Numerical Prediction of Supersonic Shock-boundary 

Layer Interaction 

by Vishal A. Bhagwandin and James DeSpirito 

ARL-TR-5602 July 2011 

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Army Research Laboratory 

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ARL-TR-5602 July 2011 

Numerical Prediction of Supersonic Shock-boundary 

Layer Interaction 

Vishal A. Bhagwandin and James DeSpirito 

Weapons and Materials Research Directorate, ARL 

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Numerical Prediction of Supersonic Shock-boundary Layer Interaction 

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14. ABSTRACT 

Shock-boundary layer interaction was numerically simulated via steady-state, Reynolds-averaged Navier-Stokes methods 

using CFD++, a general purpose fluid flow solver. The simulations emulated experiments performed in two wind tunnel 

facilities, the Institut Universitaire des Systèmes Thermiques Industriels Wind Tunnel and the University of Michigan Wind 

Tunnel. Typical flow characteristics were captured, such as the location of the incident and reflected shock waves, boundary 

layer recirculation and separation, and increase in boundary layer displacement and momentum thicknesses. The numerical 

velocity components and Reynolds stresses in the boundary-layer compared fairly well with experimental data; however, 

improvement is still needed. The sensitivity of the simulations to the choice of turbulence model and grid density was also 

demonstrated. 

15. SUBJECT TERMS

Computational fluid dynamics, shock-boundary layer interaction 

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Vishal A. Bhagwandin 

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iii 

Contents 

List of Figures iv

List of Tables v

Acknowledgments vii

1. Introduction 1

2. Simulation Methodology 3

3. Results 8

3.1 UFAST Wind Tunnel ......................................................................................................8

3.2 UMICH Wind Tunnel ...................................................................................................22

4. Summary and Conclusions 30

5. References 32

List of Symbols, Abbreviations, and Acronyms 34 

Distribution List 35 
iv 

List of Figures 

Figure 1. Schematic showing the shock-boundary layer interaction in an internal flowpath 

(9). ..............................................................................................................................................1

Figure 2. UFAST wind tunnel flowpath. ........................................................................................3

Figure 3. UMICH wind tunnel flowpath. ........................................................................................3

Figure 4. UFAST computational domain: (a) full length view, ~20.4M cells, (b) 3-D grid 

detail in test-section, and (c) 2-D grid detail in test-section. Grids generated in Pointwise 

v16.04R1. Note: only half the geometry was gridded. ..............................................................4

Figure 5. UMICH computational domain: (a) full length view, ~18M cells (b) 3-D grid detail 

in test-section, and (c) 2-D grid detail in test-section. Grids generated in Gambit v2.4.6. 

Note: only half the geometry was gridded. ................................................................................5

Figure 6. Sample residual histories of the right-hand side equations: (a) UFAST – SST and 

(b) UMICH – SST. .....................................................................................................................7

Figure 7. Three-dimensional rendering of UFAST numerical flowfield for SST turbulence 

model showing Mach number contours on the tunnel symmetry plane and oil flow 

streamlines on the tunnel lower wall. ........................................................................................9

Figure 8. Transverse profile in the interaction zone showing centerline and corner SBLIs: 

(a) experimental SPIV data (9) on a plane 2 mm above the lower wall (scale units in 

mm), (b) CFD streamline data using SST turbulence model on a plane 2 mm above the 

lower wall (scale units in m), and (c) CFD streamline data using SST turbulence model 

on the lower wall (scale units in m). ..........................................................................................9

Figure 9. Turbulence model comparison on symmetry plane: (a) u-velocity and 

(b) numerical Schlieren. ...........................................................................................................10

Figure 10. Experiment vs. CFD on symmetry plane in SBLI region: (a) u-velocity and 

(b) v-velocity. ...........................................................................................................................11

Figure 11. Turbulence model sensitivity: u-velocity axial profiles on the symmetry plane in 

the SBLI region. .......................................................................................................................12

Figure 12. Turbulence model sensitivity: v-velocity axial profiles on the symmetry plane in 

the SBLI region. .......................................................................................................................13

Figure 13. Turbulence model sensitivity: uʹvʹ axial profiles on the symmetry plane in the 

SBLI region. .............................................................................................................................14

Figure 14. Grid sensitivity: (a) u-velocity and (b) v-velocity symmetry plane contours in 

SBLI region, SST model. .........................................................................................................15

Figure 15. Grid sensitivity: u-velocity axial profiles on the symmetry plane in the SBLI 

region, SST model. ..................................................................................................................16

Figure 16. Grid sensitivity: v-velocity axial profiles on the symmetry plane in the SBLI 

region, SST model. ..................................................................................................................17
v 

Figure 17. Grid sensitivity: uʹvʹ axial profiles on the symmetry plane in the SBLI region, 

SST model. ...............................................................................................................................18

Figure 18. Pressure contours on the lower wall for the (a) full grid and (b) mirrored half grid, 

SST model. ...............................................................................................................................19

Figure 19. Spanwise profiles of pressure, skin friction and velocity using full grid, 

X=0.20 m, SST model. ............................................................................................................20

Figure 20. Spanwise profiles of pressure, skin friction and velocity using full grid, 

X=0.32 m, SST model. ............................................................................................................21

Figure 21. Spanwise profiles of pressure, skin friction and velocity using full grid, 

X=0.40 m, SST model. ............................................................................................................22

Figure 22. Three-dimensional rendering of UMICH numerical flowfield showing Mach 

number contours on the tunnel symmetry plane and oil flow streamlines on the tunnel 

lower wall, SST model, 7.75o case...........................................................................................23

Figure 23. Turbulence model comparison on symmetry plane: (a) u-velocity and 

(b) numerical Schlieren, 7.75o case. .........................................................................................24

Figure 24. Experiment vs. CFD on symmetry plane in SBLI region: (a) u-velocity and 

(b) v-velocity. ...........................................................................................................................25

Figure 25. Turbulence model sensitivity: u-velocity axial profiles on the symmetry plane in 

the SBLI region. .......................................................................................................................26

Figure 26. Turbulence model sensitivity: v-velocity axial profiles on the symmetry plane in 

the SBLI region. .......................................................................................................................27

Figure 27. Effect of shock-generator wedge angle on the test section flowfield – symmetry 

plane numerical contours of: (a) Mach number and (b) numerical Schlieren. Plots are in 

mm. ..........................................................................................................................................29

Figure28. Effect of shock-generator wedge angle on the SBLI: Mach number numerical 

contours on the symmetry plane near the lower wall. Plots are in mm. ..................................30

List of Tables 

Table 1. Test cases for SBLI workshop. .........................................................................................2

Table 2. Summary of simulation parameters. .................................................................................6

 
vi 

INTENTIONALLY LEFT BLANK. 
vii 

Acknowledgments 

The authors would like to acknowledge the organizers and participants of the American Institute 

of Aeronautics and Astronautics (AIAA) Shock Boundary Layer Interaction (SBLI) Workshop 

held in Orlando, Florida, on 8 January 2010, for provision of experimental data and intellectual 

input on the current subject. This work was supported in part by a grant of high-performance 

computing time from the U.S. Department of Defense (DoD) High Performance Computing 

Modernization Program at the ARL DoD Supercomputing Resource Center, Aberdeen Proving 

Ground (APG), Maryland. 

 
viii 

INTENTIONALLY LEFT BLANK. 
1 

1. Introduction 

The aerospace community has significant interest in the accurate numerical prediction of shockboundary layer interaction (SBLI) using computational fluid dynamics (CFD). SBLI occurs in 

both external and internal aerodynamic problems, including flow over airfoils, wings, rotors, 

ballistic projectiles, turbomachinery blades, and high-speed inlets and combustors. To facilitate 

the design and analysis of such aerospace systems, the SBLI phenomena must be isolated, 

modeled, and studied. 

Figure 1 provides a simple illustration of SBLI on a wind tunnel wall induced by supersonic flow 

over a flat plate. When the supersonic flow encounters the leading edge of the plate, a shock 

wave propagates towards the lower wall where it interacts with the boundary layer, inducing an 

abrupt pressure gradient. Capturing detailed flow physics—such as the location of incident shock 

wave, boundary layer recirculation and separation, increase in boundary layer displacement and 

momentum thicknesses, corner flow influence, and unsteadiness of the reflected shock wave, 

among others—has proven to be a challenge (1, 7–10, 15, 19, 22).

Figure 1. Schematic showing the shock-boundary layer interaction in an 

internal flowpath (9). 

Recognizing the need to study SBLI, the U.S. Air Force Research Laboratory (AFRL) hosted a 

workshop in January 2010 at the AIAA Aerospace Sciences Meeting in Orlando, Florida, where 

current high-fidelity physics-based computational methods were assessed for prediction of SBLI 

phenomena (3, 4, 6, 13). While the impetus of the workshop was to study SBLI in relation to 

inlets for supersonic vehicles, SBLI is also of interest to the U.S. Army Research Laboratory 
2 

(ARL) in the study of external aerodynamics of supersonic projectiles and missiles. A variety of 

computational approaches was presented at the SBLI Workshop to include Reynolds-averaged 

Navier-Stokes (RANS) methods; Large Eddy Simulation (LES); hybrid RANS/LES, structured, 

unstructured, and adaptive gridding techniques; various upwind numerical schemes; and various 

turbulence models. 

An experimental database for comparison purposes was derived from two wind tunnel facilities, 

viz, (i) the Institut Universitaire des Systèmes Thermiques Industriels Wind Tunnel (IUSTI) in 

Marseille, France (this data was furnished by UFAST (4, 9), a European Union consortium 

sponsoring SBLI research); and (ii) the University of Michigan Wind Tunnel (UMICH) (4, 14),

working with the Collaborative Center for Aeronautical Sciences (CCAS). 

The experimental database was comprised of four SBLI test cases—one from UFAST and three 

from UMICH. Table 1 summarizes these test cases. Results for all four cases were presented by 

the authors at the SBLI Workshop. For case 1, additional simulations and analyses were 

completed following the workshop; for case 2, no additional simulations, only further analyses, 

were completed. 

Table 1. Test cases for SBLI workshop. 

Case # Mach Shock Generator Angle Data Source 

1 2.25 8.0 UFAST 

2 2.75 7.75 UMICH 

3 2.75 10.0 UMICH 

4 2.75 12.0 UMICH 

Figure 2 shows a 2-D schematic of the UFAST wind tunnel, in which supersonic flow at Mach 

2.25 (test case 1) is delivered to a 122.1 mm × 170 mm (height × span) test section where a flat 

plate shock generator is located. The flow is fully turbulent as it approaches the test section with 

a boundary layer thickness of about 10 mm. The plate is 4 mm thick, has a sharp leading edge, is 

inclined at 8o to the tunnel axis, and spans the width of a test section. The tunnel walls have a 

0.28o divergence for a boundary layer correction. Streamwise particle image velocimetry (PIV) 

data for mean and fluctuating velocity components in the SBLI region was obtained on the 

tunnel centerline. Stereo-PIV (SPIV) data was also obtained on a plane 2 mm above and parallel 

to the tunnel lower wall. 

Figure 3 shows a 2-D schematic of the UMICH wind tunnel, in which supersonic flow at Mach 

2.75 is delivered to a 70 mm × 57 mm (height × span) test section where a wedge-shaped shock 

generator is located. The flow is fully turbulent as it approaches the test section with a boundary 

layer thickness of about 10 mm. The wedge is centered in the test section, supported by a strut, 

and spans half the tunnel. The bottom wall of the wedge is inclined at 7.75o, 10o, and 12o (test 

cases 2, 3, and 4) to the tunnel axis. SPIV data for mean and fluctuating velocity components 

was obtained on the centerline streamwise plane and several spanwise planes in the SBLI region. 
3 

SPIV inflow data was also obtained on two planes upstream of the SBLI region, one providing 

core flow data and the other boundary layer data. 

All simulations used steady-state, RANS methods and were conducted using a CFD++ v8.1.1 

flow solver—a commercial fluid flow solver by Metacomp Technologies, Inc (18). The effect of 

several turbulence models was investigated. The influence of grid resolution on the results was 

also addressed. 

Figure 2. UFAST wind tunnel flowpath. 

Figure 3. UMICH wind tunnel flowpath. 

2. Simulation Methodology 

The computational domains were three-dimensional and consisted of half the wind tunnel 

geometries, since they were symmetrical about the vertical centerline plane (or z=0 plane). The 

UFAST and UMICH computational domains, respectively, consisted of 20.4M and 18.0M 

structured, hexahedral cells generated using Pointwise v16.04R1 and Gambit v2.4.6 (see figures 

4 and 5). The UMICH grid additionally contained prism elements to facilitate gridding between 

the half-span shock generator and the tunnel side wall. To capture wall boundary layer effects, 

the initial grid wall spacing was 5x10–6 m (0.005 mm) in both cases. This ensured y+ values of 

less than 0.5 along the wind tunnel test section walls, and less than 1.0 otherwise, for 

compatibility with a solve-to-the-wall strategy. The UFAST and UMICH grids were decomposed 

for parallel processing on 136 and 120 processors, respectively. Computations were performed 

on “Harold,” a SGI Altix ICE 8200 Supercomputer, and “MJM,”a Linux Networx Advanced 
4 

Technology Cluster, both housed and managed by the ARL DoD Supercomputing Resource 

Center (DSRC) at Aberdeen Proving Ground (APG), MD. 

Figure 4. UFAST computational domain: (a) full length view, ~20.4M cells, (b) 3-D grid detail in test-section, 

and (c) 2-D grid detail in test-section. Grids generated in Pointwise v16.04R1. Note: only half the 

geometry was gridded.

(c) 

Blue region 

where grid 

points were 

doubled to 

produce the 

fine grid 

Plate Shock Generator 

(a) 

Plate Shock Generator 

Symmetry Plane 

Lower Wall 

Inflow Plane 

Outflow Plane 

(b) 
5 

Figure 5. UMICH computational domain: (a) full length view, ~18M cells, (b) 3-D grid detail in test-section, and 

(c) 2-D grid detail in test-section. Grids generated in Gambit v2.4.6. Note: only half the geometry was 

gridded. 

A grid independence study was conducted for the UFAST case using the SST turbulence model, 

where the baseline grid, termed the “medium” grid, consisted of 20.4M cells. The number of grid 

points in each x, y, z direction were halved (including the initial wall spacing) to produce the 

“coarse” grid consisting of 2.5M cells. The grid points were then doubled in each x, y, z 

direction (including the initial wall spacing), but only for a rectangular region enveloping the 

predicted SBLI zone (rendered in blue in figure 4c). This resulted in the “fine” grid that 

consisted of 47.4M cells. 

Stagnation pressure and temperature conditions, as specified by the experiments, were imposed 

at the tunnel inlets (see table 2), while a centroidal extrapolation boundary condition was used at 

the outflow planes. The symmetry planes were modeled as inviscid walls (i.e., the “symmetry” 

boundary condition in CFD++). A viscous wall function boundary was used to model the strut 

(c) 

Wedge Shock Generator

(b) 

(a) 

Wedge Shock Generator

Symmetry Plane 

Lower Wall 

Outflow Plane 

Inflow Plane 
6 

that supported the wedge-shaped shock generator in the UMICH tunnel; this maintains the same 

wall grid spacing on the symmetry plane and strut, thereby conserving grid size. All other walls 

were modeled as viscous, adiabatic walls with an initial y+ spacing of less than 1. 

Table 2. Summary of simulation parameters. 

PARAMETER UFAST UMICH 

Shock Generator Angle 8.0o 7.75o, 10o, 12o 

Mach # 2.25 2.75 

Air Inlet Total 

Conditions P0=50.5 kPa, T0=293 K P0=101 kPa, T0=293 K 

Outflow Centroidal Extrapolation 

Wall Boundaries Viscous, No-Slip, Adiabatic, except UMICH support strut 

Numerics 

3-D, compressible, steady-state, RANS; finite volume method; point-implicit time 

integration w/ local CFL ramped from 1–40; multigrid W-cycle method; implicit 

temporal smoothing; second-order inviscid spatial discretization; upwind HLLC 

scheme; TVD continuous flux limiter 

Turbulence Models 

(i) Menter’s Shear Stress Transport (SST) 

(ii) Spalart-Allmaras (SA) 

(iii) Realizable k-ε (RKE) 

(iv) Cubic k-ε (CKE) 

(v) Goldberg’s Rt Model (RT) 

(vi) Goldberg's k-ε-R (KER) 

(vii) Reynolds Stress Transport (RSM) 

(i) Menter’s Shear Stress Transport (SST) 

(ii) Spalart-Allmaras (SA) 

(iii) Realizable k-ε (RKE) 

Inlet Turbulence Intensity 0.1% 

Turbulence Length Scale 0.15175 m 

Inlet Turbulence Intensity 1%; 

Turbulence Length Scale 0.08935 m 

Computational Domain 

(Baseline Grids) 

~20.4M Hexahedral Cells; 1⁄2 Geometry 

136 Processors 

Pointwise v16.04R1 

~18M Hexahedral Cells; 1⁄2 Geometry 

120 Processors 

Gambit v2.4.6 

Δswall= 0.005 mm → y+ < 1 

Computing 

Hardware 

“Harold” SGI Altix ICE 8200 

10,752 2.8-GHz Intel Xeon Nehalem Cores 

1344 Nodes, 24 GB RAM/ Node 

4x DDR Infiniband Interconnect 

“MJM” Linux Networx Advanced Tech Cluster 

4400 3.0-GHz Intel Woodcrest Cores 

1100 Nodes, 8 GB RAM/ Node 

4x DDR Infiniband Interconnect 

CFD++ v8.1.1 (18), a commercial fluid flow solver, was used to solve the 3-D, compressible, 

steady-state, RANS equations using a finite volume method. A point-implicit time integration 

scheme with local time-stepping, defined by the CFL (Courant-Friedrichs-Lewy) number, was 

used to advance the solutions towards steady-state. The CFL number was ramped typically from 

1 to 40 over the first few hundred iterations, and remained at 40 until convergence. The multigrid 

W-cycle method with a maximum of four cycles and a maximum of 20 coarse grid levels was 

used to accelerate convergence. Implicit temporal smoothing was applied for increased stability. 

First-order spatial discretization was used for the first few hundred iterations, which then blended 
7 

over 100 iterations to second-order discretization. The inviscid flux function was an upwind 

scheme using a Harten-Lax-van Leer-Contact (HLLC) Riemann solver and Metacomp’s multidimensional Total-Variation-Diminishing (TVD) continuous flux limiter. 

Convergence was partially monitored by the reduction in the right-hand side residuals of the 

Navier-Stokes equations averaged over all cells at each iteration. The UFAST and UMICH cases 

typically demonstrated 8 and 4–5 orders of magnitude reduction in the residuals, respectively 

(see figure 6). Convergence for the UMICH case was more restricted, likely due to the prism 

elements in the test section grid. For the UFAST case, the exception was the seven-equation 

RSM turbulence model, which demonstrated only a 2–3 order of magnitude reduction in 

residuals. Also, it was ensured that the net mass and energy fluxes through the computational 

domain were driven to zero to satisfy mass and energy conservation laws. 

Figure 6. Sample residual histories of the right-hand side equations: (a) UFAST – SST and (b) UMICH – SST. 

The simulations were sensitive to the choice of turbulence model, as will be shown in section 3. 

Therefore, several turbulence models were investigated. Although some models are known to 

produce better results than others for internal flows, the authors tested almost all turbulence 

models available within CFD++ for the UFAST case, mainly for the purpose of completeness 

and future referencing. 

For the UFAST case, the turbulence models tested were the: 

1. Menter’s Shear Stress Transport (SST) 2-equation model (16, 17, 20) 

2. Spalart-Allmaras’s (SA) 1-equation model (20, 21) 

3. Realizable k-ε (RKE) 2-equation model (5, 20) 

4. Cubic k-ε (CKE) non-linear, 2-equation model (11) 

5. Goldberg’s Rt (RT) 1-equation model (11) 

6. Goldberg's k-ε-R (KER) 3-equation model (12) 

(a) (b) 
8 

7. Reynolds Stress Transport (RSM) 2nd moment closure, 7-equation model (2) 

For the UMICH case, only the first three of the aforementioned models were used. (See table 2). 

The flows were assumed to be fully turbulent— i.e., no transition modeling was used. 

The inlet turbulence intensity was chosen as 0.1% for the UFAST case (as determined by the 

experiment) and 1% for the UMICH case (as typically used for similar internal flows). The 

turbulent length scale was taken to be the square root of the cross-sectional area (as 

recommended by Metacomp (18)) of the inflow plane for the UFAST tunnel—i.e., 0.15175 m—

and the test section for the UMICH tunnel—i.e., 0.08935 m. The laminar-to-turbulent Prandtl 

number ratio was chosen as default value 0.8. Metacomp’s wall-bounded compressibility 

correction (18) was applied to realize diffusive mixing in the turbulent regions, which would 

otherwise be under-predicted in compressible flows. 

3. Results 

The following compares experimental and numerical data for the UFAST (section 3.1) and 

UMICH (section 3.2) wind tunnels. Plots include 3-D flowfield contours, numerical oil flow 

visualizations, and 2-D and 1-D velocity and perturbation velocity profiles for several turbulence 

models. Grid sensitivity is also demonstrated for the UFAST case. Except where turbulence 

models and grid densities are compared, the results are for the SST turbulence model using the 

“medium” grid, as this was the assumed baseline case. For the 2-D contour plots, the end-values 

on the legends are not necessarily the min/max values of the plotted flow variable; the legend 

end-values were, instead, altered for best visual rendering. All length scales are in meters, unless 

otherwise stated. 

3.1 UFAST Wind Tunnel 

Figure 7 provides a 3-D perspective of the flow in the tunnel test section, with Mach number 

contours on the symmetry plane and oil flow streamlines on the lower wall. Figure 8 compares 

experimental (9) and numerical oil flow streamlines on a plane 0.002 m above the lower wall. 

The adverse pressure gradient imposed by the incident shock induces a region of recirculation on 

the lower wall and thickens the boundary layer. The influence of the corner flow on the 

centerline core flow is very evident in both experiment and CFD, implying that the flow is highly 

three-dimensional. This was purported to be the result of shock waves thickening and possibly 

separating the corner flow boundary layers, which then pushes flow away from the walls towards 

the centerline, therefore altering the strength of the centerline SBLI (3). This effect will also be 

shown in the UMICH case in section 3.2. The coupling of the centerline and corner SBLIs, 

therefore, necessitates simultaneous accurate numerical prediction of both flows. However, due 

to the lack of experimental corner flow data, the CFD numerical modeling of the corner flow 

could not be verified; this, in turn, adds a source of uncertainty in predicting the centerline flow. 
9 

Note that the scales of the experimental and numerical plots in figure 8 are different. For the 

region z = –0.06–0.06 m, in the vicinity of x = 0.320 m, the centerline SBLI flow structure 

obtained via CFD (figure 8b) compares very well with that obtained via experiment (figure 8a). 

Figure 7. Three-dimensional rendering of UFAST numerical flowfield for SST turbulence model showing Mach 

number contours on the tunnel symmetry plane and oil flow streamlines on the tunnel lower wall. 

Figure 8. Transverse profile in the interaction zone showing centerline and corner SBLIs: (a) experimental SPIV data 

(9) on a plane 2 mm above the lower wall (scale units in mm), (b) CFD streamline data using SST 

turbulence model on a plane 2 mm above the lower wall (scale units in m), and (c) CFD streamline data 

using SST turbulence model on the lower wall (scale units in m). 

Although seven turbulence models were simulated (see table 2), the contour plots of figures 9 

and 10 depict only the “best” four, while the line plots of figures 11–13 depict all seven. The 

“best” turbulence models refer to those that compared best with experiment. In no particular 

order, these were the RSM, SA, Menter’s SST and Goldberg’s Rt models. 

Figure 9 compares u-velocity and the numerical Schlieren (density gradient in the axial 

direction) between different turbulence models. The plots show the shock structure—a sequence 

(a) (b) (c) 

z 

x 

Oil Flow 

Streamlines 

on Lower 

Wall 

Symmetry 

Plane Mach 

No. Contours 

Plate Incident 

Shock Wave 

Reflected 

Shock Wave

Corner

SBLI 

Flow 

Direction 

Centerline 

SBLI 
10 

of oblique shock and expansion waves—induced by the plate in the wind tunnel test section. The 

different turbulence models produce slightly different characteristics, especially where the waves 

interact with the flow near the walls. The shock emanating from the leading edge of the plate 

impinges the boundary-layer on the bottom wall in the vicinity of x≈0.3–0.35 m (the SBLI zone). 

Typical aerodynamic SBLI characteristics consistent with that observed in the experiment can be 

identified (particularly from the Schlieren plots), such as the incident and reflected shocks, a λshock, an expansion fan, and increased boundary layer thickness due to adverse pressures. 

Figure 9. Turbulence model comparison on symmetry plane: (a) u-velocity and (b) numerical Schlieren. 

Figure 10 compares u- and v-velocities in the SBLI zone between the experimental and 

numerical results. The SA and RSM models best capture the size and shape of the boundary 

layer in the interaction region, as well as the location of the interaction. The SST and RT models 

slightly over-predict the size of the interaction region; they also predict the interaction to occur 

slightly more upstream than that of the experiment. 

(a) (b) 
11 

Figure 10. Experiment vs. CFD on symmetry plane in SBLI region: (a) u-velocity and (b) v-velocity. 

Figures 11–13 compare 1-D axial profiles of u-velocity, v-velocity, and uʹvʹ Reynolds stress in 

the SBLI region. Upstream of the shock-induced pressure gradients, the velocity and perturbation 

velocity profiles are reasonably captured by all turbulence models. Progressing downstream, the 

disparity increases through the main interaction region, after which the agreement is again 

reasonable. There is increased sensitivity to turbulence modeling for the v-velocity and uʹvʹ

Reynolds stress components. The observed discrepancies in figures 11–13 are partially due to the 

variation in the prediction of the SBLI location, as shown in figure 10. This error would be 

reflected as a translation of the numerical data along the x-axis. 

(a) (b) 
Figure 11. Turbulence model sensitivity: u-velocity axial profiles on the symmetry plane in the SBLI region. 

12 
Figure 12. Turbulence model sensitivity: v-velocity axial profiles on the symmetry plane in the SBLI region. 

13 
Figure 13. Turbulence model sensitivity: uʹvʹ axial profiles on the symmetry plane in the SBLI region. 

14 
15 

Figures 14–17 show the results of a grid density study using the SST turbulence model (the 

aforementioned baseline model), where the coarse, medium, and fine grids consisted of 2.5, 20.4, 

and 47.4 million cells, respectively (as described in section 2). Figure 14 shows u- and v-velocity 

contour plots on the symmetry plane for each grid level. While the medium and fine grid results 

show qualitatively similar results, it is apparent that higher resolution of the shock-boundary 

layer interactions is obtained with each increase in grid density. 

Figure 14. Grid sensitivity: (a) u-velocity and (b) v-velocity symmetry plane contours in SBLI region, SST model. 

Figures 15–17 compare 1-D axial profiles of u-velocity, v-velocity, and uʹvʹ Reynolds stress on 

the symmetry plane for the three grid levels. For the u-velocity variable, the medium and fine 

grid plots overlay each other, while for the v-velocity and uʹvʹ Reynolds stress, the medium and 

fine grids very closely match. This indicates that grid independence was achieved with the 

medium grid and was, therefore, deemed sufficient for all other UFAST computations. 

(a) (b) 
Figure 15. Grid sensitivity: u-velocity axial profiles on the symmetry plane in the SBLI region, SST model. 

Note that the fine grid results (green) overlay that of the medium grid (blue). 

16 
Figure 16. Grid sensitivity: v-velocity axial profiles on the symmetry plane in the SBLI region, SST model. 

17 
Figure 17. Grid sensitivity: uʹvʹ axial profiles on the symmetry plane in the SBLI region, SST model. 

18 
19 

For a symmetric duct with a uniform symmetric inflow (as is the case here), it is expected that 

the flowfield downstream of the inflow would also be symmetric. This would normally be true 

for both experimental and numerical flowpaths. However, it was briefly discussed at the SBLI 

Workshop that flow asymmetries were observed in some of the UFAST experiments; these 

experiments may or may not have included the one simulated in this study, but were performed 

using the same wind tunnel. It was also suggested that this may also occur in numerical 

simulations and that the results may be physically real. Recall that all simulation results 

presented so far were for half of the UFAST flowpath, since a symmetric flowfield was assumed. 

To explore possible flow asymmetries, the baseline medium level “half” grid was mirrored to 

produce the “full” grid, and the simulation was re-run (using the SST turbulence model). 

Mirroring the grid eliminates the possibility of an asymmetric flowfield developing because of 

an asymmetric grid. These results also provide some qualitative insight into the spanwise 

flowfield profiles. 

Figure 18 compares pressure contours on the lower wall (where the SBLI occurs) for the full grid 

and half grid. The contour plot for the half grid is mirrored for better visual comparison. Figures 

19–21 compare pressure, skin friction, and u-velocity at three axial locations—x=0.20, 0.32, and 

0.40 m. The pressure and skin friction line data were extracted from the lower wall across the 

span (z-direction) of the tunnel. To verify symmetry using the line plots, the +z and –z data were 

superimposed on each other, i.e., the -z data was mirrored about the z=0 axis. All results indicate 

that, from a numerical standpoint, the flowfield using the full grid is very symmetric and 

produces the same results as the half grid. It is possible that asymmetries observed in the 

experiment may be due to spanwise asymmetries in the tunnel hardware and/or inflow 

conditions. 

Figure 18. Pressure contours on the lower wall for the (a) full grid and (b) mirrored half grid, SST model. 

Mirror

Line 
20 

 

Figure 19. Spanwise profiles of pressure, skin friction and velocity using full grid, X=0.20 m, SST model. 

 

Lower 

Wall 

Plate 

Upstream of SBLI 

Interaction Zone 
21 

 

Figure 20. Spanwise profiles of pressure, skin friction and velocity using full grid, X=0.32 m, SST model. 

 

Plate SBLI Interaction 

on Lower Wall 
22 

 

Figure 21. Spanwise profiles of pressure, skin friction and velocity using full grid, X=0.40 m, SST model. 

3.2 UMICH Wind Tunnel 

As described in section 1—Introduction, the results presented here were those presented at the 

AIAA SBLI Workshop. No further simulations, other than additional post-processing of the data, 

were completed after the Workshop. The UMICH results include three shock-generator wedge 

angles, viz, 7.75°, 10°, and 12° (test cases 2, 3, and 4). Results are first presented for the 7.75° 

case, which includes comparisons of three turbulence models, viz, SST, SA, and RKE. The effect 

of wedge angle on the flowfield is then compared via numerical contour plots in the tunnel 

section. All experimental and numerical test conditions and run parameters were the same for 

each wedge angle case. All UMICH simulations were conducted using a single grid level. 

Figure 22 provides a 3-D perspective of the flow in the tunnel test section, with Mach number 

contours on the symmetry plane and oil flow streamlines on the lower wall. Although this result 

is for the SST turbulence model with wedge angle 7.75°, it represents the general flowfield of all 

the UMICH cases. The adverse pressure gradient imposed by the incident shock induces a region 

Downstream of 

Plate & SBLI 

Interaction Zone
23 

of recirculation on the lower wall and thickens the boundary layer. Similar to the UFAST case, 

the influence of the corner flow on the centerline core flow is very evident in the CFD results, 

implying that the flow is highly three-dimensional. This flow structure is due to the coupling of 

the corner and centerline SBLIs, as described in the UFAST case (section 3.1). As previously 

stated, lack of corner flow data lends to a source of uncertainty in numerically predicting the 

centerline flow. 

Figure 22. Three-dimensional rendering of UMICH numerical flowfield showing Mach number contours on the 

tunnel symmetry plane and oil flow streamlines on the tunnel lower wall, SST model, 7.75° case. 

Figure 23 compares u-velocity and the numerical Schlieren (density gradient in the axial 

direction) between different turbulence models for the 7.75° case. The plots show the shock 

structure, a sequence of oblique shock and expansion waves induced by the wedge-shaped shock 

generator in the wind tunnel test section. The different turbulence models produce slightly 

different characteristics, especially where the waves interact with the flow near the walls. The 

choice of turbulence model seems to have a more subtle effect on the overall flowfield than for 

the UFAST case, possibly because of a higher inflow Mach number and resulting stronger shock 

waves in the test section. The shock emanating from the leading edge of the plate impinges the 

boundary-layer on the bottom wall in the vicinity of x≈0.03–0.04 m (the SBLI zone). 

Aerodynamic flow structures similar to that of the UFAST case can be identified in the SBLI 

region. 

Reflected 

Shock Wave

Incident 

Shock Wave 

Symmetry 

Plane Mach 

No. Contours Wedge 

Centerline 

SBLI 

Corner 

SBLI 

Oil Flow 

Streamlines on 

Lower Wall Flow 

Direction 
24 

Factors such as grid resolution and corner flow influence may be partially responsible for the 

discrepancies. Future studies may include a grid density study, additional turbulence models, and 

data comparison for other flow variables (such as the Reynolds stress). 

Figure 23. Turbulence model comparison on symmetry plane: (a) u-velocity and (b) numerical Schlieren, 7.75° case. 

Figure 24 compares u- and v-velocity contours in the SBLI zone between the experimental and 

numerical results for the 7.75° case. Although the extent of the interaction is under-predicted by 

the SST model, this model best captures the size and shape of the recirculation region in the 

boundary layer, as well as the location of the interaction. The SA and RKE models show smaller 

interaction regions that occur slightly more downstream (~0.002–0.003 m) than in the 

experiment. 

(a) (b) 
25 

Figure 24. Experiment vs. CFD on symmetry plane in SBLI region: (a) u-velocity and (b) v-velocity. 

Figures 25 and 26 compare 1-D axial profiles of u- and v-velocity in the SBLI zone between the 

experimental and numerical results for the 7.75° case. The CFD u-velocity profiles match very 

well with that of the experiment for the SST and SA models, and fairly so for the RKE model. 

Like the UFAST case, the v-velocity profiles are more sensitive to turbulence modeling. There is 

a larger discrepancy between the numerical and experimental results for this variable. Again, the 

SST and SA models are favored since the RKE model provides the worst comparison with 

experiment. However, improved prediction of the v-velocity variable is needed. 

(a) (b) 
Figure 25. Turbulence model sensitivity: u-velocity axial profiles on the symmetry plane in the SBLI region. 

26 
Figure 26. Turbulence model sensitivity: v-velocity axial profiles on the symmetry plane in the SBLI region. 

27 
28 

The following results compare the effect of the shock-generator wedge angle on the test section 

flowfield and the SBLI zone. These wedge angles include 7.75°, 10°, and 12°, which comprise 

test cases 2, 3, and 4 (see section 1). Figure 27 shows Mach number contours and the numerical 

Schlieren on the test section symmetry plane for each wedge angle. While the shock structures 

are similar, it is observed that a larger wedge angle induces a leading edge shock that is stronger 

and has a larger angle of incidence (angle between the shock plane and the tunnel axis). The 

Mach number contours of figure 28 shows that this incurs stronger adverse pressure gradients at 

the lower wall, producing the observed larger recirculation region. 

A larger incident shock angle, induced by a larger wedge angle, is expected to move the SBLI 

location further upstream. However, note that the leading edge of the wedge is further 

downstream for the larger wedge angles (figure 27). The net effect is to, instead, move the SBLI 

location further downstream (figure 28). 

 
29 

(a) 

(b) 

Figure 27. Effect of shock-generator wedge angle on the test section flowfield – symmetry plane numerical 

contours of: (a) Mach number and (b) numerical Schlieren. Plots are in mm. 
30 

Figure 28. Effect of shock-generator wedge angle on the SBLI: Mach number numerical 

contours on the symmetry plane near the lower wall. Plots are in mm. 

4. Summary and Conclusions 

Shock-boundary layer interaction has been numerically simulated using steady-state RANS 

methods in two wind tunnel facilities, viz, UFAST and UMICH. Typical flow characteristics 

were identified, such as the recirculation region, incident, reflected, and λ shocks, and increase in 

boundary layer thickness. The accuracy of the numerical predictions in the shock-boundary layer 

interaction region was good in some cases and poor in others, and highly dependent on 

turbulence model chosen. The prediction of the v-velocity component and Reynolds stresses 

were more sensitive than the prediction of the u-velocity component. Observed inaccuracies may 

be the result of one or more factors. Inflow turbulence conditions may not have been adequately 

modeled due to lack of data from the experiments. Also, inadequacies in the turbulence model 

formulation, such as their inherent empiricisms, can lead to errors that are difficult to 

characterize. A lack of experimental corner flow data, which was shown to be coupled to the 

centerline flow, also adds a measure of uncertainty in the numerical results. Future studies will 

attempt to address some of these issues. 

A suite of RANS turbulence models was studied, particularly for the UFAST case. Based on 

these studies, it appears that four of these models are best for internal flows where shockboundary layer interaction phenomena occur. These were the Menter’s SST, the SA, the 

Goldberg RT, and the RSM models. However, the choice of turbulence model is contingent upon 
31 

the type of problem being solved, as well as time/computing constraints. For the instance, the 

SST model is a two-equation model that takes about twice as long to produce a converged 

solution as compared to the one-equation SA or RT models. Therefore, the SA and RT models 

may be more desirable where rapid solutions are needed, as is the case in industrial applications. 

The RSM model—because it is a seven-equation model and requires very fine grid spacing near 

viscous walls, resulting in larger grid sizes—demands far more computational resources, and 

may also be less stable. Hybrid RANS/LES will be used in a future study, not only for 

comparison purposes, but to also resolve possible unsteady effects, such as resonance of the 

reflected shock wave from the boundary layer, as observed in the UFAST experiment. 

Given the large grid sizes required to quantitatively resolve the flow in the SBLI region, an 

adaptive gridding technique may offer improved accuracy, while minimizing grid sizes. In such a 

case, grid points would be concentrated in high gradient regions via an automated procedure. 

The results of this study conform to the general conclusions derived from the AIAA SBLI 

Workshop (3, 6, 13). Error margins observed in the numerical solutions submitted by all 

participants indicated that, for a single simulation methodology, good prediction of one flow 

variable does not guarantee good prediction of another. Therefore, no discernable trends could be 

elicited from the collective results of the participants. However, it was agreed that the dominant 

factors in the RANS prediction of SBLI were the choice of turbulence model and grid density. 
32 

5. References 

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1634. 

9. Dupont, P.; Debieve, J. F.; Dussauge, J. P.; Piponniau, S.; Souverein, L. J. Final Report on 

the Measurements of Shock Reflection at M=2.25. Groupe Supersonique, Laboratoire 

IUSTI, Ecole Polytechnique Universitaire de Marseille, Deliverable No 3.3.5, May 12, 2009. 

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Velocimetry Measurements of Oblique Shock Reflection with Separation. AIAA Journal

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Flow Predictions Using Linear and Nonlinear Turbulence Closures. AIAA Journal of 

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To Wall-Bounded Compressive Flows; AIAA 98-0323; January 1998. 
33 

13. Hirsch, C. Lessons Learned from the First AIAA-SWBLI Workshop CFD Simulations of Two 

Test Cases; AIAA 2010-4824, June/July 2010. 

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Journal 1992, 30 (6), 1657–1659. 

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Applications. AIAA Journal August 1994, 32 (8), 1598–1605. 

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93–2906; July 1993. 

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Aerodynamic Calculations. AIAA Journal October 2007, 45 (10), 2544–2553. 

 
34 

List of Symbols, Abbreviations, and Acronyms 

AFRL U.S. Air Force Research Laboratory 

APG Aberdeen Proving Ground 

ARL U.S. Army Research Laboratory 

CCAS Collaborative Center for Aeronautical Sciences 

CFD computational fluid dynamics 

CKE cubic k-ε

DoD Department of Defense 

DSRC DoD Supercomputing Resource Center 

HLLC Harten-Lax-van Leer-Contact 

IUSTI Institut Universitaire des Systèmes Thermiques Industriels Wind Tunnel 

KER k-ε-R 

LES Large Eddy Simulation 

PIV particle image velocimetry 

RANS Reynolds-averaged Navier-Stokes 

RKE realizable k-ε

RSM Reynolds Stress Transport 

RT Rt

SA Spalart-Allmaras 

SBLI shock-boundary layer interaction 

SPIV Stereo-PIV 

SST Shear Stress Transport 

TVD Total-Variation-Diminishing 

UMICH University of Michigan Wind Tunnel 
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