Direct numerical simulation of internal compressible flows at high Reynolds number: numerical and physical insight

This work reports results of direct numerical simulations (DNS) of compressible internal flows. For this purpose three internal flow geometries of increasing complexity are considered, namely planar channel, pipe and rectangular duct flow. The work focuses on both numerical and physical issues related to wall-bounded turbulent flows. In the first part of the work some numerical issues concerning the solution of compressible wall-bounded flows, both in Cartesian and cylindrical coordinates, are addressed. Attention is focused on the acoustic time-step limitation which, in the case of wall-bounded flows, is restrictive across all Mach numbers. For this reason we develop a semi-implicit algorithm for time-accurate simulation of the compressible Navier-Stokes (N-S) equations. The method is based on linearization of the partial convective fluxes associated with acoustic waves, in such a way to suppress, or at least mitigate the acoustic time step restriction. Together with replacement of the total energy equation with the entropy transport equation, this approach avoids the inversion of block-banded matrices involved in classical methods, which is replaced by less demanding inversion of standard banded matrices. This novel implementation, in which only Acoustic Terms are Implicit (ATI), is more efficient than previous approaches, barely requiring the inversion of a banded scalar system in each coordinate direction. All available data support higher computational efficiency than existing methods, and saving of resources ranging from 85% under low-subsonic flow conditions, to about 50% in supersonic flow. Numerical issues arising from the use of cylindrical coordinates are also discussed. We show that N-S equations in cylindrical coordinates can be conveniently recast to guarantee discrete conservation of total kinetic energy. The ATI approach is extended to the cylindrical case to deal with the severe time-step limitation in the azimuthal direction. In the second part of the work attention is focused on the effects of Mach and Reynolds number variation for the three flow geometries considered. DNS of planar channel, pipe and rectangular duct flow at bulk Mach number Mb=0.2,1.5, 3 and up to Re_tau=1000$ are presented. A long-standing topic in compressible flows is the relevant Reynolds number for comparing flow cases across the Mach number range. At this purpose, different compressibility transformations are compared to incompressible datasets at matching relevant Reynolds number. All data show that the Trettel-Larsson transformation allows excellent collapse of the compressible statistics on the incompressible ones, thus supporting the validity of semi-local scaling and Morkovin hypothesis. The size of the typical turbulent eddies is studied through spanwise spectral densities of the velocity field, which support validity of a scaling based on the local mean shear and the local friction velocity, with the main conclusion that the actual size of the eddies does not vary with the Mach number, at a fixed outer wall distance. Passive scalar transport is also studied across Mach and Reynolds number. Eventually, similarities and differences between compressible channel, pipe and rectangular duct flow are investigated.