Introduction

Turbulent motion is irregular. The irregularity is manifested through complex variations of velocity, temperature, etc. with space and time. The irregular motion is generated due to random fluctuations. At a Reynolds number less than critical, the kinetic energy of flow is not enough to sustain the random fluctuation against the viscous damping and in such laminar flow continues to exits. At somewhat higher Reynolds number than the critical Reynolds number, the kinetic energy of flow supports the growth of fluctuations and transition to turbulence is induced. The turbulence promotes improved mixing. High transfer rate of momentum, heat, and mass by fluctuating turbulent motion, are practically most important feature of turbulence.

Turbulent motion carries vorticity which is composed of eddies interacting with each other. At large Reynolds numbers, there exists a continuous transport of energy from the free stream to large eddies. From the large eddies a series of increasingly smaller eddies are formed. The smallest eddies dissipate energy and destroy themselves. The smaller eddies are influenced by the strain rate imposed by the large eddies and are stretched. The turbulence consists of a wide spectrum of eddies.

In this text, the commonly used mathematical treatments of turbulence for the analysis concerning Computational Fluids Dynamics (CFD) are described. We envisage to provide a comprehensive but brief review of the mathematical treatments used in CFD for applications in complex flow and heat transfer problems. The results due o the investigations of the author and his coworkers have been wherever possible. The methods for calculating turbulence can be grouped into the three broad categories.

• Reynolds averaged Navier-Stokes (RANS) equations of turbulence. The RANS approach includes eddy-viscosity based models, such as the k- e models and its variants on one hand and the Reynolds Stress Model (RSM) on the other. Usually the RSM consists of the second moment turbulence modeling.
  
• Large Eddy Simulation (LES) techniques.
  
• Direct Numerical Simulation (DNS) of turbulence.

Any flow whether laminar or turbulent, is fully represented by the Navier-Stokes equations. The Navier-Stokes equations can be solved on a fine enough grid with an exceptionally accurate discretization method so that both the fine scale and large scale aspects of turbulence can be calculated. This is termed as the Direct Numerical Simulation (DNS) of turbulence (Rai and Moin, 1991). However, the length-scale-range of the eddies of varying sizes and the time-scale-range of the velocity fluctuations due to the eddying motion cannot be economically resolved by ordinary discretization methods. Therefore, the Engineering Problems may be solved using Statistical Calculation Methods. Thus it is necessary to use some statistical average and a measure of the deviation from that average. Rodi (1993), and Wilcox (1993) provide excellent documentations on statistical approaches of turbulent flows. Notwithstanding the extent of the coverage of these methods, we shall focus at the equations governing the flow and the heat transfer in the turbulent regime at the first place. Subsequently, we shall describe various models and methods to solve the equations.