Staggered scheme for the compressible fluctuating hydrodynamics of multispecies fluid mixtures

We present a numerical formulation for the solution of nonisothermal, compressible Navier-Stokes equations with thermal fluctuations to describe mesoscale transport phenomena in multispecies fluid mixtures. The novelty of our numerical method is the use of staggered grid momenta along with a finite volume discretization of the thermodynamic variables to solve the resulting stochastic partial differential equations. The key advantages of the numerical scheme are that it significantly simplifies the discretization of diffusive and stochastic momentum fluxes into a more compact form, and it provides an unambiguous prescription of boundary conditions involving pressure. The staggered grid scheme more accurately reproduces the equilibrium static structure factor of hydrodynamic fluctuations in gas mixtures compared to a collocated scheme described previously by Balakrishnan et al. [Phys. Rev. E 89, 013017 (2014)]. The numerical method is tested for ideal noble gases mixtures under various nonequilibrium conditions, such as applied thermal and concentration gradients, to assess the role of cross-diffusion effects, such as Soret and Dufour, on the long-ranged correlations of hydrodynamic fluctuations, which are also more accurately reproduced compared to the collocated scheme. We numerically study giant nonequilibrium fluctuations driven by concentration gradients and fluctuation-driven Rayleigh-Taylor instability in gas mixtures. Wherever applicable, excellent agreement is observed with theory and measurements from the direct simulation Monte Carlo method.

Figure 1

An illustration of the staggered spatial discretization. (a) The thermodynamic variables such as $p, \rho , T$, and $\mathbf{Y}$ are located at the cell centers, and components of vector quantities such as ${\mathbf{v}}^{\left(x\right)}$ and ${\mathbf{j}}^{\left(x\right)}$ (red), ${\mathbf{v}}^{\left(y\right)}$ and ${\mathbf{j}}^{\left(y\right)}$ (green), and ${\mathbf{v}}^{\left(z\right)}$ and ${\mathbf{j}}^{\left(z\right)}$ (blue) are located on respective staggered grids, i.e., on the faces of the finite volume grid. (b) Random numbers ${\mathcal{Z}}^{(\mathit{F},\mathit{Q})}$ corresponding to $\tilde{\mathit{F}}$ and $\tilde{\mathit{Q}}$ are generated at the faces of the control volume centered around the location of the thermodynamic variables. (c) The random numbers $S^{xx}$ and $S^{zz}$ corresponding to the diagonal terms of $\tilde{\mathit{\Pi }}$ are generated at the faces of the staggered control volumes around ${\mathbf{j}}^{\left(x\right)}$ (red) and ${\mathbf{j}}^{\left(z\right)}$ (blue), respectively, that are are collocated at the cell centers of the finite volume grid. The random numbers $S^{xz}$ corresponding to the off-diagonal terms of $\tilde{\mathit{\Pi }}$ are generated on faces of the staggered control volumes around both ${\mathbf{j}}^{\left(x\right)}$ and ${\mathbf{j}}^{\left(z\right)}$ that are collocated on the edges of the finite volume grid. The dotted lines in panels (b) and (c) represent the six-point divergence operator.

Figure 2

Nondimensionalized static structure factors along the $k_y$ and $k_z$ axes for $k_x=0$ with the wave numbers ranging from $-3.93\times {10}^{-7}{\text{cm}}^{-1}$ to $3.93\times {10}^{-7}{\text{cm}}^{-1}$. The panels correspond to (a) $\langle \left(\delta \widehat{\rho }\right){\left(\delta \widehat{\rho }\right)}^{*}\rangle$, (b) $\langle \left(\delta \widehat{{\mathbf{j}}^{\left(x\right)}}\right){\left(\delta \widehat{{\mathbf{j}}^{\left(x\right)}}\right)}^{*}\rangle$, (c) $\langle \left(\delta \widehat{\rho E}\right){\left(\delta \widehat{\rho E}\right)}^{*}\rangle$, (d) $\langle \left(\delta \widehat{{\rho }_1}\right){\left(\delta \widehat{{\rho }_1}\right)}^{*}\rangle$, (e) $\langle \left(\delta \widehat{{\rho }_4}\right){\left(\delta \widehat{{\rho }_4}\right)}^{*}\rangle$, and (f) $\langle \left(\delta \widehat{T}\right){\left(\delta \widehat{T}\right)}^{*}\rangle$. The colored data ranges around $\pm 20\%$ from the expected theoretical value of unity. The center of the image corresponds $k_y=k_z=0$.

Figure 3

Magnitude of nondimensionalized correlations in Fourier space (see the caption in Fig. 2). The panels correspond to (a) $\langle \left(\delta \widehat{\rho }\right){\left(\delta \widehat{{\mathbf{j}}^{\left(x\right)}}\right)}^{*}\rangle$, (b) $\langle \left(\delta \widehat{\rho }\right){\left(\delta \widehat{T}\right)}^{*}\rangle$, (c) $\langle \left(\delta \widehat{{\mathbf{j}}^{\left(x\right)}}\right){\left(\delta \widehat{{\mathbf{j}}^{\left(y\right)}}\right)}^{*}\rangle$, (d) $\langle \left(\delta \widehat{{\mathbf{j}}^{\left(x\right)}}\right){\left(\delta \widehat{\rho E}\right)}^{*}\rangle$, (e) $\langle \left(\delta \widehat{{\rho }_1}\right){\left(\delta \widehat{{\rho }_4}\right)}^{*}\rangle$, and (f) $\langle \left(\delta \widehat{{\mathbf{v}}^{\left(x\right)}}\right){\left(\delta \widehat{T}\right)}^{*}\rangle$.

Figure 4

The structure factor ${\mathcal{S}}_{Y,{\mathbf{v}}_{\left|\right|}}$ as a function of wave number $k_y$ normalized by cell spacing $\mathrm{\Delta }y$ for wave vectors perpendicular to the applied concentration gradient, i.e., $k_x=0$, for various values of $L_y$. The theoretical prediction from Eq. (33) for an infinitely periodic system is shown with a dotted line.

Figure 5

Real-space spatial correlations in a quasi-1D binary fluid mixture. Results are shown for equilibrium and nonequilibrium steady states using FHD and DSMC simulations [see legend in panel (a)]. (a) Spatial correlation between the fluctuations of partial density of species 0, $\delta {\rho }_0^{†}$, at $x^{†}$ with the fluctuations $\delta {\rho }_0$ in every other cell. (b) Spatial correlation between the fluctuations of partial density of species 1, $\delta {\rho }_1^{†}$, at $x^{†}$ with the fluctuations $\delta {\rho }_0$ in every other cell. The correlations are normalized by the volume $V$ of either FHD or DSMC cell in their respective cases. The position of $x^{†}$ is shown by vertical dashed lines, and the value of the correlations at $x^{†}$ have been removed in panels (a) and (b).

Figure 6

Real-space spatial correlations in a quasi-1D binary mixture of dissimilar gases under thermal gradient. Results are shown for FHD and DSMC simulations [see legend in panel (b)]. (a) Spatial correlation between the fluctuations of temperature $\delta T^{†}$ at $x^{†}$ with the fluctuations $\delta T$ in every other cell. (b) Spatial correlation between the fluctuations of $x$-velocity $\delta {\mathbf{v}}_x^{†}$ at $x^{†}$ with the fluctuations of total density $\delta \rho$ in every other cell. The correlations are normalized by the volume $V$ of either FHD or DSMC cell in their respective cases. The position of $x^{†}$ is shown by vertical dashed lines, and the value of the correlations at $x^{†}$ have been removed in panels (a) and (b).

Figure 7

Real-space spatial correlations in a quasi-1D binary mixture of dissimilar gases under thermal gradient. Results are shown for FHD and DSMC simulations [see legend in panel (b)]. (a) Spatial correlation between the fluctuations of partial density of species 0, $\delta {\rho }_0^{†}$, at $x^{†}$ with the fluctuations $\delta {\rho }_0$ in every other cell. (b) Spatial correlation between the fluctuations of partial density of species 1, $\delta {\rho }_1^{†}$, at $x^{†}$ with the fluctuations $\delta {\rho }_0$ in every other cell. The correlations are normalized by the volume $V$ of either FHD or DSMC cell in their respective cases. The position of $x^{†}$ is shown by vertical dashed lines, and the value of the correlations at $x^{†}$ have been removed in panels (a) and (b).

Figure 8

The role of Soret and Dufour effects in the mean and fluctuating hydrodynamics of a mixture of dissimilar gases under an applied thermal gradient. Results are shown for FHD simulations with $\tilde{\chi }\ne 0$ (circles), and $\tilde{\chi }=0$ (squares) that corresponds to neglecting the Soret and Dufour effects. (a) The mean profile $\langle {\rho }_1\rangle$ of the partial density of species 1. (b) Spatial correlation between the fluctuations of partial density of species 1, $\delta {\rho }_1^{†}$, at $x^{†}$ with the fluctuations $\delta {\rho }_1$ in every other cell. The correlations are normalized by the volume $V$ of the FHD cell. The position of $x^{†}$ is shown by vertical dashed lines, and the $\delta$ function for correlation at $x^{†}$ has been removed in panel (b).

Figure 9

Rayleigh-Taylor instability. The images depict the temporal evolution of the partial density of the lighter species ${\rho }_0$. Panels (a)–(d) show the vertical cross section of ${\rho }_0$ at times $t=0, t=2.0\times {10}^{-8}\text{s}, t=3.75\times {10}^{-8}\text{s}$, and $t=5.0\times {10}^{-8}\text{s}$, respectively. Panels (e, f) show the horizontal slice at the center of the domain corresponding to panels (b–d). The data range for all the images is shown in the color bar on the top right of the figure.