Numerical investigation of turbulence in reshocked Richtmyer–Meshkov unstable curtain of dense gas

Abstract

Moderate-resolution numerical simulations of the impulsive acceleration of a dense gas curtain in air by a Mach 1.21 planar shock are carried out by solving the 3D compressible multi-species Navier–Stokes equations coupled with localized artificial diffusivity method to capture discontinuities in the flow field. The simulations account for the presence of three species in the flow field: air, $\hbox {SF}_6$ and acetone (used as a tracer species in the experiments). Simulations at different concentration levels of the species are conducted and the temporal evolution of the curtain width is compared with the measured data from the experimental studies by Balakumar et al. (Phys Fluids 20:124103–124113, 2008). The instantaneous density and velocity fields at two different times (prior and after the reshock) are compared with experimental data and show good qualitative agreement. The reshock process is studied by re-impacting the evolving curtain with the reflected shock wave. Reshock causes enhanced mixing and destroys the ordered velocity field causing a chaotic flow. The unsteady flow field is characterized by computing statistics of certain flow variables using two different definitions of the mean flow. The average profiles conditioned on the heavy gas (comprising $\hbox {SF}_6$ and acetone) and the corresponding fluctuating fields provide metrics which are more suitable to comparing with experimentally measured data. Mean profiles (conditioned on the heavy gas) of stream-wise velocity, variance of stream-wise velocity, and turbulent kinetic energy and PDF (probability distribution function) of fluctuating velocity components are computed at two different times along the flow evolution and are seen to show trend towards grid convergence. The spectra of turbulent kinetic energy and scalar energy (of mass fraction of heavy gas) show the existence of more than half decade of inertial sub-range at late times following reshock. The Reynolds stresses in the domain are reported while identifying the term that is dominant in its contribution to the Reynolds stresses.

Appendices

Appendix A: Artificial diffusivity method

The localized artificial diffusivity method formulated on a generalized multi-dimensional coordinate system is given below:

$$\begin{aligned}&\mu ^* = C_{\mu } \overline{\rho \left| \sum ^{3}_{l=1} \frac{\partial ^{4}S}{\partial \xi _{l}^{4}} \Delta \xi _{l}^{4} \Delta ^{2}_{l,\mu } \right| }, \end{aligned}$$

(29)

$$\begin{aligned} \begin{aligned}&\beta ^* = C_{\beta } \overline{\rho f_\mathrm{sw}\left| \sum ^{3}_{l=1} \frac{\partial ^{4}\nabla \cdot {\varvec{u}}}{\partial \xi _{l}^{4}} \Delta \xi _{l}^{4} \Delta ^{2}_{l,\beta } \right| },\\&f_\mathrm{sw}=H(-\nabla \cdot {\varvec{u}}) \times \frac{(\nabla \cdot {\varvec{u}})^2}{(\nabla \cdot {\varvec{u}})^2 + (\nabla \times {\varvec{u}})^2 + \varepsilon } \end{aligned} \end{aligned}$$

(30)

$$\begin{aligned} \kappa ^* = C_{\kappa } \overline{\frac{\rho c_\mathrm{s}}{T} \left| \sum ^{3}_{l=1} \frac{\partial ^{4}e}{\partial \xi _{l}^{4}} \Delta \xi _{l}^{4} \Delta _{l,\kappa } \right| }, \end{aligned}$$

(31)

$$\begin{aligned} D_k^*&= C_{D_k} \overline{c_\mathrm{s} \left| \sum ^{3}_{l=1} \frac{\partial ^{4}Y_{k}}{\partial \xi _{l}^{4}} \Delta \xi _{l}^{4} \Delta _{l, D_k} \right| }\nonumber \\&\quad +\,C_{Y_k} \overline{c_\mathrm{s} [Y_k-1]H(Y_k-1) - Y_k[1-H(Y_k)]\Delta _{Y_{k}}},\nonumber \\ \end{aligned}$$

(32)

The method is the extension of the original artificial viscosity scheme proposed by Cook [18], Kawai and Lele [20] (modification to curvilinear mesh) and Mani et al. [21] (dilatation based bulk viscosity with the length scaling modification). $C_\mu ,C_\beta ,C_\kappa ,C_{D_k}$ and $C_{Y_k}$ are dimensionless user-specified constants. $c_\mathrm{s}$ is the speed of sound, and $H$ is the Heaviside function. $\xi _{l}$ refers to generalized coordinates $\xi ,\eta$ and $\zeta$ when $l$ is 1, 2 and 3, respectively. $\varepsilon =10^{-32}$ is a small positive constant to prevent division by zero in the region where both $\nabla \cdot {\varvec{u}}$ and $\nabla \times {\varvec{u}}$ are zero. The overbar denotes an approximate truncated-Gaussian filter [19]. The $f_\mathrm{sw}$ in (30) is the switching function that is designed to help localize the artificial bulk viscosity only near shock waves.

$\Delta \xi _{l}$ and $\Delta _{l, \bullet }$ are the grid spacing in the computational space and physical space. Usually, $\Delta \xi _{l}$ is set to 1. We define the grid spacing $\Delta _{l, \bullet }$ as follows:

$$\begin{aligned}&\Delta _{l,\mu } = \left| {\varvec{\Delta }}{\varvec{x}}_{\varvec{l}} \right| , \quad \Delta _{l,\beta } = \left| {\varvec{\Delta }}{\varvec{x}}_{\varvec{l}}\cdot \frac{{\varvec{\nabla }}{\varvec{\rho }}}{ \left| {\varvec{\nabla }}{\varvec{\rho }}\right| } \right| ,\nonumber \\&\quad \Delta _{l,\kappa } = \left| {\varvec{\Delta }}{\varvec{x}}_{\varvec{l}}\cdot \frac{{\varvec{\nabla }}{\varvec{e}}}{\left| {\varvec{\nabla }}{\varvec{e}} \right| }\right| , \quad \Delta _{l,D_k} = \left| {\varvec{\Delta }}{\varvec{x}}_{\varvec{l}}\cdot \frac{{\varvec{\nabla }}{\varvec{Y}}_{\varvec{k}}}{\left| {\varvec{\nabla }}{\varvec{Y}}_{\varvec{k}}\right| } \right| . \end{aligned}$$

(33)

${\varvec{\Delta }}{\varvec{x}}_{\varvec{l}}$ is the local displacement vector along the grid line in the $\xi _{l}$ direction and is defined as

$$\begin{aligned} {\varvec{\Delta }}{\varvec{x}}_{\varvec{l}}&= \left( \frac{x_{i+1}-x_{i-1}}{2}, \frac{y_{i+1}-y_{i-1}}{2}, \frac{z_{i+1}-z_{i-1}}{2} \right) ^\mathrm{T}, \end{aligned}$$

(34)

where the nodes in the $\xi _{l}$ direction are indexed by $i$. Thus, $\Delta _{l,\beta },\Delta _{l,\kappa }$ and $\Delta _{l, D_k}$ are the grid spacing in the $\xi _{l}$ direction perpendicular to the shock waves, contact surfaces and material discontinuities. $\Delta _{Y_{k}}$ in (32) is the grid spacing in the physical space defined by $\Delta _{Y_{k}} = \frac{ \sum ^{3}_{l=1} \left| \frac{\partial ^{4}Y_k}{\partial \xi _{l}^{4}} \right| \Delta _{l, D_k}}{ \sqrt{\sum ^{3}_{l=1} \left( \frac{\partial ^{4}Y_k}{\partial \xi _{l}^{4}} \right) ^2} + \varepsilon }$.

Appendix B: Viscosity coefficient [32]

The viscosity of a pure gas species $i$ is given by:

$$\begin{aligned} \mu _i=\frac{26.69}{\varOmega _{\mu ,i}}\frac{\sqrt{M_iT}}{\sigma _i^2} \end{aligned}$$

(35)

where $\sigma _i$ is the collision diameter and $\varOmega _{\mu ,i}$ is the collision integral:

$$\begin{aligned} \varOmega _{\mu ,i}=A(T_i^*)^{-B}+C\hbox {exp}(-DT_i^*) +E\hbox {exp}(-FT_i^*) \end{aligned}$$

(36)

in which $T_i^*=T/T_{\epsilon ,i}$ and $A=1.16145$, $B=0.14874$, $C=0.52487$, $D=0.77320$, $E=2.16178$, $F=2.43787$, $T_{\epsilon ,i}=\epsilon _i/k_\mathrm{B}$ is the effective temperature characteristic of the force potential function ($k_\mathrm{B}$ is the Boltzmann constant). The values of $\sigma _i$ and $\epsilon _i$ for the different species of interest are given below. Air is regarded as a mixture of nitrogen and oxygen in the ratio of 71:29, respectively.

Appendix C: Diffusion coefficient [32]

The diffusion coefficient in a binary mixture of two species $i$ and $j$ is given as:

$$\begin{aligned} D_{ij}=\frac{0.00266}{\varOmega _{D,ij}}\frac{T^{3/2}}{p\sqrt{M_{ij}}(\sigma _{ij})^2} \end{aligned}$$

(37)

The collision integral for diffusion is given by:

$$\begin{aligned} \varOmega _{D,ij}&= A(T_{ij}^*)^{-B}+C \hbox {exp}(-DT_{ij}^*)+E\hbox {exp}(-FT_{ij}^*)\nonumber \\&+\,G\hbox {exp}(-HT_{ij}^*) \end{aligned}$$

(38)

where $M_{ij}=\frac{2}{\frac{1}{M_i}+\frac{1}{M_j}},\,A=1.06036,\, B=0.15610,\,C=0.19300,\,D=0.47635,\,E=1.03587,\, F=1.52996,\,G\!=\!1.76474$ and $H=3.89411$. $\sigma _{ij}=\frac{\sigma _i+\sigma _j}{2},\,T_{ij}^* = T/T_{\epsilon ,ij}$ and $T_{\epsilon ,ij}=\sqrt{\frac{\epsilon ^i}{k_\mathrm{B}}\frac{\epsilon ^j}{k_\mathrm{B}}}$. The value of Lennard Jones potentials $(\epsilon /k_\mathrm{B}$ and $\sigma )$ are given in Table 2. Air is regarded as a mixture of nitrogen and oxygen in the ratio of 71:29, respectively.

Table 2 Molecular properties

Appendix D: Contribution of the artificial property

The role of the artificial property in the simulated results is characterized in this section by comparing the magnitude of the artificial property in the governing equations to the corresponding molecular property of the fluid. Of particular interest is the artificial diffusivity term which is added to the RHS of the scalar mass fraction equation. The cross section averaged profiles of the artificial diffusivity is plotted with the corresponding species diffusivity (between $\hbox {air-SF}_6$) at two different times in Fig. 21. The artificial diffusivity dominates the species diffusivity in the coarse mesh simulations, while on the finer mesh, the magnitudes of the artificial and the molecular property terms remain comparable indicating that the artificial property model is active on the fine grid resolutions explored here. At early times, the molecular diffusivity in the flow domain is larger than the artificial diffusivity introduced by the localized artificial diffusivity method. Hence, good comparison of the numerical results with the experimental data suggests that the artificial property method which is active in the numerical simulations is able to robustly handle the material discontinuities in the flow field and leads to a good prediction of this unsteady multi-component compressible flow.

Fig. 21

True-average profiles of artificial and molecular species diffusivity (in kg/ms) at a $t=236.8\,\upmu \hbox {s}$ and b $t=757.6\,\upmu \hbox {s}$