An anisotropic modification of the Reynolds stresses for algebraic models of turbulence

Abstract

In this paper the development and a partially verification of an anisotropic modification of linear and nonlineare models of turbulence is presented. This modification departs from the observation that there are some regions of fluid domains in which the Reynolds stresses do not depend on the magnitude of the mean flow gradients. The model is based on the hypothesis that actions of fluctuating motion on mean motion are almost exclusively due to large turbulent eddies, and that the latter contain the same kinetic energy of the small eddies of mean flow. The contribution of large turbulent eddies to the Reynolds stresses is then modelled by a linear combination of an isotropic term and an anisotropic term. The latter is obtained by multiplying a quote of the turbulent kinetic energy by a normalised redistribution tensor. The performances of the model are tested calculating the Reynolds stresses in a fully developed turbulent channel flow, and simulating the separated turbulent flow over a backward-facing step, and a fully developed turbulent flow in a squared duct. The results show that the modified models improve the prediction of the normal Reynolds stresses in a bidimensional channel and of the reattachment point in a backward-facing step. On the other hand, these models do not supply any improvement of the secondary flow in a squared duct. INTRODUCTION Algebraic two-equation models of turbulence for Reynolds averaged Navier-Stokes equations are still, today, among the most widely used in the field of engineering design. The two major alternative to such models are the second-order closure models and large-eddy simulations (LES). The weak points of the first approach are high computational cost, due to the five more equations to be solved, and wall-bounded turbulent flows. On the other hand, LES requires much more computational time than second-order closure models, and improved models for the smallscale and the wall region are needed. For these reasons these two approaches have not become yet a useful engineering tool. The aim of this work is to develop an anisotropic modification for linear and nonlinear turbulence model which improves the prediction of normal Reynolds stresses, without increasing the number of differential equations to be solved. More precisely, the objective is to simulate the contribution to the anisotropy of the turbulent stresses observable in areas characterised by low mean velocity gradients. This phenomenon, which can be noted, for example, in experimental data of the turbulent channel flow in figure 5, is not actually described by standard, explicit algebraic models. The starting point of this model consists in the substitution of turbulent kinetic energy by an appropriate vectorial magnitude which provides information about the anistropy of the turbulent stresses. The quantity is identified by the field of velocity of the small eddies of mean motion. The kinetic energy of these small eddies is obtained by assuming that these t Dr, Center of Studies and Activities for Space (CISAS). Dr, Department of Mechanical Engineering. ' Professor, Department of Mechanical Engineering. Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. have the same kinetic energy content of the large turbulent eddies: the two fields of motion are said to be similar. It has been demonstrated that the field of motion of large turbulent eddies, or coherent structures^ can be obtained directly from measurements of instantaneous velocity once the quota of turbulent kinetic energy has been defined. The models obtained provides improvements in the predictions of normal Reynolds stresses present in the turbulent flow of a bidimensional channel; as for the separated turbulent flow in a backward-facing step appreciable improved results of mean flow and of normal Reynolds stresses have been attained compared to standard algebraic models. On the other side, as it will be shown in the following paragraphs, the model of the turbulent kinetic energy redistribution tensor does not supply any improvement in the case of the fully developed turbulent flow in a squared duct. ANISOTROPIC MODIFICATION OF THE REYNOLDS STRESSES Let us consider a field of turbulent motion of a viscous, homogenous and incompressible fluid; the instantaneous velocity v and the instantaneous pressure p can be set out as follows in a mean and fluctuating parts: v = v -f v' p-p + p (1) The evolution of the velocities and mean pressures can be described completely by the well-known Reynolds equations: