Development of a nonconservative discontinuous Galerkin formulation for simulations of unsteady and turbulent flows

Abstract

Summary In this paper, discontinuous Galerkin (DG) discretization schemes for Navier‐Stokes equations of a nonconservative form was proposed, in which the constitutional equation for energy conservation is written in terms of pressure. The primary focus is to address the treatment of nonconservative products in the pressure equation, for which we formulate four different scheme variants by using the path‐conservative scheme or solely regarding the nonconservative products as source terms. In addition to the complete description of the discretization formulations, we highlight the positivity‐preserving properties of the proposed nonconservative DG schemes. The performance of the four schemes are thoroughly assessed and tested with the consideration of a series of classical flow configurations that involve steady/unsteady, inviscid/viscous, and laminar/turbulence flow physics. It is found that two out of the four schemes are able to provide the optimal convergence and similar accuracies as the fully conservative formulation. The nonconservative formulations provide an alternative way to model fluid flows with substantial thermodynamic and compositional complexities. The present work aims to lay the theoretical foundation for further algorithmic extensions to simulations of such fluid flows of practical relevance.

A pressure‐based method for weakly compressible two‐phase flows under a Baer–Nunziato type model with generic equations of state and pressure and velocity disequilibrium

“…In addition, in density-based method, the pressure field is generally updated by means of an EOS, an operation that, in compressible multiphase flows, may generate spurious oscillations at material interfaces. 36,37 On the other hand, using the pressure rather than the density as a solution variable in the governing equations could circumvent most of the issues arising from the weak pressure-density coupling at low Mach numbers, because pressure variations are significant at all speeds. Thus, a unique reference pressure can be easily identified for the nondimensional scaling of the governing equations, as in References 38.…”

“…A further open question in the development of the numerical method here proposed concerns the momentum or velocity correction, that is whether (33) can be substituted by (37). For this reason, we have rerun the simulations presented in Figures 2 and 3 with the velocity correction, where (33) is solved instead of (37).…”

“…However, this approach is difficult to apply to nonequilibrium multiphase models, because it is not possible to define a unique, unambiguous reference speed of sound. In addition, in density‐based method, the pressure field is generally updated by means of an EOS, an operation that, in compressible multiphase flows, may generate spurious oscillations at material interfaces 36,37 …”

A pressure-based method for weakly compressible two-phase flows under a Baer-Nunziato type model with generic equations of state and pressure and velocity disequilibrium

“…However, this approach is difficult to apply to non-equilibrium multi-phase models, because it is not possible to define a unique, unambiguous reference speed of sound. In addition, in density-based method, the pressure field is generally updated by means of an EOS, an operation that, in compressible multi-phase flows, may generate spurious oscillations at material interfaces 36,37 .…”

“…A further open question in the development of the numerical method here proposed concerns the momentum or velocity correction, that is whether (33) can be substituted by (37). For this reason, we have re-run the simulations presented in Figures 2 and 3 with the velocity correction, where (33) is solved instead of (37).…”