Direct Numerical Simulation, Theories and Modelling of Wall Turbulence with a Range of Pressure Gradients

Abstract

A Direct Numerical Simulation (DNS) of Couette-Poiseuille flow is presented and analyzed in two ways. First, we test four semi-theoretical proposals for universal behaviour of respectively the velocity, the mixing length, the eddy viscosity, and the turbulence-kinetic-energy production rate. The question is which one may carry over from zero pressure gradient, for which they all agree, to finite pressure gradients, in which case they conflict. The DNS results, which are consistent between the favorable and the adverse pressure gradients, do not agree precisely with any of the theoretical proposals, but fall between those based on velocity (i.e., the log law) and on production. We do not know of a physical explanation for this trend, which has implications for theory, modelling, and wall functions. The residual effect of pressure gradients on the velocity profile, taken at y+=50, is examined in a wide range of simulations and experiments, which almost all turn out to follow a weak but definite trend to fall in adverse gradients. The second use of the data is to assess the accuracy of two turbulence models of the Reynolds-Averaged Navier-Stokes (RANS) type. These models are used in two modes. In the traditional mode, only the U velocity component is non-zero, and it depends only on the wall-normal coordinate y. In the newer mode, the V and W components are non-zero and the flow is allowed to depend on the lateral coordinate z with periodic conditions, but still not on x or t. For both pure Couette flow and the Couette-Poiseuille case considered here, this set-up allows some RANS models with nonlinear constitutive relations to generate streamwise vortices, which fill the channel. The presence of the rolls in the RANS generally improves the agreement with DNS and experiment, both qualitatively since the vortices are a known feature of such flows, and quantitatively, especially in terms of mixing in the core region of the channel. The vortices raise the skin-friction coefficient on the adverse-pressure-gradient side by as much as 30 %, depending on the chosen spacing between vortices.