Continuum breakdown in compressible mixing layers

Gas-kinetic simulations of rarefied and compressible mixing layers are performed to characterize continuum breakdown and the effect on the Kelvin-Helmholtz instability. The unified gas-kinetic scheme (UGKS) is used to perform the simulations at different Mach and Knudsen numbers. The UGKS stress tensor and heat-flux vector fields are compared against those given by the Navier-Stokes-Fourier constitutive equations. The most significant difference is seen in the shear stress and transverse heat flux. The study demonstrates the existence of two distinct continuum breakdown regimes, one at low and the other at high convective Mach numbers. Overall, at low convective Mach numbers, the deviation from continuum stress and heat flux appears to scale exclusively with the micro-macro length scale ratio given by the Knudsen number. On the other hand, at high convective Mach numbers, the deviation depends on the global micro-macro timescale ratio given by the product of Mach and Knudsen numbers. We further demonstrate that, unlike shear stresses and transverse heat flux, the deviations in normal stresses and the streamwise heat flux depend separately on Knudsen and Mach numbers. A local parameter called the gradient Knudsen number is proposed to characterize the rarefaction effects on the local momentum and thermal transport. Noncontinuum aspects of gas-kinetic stress-tensor and heat-flux behavior that Grad's 13-moment equation model reasonably captures are identified.

Figure 1

Computational domain with initial velocity profile and boundary conditions.

Figure 2

Variation of (a) $u_x(x=L_x/2,y)$ and (b) $u_y(x=L_x/2,y)$ with $y$ for ${\text{Ma}}_c=0.2$ (black), ${\text{Ma}}_c=0.8$ (red), and ${\text{Ma}}_c=1.2$ (blue) at ${\mathrm{Kn}}_0=1.0$ and $t^{*}=10$. The flow variables are normalized by the corresponding freestream values.

Figure 3

Variation of (a) $\rho (x=L_x/2,y)$ and (b) $T(x=L_x/2,y)$ for ${\text{Ma}}_c=0.2$ (black), ${\text{Ma}}_c=0.8$ (red), and ${\text{Ma}}_c=1.2$ (blue) at ${\mathrm{Kn}}_0=1.0$ and $t^{*}=10$. The flow variables are normalized by the corresponding freestream values. The legend is the same as in Fig. 2.

Figure 4

Variation of (a) $u_x(x=L_x/2,y)$ and (b) $u_y(x=L_x/2,y)$ along $y$ for ${\mathrm{Kn}}_0=0.01$ (blue), ${\mathrm{Kn}}_0=0.1$ (red), and ${\mathrm{Kn}}_0=1$ (black) for ${\text{Ma}}_c=0.2$ and $t^{*}=10$. The flow variables are normalized by the corresponding freestream values.

Figure 5

Variation of (a) $\rho (x=L_x/2,y)$ and (b) $T(x=L_x/2,y)$ along $y$ for ${\mathrm{Kn}}_0=0.01$ (blue), ${\mathrm{Kn}}_0=0.1$ (red), and ${\mathrm{Kn}}_0=1$ (black) for ${\text{Ma}}_c=0.2$ and $t^{*}=10$. The flow variables are normalized by the corresponding freestream values. The legend is the same as in Fig. 4.

Figure 6

Variation of (a) $u_x(x=L_x/2,y)$ and (b) $u_y(x=L_x/2,y)$ along $y$ for ${\mathrm{Kn}}_0=0.01$ (blue), ${\mathrm{Kn}}_0=0.1$ (red), and ${\mathrm{Kn}}_0=1$ (black) for ${\text{Ma}}_c=1.2$ and $t^{*}=10$. The flow variables are normalized by the corresponding freestream values.

Figure 7

Variation of (a) $\rho (x=L_x/2,y)$ and (b) $T(x=L_x/2,y)$ for ${\mathrm{Kn}}_0=0.01$ (blue), ${\mathrm{Kn}}_0=0.1$ (red), and ${\mathrm{Kn}}_0=1$ (black) for ${\text{Ma}}_c=1.2$ and $t^{*}=10$. The flow variables are normalized by the corresponding freestream values. The legend is the same as in Fig. 6.

Figure 8

Transverse velocity $u_y$ contour for ${\text{Ma}}_c=0.2$ and $t^{*}=10$ for (a) ${\mathrm{Kn}}_0=0.01$, (b) ${\mathrm{Kn}}_0=0.1$, and (c) ${\mathrm{Kn}}_0=1$.

Figure 9

Transverse velocity $u_y$ contour for ${\text{Ma}}_c=1.2$ and $t^{*}=10$ for (a) ${\mathrm{Kn}}_0=0.01$, (b) ${\mathrm{Kn}}_0=0.1$, and (c) ${\mathrm{Kn}}_0=1$.

Figure 10

Comparison of (a) ${\tau }_{xy}$, (b) ${\tau }_{xx}$, and (c) ${\tau }_{yy}$ obtained from moments of the particle distribution function with the Navier-Stokes and Grad's 13-moment equations for ${\mathrm{Kn}}_0=1, {\text{Ma}}_c=1.2$, and $t^{*}=25$.

Figure 11

Comparison of (a) $q_y$ and (b) $q_x$ obtained from moments of the particle distribution function with the NSF and Grad's 13-moment equations for ${\mathrm{Kn}}_0=1, {\text{Ma}}_c=1.2$, and $t^{*}=25$.

Figure 12

Maximum deviation of the shear stress ${\tau }_{xy,B}$ from the Navier-Stokes constitutive equation, ${\tau }_{xy,\text{NS}}$, with (a) ${\mathrm{Kn}}_t$ and (b) ${\text{Ma}}_c{\mathrm{Kn}}_t$ for different values of ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1.0$.

Figure 13

Maximum deviation of the heat-flux vector $q_{y,B}$ from the Navier-Stokes constitutive equation, $q_{y,\text{NS}}$, with (a) ${\mathrm{Kn}}_t$ and (b) ${\text{Ma}}_c{\mathrm{Kn}}_t$ for different values of ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1.0$.

Figure 14

Plot of the local deviation of the shear stress ${\mathrm{\Delta }}_l{\tau }_{xy}$ against ${\mathrm{Kn}}_g$ for different ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1$ at (a) $t^{*}=10$ and (b) $t^{*}=20$.

Figure 15

Plot of the local deviation of the transverse heat flux ${\mathrm{\Delta }}_l{q}_y$ against ${\mathrm{Kn}}_g$ for different ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1$ at (a) $t^{*}=10$ and (b) $t^{*}=20$.

Figure 16

Plot of the local deviation of the shear stress ${\mathrm{\Delta }}_l{\tau }_{xy}$ against ${\mathrm{Kn}}_{GLL}$ based on (a) temperature and (b) density for different ${\text{Ma}}_c$ at $t^{*}=10$.

Figure 17

Plot of the local deviation of the transverse heat flux ${\mathrm{\Delta }}_l{q}_y$ against ${\mathrm{Kn}}_{GLL}$ based on (a) temperature and (b) density for different ${\text{Ma}}_c$ at $t^{*}=10$.

Figure 18

Maximum deviation of the normal stress ${\tau }_{yy,B}$ from the Navier-Stokes constitutive equation ${\tau }_{yy,\text{NS}}$ with (a) ${\mathrm{Kn}}_t$ and (b) ${\text{Ma}}_c{\mathrm{Kn}}_t$ for different values of ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1.0$.

Figure 19

Maximum deviation of the normal stress ${\tau }_{xx,B}$ from the Navier-Stokes constitutive equation ${\tau }_{xx,\text{NS}}$ with (a) ${\mathrm{Kn}}_t$ and (b) ${\text{Ma}}_c{\mathrm{Kn}}_t$ for different values of ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1.0$.

Figure 20

Maximum deviation of the normal stress $q_{x,B}$ from the Navier-Stokes constitutive equation $q_{x,\text{NS}}$ with (a) ${\mathrm{Kn}}_t$ and (b) ${\text{Ma}}_c{\mathrm{Kn}}_t$ for different values of ${\text{Ma}}_c$ and ${\mathrm{Kn}}_0=1.0$.

Figure 21

Scatter plot of (a) normalized ${\tau }_{xy,B}$ against normalized $\partial u_x/\partial y$ and (b) normalized $q_{y,B}$ against normalized $\partial T/\partial y$ for ${\text{Ma}}_c=0.1$ (black), 0.5 (red), and 1.2 (green) for ${\mathrm{Kn}}_0=1$ at various times.

Figure 22

Scatter plot of the normalized effective viscosity (a) ${\mu }_{\text{eff},1}/\mu$ obtained from ${\tau }_{xy}$ and $\partial u_x/\partial y$ against normalized $\partial u_x/\partial y$ and (b) ${\mu }_{\text{eff},2}/\mu$ obtained from $q_y$ and $\partial T/\partial y$ against normalized $\partial T/\partial y$ for ${\text{Ma}}_c=0.1$ (black), 0.5 (red), and 1.2 (green) and ${\mathrm{Kn}}_0=1$ at various times.

Figure 23

Scatter plot of (a) normalized ${\tau }_{xx,B}$ against normalized $\partial u_y/\partial y$ and (b) normalized ${\tau }_{yy,B}$ against normalized $\partial u_y/\partial y$ for ${\text{Ma}}_c=0.1$ (black), 0.5 (red), and 1.2 (green) and ${\mathrm{Kn}}_0=1$ at various times.