Large eddy simulation of two separated hypersonic shock/turbulent boundary layer interactions

A large eddy simulation (LES) method employing a one-coefficient dynamic mixed model is used to generate data of two separated hypersonic shock/turbulent boundary layer interactions. The first of these is a Mach 7.2 turbulent boundary layer at ${\text{Re}}_{\theta }=3300$ over a ${33}^{\circ }$ ramp. The second is a Mach 9.1 turbulent boundary layer at ${\text{Re}}_{\theta }=8000$ over a ${34}^{\circ }$ ramp. We document the mean flow statistics including Reynolds stresses, turbulent amplification, turbulent kinetic energy budget, anisotropy tensor invariants, wall pressure distribution, skin friction, and wall heat transfer. Various modeling assumptions typically used in engineering applications such as the Reynolds Analogy Factor, constant turbulent Prandtl number, and Strong Reynolds Analogies are assessed for these conditions. The documentation of the two-dimensional time-averaged flow statistics of these two datasets is the main focus of this article, however, we first demonstrate the reliability of our LES method by simulating both a separated supersonic compression ramp flow and an attached hypersonic compression ramp flow and comparing to existing direct numerical simulation (DNS) data. An excellent comparison of mean flow profiles, turbulence amplification factors, spectral content, and the state of separation is achieved. The practicality of the LES over the DNS method is also emphasized by the typical factor of 32 reduction in grid size as well as a numerical time step increase of three times the DNS. Additional grid size reduction might be possible. A discussion on the accuracy of the mixed model in comparison to the eddy viscosity model is included. We found that at Mach 7, omitting the scale-similar contribution in the subgrid-scale models of shear stress and heat flux resulted in as much as a 32% increase in the separation length and 10% decrease in shear layer spreading rate.

Figure 1

Schematic of LES computational domain and simulation strategy.

Figure 2

Auxiliary boundary layer mean profiles of (a) streamwise velocity and (b) temperature normalized by the freestream value. Profiles are taken from the rescaling plane near the outlet of the box. The DNS profiles are not filtered.

Figure 3

Van Driest transformed velocity profiles at the auxiliary boundary layer rescaling plane. The DNS profiles are not filtered.

Figure 4

Turbulent kinetic energy profiles ($\text{TKE}=\langle {\widehat{u}}_i^{\prime }{\widehat{u}}_i^{\prime }\rangle /U_e^2$) at the auxiliary boundary layer rescaling plane. The DNS data is Favre filtered by the tophat filter of Eq. (12). The filter width in $i$, $j$, and $k$ is indicated by the number in parentheses.

Figure 5

The fraction of TKE contained in the subgrid scales for the M7-L boundary layer run. The dashed line shows the TKE difference in the filtered DNS compared to the unfiltered DNS. The symbols show the TKE difference in the LES as compared to the unfiltered DNS.

Figure 6

Premultiplied power spectral density of mass fluctuations $\left[{\left(\overline{\rho }\widehat{u}\right)}^{\prime }\mathrm{for}\mathrm{the}\mathrm{LES}\mathrm{and}{\left(\rho u\right)}^{\prime }\mathrm{for}\mathrm{the}\mathrm{DNS}\right]$ at the rescaling plane of the Mach 7 auxiliary boundary layers.

Figure 7

Instantaneous visualization of density in a center-span $xz$ plane of R24-M3-L.

Figure 8

Mean wall distributions of (a) skin friction vs $x/\delta$, (b) skin friction vs $x/L_{\mathrm{sep}}$, and (c) wall pressure vs $x/L_{\mathrm{sep}}$ for the Mach 3 STBLI solutions. The DNS data are not filtered.

Figure 9

Profiles of (a) mean velocity, and (b) turbulence intensities in the shear layer of the Mach 3 STBLI solutions.

Figure 10

Premultiplied power spectral plots of the time history of (a) separation point and (b) reattachment point in the narrow Mach 3 STBLI DNS and LES solutions.

Figure 11

Instantaneous visualization of density in a center-span $xz$ plane of R8-M7-L.

Figure 12

Wall distributions of (a) skin friction coefficient $C_f=2{\tau }_w/{\rho }_e{U}_e$, (b) pressure, and (c) heat transfer coefficient $C_h\equiv q_w/{\rho }_e{U}_e{c}_p(T_w-T_r)$ of the Mach 7, $8^{\circ }$ compression ramp LES and (unfiltered) DNS mean flow solutions.

Figure 13

Time- and spanwise-averaged reverse flow probability at the wall surface of the Mach 7, $8^{\circ }$ compression ramp LES and (unfiltered) DNS flow solutions.

Figure 14

Contours of averaged (a) streamwise, (b) wall-normal, and (c) cross-turbulence stresses in the Mach 7, $8^{\circ }$ compression ramp flow solutions. The color contour is the filtered DNS data, and the solid line contour is the LES data.

Figure 15

Premultiplied power spectral density of the time history of separation and reattachment in (a) R33-M7-L and (b) R34-M10-L.

Figure 16

Sample time signals of spanwise-averaged separation and reattachment positions from R33-M7-L [(a) and (b), respectively] and from R34-M10-L [(c) and (d), respectively].

Figure 17

Instantaneous snapshot of the three-dimensional turbulence and shock front in (a) the R33-M7-L and (b) the R24-M10-L simulations. An isosurface of density gradient ($\left|\mathbf{\nabla }\rho \right|=0.7$) is colored by the magnitude of streamwise velocity. The flow direction is from the bottom left to top right of the image.

Figure 18

Instantaneous snapshot of density in an $xz$ plane through the center of the spanwidth of (a) the R33-M7-L and (b) the R34-M10-L.

Figure 19

Time- and spanwise-averaged numerical schlieren $[\mathrm{NS}=0.76\exp (-1.3|\mathbf{\nabla }\overline{\rho }|/|\mathbf{\nabla }\overline{\rho }{|}_{\max }\left)\right]$ of (a) R33-M-L and (b) R34-M10-L.

Figure 20

Postshock profiles of (a) Mach number, (b) temperature, (c) total pressure, and (d) static pressure at the outlet plane of the computational domain. Profiles are normalized by the inviscid oblique shock solution. The dashed lines are the Mach 7 solution and the solid lines the Mach 10.

Figure 21

Contours of (a), (b) Favre-averaged streamwise velocity $\tilde{u}$, (c), (d) wall-normal velocity $\tilde{w}$, (e), (f) temperature $\tilde{T}$, and (g, h) Reynolds-averaged density $\langle \rho \rangle$ normalized by the freestream values. R33-M7-L are plotted on the left and R34-M10-L on the right.

Figure 22

Individual profiles of (a), (e) Favre-averaged streamwise velocity $\tilde{u}$, (b), (f) wall-normal velocity $\tilde{w}$, (c), (g) temperature $\tilde{T}$, and (d), (h) Reynolds-averaged density $\langle \rho \rangle$ normalized by the freestream values. The top row is R33-M7-L and the bottom row is R35-M10-L. Profile locations are upstream $x^{\prime }/L_{\mathrm{sep}}=-1.3$ (solid), separation $x^{\prime }/L_{\mathrm{sep}}=-0.65$ (dash-dot), corner $x^{\prime }/L_{\mathrm{sep}}=0$ (dash-dot-dot), and downstream $x^{\prime }/L_{\mathrm{sep}}=1.9$ (dashed).

Figure 23

Time- and spanwise-averaged distributions of (a) wall pressure, (b) skin friction, and (c) heat transfer for R33-M7-L and R24-M10-L. The horizontal dashed lines in (a) indicate the inviscid shock pressure.

Figure 24

Contours of mean turbulence stress components (a), (b) $\sqrt{\widetilde{u^{″}{u}^{″}}}$, (c), (d) $\sqrt{\widetilde{v^{″}{v}^{″}}}$, (e), (f) $\sqrt{\widetilde{w^{″}{w}^{″}}}$, and (g), (h) $\widetilde{u^{″}{w}^{″}}$ normalized by the freestream values. R33-M7-L are plotted on the left and R34-M10-L on the right.

Figure 25

Individual profiles of turbulence stress components (a), (e) $\sqrt{\widetilde{u^{″}{u}^{″}}}$, (b), (f) $\sqrt{\widetilde{v^{″}{v}^{″}}}$, (c), (g) $\sqrt{\widetilde{w^{″}{w}^{″}}}$, and (d), (h) $\widetilde{u^{″}{w}^{″}}$ normalized by the freestream values. The top row is R33-M7-L, and the top row is R34-M10-L. The top row is R33-M7-L, and the bottom row is R35-M10-L. Profile locations are upstream $x^{\prime }/L_{\mathrm{sep}}=-1.3$ (solid), separation $x^{\prime }/L_{\mathrm{sep}}=-0.65$ (dash-dot), corner $x^{\prime }/L_{\mathrm{sep}}=0$ (dash-dot-dot), and downstream $x^{\prime }/L_{\mathrm{sep}}=1.9$ (dashed).

Figure 26

Lumley triangles for wall-normal profiles (a) in the incoming boundary layer $x^{\prime }/L_{\mathrm{sep}}=-1.3$, (b) at mean separation $x^{\prime }/L_{\mathrm{sep}}=-0.65$, (c) at the corner $x^{\prime }/L_{\mathrm{sep}}=0$, and (d) in the downstream boundary layer $x^{\prime }/L_{\mathrm{sep}}=1.9$.

Figure 27

Profiles of TKE budgets in (a), (b) the upstream boundary layer $x^{\prime }/L_{\mathrm{sep}}=-1.3$, (c), (d) at mean separation $x^{\prime }/L_{\mathrm{sep}}=-0.65$, (e), (f) at the corner $x^{\prime }/L_{\mathrm{sep}}=0$, and (g), (h) in the downstream boundary layer $x^{\prime }/L_{\mathrm{sep}}=1.9$. The budget profiles for R33-M7-L are on the left and R34-M10-L on the right.

Figure 28

Streamwise distributions of (a) the (inverse) Reynolds Analogy Factor and (b) the QP85 law of Back and Cuffel [47] for R33-M7-L and R24-M10-L.

Figure 29

Upstream ($x^{\prime }/L_{\mathrm{sep}}=-1.3$) and downstream boundary ($x^{\prime }/L_{\mathrm{sep}}=1.9$) layer profiles of (a) turbulent Prandtl number $P{r}_t$, (b) Strong Reynolds Analogy (SRA) phase relation, (c) SRA magnitude relation, and (d) modified SRA magnitude relation.

Figure 30

Skin friction comparison of DMM and DEV solutions of R33-M7-L.

Figure 31

Comparison of DMM and DEV upstream boundary layer solutions of R33-M7-L (a) van Driest transformed velocity and (b) turbulence intensities.

Figure 32

Comparison collapse of shear layer profiles: (a) streamwise velocity, (b) streamwise turbulence intensity, and (c) turbulence shear stress between the DMM and DEV solutions of R33-M7-L.

Figure 33

Comparison of DMM and DEV solutions of R33-M7-L (a) TKE budgets scaled by the mixing layer thickness ${\delta }_w$ and (b) the upstream boundary layer TKE budgets scaled by $z_{\tau }$ and $u_{\tau }$. Legend as in Fig. 27.