Chapter 11 - Numerical treatment of incompressible turbulent flow

Abstract

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 modeling this energy exchange, it should not be disturbed by nonphysical, 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.