Implicit largeeddy simulations of Mach 2.05 turbulent boundary layer interactions with oblique impinging shockwaves were carried out for shock generator angles of 8° and 9°. Both the streamwise extent of the separated region and the intensity of the velocity fluctuations are augmented when the strength of the impinging oblique shockwave is increased from 8° to 9°. Temporal Fourier transforms of the spanwiseaveraged wallpressure coefficient indicate lowfrequency unsteadiness at separation and midfrequency content downstream of reattachment. The wallpressure fluctuations were analyzed with the proper orthogonal decomposition. The modal analysis reveals pronounced 3D lowfrequency wallpressure fluctuations for the stronger interaction. Overall, the present findings provide advanced perspectives on lowfrequency wallpressure fluctuations in turbulent shockwave boundary layer interactions that may lead to spanwise nonuniformity of the separation and shedding with possible implications for the design of structural panels on highspeed vehicles.
NOMENCLATURE
Normal characters
 A

Amplitude
 a_{i}

Time coefficient
 c_{f}

Skinfriction coefficient
 c_{p}

Pressure coefficient
 f

Frequency
 H

Shape factor
 k

Spanwise mode number
 L

Length scale
 M

Mach number
 p

Pressure
 Pr

Prandtl number
 Q

Vortex identification criterion
 Re

Reynolds number
 T

Temperature
 t

Time
 u, v, w

Velocities
 $ u \tau $

Friction velocity
 x, y, z

Coordinates
Greek symbols
Subscripts
Superscripts
I. INTRODUCTION
Turbulent shockwave boundary layer interactions (SBLIs) occur for both external and internal flows and can exhibit a lowfrequency unsteadiness.^{1} The associated wallpressure fluctuations can excite structural vibrations of thin wall panels.^{2–5} Although different explanations for the occurrence of the lowfrequency unsteadiness have been offered, no consensus has been reached with regard to the underlying physical mechanisms. Compression ramp experiments for M = 2, 3, and 5 suggested that unsteadiness of the incoming turbulent boundary layer had a strong effect on the temporal behavior of the separation region.^{6–8} In particular, the approach boundary layer wallpressure fluctuations were shown to be strongly correlated with the separation shock motion.^{6,7} Particle image velocimetry (PIV) and planar laser scattering (PLS) measurements for a Mach 2 compression ramp interaction provided insight into the temporal behavior of the separation and the spanwise coherent structures in the turbulent approach boundary layer.^{8} On the other hand, Speaker and Ailman^{9} reported a low correlation between the approach boundary layer pressure fluctuations and the separation region for a M = 3.45 impinging shock SBLI. An improved understanding of the origin of the lowfrequency unsteadiness could lay the foundation for engineering models that predict its occurrence and intensity and provide a basis for the development of strategies aimed at its control. Knowledge of the modal structure of the wallpressure fluctuations may guide the design of wall panels that can better withstand the aerodynamic loads. In this paper, results from implicit largeeddy simulations (ILES) of two SBLI with identical freestream conditions and approach flow boundary layer properties but different interaction strengths are compared with respect to the lowfrequency unsteadiness and modal content of the wallpressure fluctuations.
A brief review of the literature on turbulent SBLI as relevant for this paper is provided. Pirozzoli and Grasso^{10} associated the lowfrequency unsteadiness with a largescale oscillatory motion of the separation. Humble et al.^{11} observed spanwise deformations referred to as “rippling pattern” for a Mach 2.1 SBLI. Priebe and Martin^{12} found that the lowfrequency unsteadiness is more strongly related to a “breathing motion” of the separation bubble than to unsteadiness of the incoming boundary layer. Largeeddy simulations (LES) of a Mach 2.05 SBLI by Morgan et al.^{13} indicated that spanwise deformations of the separation line are related to streamwise unsteadiness of the reflected shockwave. They also found spanwisemodulated lowfrequency acoustic waves with a Strouhal number of $ S t \u2248 0.03$. Grilli et al.^{14} performed an LES of a Mach 2.88 compression ramp flow and observed unsteadiness with $ S t \u2248 0.023$ near separation and $ 0.36 < S t < 5$ near reattachment. The lowfrequency unsteadiness at separation is in good agreement with the range reported in the literature, e.g., Dussauge et al.^{15} Clemens and Narayanaswamy^{16} propose that an instability of the separation bubble is the dominant contributor to the lowfrequency unsteadiness. Rainbow schlieren deflectometry measurements for a Mach 3.1 oblique SBLI by Chaganti et al.^{17} revealed a dimensionless frequency of the reflected shock foot in the range $ 0.04 < S t < 0.06$. Adler and Gaitonde^{18} investigated Mach 2 compression ramp flows with 22.5° and 37.5° sweep angle and found that the intensity of the lowfrequency unsteadiness is related to the strength of the recirculation. Vanstone and Clemens^{19} analyzed Mach 2 swept SBLI with the proper orthogonal decomposition (POD)^{20,21} and found indications that the boundary layer superstructures are responsible for the midfrequency unsteadiness ( $ 0.01 < S t < 0.1$). The highfrequency unsteadiness was attributed to crossflow structures. Threadgill and Bruce^{22} investigated various Mach 2 oblique shock and compression ramp SBLIs experimentally. The lowfrequency peaks for oblique shock SBLIs with shock generator angles of ϑ = 7°, 8°, 9°, and 10° were in the range $ 0.024 < S t < 0.03$. Gross et al.^{23} found lowfrequency “rippling” structures along the separation shock foot for a Mach 2.3 oblique turbulent SBLI. Observed midfrequency unsteadiness ( $ 0.6 < S t < 1.6$) was attributed to the shedding of spanwise coherent structures.
In summary, the occurrence of the lowfrequency unsteadiness for turbulent SBLI is widely accepted. Several experiments and simulations revealed lowfrequency 3D flow deformations at separation and/or reattachment. This paper makes a connection between the lowfrequency unsteadiness and the wallpressure fluctuations near the mean reattachment location, which is of relevance for the design of thin wall panels.
The freestream conditions and approach flow boundary layer properties for the present simulations are identical to those of earlier LES by Morgan et al.^{13} and Lee and Gross.^{24} A schematic of the oblique interaction and the placement of the computational domain is provided in Fig. 1. Two cases are being considered. The shock generator angle for a baseline case is ϑ = 8°. For the second case with stronger interaction, the shock generator angle is increased to ϑ = 9° and the spanwise extent of the computational domain is doubled. The mean flow and turbulent statistics for both simulations are compared. The unsteady wallpressure data are analyzed with Fourier transforms and with the proper orthogonal decomposition (POD).^{20,21} The pressure fluctuations were analyzed because of their significance for the unsteady loading of surface panels, which can lead to vibration as previously demonstrated for a flat plate by Shinde et al.^{25} Finally, a brief summary and conclusions are offered.
II. METHODOLOGY
A. Governing equations and discretization
The compressible Navier–Stokes equations were solved with a finitevolume Navier–Stokes code by Gross and Fasel.^{26} The advection upstream splitting (AUSM^{+}up) scheme by Liou^{27} was employed to compute the convective fluxes. The fluxes were interpolated to the cell faces with a ninthorderaccurate weighted essentially nonoscillatory (WENO) scheme. The discretization of the viscous terms was fourthorderaccurate. The implicit secondorderaccurate trapezoidal rule was employed for time integration. The upwind discretization removes enough energy at the smallest scales to obviate the need for an explicit subgrid stress model. Since no wall model was employed, the present simulations can be classified as wallresolved implicit largeeddy simulations (ILES). The pressure was obtained from the ideal gas equation, $ p = \rho R T$, and the viscosity was obtained from Sutherland's law.
B. Normalization and freestream conditions
All flow quantities, such as velocities, density, pressure, temperature, and time, were normalized with the reference quantities shown in Table I. The asterisk indicates dimensional quantities. The reference Mach number and Prandtl number for the present simulations were M = 2.05 and Pr = 0.72. The approach flow freestream conditions and boundary layer properties (displacement and momentum thickness, $ \delta *$ and ϑ, and corresponding Reynolds numbers) were identical to those of Morgan et al.^{13} and are listed in Table II. The unit Reynolds number was 1 × 10^{6} per meter, and the reference length scale was $ L ref * = 0.02836 \u2009 m$.
Velocity .  Density .  Pressure .  Temperature .  Time .  Length . 

$ u \u221e *$  $ \rho \u221e *$  $ \rho \u221e * u \u221e * 2$  $ T \u221e *$  $ L ref * / u \u221e *$  $ L ref *$ 
Turbulence kinetic energy  Turbulence dissipation rate  
$ u \u221e * 2$  $ u \u221e * 3 / L ref *$ 
Velocity .  Density .  Pressure .  Temperature .  Time .  Length . 

$ u \u221e *$  $ \rho \u221e *$  $ \rho \u221e * u \u221e * 2$  $ T \u221e *$  $ L ref * / u \u221e *$  $ L ref *$ 
Turbulence kinetic energy  Turbulence dissipation rate  
$ u \u221e * 2$  $ u \u221e * 3 / L ref *$ 
M .  $ T \u221e *$ (K) .  $ p \u221e *$ (Pa) .  $ R e \delta *$ .  $ R e \u03d1$ .  $ \delta *$ (mm) .  ϑ (mm) . 

2.05  155.39  32 631  5700  1800  0.163  0.051 
M .  $ T \u221e *$ (K) .  $ p \u221e *$ (Pa) .  $ R e \delta *$ .  $ R e \u03d1$ .  $ \delta *$ (mm) .  ϑ (mm) . 

2.05  155.39  32 631  5700  1800  0.163  0.051 
C. Generation of turbulent boundary layer
D. Computational grids
The computational grids for the present simulations are illustrated in Figs. 3 and 4. The grid lines are equidistantly spaced in the streamwise and spanwise directions. In the wallnormal direction, grid lines were clustered near the wall such that the nearwall turbulence is well resolved. This is a requirement for wallresolved LES.
The number of cells and the domain extent in the streamwise, wallnormal, and spanwise directions are provided in Table III. The present grid resolution was chosen based on an earlier grid convergence study for the 8° shock generator case by Lee and Gross.^{24} The domain width was chosen to be approximately two times the separation length. The streamwise extent of the separated region for case 2 is about two times longer than for case 1. Therefore, the spanwise domain extent for case 2 was chosen to be double the spanwise domain extent for case 1.
Case .  Shock generator angle .  Number of cells .  Domain extent . 

1  $ \u03d1 = 8$°  $ 698 \xd7 168 \xd7 228$  $ 0.548 \xd7 0.188 \xd7 0.089$ 
2  $ \u03d1 = 9$°  $ 750 \xd7 168 \xd7 456$  $ 0.589 \xd7 0.188 \xd7 0.179$ 
Case .  Shock generator angle .  Number of cells .  Domain extent . 

1  $ \u03d1 = 8$°  $ 698 \xd7 168 \xd7 228$  $ 0.548 \xd7 0.188 \xd7 0.089$ 
2  $ \u03d1 = 9$°  $ 750 \xd7 168 \xd7 456$  $ 0.589 \xd7 0.188 \xd7 0.179$ 
E. Boundary conditions
Dirichlet boundary conditions were applied at the inflow and freestream boundary, and Neumann boundary conditions were employed at the outflow boundary. Flow periodicity was enforced in the spanwise direction. Noslip and nopenetration conditions were employed at the wall. The wall was considered adiabatic.
F. Generation of oblique shockwaves
For case 1, the same shock generator angle as in Morgan et al.^{13} was chosen, ϑ = 8°. For case 2, the shock generator angle was increased to ϑ = 9°. This resulted in a stronger oblique shockwave and a larger separation. The shock angles, σ, and the before and aftershock conditions for the impinging and reflected shockwaves were obtained from the Rankine–Hugoniot relations and are listed in Table IV. The flow states upstream and downstream of the oblique shockwave were enforced at the freestream boundary such that the inviscid shock impingement point was at x = 0.4.
Approach flow .  After impinging shock .  After reflected shock .  

Cases 1 and 2 .  .  Case 1 .  Case 2 .  .  Case 1 .  Case 2 .  
.  .  ϑ (°) .  8 .  9 .  .  8 .  9 . 
σ (°) .  36.499 .  37.517 .  .  43.055 .  45.399 .  
M_{1}  2.0380  M_{2}  1.7493  1.7127  M_{3}  1.4726  1.3992 
u_{1}  0.9968  $ u 2 / u 1$  0.9029  0.8916  $ u 3 / u 1$  0.8165  0.7875 
v_{1}  0  $ v 2 / u 1$  −0.1269  −0.1412  $ v 3 / u 1$  0  0 
w_{1}  0  $ w 2 / u 1$  0  0  $ w 3 / u 1$  0  0 
p_{1}  0.1732  $ p 2 / p 1$  1.5477  1.6304  $ p 3 / p 1$  2.3173  2.5570 
T_{1}  1.0053  $ T 2 / T 1$  1.1195  1.1538  $ T 3 / T 1$  1.2769  1.3156 
Approach flow .  After impinging shock .  After reflected shock .  

Cases 1 and 2 .  .  Case 1 .  Case 2 .  .  Case 1 .  Case 2 .  
.  .  ϑ (°) .  8 .  9 .  .  8 .  9 . 
σ (°) .  36.499 .  37.517 .  .  43.055 .  45.399 .  
M_{1}  2.0380  M_{2}  1.7493  1.7127  M_{3}  1.4726  1.3992 
u_{1}  0.9968  $ u 2 / u 1$  0.9029  0.8916  $ u 3 / u 1$  0.8165  0.7875 
v_{1}  0  $ v 2 / u 1$  −0.1269  −0.1412  $ v 3 / u 1$  0  0 
w_{1}  0  $ w 2 / u 1$  0  0  $ w 3 / u 1$  0  0 
p_{1}  0.1732  $ p 2 / p 1$  1.5477  1.6304  $ p 3 / p 1$  2.3173  2.5570 
T_{1}  1.0053  $ T 2 / T 1$  1.1195  1.1538  $ T 3 / T 1$  1.2769  1.3156 
G. Time intervals for obtaining mean flow and statistics
Both cases were first advanced in time from t_{0} to t_{1} to obtain statistically independent stationary flows. Then, time averages were computed over time intervals of $ t 2 \u2212 t 1$. The mean flows were averaged in the spanwise direction. Based on the averaged mean flows, statistical quantities were computed for time intervals of $ t 3 \u2212 t 2$. A summary of the time intervals for the present simulations is provided in Table V. The computational time step for both cases was $ \Delta t = 0.0005$.
Case .  Initial transient .  Time average .  Statistics .  Wall data capture .  N_{f} . 

t_{0}, t_{1}, and $ t 1 \u2212 t 0$ .  t_{1}, t_{2}, and $ t 2 \u2212 t 1$ .  t_{2}, t_{3}, and $ t 3 \u2212 t 2$ .  t_{1}, t_{3}, and $ t 3 \u2212 t 1$ .  
1  −15, 0, and 15  0, 12, and 12  12, 48, and 36  0, 48, and 48  1096 
2  −68, 0, and 68  0, 48, and 48  0, 48, and 48  0, 48, and 48  1178 
Case .  Initial transient .  Time average .  Statistics .  Wall data capture .  N_{f} . 

t_{0}, t_{1}, and $ t 1 \u2212 t 0$ .  t_{1}, t_{2}, and $ t 2 \u2212 t 1$ .  t_{2}, t_{3}, and $ t 3 \u2212 t 2$ .  t_{1}, t_{3}, and $ t 3 \u2212 t 1$ .  
1  −15, 0, and 15  0, 12, and 12  12, 48, and 36  0, 48, and 48  1096 
2  −68, 0, and 68  0, 48, and 48  0, 48, and 48  0, 48, and 48  1178 
III. RESULTS
A. Approach boundary layer analysis
The nearwall gridline spacing and time step in wall units are plotted in Fig. 6. The van Driest transformed gridline spacing upstream of the interaction near $ x \u2248 0.2$ is $ \Delta x + \u2248 14.3 , \u2009 \Delta y + \u2248 0.45$, and $ \Delta z + \u2248 7.0$ sufficient for wallresolved LES as shown by Lee and Gross.^{24} The time step in wall units is 0.43 and, thus, small enough for LES according to Choi and Moin.^{33}
In Fig. 7(a), distributions of the displacement thickness, $ \delta *$, momentum thickness, ϑ, and incompressible shape factor, H_{incomp}, are plotted vs the xcoordinate. For both cases, upstream of separation at x = 0.2, the incompressible shape factor obtains a value of $ H incomp \u2248 1.4$. The displacement and momentum thickness based Reynolds numbers at x = 0.2 are approximately $ R e \delta * = 5700$ and $ R e \theta = 1800$ [Fig. 7(b)].
The vanDriest transformed profiles for the two cases are nearly identical and in good agreement with the relationships for the viscous sublayer and loglayer (Fig. 9). The discontinuity near $ y + \u2248 10 3$ is due to the impinging oblique shockwave.
B. Mean flow analysis
The spanwiseaveraged mean flows are discussed first. Contours of the temporal and spanwise averaged streamwise velocity and temperature are provided in Fig. 10. The contours of the streamwise velocity reveal a mostly longer and somewhat taller separation for case 2 [Fig. 10(a)]. Also for case 2, the flow inside and downstream of the interaction is hotter [Fig. 10(b)]. The streamwise and wallnormal velocity profiles for case 1 are in a good agreement with Morgan et al.^{13} (Fig. 11).
Streamwise distributions of the wallpressure and skinfriction coefficient, c_{p} and c_{f}, are plotted in Fig. 12. For case 2, the c_{p}increase across the interaction is larger. The pressure rise across the interaction is in good agreement with Threadgill and Bruce.^{22} The c_{f}distributions reveal the downstream extent of the separated flow regions and the reverse flow intensity. Compared to case 1, for case 2 the flow separates more upstream and reattaches farther downstream. The peak c_{f} value for case 2 is $ c f \u2248$ −0.000 87 and slightly (6%) higher than for case 1 ( $ c f \u2248$ −0.000 82). Different from case 1, for case 2 the skinfriction coefficient dips down prior to reattachment.
The separation and reattachment locations, x_{s} and x_{r}, and the separation length, $ L s = x s \u2212 x r$, are summarized in Table VI. The separation length for case 2 is $ L s = 0.0876$ and 78% larger than for case 1, $ L s = 0.0491$.
Case .  ϑ (°) .  x_{s} .  x_{r} .  L_{s} . 

1  8  0.3363  0.3855  0.0491 
2  9  0.3138  0.4014  0.0876 
Case .  ϑ (°) .  x_{s} .  x_{r} .  L_{s} . 

1  8  0.3363  0.3855  0.0491 
2  9  0.3138  0.4014  0.0876 
Contours of the Reynolds normal stresses are provided in Fig. 13. In general, for case 2 stronger velocity fluctuations are observed throughout the interaction. For both cases, the large streamwise velocity fluctuations at separation [Fig. 13(a)] imply a streamwise motion of the separation shock foot. The wallnormal velocity fluctuations [Fig. 13(b)] increase throughout the interaction. The spanwise velocity fluctuations [Fig. 13(c)] are largest near separation, which may imply a rippling of the separation line. Downstream of the interaction, the velocity fluctuations are decaying as the turbulent boundary layer recovers its equilibrium state.
The Reynolds shear stress, $ u \u2032 v \u2032 \xaf$, is large in the upstream part of the interaction and downstream of reattachment (Fig. 14). A large Reynolds shear stress downstream of separated flow regions often implies the shedding of spanwise coherent structures as, for example, stated by Chen et al.^{35} The shedding of spanwise coherent structures would explain the initial dip of the skinfriction coefficient immediately downstream of separation (Fig. 12).
C. Unsteady flow analysis
1. Instantaneous flow visualizations
Instantaneous flow visualizations for both cases are provided in Fig. 15. Shown are isosurfaces of the Qvortex identification criterion by Hunt,^{36} flooded by the streamwise velocity, and numerical schlieren images (contours of the magnitude of the density gradient, $  \u2207 \rho $) for the z = 0 plane. The Q surfaces reveal the turbulent flow structures and the schlieren images reveal the shock system. For both cases, the turbulent approach boundary layer is wellresolved as indicated by the multiple hairpin vortices which are a trademark of turbulent boundary layers. The separation bubble and interaction region can be discerned from the images in the right column of Fig. 15.
2. Analysis of unsteady wallpressure data
Similar observations were made by Threadgill and Bruce^{22} for SBLI with $ \u03d1 =$ 8° and 9° shock generator angle. Their premultiplied power spectral density spectra revealed a significant peak with $ S t L s \u2248 0.025$ for ϑ = 8° and a band of peaks with $ 0.03 < S t L s < 0.065$ for ϑ = 9°. Premultiplied power spectra of the wallpressure coefficient for the location of the strongest wallpressure fluctuations (x = 0.349 for case 1 and x = 0.328 for case 2) compare favorably with data by Threadgill and Bruce^{22} (amplitude scaled by 0.01) for the nearest measurement location (Fig. 18).
3. Proper orthogonal decomposition of wallpressure coefficient
The wallpressure coefficient data were also analyzed with the POD. The temporal and spanwise average of the data were subtracted before the POD analysis. Two regions at separation and reattachment were selected that cover streamwise intervals over which the lowfrequency Fourier mode amplitudes in Fig. 16 are significant. The start and end points for the 2D POD analysis are provided in Table VII. The regions for the POD analysis of the wallpressure coefficient are outlined in Fig. 19.
Case .  Subdomain .  x_{0} .  x_{1} . 

1  Region 1: near separation  0.3097  0.3529 
Region 2: near reattachment  0.3529  0.4298  
2  Region 1:near separation  0.2901  0.3599 
Region 2: near reattachment  0.3599  0.4400 
Case .  Subdomain .  x_{0} .  x_{1} . 

1  Region 1: near separation  0.3097  0.3529 
Region 2: near reattachment  0.3529  0.4298  
2  Region 1:near separation  0.2901  0.3599 
Region 2: near reattachment  0.3599  0.4400 
The POD results for region 1 are discussed first. The eigenvalue magnitudes for case 2 are more than two times larger than for case 1 (Fig. 20). For case 1, a weak spectral gap between modes 2 and 3 suggests that modes 0–2 are more significant than the higher modes. For case 2, modes 0–3 are more significant than the higher modes. Compared to case 1, for case 2 the spectral gap between modes 0 and 1 is smaller.
Spectra of the POD time coefficients for region 1 are presented in Fig. 21. The mode 0 spectrum has a peak at $ S t L s \u2248 0.03$ in agreement with the Fourier spectra in Fig. 10. The corresponding eigenfunction is highly 2D (Fig. 22). As mode 0 is not paired with another mode with comparable time signature and mode shape, mode 0 is a stationary (i.e., nontraveling) pressure fluctuation. The mode shapes 1 and 2 suggest a pressure disturbance downstream of separation that is traveling in the spanwise direction. The associated time coefficient spectra have no dominant peaks. Mode 3 is again 2D, and mode 4 is a higher harmonic of modes 1 and 2.
As for case 1, for case 2 mode 0 is stationary and not traveling. However, different from case 1, mode 0 is noticeably 3D [Fig. 22(b)]. Modes 1 and 3 form a pair that captures a wave that is traveling in the spanwise direction. Mode 2 shares similarities with modes 0 and 1 and 3 and may facilitate an energy transfer between the modes. The modes 0 and 1 time coefficient spectra for case 2 feature multiple lowfrequency peaks in the range $ 0.02 < S t L s < 0.04$ [Fig. 21(b)] that are in good agreement with the most significant peaks at separation in Fig. 16(b).
The wallpressure distributions were reconstructed from the first six POD modes. Samples of the instantaneous reconstructed wallpressure distributions for positive and negative maxima of individual time coefficients (Fig. 23) are visualized in Fig. 24. As the temporal and spanwise averages were removed, the reconstructed c_{p} value can go negative. The samples for both cases exhibit spanwise traveling pressure fluctuations. Such spanwise “ripples” were also observed in experiments by Threadgill and Bruce.^{22} Over time, the ripples are superimposed with strong (large mode 0 eigenvalue, Fig. 20) spanwisecoherent lowfrequency pressure fluctuations (mode 0 shape, Fig. 22).
The POD eigenvalues for region 2 are provided in Fig. 25. Spectra of the time coefficients are plotted in Fig. 26, and the associated mode shapes are provided in Fig. 27. Different from region 1, the dominant modes at reattachment for both cases are 2D. For case 1, modes 0 and 1 have midfrequency content at $ 0.2 < S t L s < 0.4$. For case 2, the mode 0 and 1 midfrequency content is shifted to approximately $ 0.3 < S t L s < 0.5$ and mode 2 has lowfrequency content at $ S t L s = 0.03$. In agreement with Fig. 16, this suggests that for the stronger interaction, the lowfrequency dynamics govern a larger part of the bubble that extends to reattachment. Interestingly, mode 2 is 3D for case 2 but not for case 1.
IV. CONCLUSIONS
Implicit largeeddy simulations of $ M =$ 2.05 shockwave turbulent boundary layer interactions were carried out for two different shock generator angles, $ \u03d1 =$ 8° (case 1) and $ \u03d1 =$ 9° (case 2). For $ \u03d1 =$ 9° (case 2), the spanwise extent of the computational domain was doubled compared to $ \u03d1 =$ 8° (case 1) such that the spanwise extent of the domain was roughly equal to two times the separation length. The freestream conditions and approach boundary layer properties were identical to those of Morgan et al.^{13} and Lee and Gross.^{24}
By comparing with case 1, for case 2, the downstream and wallnormal extent of the separated flow region are substantially increased and the Reynolds stresses are larger. The unsteady flow data (wallpressure coefficient) were analyzed with Fourier transforms and the proper orthogonal decomposition (POD). The Fourier transforms revealed lowfrequency content at separation and highfrequency content downstream of reattachment. The lowfrequency content for case 1 ( $ \u03d1 =$ 8°) is in the range $ 0.02 < S t L s < 0.05$ and, thus, in good agreement with Dussauge et al.^{15} For case 2 ( $ \u03d1 =$ 9°), multiple stronger lowfrequency peaks are observed for $ 0.02 < S t L s < 0.07$. Downstream of reattachment, the present simulations indicate broad frequency content with $ 0.2 < S t L s < 1$. The POD for a region near separation revealed lowfrequency unsteadiness that is 2D for the weaker interaction and 3D for the stronger interaction. The POD also captured spanwise traveling structures at separation that may be related to an observed rippling in experiments.^{22} At reattachment, the first two modes for both cases are 2D. The third mode is 2D for the weaker interaction and 3D for the stronger interaction. As a result of modesuperposition, for some time instances, the wallpressure distribution at reattachment can be 3D. The present results support the notion that 3D effects become more important with increasing strength of the interaction. The results also provide supporting evidence of the rippling behavior at separation. In the context of flight hardware with thin surface panels that may be softened by aerothermal heating, understanding the unsteady pressure loads is important for predicting the onset of panel flutter.
ACKNOWLEDGMENTS
The simulations were carried locally at the New Mexico State University Information and Communication Technologies Center.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Sunyoung Lee: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (supporting); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (supporting). Andreas Gross: Conceptualization (equal); Data curation (equal); Formal analysis (supporting); Investigation (supporting); Methodology (lead); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (lead).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.