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

In this work, we study the large-scale structure of homogeneous flows displaying high density contrasts and small turbulent Mach numbers. Following Batchelor & Proudman (PTRSA 258, 1956), we draw an analogy between this configuration and that of a single isolated eddy displaying density non-uniformities. 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 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.