Home > Direct Numerical Simulation of Supersonic Turbulent Boundary Layer over a Compression Ramp

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M. Wu

∗

and M. P. Martin

†

DOI: 10.2514/1.27021

= optimal weight for stencil

= nondimensional value

interaction (STBLI) are not fully understood, including the dynamics of shock unsteadiness, turbulence amplification and mean flow modification induced by shock distortion, separation and reattachment criteria as well as the unsteady heat transfer near the separation and reattachment points, and the generation of turbulent mixing layers and underexpanded jets in the interaction region, especially when they impinge on a surface. Yet, STBLI problems are of great importance for the efficient design of scramjet engines and control surfaces in hypersonic vehicles. A more profound understanding of STBLI will lead to flow control methodologies and novel hypersonic vehicle designs. Different canonical configurations have been used in STBLI studies. The compression ramp configuration has been studied extensively experimentally, and there are numerous experimental data available for this configuration. For example, Settles et al. [1–3] studied 2-D/3-D compression ramp and sharp fin STBLI problems in detail. Dolling et al. [4,5] studied the unsteadiness for compression ramp configurations, and Selig [6] studied the unsteadiness of STBLI and its control for a 24 deg compression ramp. Recently, Bookey et al. [7] performed experiments on a 24 deg compression ramp configuration with flow conditions accessible for direct numerical simulation (DNS) and large eddy simulation (LES), which provides valuable data for the validation of our simulations. In contrast with numerous experimental data, there are few detailed numerical simulations such as DNS and LES. Numerical simulations of STBLI have been mainly confined to Reynolds averaged Navier–Stokes simulation (RANS) due to the limitation of computational resources. However, RANS is shown not capable of predicting the wall pressure or the heat flux within a satisfactory accuracy for shock interactions. Settles et al. [2] compared experimental results with those of a one-equation model RANS for the compression ramp configuration and showed that there were significant differences in the wall-pressure distribution when the flow was separated. Zheltovodov [8] showed that the state-of-the-art RANS models do not give accurate predictions for strong STBLI. The unsteady nature of STBLI problems is believed to account for the discrepancies between RANS and experiments. DNS and LES of STBLI have existed for less than a decade. Knight et al. [9] compiled a summary of existing LES for the compression ramp configuration and concluded that LES did not predict the wall pressure or the separation length accurately in separated flows. In 2000, Adams [10] performed the first DNS for an 18 deg compression ramp flow at

Received 3 August 2006; revision received 27 December 2006; accepted for publication 27 December 2006. Copyright © 2007 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/07 $10.00 in correspondence with the CCC.

∗Graduate Student, Mechanical and Aerospace Engineering Department.

Student Member AIAA.

†Assistant Professor, Mechanical and Aerospace Engineering Department.

Member AIAA.

AIAA JOURNAL Vol. 45, No. 4, April 2007

879

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The governing equations are the nondimensionalized conserva- tive form of the continuity, momentum, and energy equations in curvilinear coordinates. The working fluid is air, which is assumed to be a perfect gas.

��

��

��

��

9 >>>>>>>>>>= >>>>>>>>>>; (4) and

q (5) In curvilinear coordinates, flux terms

1

2 3

(6) The heat flux terms

1

(7) The dynamic viscosity is computed by Sutherland��s law:

1,

Incoming boundary layer thickness

Figure 1 shows an inviscid flow schematic for the present STBLI configuration. The incoming flow conditions are listed in Table 1, including the reference experiment of Bookey et al. [7] for the same flow. To minimize numerical errors in the computation of Jacobian matrices, we generate the grid using analytical transformations. Details about the transformation can be found in Wu and Martin [12]. A sample grid is plotted in Fig. 2. The grid is clustered near the corner in the streamwise direction and near the wall in the wall-normal direction. The size of the computational domain is shown in Fig. 3. There are 9

Shock Flow 24o

WU AND MARTIN

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A third-order accurate low-storage Runge–Kutta method is used for the time integration, and a fourth-order accurate central standard finite difference scheme is used to compute the viscous flux terms. The incoming boundary layer is generated as in Martin [20]. The rescaling method developed by Xu and Martin [21] is used to generate the inflow condition. The recycling station is located at 4

2

ˆ X

(9) where

The weights are determined by the smoothness on each candidate stencil, where the smoothness is measured by

Z

��

(11) This means that in smooth regions for a well-resolved flowfield [meaning

discontinuity,

7�� 9�� 4.5�� 2.2�� 5 ��

x/�� z/��

-5 0 5 0 2 4 6 8

∆x/�� 〈u ��(x)u ��(x+ ∆ x)〉/u

rm s2

0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1

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thresholds for the limiters. It is found that the relative limiter is more general and less problem dependent [22]. In contrast, the relative limiter defined by Eq. (13) is method dependent, that is, WENO methods with different candidate stencil sizes have different threshold values in the relative limiter. Thus, we define an alternative relative limiter:

(15) Figure 6 plots the results at

��1

P

��1

x* ��*

-0.5 0 0.5 1 1.5 2 2.5 3.0 3.5 4.0 4.5 exact no limiter relative limiter

WU AND MARTIN

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RL guarantees enough dissipation

whenever max��TV

RL ˆ 5, and

of WENO methods including limiters for DNS of compressible turbulence is given in Taylor et al. [22].

DNS statistics are gathered using 300 flowfields with time intervals equal to 1

‡

which corresponds to the error bar in Fig. 10. The empirical envelope is from Zheltovodov et al. [26] who correlated the size of the separation bubble for a large set of experimental data. The characteristic length is defined as [26]

ky�� E

u��

(k

y,x

,z )/U

�� 2

50 100 150200 10-8 10-7 10-6 10-5 10-4 10-3 (k��)-5/3

x/�� P

w

/P

��

-5 0 5 10 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Bookey et al. DNS

Re�� L se

p

/L

c

104 105 106 107 0 5 10 15 20 25 30 Empirical envelope Bookey et al. DNS

‡Smits, A., private communication, 2006. ��Smits, A., private communication, 2006.

WU AND MARTIN

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Figure 14 is an instantaneous isosurface contour plot of the magnitude of pressure gradient jr

z/�� 〈u 〉/U

��

0 0.5 1 1.5 0.0 0.2 0.4 0.6 0.8 1.0 DNS Bookey et al.

z/�� 〈u 〉/U

e

0 0.5 1 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 DNS Bookey et al.

z/�� (�� u)��

rms

/��

��

U

��

0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 -8.0�� -4.1�� 2.0�� 4.2�� 6.1�� 6.9��

z+ 〈u 〉 VD

100 101 102 103 0 5 10 15 20 25 30 -8.0�� -4.1�� 2.0�� 4.2�� 6.1�� 6.9�� 2.44log(z+)+5.1

x/�� z/��

-5 0 5 0 2 4 6 8

x/�� z/��

-5 0 5 0 2 4 6 8

WU AND MARTIN

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As shown in Fig. 13, streamwise velocity profiles change greatly throughout the interaction region. Figure 17 plots three velocity profiles at different streamwise locations using outer scales. For the profile at

rms ~

1†

rms

~

z/�� 〈u 〉/U

��

0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 -8.0�� -1.9�� 6.1��

z/�� 〈�� u ��u ��〉/〈�� 〉U

�� 2

10-2 10-1 100 0.00 0.05 0.10 -8.0�� -4.1�� -1.9�� 1.0�� 4.2�� 6.1��

z/�� 〈�� v ��v ��〉/〈�� 〉U

�� 2

10-2 10-1 100 0.00 0.01 0.02 0.03 0.04 0.05 -8.0�� -4.1�� -1.9�� 1.0�� 4.2�� 6.1��

z/�� 〈�� w ��w ��〉/〈�� 〉U

�� 2

10-2 10-1 100 0.00 0.01 0.02 0.03 0.04 -8.0�� -4.1�� -1.9�� 1.0�� 4.2�� 6.1��

z/�� 〈�� u ��w ��〉/〈�� 〉U

�� 2

10-2 10-1 100 0.00 0.01 0.02 0.03 0.04 0.05 -8.0�� -4.1�� -1.9�� 1.0�� 4.2�� 6.1��

x/�� z/��

-5 0 5 0 2 4 6 8 10 0.01 0.02 0.04 0.05 0.07 0.08 0.10

1

0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 T��rmsu/((��-1)M2u��rmsT) -RuT ~ ~

z/��

0 0.2 0.4 0.6 0.8 1 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 T��rmsu/((��-1)M2u��rmsT) -RuT ~ ~

z/��

0 0.2 0.4 0.6 0.8 1 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 T��rmsu/((��-1)M2u��rmsT) -RuT ~ ~

z/��

0 0.2 0.4 0.6 0.8 1 0.0 0.5 1.0 1.5 2.0 T��rmsu/((��-1)M2u��rmsT) -RuT ~ ~

WU AND MARTIN

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Experiments have shown evidence of large scale, slow shock motion. Ganapathisubramani et al. [19] proposed that very long structures of uniform momentum in the incoming boundary layer are responsible for the slow motion. There have been many experimental studies on the turbulent structure of supersonic boundary layers [32– 36]. In particular, Ganapathisubramani et al. [37] have shown evidence of the existence of very long structures in supersonic boundary layers. For the signal length that they considered, they observed structures as long as 8

x/�� y/��

0 50 100 150 200 0 1 2

510 480 450 420 390

u(m/s)

tU��/�� P

w

/P

��

0 100 200 300 1.0 1.5 2.0 2.5 -6.9�� -2.98�� (mean separation point) -2.18��

WU AND MARTIN

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f��/U�� E

p/p �� 2

10-2 10-1 100 101 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 -6.9�� -2.98�� -2.18��

x/�� In te rm itte n c y fu n c tio n

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 0 0.2 0.4 0.6 0.8 1

tU��/�� ��u/��

��

U

��

0 100 200 300 0.8 1.0 1.2 1.4 1.6 1.8 2.0

upstream of the shock (x=-2.9��) inside shock motion region (x=0.8��) downstream of the shock (x=1.5��)

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A DNS of a 24 deg compression ramp configuration is performed. Applying limiters to the smoothness measurement in the WENO scheme reduces the numerical dissipation. In particular, using a combination of absolute and relative limiters is very effective. The DNS data predict the experiments with a satisfactory accuracy for the upstream boundary layer, mean wall-pressure distribution, size of the separation bubble, velocity profile downstream of the interaction, and mass-flux turbulence intensity amplification. Numerical schlieren and 3-D isosurfaces of jr

This work is supported by the U.S. Air Force Office of Scientific Research under grants AF/F49620-02-1-0361 and AF/9550-06-1-

f��/U�� E

��u

/(�� U

��

)2

10-2 10-1 10-6 10-5 10-4 10-3 10-2 upstream of the shock inside shock motion region downstream of the shock

-0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 DNS Weiss & Chokani

x/�� p �� rms /P

w

-5 0 5 0.04 0.06 0.08 0.10 0.12 0.14

WU AND MARTIN

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[1] Settles, G. S., Vas, I. E., and Bogdonoff, S. M., ��Details of a Shock- Separated Turbulent Boundary Layer at a Compression Corner,��

N. Clemens

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