Density variations in weakly compressible flows

Density variations in real fluids are related to both pressure and entropy variations, even in the incompressible limit. The behavior of density variations depends crucially on the relative sizes of the pressure, temperature, and entropy fluctuations. It is shown how this arises from a formal asymptotic expansion of the compressible Navier-Stokes equations about a uniform state. Direct numerical simulations of the full compressible equations verify the consistency of different asymptotic regimes. In the case of turbulent flow, the Kolmogorov-Obukhov theory allows one to predict inertial range scalings for the density fluctuation power spectra in various situations. The results may be relevant to observations of the interstellar medium.