**Research and Numerical Algorithms in Computational Fluid Dynamics Simulations**

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Sharaf F. Al-Sharif

*Saudi Arabia*

**Reynolds Stress Transport Modelling**

*Center of Excellence in Desalination Technology, King Abdulaziz University, Jeddah*

The Reynolds–averaged Navier–Stokes (RANS) approach is the most commonly employed approach in CFD for industrial applications, and is likely to continue to be so for the foreseeable future. The need to handle complex wall-bounded flows, and the need to evaluate large numbers of design variations usually prohibits high-fidelity approaches such as direct numerical simulation (DNS), and large-eddy simulation (LES). The application of Reynolds–averaging to the equations of motion introduces a set of unclosed terms, the *Reynolds Stresses*, into the mean flow momentum equations, and turbulence models are needed to provide closure of these terms before the set of equations can be solved. Within the framework of RANS approaches, a hierarchy of modelling schemes exists based on the level of sophistication in which these unclosed terms are modelled. In Reynolds stress transport (RST) modelling, rather than assuming a direct (linear or non–linear) link between the Reynolds stresses and mean strain, a separate transport equation for each of the stress components is solved. This in principle provides a number of advantages over other RANS models, which

This chapter aims to provide a general introduction and overview of Reynolds Stress transport modelling. The first section will provide a brief historical background on the development of this class of models. Next, the theoretical background and rationale underlying the most common modelling practises within this framework are presented. This is followed by a discussion of some numerical implementation issues specific to RST modelling within the context of the finite volume method. Finally the chapter is closed with some concluding

Early work leading to the development of Reynold-stress transport (RST) modelling was mainly theoretical, due to the relative complexity of this level of modelling compared to the available computational capabilities of the time. Chou (1945) constructed a formal solution to the fluctuating pressure Poisson equation that is the basis for current models of the pressure–strain-rate correlation. Later Rotta (1951), laid the foundation for Reynolds stress transport modelling by being the first to develop a closed model of all the terms in the exact equation (Speziale, 1991). Because of limited computational capability at the time, successful computations were not carried out until several decades later (Speziale, 1991). Another important development came when the continuum mechanics community speculated on the potential similarity between turbulent flow and the flow of non-Newtonian fluids (Gatski,

**1. Introduction**

will be reviewed here.

**2. Development of RST modelling**

remarks.
