Molecular viscosity and diffusivity effects in transitional and shock-driven mixing flows

This paper investigates the importance of molecular viscosity and diffusivity for the prediction of transitional and shock-driven mixing flows featuring high and low Reynolds and Mach number regions. Two representative problems are computed with implicit large-eddy simulations using the inviscid Euler equations (EE) and viscous Navier-Stokes equations (NSE): the Taylor-Green vortex at Reynolds number $\mathrm{Re}=3000$ and initial Mach number $\mathrm{Ma}=0.28$, and an air-${\mathrm{SF}}_6$-air gas curtain subjected to two shock waves at $\mathrm{Ma}=1.2$. The primary focus is on differences between NSE and EE predictions due to viscous effects. The outcome of the paper illustrates the advantages of utilizing NSE. In contrast to the EE, where the effective viscosity decreases upon grid refinement, NSE predictions can be assessed for simulations of flows with transition to turbulence at prescribed constant $\mathrm{Re}$. The NSE can achieve better agreement between solutions and reference data, and the results converge upon grid refinement. On the other hand, the EE predictions do not converge with grid refinement, and can only exhibit similarities with the NSE results at coarse grid resolutions. We also investigate the effect of viscous effects on the dynamics of the coherent and turbulent fields, as well as on the mechanisms contributing to the production and diffusion of vorticity. The results show that nominally inviscid calculations can exhibit significantly varying flow dynamics driven by changing effective resolution-dependent Reynolds number, and highlight the role of viscous processes affecting the vorticity field. These tendencies become more pronounced upon grid refinement. The discussion of the results concludes with the assessment of the computational cost of inviscid and viscous computations.

Figure 1

Initial flow fields of the selected test cases.

Figure 2

Temporal evolution of the number of grid cells ($N_c\times {10}^{-6}$) of gas curtain simulations for EE and NSE on different grids $g_i$.

Figure 3

Temporal evolution of kinetic energy, $k$, for EE and NSE on different grid resolutions.

Figure 4

Temporal evolution of kinetic energy, $k$, and respective numerical uncertainty, $U_n\left(k\right)$, for EE and NSE on grids with $N_c={512}^3$ and ${1024}^3$.

Figure 5

Temporal evolution of dissipation, $\varepsilon$, for EE and NSE on different grid resolutions.

Figure 6

Vortical structures of the TGV flow for EE and NSE on $N_c={512}^3$ at $t=7$. Structures identified with the ${\lambda }_2$ criterion [83].

Figure 7

Temporal evolution of kinetic energy, $k$, and dissipation, $\varepsilon$, for viscous (NSE) and inviscid (EE) RANS BHR-2 simulations on the finest grid.

Figure 8

Temporal evolution of the numerical Reynolds number, ${\mathrm{Re}}_{\mathrm{n}}$, for EE and NSE on different grid resolutions.

Figure 9

Temporal evolution of the enstrophy magnitude, $\mathrm{\Omega }$, for EE and NSE on different grid resolutions.

Figure 10

Isosurfaces of the normalized $x_3$ vorticity field ($|{\omega }_3|=1.7$) colored by the vorticity magnitude for distinct models: results at $t=20$ with $N_c={512}^3$.

Figure 11

Temporal evolution of the norm of the vorticity equation terms (${\mathcal{T}}_S$, ${\mathcal{T}}_D$, and ${\mathcal{T}}_V$) for EE and NSE.

Figure 12

Temporal evolution of the turbulence kinetic energy spectrum, $E\left(k\right)$, for EE and NSE. Grid resolution is $N_c={512}^3$.

Figure 13

Temporal evolution of the ${\text{SF}}_6$ intensity, $I_{{\text{SF}}_6}$, for EE and NSE on the finest grid. Experiments are taken from Balakumar et al. [37].

Figure 14

Visualization of the Richtmyer-Meshkov structure and projectiles at $t=220\mu \mathbf{s}$ for EE and NSE on the finest grid. Experiments are taken from Balakumar et al. [37].

Figure 15

Fraction of the gas curtain with a local mass concentration of ${\mathrm{SF}}_6$, $c_{{\mathrm{SF}}_6}$, within selected ranges for EE and NSE at different time instants and grid resolutions.

Figure 16

Temporal evolution of the ${\text{SF}}_6$ intensity, $I_{{\text{SF}}_6}$, for EE and NSE on $g_2$.

Figure 17

Temporal evolution of the mixing-layer width, $w$, for EE and NSE on different grid resolutions. Experiments are taken from Balakumar et al. [37].

Figure 18

Temporal evolution of the flow kinetic energy, $k$, for EE and NSE on $g_3$.

Figure 19

Difference between the variance of $c_{{\mathrm{SF}}_6}$, $\mathrm{var}\left({\mathrm{c}}_{{\mathrm{SF}}_6}\right)$, for NSE and EE on $g_3$ at different instants. Results are shown as a percentage of ${\left[\mathrm{var}\left({\mathrm{c}}_{{\mathrm{SF}}_6}\right)\right]}_{EE}$.

Figure 20

Temporal evolution of the numerical Reynolds number, ${\mathrm{Re}}_{\mathrm{n}}$, for EE and NSE on different grid resolutions.

Figure 21

Temporal evolution of the vorticity magnitude, $\omega$, for EE and NSE on the finest grid.

Figure 22

Temporal evolution of the norm of the vorticity equation terms (${\mathcal{T}}_S$, ${\mathcal{T}}_D$, ${\mathcal{T}}_B$, and ${\mathcal{T}}_V$) for EE and NSE on grid $g_3$.

Figure 23

Temporal evolution of the norm of the vorticity equation terms (${\mathcal{T}}_S$, ${\mathcal{T}}_D$, ${\mathcal{T}}_B$, and ${\mathcal{T}}_V$) for NSE on grid $g_3$.

Figure 24

Temporal evolution of the turbulence kinetic energy spectrum, $E\left(k\right)$, for EE and NSE on grid $g_3$.

Figure 25

Spatial resolution of grid $g_3$ at $t=220\mu \mathrm{s}$ for EE and NSE.