Compressibility effects on homogeneous isotropic turbulence using Schur decomposition of the velocity gradient tensor

The study of compressibility effects on the dynamics and the structure of turbulence is an important, but difficult, topic in turbulence modeling. Taking advantage of a recently proposed Schur decomposition approach (Keylock, C. J., The Schur decomposition of the velocity gradient tensor for turbulent flows, Journal of Fluid Mechanics, 2018) to decompose the velocity gradient tensor into its normal and non-normal parts, here we evaluate the influence of the compressibility on some statistical properties of the turbulent structures. We perform a set of direct numerical simulations of decaying compressible turbulence at six turbulent Mach numbers between Mt = 0.12 and Mt = 0.89 and a Reynolds number based on the Taylor micro-scale of Ret = 100. All the simulations have been carried out using an improved seventh-order accurate WENO scheme to discretize the non-linear advective terms and an eight-order accurate centered finite difference scheme is retained for the diffusive terms. In the double decomposition, the normal parts of the velocity gradient tensor (represented by the eigenvalues) are separated explicitly from non-normal components. The two-dimensional space defined by the second and third invariants of the velocity gradient tensor is subdivided into six regions and the contribution of each regional term to the Schur decomposition of the velocity gradient tensor is analyzed. Our preliminary findings show the difficulty of understanding the non-local effects without taking into account both the normal contribution (represented by the eigenvalues) and the non-normal component computed with of the Schur decomposition.