The pressure–dilatation correlation in compressible flows

Simulations of simple compressible flows have been performed to enable the direct estimation of the pressure–dilatation correlation. The generally accepted belief that this correlation may be important in high-speed flows has been verified by the simulations. The pressure–dilatation correlation is theoretically investigated by considering the equation for fluctuating pressure in an arbitrary compressible flow. This leads to the isolation of a component of the pressure–dilatation that exhibits temporal oscillations on a fast time scale. Direct numerical simulations of homogeneous shear turbulence and isotropic turbulence show that this fast component has a negligible contribution to the evolution of turbulent kinetic energy. Then, an analysis for the case of homogeneous turbulence is performed to obtain a formal solution for the nonoscillatory pressure–dilatation. Simplifications lead to a model that algebraically relates the pressure–dilatation to quantities traditionally obtained in incompressible turbulence closures. The model is validated by direct comparison with the simulations.