Direct numerical simulation and Reynolds-averaged Navier-Stokes modeling of the sudden viscous dissipation for multicomponent turbulence

Simulations of a turbulent multicomponent fluid mixture undergoing isotropic deformations are carried out to investigate the sudden viscous dissipation. This dissipative mechanism was originally demonstrated using simulations of an incompressible single-component fluid [S. Davidovits and N. J. Fisch, Phys. Rev. Lett. 116, 105004 (2016)]. By accounting for the convective and diffusive transfer of various species, the current work aims to increase the physical fidelity of previous simulations and their relevance to inertial confinement fusion applications. Direct numerical simulations of the compressed fluid show that the sudden viscous dissipation of turbulent kinetic energy is unchanged from the single-component scenario. More importantly, the simulations demonstrate that the mass fraction variance and covariance for the various species also exhibit a sudden viscous decay. Reynolds-averaged Navier-Stokes simulations were carried out using the $k-l$ model to assess its ability to reproduce the sudden viscous dissipation. Results show that the standard $k-l$ formulation does not capture the sudden decay of turbulent kinetic energy, mass-fraction variance, and mass-fraction covariance for simulations with various compression and expansion rates, or different exponents for the power-law model of viscosity. A new formulation of the $k-l$ model that is based on previous improvements to the $k-\epsilon$ family of models is proposed, which leads to consistently good agreement with the direct numerical simulations for all the isotropic deformations under consideration.

Figure 1

Dissipation of the mass-fraction variance for the five different species simulated, compared against the target values used by the linear forcing mechanism. $t$ is time and ${\tau }_0$ is the eddy turnover time.

Figure 2

Evolution of (a) turbulent kinetic energy $k$, (b) mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$, and (c) mass-fraction covariance of tritium and oxygen ${\upsilon }_{\text{TO}}$, for direct numerical simulations of isotropic compressions. The subscript “0” indicates the initial value. The $1/L^2$ scaling in (a) follows from rapid distortion theory (RDT) [26].

Figure 3

Evolution of (a) TKE dissipation and (b) mass-fraction-variance dissipation, for direct numerical simulations of isotropic compressions. The subscript “0” indicates the initial value.

Figure 4

Evolution of (a) turbulent kinetic energy $k$ and (b) mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$, for direct numerical simulations of isotropic compressions. The vertical dashed line corresponds to the point in time at which production and dissipation of turbulent kinetic energy are equal. The diagonal dashed lines serve as fiducials, with a slope of 2.8 in (a) and 3.3 in (b). The subscript “0” indicates the initial value.

Figure 5

Evolution of (a) turbulent kinetic energy $k$, (b) mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$, and (c) mass-fraction covariance of tritium and oxygen ${\upsilon }_{\text{TO}}$, for direct numerical simulations of isotropic compressions. $n$ is the power-law exponent, and the subscript “0” indicates the initial value.

Figure 6

Evolution of (a) turbulent kinetic energy $k$, (b) mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$, and (c) mass-fraction covariance of tritium and oxygen ${\upsilon }_{\text{TO}}$, for direct numerical simulations of isotropic expansions. The subscript “0” indicates the initial value. The $1/L^2$ scaling in (a) follows from rapid distortion theory (RDT) [26].

Figure 7

Evolution of turbulent kinetic energy $k$ on the top row, mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$ on the middle row, and mass-fraction covariance of tritium and oxygen ${\upsilon }_{\text{TO}}$ on the bottom row, for RANS simulations of isotropic compressions. Left column, original $k-l_d$; right column, modified $k-l_d$. The subscript “0” indicates the initial value. The $1/L^2$ scaling in (a) and (b) follows from rapid distortion theory (RDT) [26].

Figure 8

Evolution of turbulent kinetic energy $k$ on the top row, mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$ on the middle row, and mass-fraction covariance of tritium and oxygen ${\upsilon }_{\text{TO}}$ on the bottom row, for RANS simulations of isotropic compressions. Left column, original $k-l_d$; right column, modified $k-l_d$. $n$ is the power-law exponent, and the subscript “0” indicates the initial value.

Figure 9

Evolution of turbulent kinetic energy $k$ on the top row, mass-fraction variance of deuterium ${\upsilon }_{\text{DD}}$ on the middle row, and mass-fraction covariance of tritium and oxygen ${\upsilon }_{\text{TO}}$ on the bottom row, for RANS simulations of isotropic expansions. Left column, original $k-l_d$; right column, modified $k-l_d$. The subscript “0” indicates the initial value. The $1/L^2$ scaling in (a) and (b) follows from rapid Ddortion theory (RDT) [26].