Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows

The form of convective terms for compressible flow equations is discussed in the same way as for an incompressible one by Morinishi et al. [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys. 124 (1998) 90], and fully conservative finite difference schemes suitable for shock-free unsteady compressible flow simulations are proposed. Commutable divergence, advective, and skew-symmetric forms of convective terms are defined by including the temporal derivative term for compressible flow. These forms are analytically equivalent if the continuity is satisfied, and the skew-symmetric form is secondary conservative without the aid of the continuity, while the divergence form is primary conservative. The relations between the present and existing quasi-skew-symmetric forms are also revealed. Commutable fully discrete finite difference schemes of convection are then derived in a staggered grid system, and they are fully conservative provided that the corresponding discrete continuity is satisfied. In addition, a semi-discrete convection scheme suitable for compact finite difference is presented based on the skew-symmetric form. The conservation properties of the present schemes are demonstrated numerically in a three-dimensional periodic inviscid flow. The proposed fully discrete fully conservative second-order accurate scheme is also used to perform the DNS of compressible isotropic turbulence and the simulation of open cavity flow.