A stabilized finite element method based on SGS models for compressible flows

We present an appropriate extension of the stabilized finite element formulation, introduced in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823] for the prediction of incompressible flows, aimed at compressible flows. The stabilized formulation is the so-called variable subgrid scale method (V-SGS) based on an approximation of the class of subgrid scale models (SGS) derived from the Hughes variational multiscale method (Hughes-VMS) introduced in [T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401]. It is characterized by a variable stabilization parameter within the domain of element.We also propose an innovative procedure for computing the stabilization parameter by using the one-dimensional element Green’s functions corresponding to an advective–diffusive differential operator. The stabilization parameter is defined as the sum of two components, the first providing its mean value and the second its element-wise space dependency. On this basis the present work proposes the stabilization matrix for the multidimensional advective–diffusive system of equations governing compressible flows. The procedure developed for computing the stabilization parameter can be applied to other differential operators as it has been shown in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823]. The proposed stabilization device is validated by simulation of inviscid and viscous supersonic flow configurations.