Turbulence in a hypersonic compression ramp flow

A hypersonic shock wave/turbulent boundary layer interaction (STBLI) is investigated using direct numerical simulation (DNS). The geometry is an $8^{\circ }$ compression ramp, and the flow conditions upstream of the interaction are Mach 7.2 and ${\text{Re}}_{\theta }=3500$. Consistent with experiments at similar conditions, the flow is found to be attached in the mean, although the DNS shows that the probability of observing reversed flow on an instantaneous basis is significant. Due to the high Mach number of the flow combined with a low deflection angle, the shock angle is shallow and the shock is immersed in the boundary layer for a streamwise distance equal to several incoming boundary layer thicknesses downstream of the compression corner. The instantaneous flow structure observed in the DNS is in good qualitative agreement with filtered Rayleigh scattering images obtained experimentally that are available in the literature. The behavior of the turbulence is described based on the evolution of the Reynolds stresses, the anisotropy tensor, the wall pressure spectra, and the turbulence kinetic energy budget through the interaction. The various Reynolds stress components are found to be amplified by factors of 1.8–2.5. The heat transfer through the interaction is also investigated, as well as the relationship between the velocity and temperature fields. At the corner and for a significant distance downstream of the corner, the Reynolds analogy factor lies above values typically observed in zero-pressure-gradient hypersonic boundary layers. A common heat-transfer–pressure scaling describes the behavior observed in the DNS more accurately but with some departures near the corner. In the present attached STBLI, the strong Reynolds analogy, including the assumption of a constant turbulent Prandtl number around unity, is satisfied reasonably well in the interaction, although there are significant departures in the near-wall region.

Figure 1

Comparison between Sutherland's and Keyes' viscosity model: (a) model-predicted viscosity vs temperature, and (b) relative difference between the two models, ${\mathrm{\Delta }}_{\mu }=\frac{{\mu }_{\text{Keyes}}-{\mu }_{\text{Sutherland}}}{{\mu }_{\text{Sutherland}}}$, vs temperature.

Figure 2

Numerical Schlieren visualization of the Mach 7.2, zero-pressure-gradient boundary layer (auxiliary DNS).

Figure 3

Upstream boundary layer profiles. (a) $\overline{\rho }$ (green), $\tilde{u}$ (black), and $\tilde{T}$ (red); (b) $\frac{\sqrt{\widetilde{u^{″}{u}^{″}}}}{u_{\tau }}$ (black), $\frac{\sqrt{\widetilde{v^{″}{v}^{″}}}}{u_{\tau }}$ (red), and $\frac{\sqrt{\widetilde{w^{″}{w}^{″}}}}{u_{\tau }}$ (green).

Figure 4

(a) Time- and spanwise-averaged skin-friction coefficient in the upstream boundary layer (auxiliary DNS) as a function of the streamwise coordinate; (b) relative difference between the van Driest II prediction and the DNS, ${\mathrm{\Delta }}_{Cf}=\frac{C_{f,\text{VDII}}-C_{f,\text{DNS}}}{C_{f,\text{DNS}}}$.

Figure 5

Van-Driest-transformed velocity profiles in the upstream boundary layer.

Figure 6

Spectra in the upstream boundary layer at $x/{\delta }_{\text{ref}}=26.0$: (a) wall pressure, (b)–(d) streamwise velocity, momentum, and Morkovin-type scaled velocity at $z^{+}=15$ (b), $z^{+}=50$ (c), and $z/{\delta }_{\text{ref}}=0.5$ (d). The nominal frequency of the rescaling boundary condition, $f_{\text{resc}}=\frac{U_{\infty }}{26\delta }$, is shown on the spectra.

Figure 7

Numerical Schlieren visualizations of the $8^{\circ }$ compression-ramp STBLI (principal DNS).

Figure 8

Instantaneous, three-dimensional flow structure. Isosurface of the magnitude of the density gradient, colored by the wall-normal coordinate.

Figure 9

Filtered Rayleigh scattering images of the $8^{\circ }$ compression-ramp interaction. Reproduced from Bookey et al. [2]; see also Schreyer et al. [4].

Figure 10

Time- and spanwise-averaged pressure gradient magnitude.

Figure 11

Time- and spanwise-averaged surface quantities through the interaction: (a) wall pressure, (b) first derivative (solid) and second derivative (dashed) of the wall pressure with respect to the streamwise coordinate $x$, (c) skin-friction coefficient, and (d) Stanton number.

Figure 12

Instantaneous contours of the skin-friction coefficient $C_f$. The red line is the $C_f=0$ contour.

Figure 13

Spanwise-averaged probability ${\overline{\gamma }}_u$ of flow reversal: (a) at the first grid point above the wall, and (b) in the streamwise–wall-normal plane.

Figure 14

Probability of flow reversal: (a) isosurface ${\gamma }_{u,\mathrm{\Delta }T}=0.01$, and (b) isosurface ${\gamma }_{u,\mathrm{\Delta }T}=0.2$.

Figure 15

Profiles of $\overline{\rho }$ (green), $\tilde{u}$ (black), $\tilde{w}$ (blue), and $\tilde{T}$ (red) through the interaction (a) $x/\delta =-4.0$, (b) $x/\delta =0.0$, (c) $x/\delta =1.0$, (d) $x/\delta =2.0$, (e) $x/\delta =4.0$, and (f) $x/\delta =6.0$.

Figure 16

Near-wall detail of profiles of streamwise velocity $\tilde{u}$ at $x/{\delta }_{\text{ref}}=-4$ (red), $x/{\delta }_{\text{ref}}=0$ (green), and $x/{\delta }_{\text{ref}}=1$ (blue).

Figure 17

Comparison of the mean velocity profiles in the $(x,y,z)$ coordinate system aligned with the upstream wall and the $(x_r,y,z_r)$ coordinate system aligned with the ramp: (a) $\tilde{u}$ (solid) vs $\widetilde{u_r}$ (dashed), and (b) $\tilde{w}$ (solid) vs $\widetilde{w_r}$ (dashed). The streamwise locations shown are $x/{\delta }_{\text{ref}}=1$ (red), $x/{\delta }_{\text{ref}}=2$ (blue), and $x/{\delta }_{\text{ref}}=4$ (green).

Figure 18

Time- and spanwise-averaged contours of the (a) streamwise velocity fluctuation intensity $\sqrt{\widetilde{u^{″}{u}^{″}}}/u_{\tau ,\text{ref}}$, (b) spanwise velocity fluctuation intensity $\sqrt{\widetilde{v^{″}{v}^{″}}}/u_{\tau ,\text{ref}}$, (c) wall-normal velocity fluctuation intensity $\sqrt{\widetilde{w^{″}{w}^{″}}}/u_{\tau ,\text{ref}}$, and (d) Reynolds shear stress $\widetilde{u^{″}{w}^{″}}/u_{\tau ,\text{ref}}^2$. The normalization is by the upstream friction velocity obtained at the recycling station of the auxiliary DNS, which is equivalent to the inlet of the principal DNS.

Figure 19

Time- and spanwise-averaged field of the Favre-averaged turbulent kinetic energy. The three streamlines used in Fig. 23 are also shown.

Figure 20

Reynolds stress profiles at several locations through the interaction (a) $\widetilde{u_r^{″}{u}_r^{″}}/u_{\tau ,\text{ref}}^2$, (b) $\widetilde{v_r^{″}{v}_r^{″}}/u_{\tau ,\text{ref}}^2$, (c) $\widetilde{w_r^{″}{w}_r^{″}}/u_{\tau ,\text{ref}}^2$, and (d) $\widetilde{u_r^{″}{w}_r^{″}}/u_{\tau ,\text{ref}}^2$.

Figure 21

Lumley triangles. Each subfigure shows the trace of the invariant pair $(-\text{II,III})$ of the Reynolds stress anisotropy tensor as a function of the wall-normal coordinate, $z$, for a fixed streamwise location, $x$: (a) $x/\delta =-4.0$, (b) $x/\delta =-0.25$, (c) $x/\delta =0.0$, and (d) $x/\delta =0.5$.

Figure 22

Lumley triangles as in Fig. 21 but at the streamwise locations: (a) $x/\delta =1.0$, (b) $x/\delta =2.0$, (c) $x/\delta =3.0$, and (d) $x/\delta =5.0$.

Figure 23

Traces of the invariant pair $(-\text{II,III})$ of the Reynolds stress anisotropy tensor along three mean-flow streamlines: (a) along the streamline passing through $z^{+}=1$ at $x/\delta =-4$, (b) along the streamline passing through $z^{+}=12$ at $x/\delta =-4$, and (c) along the streamline passing through $z/\delta =0.2$ at $x/\delta =-4$. See also Fig. 19 for the location of the streamlines.

Figure 24

Premultiplied spectra of wall pressure fluctuations at several streamwise locations.

Figure 25

TKE budget at various streamwise locations: (a) $x/\delta =-4.0$, (b) $x/\delta =-0.25$, (c) $x/\delta =0.0$, and (d) $x/\delta =0.5$.

Figure 26

TKE budget at various streamwise locations: (a) $x/\delta =1.0$, (b) $x/\delta =2.0$, (c) $x/\delta =3.0$, and (d) $x/\delta =5.0$.

Figure 27

TKE budget at $x/\delta =5.0$ and on a larger wall-normal scale than in Fig. 26 to show the shock region.

Figure 28

Surface heat-transfer scalings through the interaction: (a) Reynolds analogy and (b) QP85 scaling.

Figure 29

Profiles of the SRA relations at various streamwise locations. SRA phase $R_{uT}=-\frac{\widetilde{u^{″}{T}^{″}}}{u_{\text{rms}}^{″}{T}_{\text{rms}}^{″}}$ (solid), SRA magnitude $\frac{1}{(\gamma -1)M^2}\frac{\tilde{u}}{u_{\text{rms}}^{″}}\frac{T_{\text{rms}}^{″}}{\tilde{T}}$ (dashed), and modified SRA magnitude $\frac{1}{(\gamma -1)M^2}\frac{\tilde{u}}{u_{\text{rms}}^{″}}\frac{T_{\text{rms}}^{″}}{\tilde{T}}{\text{Pr}}_t\left(1-\frac{\partial \widetilde{T_t}}{\partial \tilde{T}}\right)$ (dash-dot). (a) $x/\delta =-4.0$, (b) $x/\delta =0.0$, (c) $x/\delta =2.0$, and (d) $x/\delta =4.0$.

Figure 30

Profiles of the turbulent Prandtl number ${\text{Pr}}_t=\frac{\overline{\rho u^{″}{w}^{″}}}{\overline{\rho T^{″}{w}^{″}}}\frac{\partial \tilde{T}/\partial z}{\partial \tilde{u}/\partial z}$ at various streamwise locations: (a) $x/\delta =-4.0$, (b) $x/\delta =0.0$, (c) $x/\delta =2.0$, and (d) $x/\delta =4.0$.

Figure 31

${\text{Pr}}_t$ profiles on a semilog scale to highlight the near-wall behavior: (a) $x/\delta =-4.0$ and (b) $x/\delta =4.0$.