Permanence of large eddies in decaying variable-density homogeneous turbulence with small Mach numbers

In this paper, we study the large-scale structure of decaying homogeneous flows displaying high-density contrasts and small turbulent Mach numbers. Following Batchelor and Proudman [Batchelor and Proudman, The large-scale structure of homogeneous turbulence, Phil. Trans. R. Soc. A 248, 369 (1956)], we draw an analogy between this configuration and that of a single isolated eddy displaying density nonuniformities. By doing so, we are able to highlight the crucial role played by the solenoidal component of the momentum in the preservation of initial conditions. In particular, we show that the large-scale initial conditions of the spectrum of the solenoidal momentum are preserved provided its infrared exponent is smaller than 4. This condition is reminiscent of the one usually derived in constant-density flow, except that it does not apply to the velocity spectrum. The latter is actually shown to be impermanent for infrared exponents larger than or equal to 2. The consequences of these properties on the self-similar decay of the flow are discussed. Finally, these predictions are verified by performing large eddy simulations of homogeneous isotropic turbulence.

Figure 1

Schematic representation of a single variable-density eddy.

Figure 2

Schematic representation of a superposition of independent eddies with and without linear impulse.

Figure 3

Time evolution of (a) the kinetic energy and (b) the density variance for the first series of simulations.

Figure 4

Evolution of the spectrum $Q^{\text{so}}$ of the solenoidal momentum with ${Q^{\text{so}}}^{\left(0\right)}$ given by Eq. (32) and for different values of $s_q$. (a) $s_q=1$, (b) $s_q=2$, (c) $s_q=3$, (d) $s_q=4$, and (e) $s_q=10$.

Figure 5

Evolution of the velocity spectrum $E$ with ${Q^{\text{so}}}^{\left(0\right)}$ given by Eq. (32) and for different values of $s_q$. (a) $s_q=1$, (b) $s_q=2$, (c) $s_q=3$, (d) $s_q=4$, and (e) $s_q=10$.

Figure 6

Evolution of the spectrum $Q^{\text{di}}$ of the dilatational momentum with ${Q^{\text{so}}}^{\left(0\right)}$ given by Eq. (32) and for different values of $s_q$. (a) $s_q=1$, (b) $s_q=2$, (c) $s_q=3$, (d) $s_q=4$, and (e) $s_q=10$.

Figure 7

Evolution of the spectrum $Q^{\text{so}}$ of the solenoidal momentum with $E^{\left(0\right)}$ given by Eq. (34), for $s_e=6$ and for different density contrasts. (a) $\sqrt{\overline{{\rho }^{\prime 2}}}/\overline{\rho }=0.02$ and (b) $\sqrt{\overline{{\rho }^{\prime 2}}}/\overline{\rho }=0.75$.

Figure 8

Evolution of the velocity spectrum $E$ with $E^{\left(0\right)}$ given by Eq. (34), for $s_e=6$ and for different density contrasts. (a) $\sqrt{\overline{{\rho }^{\prime 2}}}/\overline{\rho }=0.02$ and (b) $\sqrt{\overline{{\rho }^{\prime 2}}}/\overline{\rho }=0.75$.

Figure 9

Absolute value of the nonlinear transfer terms $T$ of $Q^{\text{so}}$ (a) and $T_E$ of $E$ (b). The result is displayed at time $t=2{\tau }_0$ for the series of two simulations for which $E$ is imposed at initial times.

Figure 10

Instantaneous and averaged measures of the decay exponent of the kinetic energy $\mathcal{K}$. (a) Evolution of $n_k^{\mathrm{simu}}\left(t\right)$ and (b) Comparison between $\langle n_k^{\mathrm{simu}}\rangle$ and formula (31).

Figure 11

Material point $M$ of mass $m$ moving with respect to the origin $O$ with velocity $\mathit{u}$.

Figure 12

Schematic representation of the pressure scaling.

Figure 13

Schematic representation of the nonlinear transfer term.

Figure 14

Schematic representation of $Q^{\text{so}}$ when $E\propto k^{s_e}$ at small wave numbers.