34th AIAA Fluid Dynamics Conference and Exhibit,June 28–July 1, Portland, Oregon

Direct Numerical Simulation of Shockwave/Turbulent

Boundary Layer Interaction

Minwei Wu∗

and M. Pino Martin†

Princeton University, Princeton, NJ 08540

Direct numerical simulations of two canonical configurations, a 24-degree compression

ramp and shock impinging a wall with reflection, are performed to study shockwave and

turbulent boundary layer interactions. The Mach number for the incoming boundary

layers is 2.9 and the Reynolds number is 2400 based on the momentum thickness of the

incoming flow. The shock unsteadiness, turbulence amplification, and changes of length

scales through the shockwave and turbulent boundary layer interaction are studied in this

paper.

I. Introduction

Many aspects of shock/turbulent boundary layer interaction (STBLI) are not fully understood, includingthe dynamics of shock unsteadiness; turbulence amplification and mean flow modification induced by shockdistortion; separation and reattachment criteria as well as the unsteady heat transfer near the separationand reattachment points; and the generation of turbulent mixing layers and under-expanded jets in theinteraction region, especially when they impinge on a surface. Accurate prediction and effective means tocontrol the interaction regions can only be achieved by understanding the fundamental physics governingthe STBLI problems.Much experimental and theoretical work has been done on STBLI. Settles and Dolling did experiments

on compression ramp and swept fin to study 2D and 3D STBLI problems.1–5 Smits did experiments oncompression ramps of different turning angles.6 Andreopolous studied the shock unsteadiness of compressionramp flow.7 Theoretical work includes rapid distortion theory (RDT) proposed by Batchelor8 and linearinteraction analysis (LIA) performed by Anyiwo and Bushnell.9 There are few numerical simulations ofSTBLI problems. Some efforts on RANS and LES have been taken. However, there are no accurate orvalidated RANS or LES models for STBLI problems.10 Due to the limitations of computer capability andnumerical methods, no direct numerical simulation (DNS) of STBLI problems had been presented until thefirst DNS of an 18-degree compression ramp at Ma = 3 and Reθ = 1685 was performed by Adams.

11 In thispaper, we present new DNS data for two STBLI cases: (1) 24-degree compression ramp case and (2) shockimpinging and reflection case. Our motivation is to provide accurate data for studying the flow physics ofSTBLI, which are explored in this paper.

II. Governing Equations

The governing equations are the conservative form of the continuity, momentum and energy equations incurvilinear coordinates. The working fluid is air, which is assumed to be a perfect gas.

∂~U

∂t+∂ ~F

∂ξ+∂ ~G

∂η+∂ ~H

∂ζ= 0 (1)

∗Graduate Student, Student Member AIAA.†Assistant Professor, Member AIAA.Copyright c© 2004 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free

license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by thecopyright owner.

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American Institute of Aeronautics and Astronautics Paper 2004–2145

34th AIAA Fluid Dynamics Conference and Exhibit28 June - 1 July 2004, Portland, Oregon

AIAA 2004-2145

Copyright © 2004 by Minwei Wu. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

where

~U = J

ρ

ρu

ρv

ρw

ρe

, ~F = ~Fc + ~Fv (2)

and

~Fc = Jrξ

ρu′

ρuu′ + psx

ρvu′ + psy

ρwu′ + psz(ρe+ p)u′

, ~Fv = −Jrξ

0

σxxsx + σxysy + σxzszσyxsx + σyysy + σyzsz

σzxsx + σzysy + σzzsz

(σxxu+ σxyv + σxzw)sx+

(σyxu+ σyyv + σyzw)sy+

(σzxu+ σzyv + σzzw)sz−

qxsx − qysy − qzsz

(3)

sx = ξx/rξ, u′ = usx + vsy + wsz, rξ =

√

ξ2x + ξ2

y + ξ2z (4)

In the curvilinear coordinate system F, G and H are functionally equivalent. Thus equations (4) through(8) apply in the particular curvilinear direction. σij is the shear stress tensor given by Newtonian linearstress-strain relation:

σij = 2µSij −2

3δijSkk (5)

The heat flux terms qj are given by Fourier law:

qj = −k∂T

∂xj(6)

And the total energy per unit mass e is given by:

e = cvT +1

2uiui (7)

III. Flow Configurations

Two canonical configurations have been chosen to study STBLI: (1) compression ramp; (2) reflectedshock with separation and turbulent slip layer. Figure 1 shows the sketches of these two configurations. Theturning angle of the compression ramp is 24 degrees. The angle of the wedge that is used to generate theoblique shock in the reflected shock configuration is 12 degrees.

Shock

Flow

24o

(a)

Shock

Flow12

o

(b)

Figure 1. Configurations for the (a) compression ramp case and (b) reflected shock case.

Numerical errors associated with the discrete evaluation of the Jacobian matrices might be amplifiedthrough the simulation. Therefore we use analytical transformations to generate the grids and to minimizethese errors. For the compression ramp case, the transformations are chosen to make the grid clustered nearthe wall in the wall-normal direction and near the ramp corner in the streamwise direction. For the reflectedshock case, the grid is made to be clustered near both the lower and upper walls. In the streamwise direction,

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x(mm)

y(m

m)

0 2 4 6 8 100

2

4

6

(a)

x(mm)

y(m

m)

0 2 4 6 8 10 120

2

4

6

8

(b)

Figure 2. Grids for the (a) compression ramp case and (b) reflected shock case.

the grid is nearly equally spaced. Figure 2 shows sample grids for the compression ramp and reflected shockcases. These transformations are discussed in detail in the paper of Martin, Xu and Wu.12

Figures 3a and 3b show the computational domains for the compression ramp and reflected shock cases,respectively. The length in the wall-normal direction is about 4.5δ for both cases, where δ is the thicknessof the incoming boundary layer. In the spanwise direction, we have 2δ which might be small. We plan tostudy the influence of the spanwise domain size in the near future. For the ramp case, the ramp corner is7δ away from the inlet. The length along the ramp is 6δ. As for the reflected shock case, the wedge used togenerated the oblique shock is placed at 5δ downstream from the inlet. The total length in the streamwisedirection is about 15.8δ. The dashed parallelograms indicate the locations of the rescaling stations. Theyare 4δ downstream of the inlet in both cases. The rescaling method is described in the next section.

6δ

7δ4δ2δ

4.5δ

(a)

4.5δ

15.8δ

2δ

5δ

4δ

4δ

(b)

Figure 3. Computational domains for the (a) compression ramp case and (b) reflected shock case.

IV. Numerical Method and Boundary Conditions

A 3rd-order, bandwidth-optimized WENO (Weighted Essential Non-oscillatory) scheme13 is used to ap-proximate the convective flux terms in the governing equations. This scheme has been designed for highbandwidth and low dissipation, while being a shock capturing scheme. These properties are necessary forthe DNS of turbulent flows at the Mach numbers that we consider. A 4th-order standard central schemeis used to compute the viscous terms. As for the time integration, we use a 2nd-order DP-LUR (DataParallel Lower-Upper Relaxation) method,14 which is based on the DP-LUR method of Candler et al.15

This method gives about 4 times faster turn-around simulation time than a 3rd-order Runge-Kutta method.The combined numerical algorithm has been shown to give accurate results for the DNS of compressibleturbulence.16–18

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Initial conditions

Prescribing and controlling the initial flow conditions computationally has received a great deal of at-tention. The initialization procedure is sketched in Fig. 4. The first step is to get an initial flow field forthe incoming boundary layer. A RANS calculation is performed to get the mean flow quantities. Thenthe fluctuations are obtained by transforming the turbulence of a Mach 0.3 turbulent boundary layer DNSdata to that of the desired Mach number using Morkovin’s scaling laws and the strong Reynolds analogy(SRA). We run the incoming boundary layer during a DNS until it reaches a stationary state. After that,we interpolate the flow field of the incoming boundary layer onto the inlet of the grids for the two STBLIcases. Then, the last profile of the inlet is copied to the rest of the computational domain to get the initialflow field for the STBLI cases. The initialization procedure for the STBLI is also discussed in the paper ofMartin et al12 and details about the initial transformations for the incoming boundary layer data can befound in Martin’s 2003 and 2004 papers.16,18

rescaling

M>1 TBL

RANS-TBL

M=0.3 TBL

fluctuations

M>1 TBL

DNS/LES

DNS/LES

DNS/LES-STBLI

x

z

shock

transformed

Figure 4. Initialization procedure for STBLI simulations.

Boundary conditions

To get a continuous incoming turbulent flow and maintain the upstream inflow conditions, a rescalingmethod for compressible flows was developed by Xu and Martin.19 The main idea of the rescaling methodis to take a profile at some place downstream of the computational domain and rescale it using scalinglaws, then put it back to the inlet to get continuous inflow data.19 At the outlet, we use the sponge layertechnique20 together with supersonic exit conditions to minimize flow reflections. The sponge layer is about1δ in length at the outlet. Inside the sponge layer region, a term is added to the RHS of the governingequations.

~Z = −σ(x)(~U − ~U0) (8)

where

σ(x) = As(Ns + 1)(Ns + 2)(x− xs)

Ns(Lx − x)

(Lx − xs)Ns+2(9)

Typically As=3 and Ns=4. At the top boundary, we also use supersonic exit boundary conditions. In thespanwise direction, periodic boundary conditions are used. The wall boundaries are isothermal and Tw isset to 633.39K for both cases.

V. Accuracy of the DNS Data

Table 1 shows the flow conditions of the incoming boundary layers for the two simulations. The Machnumber is 2.9 and Reθ is about 2400. This Reynolds number is much smaller than those of existing experi-mental data, making it difficult to draw conclusions from direct comparison. However, experiments at theseconditions are being carried out in the Princeton Gas Dynamics Lab.21

For DNS of turbulent boundary layers, the computational domain size in the streamwise direction mustbe large enough to contain a good statistical sample of the large structures. The continuous generation

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Table 1. Incoming boundary layer flow conditions.

Ma Reθ ρe(kg/m3) Te(K) Tw(K)

2.9 2400 0.708 250.6 633.39

of inflow data requires the selection of a rescaling location inside the computational domain (upstream ofthe interaction region). To access how close to the inlet we can choose the rescaling station, we considerthree turbulent boundary layer simulations using different locations for the rescaling station. Namely, therescaling stations are of 0.3δ, 0.6δ and 1.2δ away from the outlet for each case. The computational domainis 4δ × 2δ × 14δ in streamwise, spanwise and wall-normal directions. The incoming Mach number is 2.9.Reθ is 5600. Figure 5 shows the energy spectra of u at z

+ = 10 for these three cases. Virtually, there is nodifference in the results. Therefore we conclude that we can push the rescaling station up to 2.8δ away fromthe inlet without affecting our results. In our DNS the rescaling stations are chosen to be 4δ away from theinlet.

kδ/lref

Eu/<

u>2

50 100 15020010-8

10-7

10-6

10-5

10-4

10-3 0.3δ0.6δ1.2δ

Figure 5. Streamwise velocity energy spectra from DNS using different rescaling stations.

Figure 6 plots the Van-Driest transformed velocity profiles for the incoming flow with different grid res-olutions. These simulations are performed only for the inlet of the computational domain of the ramp case,which is 5δ in length. The number of grid points in the streamwise and spanwise directions are also given.We observe that for the finest grid case, the profile in the log region lies right on top of the log law. The DNS

z+

⟨u⟩ V

D

100 101 102 1030

5

10

15

20

25

30150×128150×256300×128300×2562.44log(z+)+5.5

Figure 6. Van-Driest transformed velocity profile for the incoming boundary layer DNS with varying gridresolution.

cases presented in this paper use the coarsest resolution in Fig. 6. So these cases might NOT be consideredfully resolved. Nevertheless, the error will be quantified against theory and experimental data, as it becomes

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available. Finer grid simulations are possible. For the ramp case, we use 412 × 128 × 96 grid points instreamwise, spanwise and wall-normal direction. The corresponding resolution is:

∆x+ = 11.3, ∆y+ = 5.6, ∆z+1 = 0.12 (10)

For the reflected shock case, the number of grid points is 412× 128× 112. And the corresponding resolutionis:

∆x+ = 13.8, ∆y+ = 5.7, ∆z+1 = 0.17 (11)

The resolution has been chosen considering the available computational resources and simulation turn-aroundtime. As we will also mention in the next section, Fig. 7 shows the skin friction along the streamwise directionfor the 24-degree ramp. The square symbol indicates the theoretical value given by Van-Driest II formula.This formula gives the best prediction for the skin friction coefficient at the Reynolds number level that weconsider.22 Comparing with the value given by this formula, the error in Cf is within 5% for the DNS data.

VI. DNS Data Analysis

Table 2 shows the properties of the incoming boundary layers for the two cases.

Table 2. Incoming boundary layer parameters.

Case δ(mm) δ∗(mm) θ(mm) Cf

Ramp 0.85 0.255 0.063 2.04×10−3

Reflect 0.85 0.275 0.064 1.95×10−3

Compression ramp case

Figure 7a plots the skin friction coefficient for the compression ramp case. The black square at the inletis the value given by Van-Driest II formula. Negative value of Cf indicates that the flow separates. Theseparation bubble is about 2δ in length. This number is close to the one given in Settles’s experiments.2

Figure 7b plots the normalized wall pressure. The thick line indicates the pressure rise given by the inviscidtheory. At the outlet, the DNS wall pressure is smaller than the theoretical inviscid value. Notice that thewall pressure keeps rising after the ramp corner. Thus, the flow experiences further compression after theshock.

x/δ

Cf

-5 0 5

-0.001

0

0.001

0.002

0.003

0.004

DNSTheory

(a)

x/δ

Pw/P

w0

-5 0 5

1.5

2.0

2.5

3.0

3.5

4.0

4.5 Inviscid theory

(b)

Figure 7. (a) Skin friction coefficient and (b) wall pressure for the compression ramp DNS.

Figure 8 plots a sequence of |∇ρ| contour plots to illustrate the shock unsteady motion. Time increasesfrom (a) to (f). The interval between each frame is about 2δ/U∞. The shock foot does not penetrate deepinto the incoming boundary layer, as it has been observed in experiments.6 This might be due to a low

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0 50

2

4

6

(a)

0 50

2

4

6

(b)

0 50

2

4

6

(c)

0 50

2

4

6

(d)

0 50

2

4

6

(e)

0 50

2

4

6

(f)

Figure 8. Sequence of |∇ρ| contours for the compression ramp DNS (coordinate unit in δ).

Reynolds number effect. Adams used a Mach number trace to get the frequencies of the shock oscillationand bursting events in the incoming boundary layer.11 For nearly adiabatic conditions, when low-speed,high-temperature fluid ejects from the wall to the outer edge of the boundary layer, low Mach number spotscan be detected. Therefore low Mach number spots may indicate bursting events. We follow this criteriato identify bursting events in the compression ramp case. Figure 9a shows the locations where the burstingand shock oscillation frequencies are being measured on contours of |∇ρ|. Figure 9b plots the Mach numbersignal at a position 0.5δ away from the wall and 1δ away from the inlet. The indicator is set to ±1 when theMach number is greater or less than the mean value by an amount of a half standard deviation of the signal.Otherwise it is set to zero. Figure 9c plots the Mach number trace measured near the outlet. The indicatoris obtained by the same method as above. The frequencies of the shock motion and the bursting events ofthe incoming boundary layer that are given by this procedure are both about 0.14U∞/δ. These numbers arevery close to those given by Adams11 and Andreopolous,7 who proposed that the shock motion is driven bythe bursting events in the incoming boundary layer.

x/δ

z/δ

-5 0 50

2

4

6

8

10

(a)

tU∞/δ10 20 30 40-1.0

0.0

1.0

2.0Maindicator

(b)

tU∞/δ10 20 30 40-1.0

0.0

1.0

2.0

3.0

Maindicator

(c)

Figure 9. (a) |∇ρ| contours and locations of Mach number measurements; (b)Mach number trace and indicatormeasured to get the frequency of the bursting events; (c)Mach number trace and indicator measured to getthe frequency of the shock motion.

Figure 10 plots the iso-surface of |∇p|. The wrinkled nature of the shock is apparent. In the spanwisedirection, the shock has about one wave length inside the computational domain, indicating the domain sizeof 2δ in the spanwise direction might not be large enough. Figure 11 plots the mass flux turbulent intensityfor the 24-degree ramp case at different streamwise locations. Open and closed symbols indicate upstreamand downstream to the corner locations, respectively. Downstream of the interaction, the maximum of themass flux turbulent intensity is amplified by a factor of about 5. Selig23 measured a factor of about 4.8in his experiments of a 24-degree compression ramp at Ma = 2.9 and Reθ = 70, 000. Figure 12 plots the

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Figure 10. Iso-surface of |∇p| = 3× 108(N/m3) for the compression ramp DNS.

z+

⟨ρu’

⟩/ρ∞U

∞

10-1 100 101 102 1030

0.1

0.2

0.3

0.4

0.5

0.6-5.8δ-2.4δ2.5δ3.8δ4.7δ

Figure 11. Mass flux turbulence intensity at different streamwise locations for the compression ramp DNS.

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Reynolds stresses at different streamwise locations. After the interaction all components of the Reynoldsstresses are significantly amplified. The < u′u′ > component is amplified by a factor of about 3 and the othercomponents are amplified by a factor of about 20. Notice that the Reynolds stress in the spanwise direction isalso amplified, indicating that the interaction has spanwise effects, although this is a two dimensional STBLIcase. Figure 13 plots the Van-Driest transformed mean velocity profile at different streamwise locations.

z+

ρ⟨u

’u’⟩/

ρ ∞U

∞2

10-1 100 101 102 1030.00

0.02

0.04

0.06

0.08

0.10

0.12-5.8δ-2.4δ2.5δ3.8δ4.7δ

(a)

z+

-ρ⟨

u’w

’⟩/ρ ∞

U∞2

10-1 100 101 102 1030.00

0.01

0.02

0.03

0.04-5.8δ-2.4δ2.5δ3.8δ4.7δ

(b)

z+

ρ⟨v

’v’⟩/

ρ ∞U

∞2

10-1 100 101 102 1030.00

0.02

0.04

0.06-5.8δ-2.4δ2.5δ3.8δ4.7δ

(c)

z+

ρ⟨w

’w’⟩/

ρ ∞U

∞2

10-1 100 101 102 1030.00

0.02

0.04

-5.8δ-2.4δ2.5δ3.8δ4.7δ

(d)

Figure 12. Reynolds stresses at different streamwise locations for the compression ramp DNS.

There is a characteristic ‘dip’ in the log region of the mean velocity profile after the corner, which indicatesthat after the interaction the karman constant increases with distance from the wall. Smits also observedthe same trend in experiments with Ma = 2.9 and Reδ = 1, 640, 000.

6

In the experiments of Smits,6 he used the Morkovin’s SRA (strong Reynolds analogy) to deduce densityfluctuations from velocity fluctuations. The SRA relations are shown in equations 12 and 13. Tilde in theequations denotes Favre average.

√

T ′2

T= (γ − 1)M2

√

u′2

u(12)

RuT =−u′T ′

√

u′2√

T ′2= constant (13)

It is known that SRA relations may not be satisfied inside the interaction region. To investigate this

statement, we calculateT ′

rmsu

(γ−1)M2u′

rmsTand RuT at different streamwise locations and they are shown in

Figure 14. These two numbers should be 1 if SRA is satisfied assuming RuT = 1. Experimental dataindicate that RuT > 0.75.24 Upstream of the separation region, Fig. 14a, the SRA are satisfied. Figures 14band 14c show the data inside the interaction region. We observe that the SRA can not be applied in thisregion. The location of the last plot in Fig. 14d is 4.7δ away from the ramp corner which is very close tothe outlet. The two quantities show a trend of going back to their values upstream of the interaction. Butthey still deviate from the SRA relations. Therefore SRA relations can not be applied inside and in a quitelarge region downstream of the interaction region.Reflected shock case

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z+

⟨u⟩ V

D10-1 100 101 102 1030

5

10

15

20

25

30

35

40-5.8δ-2.4δ2.5δ3.8δ4.7δ2.44log(z+)+5.5

Figure 13. Van-Driest transformed velocity profile at different streamwise locations for the compression rampDNS.

z/δ0 0.2 0.4 0.6 0.8 1

0.6

0.8

1.0

1.2

1.4 T’rmsu/((γ-1)M2u’rmsT)RuT

~~

(a)

z/δ0 0.2 0.4 0.6 0.8 1

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

(b)

z/δ0 0.2 0.4 0.6 0.8 1

0.0

0.5

1.0

1.5

2.0

(c)

z/δ0 0.2 0.4 0.6 0.8 1

0.0

0.5

1.0

1.5

(d)

Figure 14. SRA Equations 12 and 13 at different streamwise locations for the compression ramp DNS: (a)x=-2.4δ; (b) x=-0.1δ; (c) x=1.2δ; (d) x=4.7δ.

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There are few experimental results available and no existing DNS data for this case. Figure 15 shows acontour plot of |∇ρ|. The impinging and reflected shocks can be clearly seen in this figure. Figure 16 plots the

x/δz/

δ

0 5 10 150

2

4

6

8

Figure 15. Contour of |∇ρ| for the reflected shock DNS.

skin friction coefficient and wall pressure for the reflected shock case. The error in Cf at the inlet is within5%. The separation bubble is about 4δ in length, which is two times larger than that for the compressionramp case. The thick line in the wall pressure plot indicates the pressure rise given by the inviscid theory.We observe that the pressure rise of the DNS result is smaller than that of the inviscid theory. Notice thatthe pressure drop near the outlet is due to the expansion fan generated at the second corner of the upperwedge.

x/δ

Cf

0 5 10 15

-0.001

0

0.001

0.002

0.003

0.004

DNSTheory

(a)

x/δ

Pw/P

w0

5 10 151.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5 Inviscid theory

(b)

Figure 16. (a) Skin friction coefficient and (b) wall pressure for the reflected shock DNS.

Figure 17 plots a sequence of contour plots of |∇ρ|. The impinging shock is quite straight. Whereasthe reflected shock has a flapping character. The amplitude of the shock motion is smaller than that of thecompression ramp case. Figure 18 plots an iso-surface of |∇p|. We observe that the 3D shock structure isnot as wrinkled as that of the compression ramp case. Figure 19 plots the mass flux turbulent intensity atdifferent streamwise locations for the reflected shock case. Open symbols indicate locations upstream of theinteraction. The numbers in the legend are the streamwise coordinates of the locations with the origin atthe inlet of the computational domain. The mass flux turbulent intensity is amplified by a factor of 3. Thisfactor is less than that found in the ramp case. It should be noted that the pressure rise for the reflectedshock case is about 1.1 times smaller than that of the compression ramp case. Also in the ramp case, thereis a concave streamline curvature effect, which contributes to the amplification of turbulence level.25

Four components of the Reynolds stresses are also plotted in Fig. 20. The < u′u′ > component isamplified by a factor of about 3. The other components are amplified by a factor of about 15, which issmaller than that in the ramp case. Figure 21 plots the Van-Driest transformed mean velocity profile. Theprofiles downstream of the interaction show a same trend as in the ramp case. i.e. in the log region there isa decreasing in the slope of the mean velocity profile.

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5 10 150

2

4

6(a)

5 10 150

2

4

6(b)

5 10 150

2

4

6(c)

5 10 150

2

4

6(d)

5 10 150

2

4

6(e)

5 10 150

2

4

6(f)

Figure 17. Sequence of |∇ρ| contours for the reflected DNS (coordinate unit in δ).

Figure 18. Iso-surface of | 5p |= 5× 108(N/m3) for the reflected shock DNS.

z+

⟨ρu’

⟩/ρ∞U

∞

100 101 102 103

0.05

0.1

0.15

0.2

0.25

0.3

0.351.2δ8.2δ10δ13δ14δ

Figure 19. Mass flux turbulence intensity at different streamwise locations for the reflected shock DNS.

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American Institute of Aeronautics and Astronautics Paper 2004–2145

z+

ρ⟨u

’u’⟩/

ρ ∞U

∞2

10-1 100 101 102 1030

0.01

0.02

0.03

0.04

0.05

0.06

0.071.2δ8.2δ10δ13δ14δ

(a)

z+

-ρ⟨

u’w

’⟩/ρ ∞

U∞2

10-1 100 101 102 1030

0.005

0.01

0.015

0.02 1.2δ8.2δ10δ13δ14δ

(b)

z+

ρ⟨v

’v’⟩/

ρ ∞U

∞2

10-1 100 101 102 1030

0.01

0.02

0.03

0.041.2δ8.2δ10δ13δ14δ

(c)

z+

ρ⟨w

’w’⟩/

ρ ∞U

∞2

10-1 100 101 102 1030

0.01

0.02

0.031.2δ8.2δ10δ13δ14δ

(d)

Figure 20. Reynolds stresses at different streamwise locations for the reflected shock DNS.

z+

⟨u⟩ V

D

100 101 102 103

0

10

20

30

40

501.2δ8.2δ10δ13δ14δ2.44log(z+)+5.5

Figure 21. Van-Driest transformed velocity profile at different streamwise locations for the reflected shockDNS.

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We have not measured the shock oscillation frequency for this case due to the relative smaller amplitudeof shock motion and lack of samples.

VII. Conclusion

Our DNS results for the compression corner and reflected shock cases show that the frequency of shockoscillation could be related to the bursting events inside the incoming boundary layer. Andreopoulos7 andAdams11 also had the same conclusion about shock motion for the compression ramp case. The turbulenceis amplified through STBLI. The mass flux turbulence intensity is amplified by a factor of 5 and 3 in theramp and reflected shock cases, respectively. The amplification factor for the ramp case is about the sameas that predicted in Selig’s experiments.23 Due to the very different Reynolds numbers, this agreement mustbe verified by comparing experimental and DNS data at the same flow conditions. The Reynolds stressesare amplified by a factor of 3-20 and 3-15 after the interactions for the two cases, respectively. There area few mechanisms that account for turbulence amplification. Across the shock, the turbulence level will beincreased due to the Rankine-Hugoniot jump conditions and nonlinear coupling of turbulence, vorticity andentropy waves.9 The unsteady shock motion will also pump energy from the mean flow to the turbulentfluctuations. In the ramp case, the concave streamline curvature can also make the flow unstable and amplythe turbulence level.25 The turbulence length scales also change as a result of STBLI. We found that afterthe interactions, the karman constant increases at a rate greater than ky near the wall. This confirms theresults shown by Smits and Muck.6 The present data will be compared against experimental data21 at thesame conditions.

Acknowledgments

We would like to thank NASA Ames for the use of the DPLR CFD code to generate the turbulent meanflow. This work is supported by the Air Force Office of Scientific Research under grant AF/F49620-02-1-0361.

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