Direct numerical simulation of compressible turbulence in a counter-flow channel configuration

Counter-flow configurations, whereby two streams of fluid are brought together from opposite directions, are highly efficient mixers due to the high turbulence intensities that can be maintained. In this paper, a simplified version of the problem is introduced that is amenable to direct numerical simulation. The resulting turbulent flow problem is confined between two walls, with one nonzero mean velocity component varying in the space direction normal to the wall, corresponding to a simple shear flow. Compared to conventional channel flows, the mean flow is inflectional and the maximum turbulence intensity relative to the maximum mean velocity is nearly an order of magnitude higher. The numerical requirements and turbulence properties of this configuration are first determined. The Reynolds shear stress is required to vary linearly by the imposed forcing, with a peak at the channel centerline. A similar behavior is observed for the streamwise Reynolds stress, the budget of which shows an approximately uniform distribution of dissipation, with large contributions from production, pressure-strain, and turbulent diffusion. A viscous sublayer is obtained near the walls and with increasing Reynolds number small-scale streaks in the streamwise momentum are observed, superimposed on the large-scale structures that buffet this region. When the peak local mean Mach number reaches 0.55, turbulent Mach numbers of 0.6 are obtained, indicating that this flow configuration can be useful to study compressibility effects on turbulence.

Figure 1

Two-dimensional schematic of the three-dimensional counter-flow channel configuration. The streamwise and spanwise (in-plane, in the $z$ direction) boundaries are periodic. The top and bottom boundaries are isothermal no-slip walls. The red arrows show the theoretical directions of the mean flow in the upper and lower halves of the channel. Profiles of the hyperbolic tangent forcing term ($c_1$) and the mean streamwise velocity are shown by the black and blue arrows, respectively. The black dashed line shows the channel centerline at $y=0$. An alternative coordinate in the $y$ direction relative to the wall will be denoted $\tilde{y}$ (with $\tilde{y}=0$ is at the wall) and will be used to compute the normalized wall distance (${\tilde{y}}^{+}$).

Figure 2

Instantaneous field of the streamwise velocity fluctuation ($u^{\prime }=u-\langle u\rangle$) on $x\text{−}z$ planes at different channel heights for the counter-flow cases with $M=0.1, \text{Re}=400$, and a $24\times 2\times 12$ domain size at $t=1400$: (a) $y=0.0$, (b) $y\approx 0.791$ (${\tilde{y}}^{+}\approx 27$), and (c) $y\approx 0.905$ (${\tilde{y}}^{+}\approx 12$).

Figure 3

Instantaneous field of the streamwise velocity ($u$) on various planes for the counter-flow case with $M=0.1, \text{Re}=400$, and a $24\times 2\times 12$ domain size at $t=1400$. The red lines show the outlines of the $x\text{−}z$ plane cutting through the channel centerline (the figure shows the full height of the channel from $y=-1$ to $y=1$).

Figure 4

(a) Isosurfaces of the $Q$ criterion with an isovalue of $Q_{iso}=200$ colored by the magnitude of the streamwise velocity ($u$) for the counter-flow case with $M=0.1, \text{Re}=400$, and a $24\times 2\times 12$ domain size at $t=1400$. (b) Zoomed view of the region within the dashed rectangle in (a). The legend limits are cut off for a better visualization. The light violet lines in (a) show the outlines of the $x\text{−}z$ plane cutting through the channel centerline (the figure shows the full height of the channel from $y=-1$ to $y=1$).

Figure 5

The two-point correlations of $u^{\prime }, v^{\prime }$, and $w^{\prime }$ at different channel heights for the counter-flow case with a $24\times 2\times 12$ domain size: (a) $y=0.0$, (b) $y\approx 0.12$, (c) $y\approx 0.35$, (d) $y\approx 0.78$, (e) $y\approx 0.90$, and (f) $y\approx 0.997$ (first grid point above the wall).

Figure 6

Mean velocity and Reynolds stresses of the counter-flow cases with $M=0.1$ and $\text{Re}=400$ on different domain sizes and grid resolutions. Please note that for the ${\langle u\rangle }^{+}$ panel, the data are averaged over the two halves of the channel and ${\tilde{y}}^{+}$ is relative to the wall. The markers are shown for every ten grid points.

Figure 7

A direct comparison between the exact solution of $\langle \rho uv\rangle$ based on Eq. (9) and the present DNS with $M=0.1$ and $\text{Re}=400$ (case 4 of Table 1). The markers are shown for every ten grid points.

Figure 8

Energy budget of the streamwise normal stress $\langle u^{\prime }{u}^{\prime }\rangle$ and its terms for the cases with $M=0.1, \text{Re}=400$ and (left) domain size of $6\times 2\times 3$ and (right) domain size of $12\times 2\times 6$. The markers are shown for every ten grid points.

Figure 9

Mean velocity, temperature, density, and Mach number profiles and the turbulent Mach number ($M_t$) profile of the counter-flow cases with $M=0.1$ and $\text{Re}=400$, 800, and 1600. Please note that for the ${\langle u\rangle }^{+}$ panel, the data are averaged over the two halves of the channel and ${\tilde{y}}^{+}$ is relative to the wall. The markers are shown for every 10, 20, and 40 grid points for the cases with $\text{Re}=400$, 800, and 1600, respectively.

Figure 10

Reynolds stresses for the counter-flow cases with $M=0.1$ and $\text{Re}=400$, 800, and 1600. The markers are shown for every 10, 20, and 40 grid points for the cases with $\text{Re}=400$, 800, and 1600, respectively.

Figure 11

Instantaneous field of the streamwise velocity ($u$) on $x\text{−}z$ planes at different channel heights for the counter-flow cases with $M=0.1$ and $\text{Re}=400$ (left), $\text{Re}=800$ (middle), and $\text{Re}=1600$ (right) at $t=400$: (a) $y=0.0$ (channel center) and (b) ${\tilde{y}}^{+}=12$.

Figure 12

Mean velocity, temperature, density, and Mach number profiles and the turbulent Mach number ($M_t$) profile of the counter-flow cases with $M=0.1$ and 0.4 ($\text{Re}=400$). Please note that for the ${\langle u\rangle }^{+}$ panel, the data are averaged over the two halves of the channel and ${\tilde{y}}^{+}$ is relative to the wall. The markers are shown for every ten grid points.

Figure 13

Favre Reynolds stresses for the counter-flow cases with $M=0.1$ and 0.4 ($\text{Re}=400$). The markers are shown for every ten grid points.

Figure 14

Vorticity fluctuation and the components of the vorticity turbulent fluctuations of the counter-flow cases with $M=0.1$ and 0.4 ($\text{Re}=400$). The markers are shown for every ten grid points.

Figure 15

Probability density of the dilatation on an $x\text{−}z$ plane at $y\approx 0.51$ and $t=1400$ for the counter-flow cases with $M=0.1$ and 0.4 ($\text{Re}=400$).