Numerical Treatment of incompressible Turbulent Flow

Turbulence features a subtle balance between the energy input at the large scales of motion, the transfer of energy to smaller and smaller scales and the energy dissipation at the smallest scales. When numerically modelling this energy exchange, it should not be disturbed by non-physical, numerical effects. In this vein, energy-conserving discretizations have been developed, which do not dissipate energy artificially and are inherently stable because they cannot spuriously generate kinetic energy. In this chapter, we consider these discretizations, with focus on finite-volume methods. Mathematically,
the conservation of energy is a consequence of the symmetries of the differential operators in the incompressible Navier-Stokes equations. When the spatial discretization method preserves these symmetries, then the energy of the discrete system is conserved upon exact time integration. The latter is complicated by the algebraic incompressibility constraint. Together with the pressure gradient, it describes acoustical pressure waves with an infinite propagation speed, hence this acoustical part of the flow equations requires implicit time integration, whereas the remaining parts may be integrated explicitly. Often it is too expensive to calculate all scales of motion, and coarse-grained models are used. To describe the dynamics of the larger eddies, the Navier-Stokes equations are filtered in space and the effect of the turbulence that is filtered out is modelled. Discretizing the resulting equations effectively introduces a second filter, by providing a
good approximation for the low wavenumbers and suppressing the high wavenumbers. Consequently, modeling and discretization errors are entangled. In this chapter, this entanglement is analyzed for a finite-volume discretization. Because of the entanglement of discretization and modeling errors, it makes sense to use a unified idea about the nature of both the discretization (of the resolved part) and the subgrid model (for the unresolved part). Here, the discrete approximation is done so that fundamental properties (such as symmetries) are preserved. The subgrid modeling should then also be like that. We focus
on asking how much dissipation the subgrid model should provide to counterbalance the nonlinear production of the small, unresolved scales of motion. This yields a minimumdissipation condition.