Numerical calculation of the Reynolds stress and turbulent heat fluxes

Numerical techniques for calculating the Reynolds stress and heat flux in turbulent-flow problems are developed and discussed, with a focus on low-cost methods based on the mean-flow equations. The role of turbulence modeling in computational fluid dynamics is reviewed, and the history of turbulence models since the first efforts by Kolmogorov (1942) and Prandtl (1945) is traced. The governing PDEs are given; the boundary conditions are formulated; the computer memory and CPU-time required to solve the problem numerically are considered; and a three-level-splitting scheme and fourth-order-compact schemes (for both PDEs and ODEs) are presented. It is concluded that economy and accuracy are achieved by employing high-order methods on coarse meshes, especially when the high-order finite-difference formulas are generated automatically inside the program.