′ ,

Πt = − ∂

∂zw′′p′ , Πd = p′

∂u′′i

∂xi

,

φdif =∂

∂zu′′

i σ′i2 , φdis = σ′

ij

∂u′′i

∂xj

,

ST = −w′′∂p

∂z+ u′′

i

∂σij

∂xj

− ρ k∂w

∂z, (1)

and P is the production due to the mean gradients, Tis the redistribution or transport of turbulent kineticenergy, Πt is the pressure diffusion, Πd is the pres-sure dilatation, φdif is the viscous diffusion, φdis isthe viscous dissipation, and the ST represents a groupof small terms. The first two appear due to the differ-ence between the Favre and Reynolds averaging, thethird one is the dilatation-production term.

z+

0 10 20 30 40 50 60

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

(a)

ζ+

0 10 20 30 40 50 60

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

(b)

Fig. 11 Turbulent kinetic energy budget (a) Tra-ditional coordinate transformation; (b) Mathemat-ical coordinate transformation including the vari-ation of the thermodynamic variables; pro-duction; viscous diffusion; pressurediffusion; transport; viscous dissipa-tion. The lines denote the adiabatic case, lines andsymbols the isothermal one. The variables havebeen made nondimensional using uτσw/zτ , whereσw = ρwu2

τ .

Figure 11a shows the terms in the budget of theturbulent kinetic energy for the adiabatic and isother-mal wall simulations under the traditional coordinatetransformation. Figure 11b shows a better collapse ofthe data when using a coordinate transformation thattakes into account the variation in density and temper-ature across the layers. This transformation is givenby

ζ+ =

∫ z

0

〈ρ〉uτ

〈µ〉 dz.

The magnitude of turbulent kinetic energy production,viscous diffusion and viscous dissipation are larger forthe adiabatic case. The location of the maximum andminimum values for the turbulent kinetic energy trans-port term is nearly the same for both simulations whenplotted versus ζ+. Similar to incompressible flows,the rest of the terms in the turbulent kinetic energytransport equation are negligible for both cases. Beingconsistent with the supersonic simulations of Huang3

and Coleman,29 we find that the dissipation is nearlysolenoidal.

Let us now consider the vorticity in the flow field.The total change of vorticity can be written as

D~ω

Dt= (~ω · ∇) ~u − ~ω (∇ · ~u) + ∇T ×∇S ,

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ζ+0 10 20 30 40 50 60 70

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

adiabaticisothermal

Fig. 12 Normalized amplitude of vorticity produc-tion terms for the adiabatic and isothermal sim-ulations. stretching/tilting and com-pressibility term. The variables have been madenondimensional using uτ and zτ .

where ~ω and ~u are the vorticity and velocity vectorsrespectively, T is the temperature and S is the entropy.The first term on the right hand side is the productionof vorticity due to vortex stretching and tilting mecha-nisms, the second term is due to the compressibility ofthe flow, and the third term is due to the change in thethermodynamic variables and includes the baroclinictorques.

Figure 12 shows the magnitude of the stretch-ing/tilting and compressibility terms for the simu-lations. The compressibility term is dominant nearζ+ = 12 for the adiabatic simulation, and negligiblefor the isothermal case. For both simulations, thebaroclinic torque is negligible in comparison to thestretching/tilting mechanism.

The different wall temperature conditions have dif-ferent effects on the mean variables, which in turn leadto differences in the turbulent structures. In particu-lar, the wall-cooling is a sink of energy, thus there ismore energy dissipated for the isothermal wall simula-tion than for the adiabatic case. It is the difference inthe mean velocity gradient that affects the location ofthe buffer-region structure and the turbulence produc-tion and dissipation mechanisms. For the isothermalcase, the increased magnitude of the stretching/tiltingterm relative to the compressible term indicates thatthe structures are more elongated than for the adi-abatic simulation. Also, since the vortex stretch-ing/tilting and turbulent kinetic energy mechanismstake place farther from the wall, the near-wall regionis more quiescent for the isothermal simulation.

These results are consistent with the flow structuretopology shown in Fig. 13, where the structures arevisualized using iso-surfaces of the second invariantof the velocity gradient tensor Φ.30 In Figures 13aand 13b the domain is given in δ units to identify thelarge structures for the adiabatic and isothermal simu-lations. Comparing these two figures, we observe that

Fig. 13 Flow structure visualized using iso-surfaces of Φ with 0.001% of the maximum valuefor the (a) adiabatic and (b) isothermal simulationswith coordinates in δ units.

when plotted in this fashion the structures are largerfor the adiabatic simulation.

Figures 14a and 14b plot iso-surfaces of Φ in a do-main that is normalized in wall units, x+

i = xi/zτ . Weobserve that the structures are larger for the isother-mal simulation. This result is consistent with the factthat δ+ increases with decreasing wall temperature.For the adiabatic simulation, the turbulence structureslook similar when plotted in a domain measured ineither δ or zτ units. In contrast, for the isothermalsimulation, the flow structure topology differs substan-tially depending on whether we choose freestream orwall units to nondimensionalize the domain.

Conclusions

We present an initialization procedure that leads toshort simulation transients, which is necessary to con-trol the final skin friction and Reθ for the simulation.We applied this procedure to initialize adiabatic andisothermal turbulent boundary layers. We are in theprocess of testing the repeatability of the initializationin Mach 3 and Mach 8 boundary layers.

The preliminary Mach 4 adiabatic and isothermalturbulent boundary layer data are presented. Thecomparison with available experimental and computa-tional data shows that the adiabatic simulation dataare accurate, and the resolution for the isothermalcase should be and will be increased for a more accu-

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Fig. 14 Flow structure visualized using iso-surfaces of Φ with 0.001% of the maximum valuefor the (a) adiabatic and (b) isothermal simulationswith coordinates in zτ units.

rate representation of the friction velocity. Ensuringthat periodic boundary conditions are adequate re-quires performing simulations in short computationaldomains. At present, we have not assessed the ad-equacy of the large scale computational sample andwhether the amount of energy computed representsthat of a statistical sample containing a large enoughnumber of structures. We are conducting this studyby relaxing the periodic boundary condition assump-tion and using the inflow boundary conditions of Xu& Martin.11

The bursting events that are present in a turbulentboundary layer bring low-momentum fluid from thenear wall toward the boundary layer edge, and large-scale compression regions form where the low-speedfluid meets the incoming freestream. We find that thecompressibility ratio is a good index of the location ofthese regions and that shock waves are present in theMach 4 boundary layers. The average Mach number ofthe shocks is about 1.10 and the wall-normal dimen-sion is about 0.03δ. As observed experimentally,7, 27

the shock structures have no effect on the overall dy-namics of the turbulent boundary layers.

The effect of wall temperature is studied. We find abetter collapse of the turbulent kinetic energy and vor-ticity budgets when the wall coordinate is modified totake into account the changes in the thermodynamicvariables. The cold-wall boundary condition for theisothermal simulation is a sink of energy. Thus, themagnitude of the turbulence mechanisms such as pro-duction and dissipation of turbulent kinetic energy aresmaller than those in the adiabatic simulation. Thevorticity budgets show an increased magnitude of thestretching/tilting term relative to the compressibility

term for the isothermal simulation. This is consis-tent with the structures being more elongated for theisothermal case. As δ+ increases with decreasing walltemperature, the structures plotted in wall units arelarger for the isothermal case.

These data will be available to the community.

Acknowledgments

We would like to thank NASA Ames for the use ofthe DPLR CFD code to generate the turbulent meanflow, and to Michael Wright at ELORET for his guid-ance in how to use this code. We would also like tothank Datta Gaitonde for making us aware of existingsemi-empirical data for the validation of the Mach 4boundary layer simulations. This work was supportedby the Air Force Office of Scientific Research undergrant AF/F49620-02-1-0361 and the National ScienceFoundation under grant # CTS-0238390.

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