Implicit large-eddy simulations of Mach 2.05 turbulent boundary layer interactions with oblique impinging shock-waves 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 shock-wave is increased from 8° to 9°. Temporal Fourier transforms of the spanwise-averaged wall-pressure coefficient indicate low-frequency unsteadiness at separation and mid-frequency content downstream of reattachment. The wall-pressure fluctuations were analyzed with the proper orthogonal decomposition. The modal analysis reveals pronounced 3D low-frequency wall-pressure fluctuations for the stronger interaction. Overall, the present findings provide advanced perspectives on low-frequency wall-pressure fluctuations in turbulent shock-wave boundary layer interactions that may lead to spanwise non-uniformity of the separation and shedding with possible implications for the design of structural panels on high-speed vehicles.

Normal characters
A

Amplitude

ai

Time coefficient

cf

Skin-friction coefficient

cp

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 τ

Friction velocity

x, y, z

Co-ordinates

Greek symbols
γ

Ratio of specific heats

δ

Boundary layer thickness

δ *

Displacement thickness

ε

Turbulent dissipation rate

ϑ

Shock generator angle, momentum thickness

λ

Eigenvalue

μ

Dynamic viscosity

ν

Kinematic viscosity

ρ

Density

σ

Shock angle

τ

Shear stress

Subscripts
i

Mode number

incomp

Incompressible

r

Reattachment

ref

Reference conditions

s

Separation

T

Turbulent

w

Wall

Approach flow freestream conditions

Superscripts
+

In wall units

*

Dimensional quantity

Fluctuation

Turbulent shock-wave boundary layer interactions (SBLIs) occur for both external and internal flows and can exhibit a low-frequency unsteadiness.1 The associated wall-pressure fluctuations can excite structural vibrations of thin wall panels.2–5 Although different explanations for the occurrence of the low-frequency 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 wall-pressure 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 Ailman9 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 low-frequency 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 wall-pressure fluctuations may guide the design of wall panels that can better withstand the aerodynamic loads. In this paper, results from implicit large-eddy 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 low-frequency unsteadiness and modal content of the wall-pressure fluctuations.

A brief review of the literature on turbulent SBLI as relevant for this paper is provided. Pirozzoli and Grasso10 associated the low-frequency unsteadiness with a large-scale 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 Martin12 found that the low-frequency unsteadiness is more strongly related to a “breathing motion” of the separation bubble than to unsteadiness of the incoming boundary layer. Large-eddy 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 shock-wave. They also found spanwise-modulated low-frequency acoustic waves with a Strouhal number of S t 0.03. Grilli et al.14 performed an LES of a Mach 2.88 compression ramp flow and observed unsteadiness with S t 0.023 near separation and 0.36 < S t < 5 near reattachment. The low-frequency unsteadiness at separation is in good agreement with the range reported in the literature, e.g., Dussauge et al.15 Clemens and Narayanaswamy16 propose that an instability of the separation bubble is the dominant contributor to the low-frequency 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 Gaitonde18 investigated Mach 2 compression ramp flows with 22.5° and 37.5° sweep angle and found that the intensity of the low-frequency unsteadiness is related to the strength of the recirculation. Vanstone and Clemens19 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 mid-frequency unsteadiness ( 0.01 < S t < 0.1). The high-frequency unsteadiness was attributed to cross-flow structures. Threadgill and Bruce22 investigated various Mach 2 oblique shock and compression ramp SBLIs experimentally. The low-frequency 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 low-frequency “rippling” structures along the separation shock foot for a Mach 2.3 oblique turbulent SBLI. Observed mid-frequency unsteadiness ( 0.6 < S t < 1.6) was attributed to the shedding of spanwise coherent structures.

In summary, the occurrence of the low-frequency unsteadiness for turbulent SBLI is widely accepted. Several experiments and simulations revealed low-frequency 3D flow deformations at separation and/or reattachment. This paper makes a connection between the low-frequency unsteadiness and the wall-pressure 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 wall-pressure 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.

FIG. 1.

Schematic diagram of the present simulations.

FIG. 1.

Schematic diagram of the present simulations.

Close modal

The compressible Navier–Stokes equations were solved with a finite-volume Navier–Stokes code by Gross and Fasel.26 The advection upstream splitting (AUSM+-up) scheme by Liou27 was employed to compute the convective fluxes. The fluxes were interpolated to the cell faces with a ninth-order-accurate weighted essentially non-oscillatory (WENO) scheme. The discretization of the viscous terms was fourth-order-accurate. The implicit second-order-accurate trapezoidal rule was employed for time integration. The upwind discretization removes enough energy at the smallest scales to obviate the need for an explicit sub-grid stress model. Since no wall model was employed, the present simulations can be classified as wall-resolved implicit large-eddy simulations (ILES). The pressure was obtained from the ideal gas equation, p = ρ R T, and the viscosity was obtained from Sutherland's law.

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, δ * 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 × 106 per meter, and the reference length scale was L ref * = 0.02836 m.

TABLE I.

Quantities for normalization.

Velocity Density Pressure Temperature Time Length
u *  ρ *  ρ * u * 2  T *  L ref * / u *  L ref * 
Turbulence kinetic energy  Turbulence dissipation rate 
u * 2      u * 3 / L ref * 
Velocity Density Pressure Temperature Time Length
u *  ρ *  ρ * u * 2  T *  L ref * / u *  L ref * 
Turbulence kinetic energy  Turbulence dissipation rate 
u * 2      u * 3 / L ref * 
TABLE II.

Freestream conditions and boundary layer properties for x = 0.2.

M T * (K) p * (Pa) R e δ * R e ϑ δ * (mm) ϑ (mm)
2.05  155.39  32 631  5700  1800  0.163  0.051 
M T * (K) p * (Pa) R e δ * R e ϑ δ * (mm) ϑ (mm)
2.05  155.39  32 631  5700  1800  0.163  0.051 
The synthetic eddy model28 (SEM) in the stream function formulation proposed by Poletto et al.29 was employed to initialize the turbulent boundary layer. For the SEM, the eddies are generated based on random number sequences. Different from recycling and rescaling methods, this removes undesirable streamwise correlations of the turbulence. As confirmed by Lee and Gross,30 the incompressible SEM can efficiently generate supersonic turbulent boundary layers. Inflow profiles required for generating the SEM “eddy particles” were obtained from a precursor Reynolds-averaged Navier–Stokes (RANS) calculation with the k-ω turbulence model.31 Profiles of the streamwise velocity, u(y), temperature, T(y), turbulent kinetic energy, k(y), and turbulence dissipation rate, ε ( y ), for the RANS solution are provided in Fig. 2(a). The van Driest transformed velocity profile,
u V D = 0 u + ρ ρ w d u + ,
(1)
y + = y u τ ν w ,
(2)
is in good agreement with the relationship for the viscous sublayer, u + = y + [Fig. 2(b)]. The log-layer ( u + = 5.2 + ln y + / 0.41) slope is slightly over-predicted, which can be attributed to the low Reynolds number conditions. The mismatch is no concern for the present simulations as the RANS solution is used solely for seeding the synthetic eddies.
FIG. 2.

Inflow profiles for SEM. (a) Streamwise velocity, temperature, turbulent kinetic energy, and turbulent dissipation rate. (b) Van Driest transformed velocity profile.

FIG. 2.

Inflow profiles for SEM. (a) Streamwise velocity, temperature, turbulent kinetic energy, and turbulent dissipation rate. (b) Van Driest transformed velocity profile.

Close modal

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 wall-normal direction, grid lines were clustered near the wall such that the near-wall turbulence is well resolved. This is a requirement for wall-resolved LES.

FIG. 3.

Planar views (x–y plane) of computational grids for (a) case 1 and (b) case 2. For clarity, only every fourth grid line is shown. Red lines represent inviscid impinging and reflected shock-waves.

FIG. 3.

Planar views (x–y plane) of computational grids for (a) case 1 and (b) case 2. For clarity, only every fourth grid line is shown. Red lines represent inviscid impinging and reflected shock-waves.

Close modal
FIG. 4.

Perspective views of computational grids for (a) case 1 and (b) case 2. For clarity, only every fourth grid line is shown.

FIG. 4.

Perspective views of computational grids for (a) case 1 and (b) case 2. For clarity, only every fourth grid line is shown.

Close modal

The number of cells and the domain extent in the streamwise, wall-normal, 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.

TABLE III.

Number of cells and domain extent in streamwise × wall-normal × spanwise direction.

Case Shock generator angle Number of cells Domain extent
ϑ = 8°  698 × 168 × 228  0.548 × 0.188 × 0.089 
ϑ = 9°  750 × 168 × 456  0.589 × 0.188 × 0.179 
Case Shock generator angle Number of cells Domain extent
ϑ = 8°  698 × 168 × 228  0.548 × 0.188 × 0.089 
ϑ = 9°  750 × 168 × 456  0.589 × 0.188 × 0.179 

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. No-slip and no-penetration conditions were employed at the wall. The wall was considered adiabatic.

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 shock-wave and a larger separation. The shock angles, σ, and the before- and after-shock conditions for the impinging and reflected shock-waves were obtained from the Rankine–Hugoniot relations and are listed in Table IV. The flow states upstream and downstream of the oblique shock-wave were enforced at the freestream boundary such that the inviscid shock impingement point was at x = 0.4.

TABLE IV.

Before- and after-shock conditions.

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
M1  2.0380  M2  1.7493  1.7127  M3  1.4726  1.3992 
u1  0.9968  u 2 / u 1  0.9029  0.8916  u 3 / u 1  0.8165  0.7875 
v1  v 2 / u 1  −0.1269  −0.1412  v 3 / u 1 
w1  w 2 / u 1  w 3 / u 1 
p1  0.1732  p 2 / p 1  1.5477  1.6304  p 3 / p 1  2.3173  2.5570 
T1  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
M1  2.0380  M2  1.7493  1.7127  M3  1.4726  1.3992 
u1  0.9968  u 2 / u 1  0.9029  0.8916  u 3 / u 1  0.8165  0.7875 
v1  v 2 / u 1  −0.1269  −0.1412  v 3 / u 1 
w1  w 2 / u 1  w 3 / u 1 
p1  0.1732  p 2 / p 1  1.5477  1.6304  p 3 / p 1  2.3173  2.5570 
T1  1.0053  T 2 / T 1  1.1195  1.1538  T 3 / T 1  1.2769  1.3156 
The inviscid pressure ratio across the entire shock system, p 3 / p 1, is 2.3173 for case 1 and 2.5570 for case 2, respectively. Whether the flow is separated or not can be assessed with a criterion by Souverein et al.,32,
S e * = 2 k γ M 2 ( P post P pre 1 ) ,
(3)
which was derived based on control volume analysis. In Fig. 5, the non-dimensional interaction length,
L * = L δ i n * sin ( σ ) sin ( ϑ ) sin ( σ ϑ ) ,
(4)
is plotted vs the separation criterion with k = 3. The approximate regions of attached, incipient, and separated flow as proposed by Souverein et al.32 are indicated in the figure. The present cases 1 and 2 fall inside the separated flow regime.
FIG. 5.

Souverein et al.32 separation criterion.

FIG. 5.

Souverein et al.32 separation criterion.

Close modal

Both cases were first advanced in time from t0 to t1 to obtain statistically independent stationary flows. Then, time averages were computed over time intervals of t 2 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 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 Δ t = 0.0005.

TABLE V.

Time intervals and number of timesteps per “flow-through” time.

Case Initial transient Time average Statistics Wall data capture Nf
t0, t1, and t 1 t 0 t1, t2, and t 2 t 1 t2, t3, and t 3 t 2 t1, t3, and t 3 t 1
−15, 0, and 15  0, 12, and 12  12, 48, and 36  0, 48, and 48  1096 
−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 Nf
t0, t1, and t 1 t 0 t1, t2, and t 2 t 1 t2, t3, and t 3 t 2 t1, t3, and t 3 t 1
−15, 0, and 15  0, 12, and 12  12, 48, and 36  0, 48, and 48  1096 
−68, 0, and 68  0, 48, and 48  0, 48, and 48  0, 48, and 48  1178 
The wall-pressure coefficient was recorded from t1 to t3. Since the approach flow velocity is normalized to one, the time required for the freestream to pass through the entire computational domain is identical to the downstream extend of the computational domain, Lx. By dividing this convection time by the computational time step, Δ t,
N f = L x Δ t ,
(5)
the number of timesteps for one “flow-through time” is obtained. This number is also provided in Table V.

The near-wall grid-line spacing and time step in wall units are plotted in Fig. 6. The van Driest transformed grid-line spacing upstream of the interaction near x 0.2 is Δ x + 14.3 , Δ y + 0.45, and Δ z + 7.0 sufficient for wall-resolved 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 

FIG. 6.

Near-wall grid-line spacing and time step in wall units.

FIG. 6.

Near-wall grid-line spacing and time step in wall units.

Close modal

In Fig. 7(a), distributions of the displacement thickness, δ *, momentum thickness, ϑ, and incompressible shape factor, Hincomp, are plotted vs the x-co-ordinate. For both cases, upstream of separation at x = 0.2, the incompressible shape factor obtains a value of H incomp 1.4. The displacement and momentum thickness based Reynolds numbers at x = 0.2 are approximately R e δ * = 5700 and R e θ = 1800 [Fig. 7(b)].

FIG. 7.

(a) Displacement and momentum thickness as well as incompressible shape factor. (b) Displacement and momentum thickness based Reynolds number.

FIG. 7.

(a) Displacement and momentum thickness as well as incompressible shape factor. (b) Displacement and momentum thickness based Reynolds number.

Close modal
In Fig. 8, the skin-friction coefficient,
c f , x = μ u y q ,
(6)
computed from the temporal and spanwise averages of the data are plotted against the momentum thickness-based Reynolds number. After an initial adjustment, for both cases the van Driest transformed skin-friction coefficient34 is matched with good accuracy, which implies that the turbulent boundary layer is well predicted. The skin-friction coefficient begins to drop upstream of the time-mean separation location (dashed lines in Fig. 8). This drop occurs near R e θ = 1950 for case 1 and 1850 for case 2. The behavior can be explained by the unsteady nature of the separation and the upstream influence through the subsonic part of the boundary layer.
FIG. 8.

Skin-friction coefficient vs momentum thickness based Reynolds number. Mean separation lines are indicated by dashed lines.

FIG. 8.

Skin-friction coefficient vs momentum thickness based Reynolds number. Mean separation lines are indicated by dashed lines.

Close modal

The van-Driest transformed profiles for the two cases are nearly identical and in good agreement with the relationships for the viscous sublayer and log-layer (Fig. 9). The discontinuity near y + 10 3 is due to the impinging oblique shock-wave.

FIG. 9.

Van Driest transformed velocity profiles.

FIG. 9.

Van Driest transformed velocity profiles.

Close modal

The spanwise-averaged 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 wall-normal velocity profiles for case 1 are in a good agreement with Morgan et al.13 (Fig. 11).

FIG. 10.

Temporal and spanwise averages. Contours of (a) streamwise velocity ( 0 < u < 1) and (b) temperature ( 1 < T < 1.7) for case 1 (left column) and case 2 (right column).

FIG. 10.

Temporal and spanwise averages. Contours of (a) streamwise velocity ( 0 < u < 1) and (b) temperature ( 1 < T < 1.7) for case 1 (left column) and case 2 (right column).

Close modal
FIG. 11.

Comparison of u- and v-velocity profiles with Morgan et al.13 for x = 0.3293, 0.329, 0.3765, 0.4, and 0.4236 [dashed lines in Fig. 10(a)].

FIG. 11.

Comparison of u- and v-velocity profiles with Morgan et al.13 for x = 0.3293, 0.329, 0.3765, 0.4, and 0.4236 [dashed lines in Fig. 10(a)].

Close modal

Streamwise distributions of the wall-pressure and skin-friction coefficient, cp and cf, are plotted in Fig. 12. For case 2, the cp-increase across the interaction is larger. The pressure rise across the interaction is in good agreement with Threadgill and Bruce.22 The cf-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 cf value for case 2 is c f −0.000 87 and slightly (6%) higher than for case 1 ( c f −0.000 82). Different from case 1, for case 2 the skin-friction coefficient dips down prior to reattachment.

FIG. 12.

Streamwise distributions of wall-pressure and skin-friction coefficient. Symbols are measurements for similar interaction by Threadgill and Bruce.22 

FIG. 12.

Streamwise distributions of wall-pressure and skin-friction coefficient. Symbols are measurements for similar interaction by Threadgill and Bruce.22 

Close modal

The separation and reattachment locations, xs and xr, and the separation length, L s = x s 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.

TABLE VI.

Location of separation, xs, and reattachment, xr, as well as separation length, Ls.

Case ϑ (°) xs xr Ls
0.3363  0.3855  0.0491 
0.3138  0.4014  0.0876 
Case ϑ (°) xs xr Ls
0.3363  0.3855  0.0491 
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 wall-normal 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.

FIG. 13.

Contours of spanwise-averaged Reynolds normal stress in (a) streamwise ( 0 < u u ¯ < 0.04), (b) wall-normal ( 0 < v v ¯ < 0.008), and (c) spanwise ( 0 < w w ¯ < 0.0135) direction for case 1 (left column) and case 2 (right column).

FIG. 13.

Contours of spanwise-averaged Reynolds normal stress in (a) streamwise ( 0 < u u ¯ < 0.04), (b) wall-normal ( 0 < v v ¯ < 0.008), and (c) spanwise ( 0 < w w ¯ < 0.0135) direction for case 1 (left column) and case 2 (right column).

Close modal

The Reynolds shear stress, u v ¯, 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 skin-friction coefficient immediately downstream of separation (Fig. 12).

FIG. 14.

Contours of spanwise-averaged Reynolds shear stress, τ x y ( 0.0052 < u v ¯ < 0.0002), for case 1 (left column) and case 2 (right column).

FIG. 14.

Contours of spanwise-averaged Reynolds shear stress, τ x y ( 0.0052 < u v ¯ < 0.0002), for case 1 (left column) and case 2 (right column).

Close modal

1. Instantaneous flow visualizations

Instantaneous flow visualizations for both cases are provided in Fig. 15. Shown are iso-surfaces of the Q-vortex identification criterion by Hunt,36 flooded by the streamwise velocity, and numerical schlieren images (contours of the magnitude of the density gradient, | ρ |) 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 well-resolved 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.

FIG. 15.

Iso-surfaces of Q = 1000 flooded by streamwise velocity and numerical Schlieren images for (a) case 1 and (b) case 2.

FIG. 15.

Iso-surfaces of Q = 1000 flooded by streamwise velocity and numerical Schlieren images for (a) case 1 and (b) case 2.

Close modal

2. Analysis of unsteady wall-pressure data

The wall-pressure coefficient was recorded at time intervals of Δ t = 0.01. The spanwise average of the wall-pressure coefficient was Fourier-transformed in time,
c ̂ p ( f ) = F [ c p ( t ) ] .
(7)
The frequency was normalized by the separation length and approach flow velocity,
S t L s = f × L s u .
(8)
The wall-pressure amplitude spectra are plotted in Fig. 16 using a log-scale ordinate. The vertical red lines indicate the mean separation and reattachment locations. For case 1, the most significant low-frequency peak at separation occurs at S t L s 0.03 which is in good agreement with the literature.15 For case 2, multiple low-frequency peaks are observed at separation in the range 0.02 < S t L s < 0.07. For cases 1 and 2, the sampling period captures 29.4 and 16.6 periods, respectively, of the low-frequency unsteadiness with S t L s = 0.03. Low-frequency content with S t L s 0.03 is also observed at reattachment and downstream of reattachment. For case 2, it takes longer for the low-frequency unsteadiness to die down downstream of reattachment. For both of the present ILES, broad mid-frequency content with 0.1 < S t L s < 1 is observed downstream of reattachment (Fig. 16). For the mean separation location, the spectra were also computed with the maximum entropy method by Martini et al.37 Results for both the entire sampling period and half of the sampling period demonstrate statistical convergence (Fig. 17).
FIG. 16.

Spectra of wall-pressure coefficient fluctuations for (a) case 1 and (b) case 2. Red lines indicate mean separation and reattachment locations.

FIG. 16.

Spectra of wall-pressure coefficient fluctuations for (a) case 1 and (b) case 2. Red lines indicate mean separation and reattachment locations.

Close modal
FIG. 17.

Maximum entropy method spectra for mean separation location.

FIG. 17.

Maximum entropy method spectra for mean separation location.

Close modal

Similar observations were made by Threadgill and Bruce22 for SBLI with ϑ = 8° and 9° shock generator angle. Their pre-multiplied power spectral density spectra revealed a significant peak with S t L s 0.025 for ϑ = 8° and a band of peaks with 0.03 < S t L s < 0.065 for ϑ = 9°. Pre-multiplied power spectra of the wall-pressure coefficient for the location of the strongest wall-pressure fluctuations (x = 0.349 for case 1 and x = 0.328 for case 2) compare favorably with data by Threadgill and Bruce22 (amplitude scaled by 0.01) for the nearest measurement location (Fig. 18).

FIG. 18.

Pre-multiplied power spectra for case 1 (x = 0.349, upper row) and case 2 (x = 0.328, bottom row).

FIG. 18.

Pre-multiplied power spectra for case 1 (x = 0.349, upper row) and case 2 (x = 0.328, bottom row).

Close modal

3. Proper orthogonal decomposition of wall-pressure coefficient

The wall-pressure 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 low-frequency 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 wall-pressure coefficient are outlined in Fig. 19.

TABLE VII.

Streamwise extent of sub-domains for POD analysis of wall-pressure coefficient.

Case Sub-domain x0 x1
Region 1: near separation  0.3097  0.3529 
  Region 2: near reattachment  0.3529  0.4298 
Region 1:near separation  0.2901  0.3599 
  Region 2: near reattachment  0.3599  0.4400 
Case Sub-domain x0 x1
Region 1: near separation  0.3097  0.3529 
  Region 2: near reattachment  0.3529  0.4298 
Region 1:near separation  0.2901  0.3599 
  Region 2: near reattachment  0.3599  0.4400 
FIG. 19.

Instantaneous wall-pressure coefficient contours for (a) case 1 and (b) case 2 and outlines of regions for POD analysis.

FIG. 19.

Instantaneous wall-pressure coefficient contours for (a) case 1 and (b) case 2 and outlines of regions for POD analysis.

Close modal

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.

FIG. 20.

Region 1: POD eigenvalues.

FIG. 20.

Region 1: POD eigenvalues.

Close modal

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 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., non-traveling) 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.

FIG. 21.

Region 1: Fourier transformed time coefficients for (a) case 1 and (b) case 2.

FIG. 21.

Region 1: Fourier transformed time coefficients for (a) case 1 and (b) case 2.

Close modal
FIG. 22.

Region 1: POD mode wall-pressure coefficient contours, 0.01 < c p < 0.01. (a) Case 1 and (b) case 2. Black dashed lines indicate mean separation.

FIG. 22.

Region 1: POD mode wall-pressure coefficient contours, 0.01 < c p < 0.01. (a) Case 1 and (b) case 2. Black dashed lines indicate mean separation.

Close modal

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 low-frequency 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 wall-pressure distributions were reconstructed from the first six POD modes. Samples of the instantaneous reconstructed wall-pressure 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 cp 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) spanwise-coherent low-frequency pressure fluctuations (mode 0 shape, Fig. 22).

FIG. 23.

Region 1: Time coefficients and time instances for reconstruction of wall-pressure distribution. (a) Case 1 and (b) case 2.

FIG. 23.

Region 1: Time coefficients and time instances for reconstruction of wall-pressure distribution. (a) Case 1 and (b) case 2.

Close modal
FIG. 24.

Region 1: reconstructed wall-pressure distributions for (a) case 1 and (b) case 2. Black dashed lines indicate mean separation.

FIG. 24.

Region 1: reconstructed wall-pressure distributions for (a) case 1 and (b) case 2. Black dashed lines indicate mean separation.

Close modal

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 mid-frequency content at 0.2 < S t L s < 0.4. For case 2, the mode 0 and 1 mid-frequency content is shifted to approximately 0.3 < S t L s < 0.5 and mode 2 has low-frequency content at S t L s = 0.03. In agreement with Fig. 16, this suggests that for the stronger interaction, the low-frequency 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.

FIG. 25.

Region 2: POD eigenvalues.

FIG. 25.

Region 2: POD eigenvalues.

Close modal
FIG. 26.

Region 2: Fourier transformed time coefficients for (a) case 1 and (b) case 2.

FIG. 26.

Region 2: Fourier transformed time coefficients for (a) case 1 and (b) case 2.

Close modal
FIG. 27.

Region 2: POD mode wall-pressure coefficient contours, 0.01 < c p < 0.01. (a) Case 1 and (b) case 2. Black dashed lines indicate mean reattachment.

FIG. 27.

Region 2: POD mode wall-pressure coefficient contours, 0.01 < c p < 0.01. (a) Case 1 and (b) case 2. Black dashed lines indicate mean reattachment.

Close modal

Using the same procedure as for case 1, the pressure fields were reconstructed from the first six POD modes for the time instances indicated in Fig. 28. The visualizations in Fig. 29 reveal that even though the dominant modes are 2D, strong 3D pressure fluctuations are possible at reattachment.

FIG. 28.

Region 2: Time coefficients and time instances for reconstruction of wall-pressure distribution. (a) Case 1 and (b) case 2.

FIG. 28.

Region 2: Time coefficients and time instances for reconstruction of wall-pressure distribution. (a) Case 1 and (b) case 2.

Close modal
FIG. 29.

Region 2: reconstructed wall-pressure distributions for (a) case 1 and (b) case 2. Black dashed lines indicate mean reattachment.

FIG. 29.

Region 2: reconstructed wall-pressure distributions for (a) case 1 and (b) case 2. Black dashed lines indicate mean reattachment.

Close modal

Implicit large-eddy simulations of M = 2.05 shock-wave turbulent boundary layer interactions were carried out for two different shock generator angles, ϑ = 8° (case 1) and ϑ = 9° (case 2). For ϑ = 9° (case 2), the spanwise extent of the computational domain was doubled compared to ϑ = 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 wall-normal extent of the separated flow region are substantially increased and the Reynolds stresses are larger. The unsteady flow data (wall-pressure coefficient) were analyzed with Fourier transforms and the proper orthogonal decomposition (POD). The Fourier transforms revealed low-frequency content at separation and high-frequency content downstream of reattachment. The low-frequency content for case 1 ( ϑ = 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 ( ϑ = 9°), multiple stronger low-frequency 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 low-frequency 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 mode-superposition, for some time instances, the wall-pressure 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 aero-thermal heating, understanding the unsteady pressure loads is important for predicting the onset of panel flutter.

The simulations were carried locally at the New Mexico State University Information and Communication Technologies Center.

The authors have no conflicts to disclose.

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).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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