Abstract: The paper reviews the problem of making numerical predictions of turbulent flow. It advocates that computational economy, range of applicability and physical realism are best served at present by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy k and its dissipation rate ϵ, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour. The width of applicability of the model is demonstrated by reference to numerical computations of nine substantially different kinds of turbulent flow.

Abstract: A general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction. Such flows give rise to parabolic differential equations and so can be called three-dimensional parabolic flows. The procedure can be regarded as a boundary-layer method, provided it is recognised that, unlike earlier published methods with this name, it takes full account of the cross-stream diffusion of momentum, etc., and of the pressure variation in the cross-stream plane. The pressure field is determined by: first calculating an intermediate velocity field based on an estimated pressure field; and then obtaining appropriate correction so as to satisfy the continuity equation. To illustrate the procedure, calculations are presented for the developing laminar flow and heat transfer in a square duct with a laterally-moving wall.

Abstract: The vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10,000 and meshes consisting of as many as 257 × 257 points. For Re = 1000, the (129 × 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional grid-clustering coordinate transformations.

Abstract: A non-iterative method for handling the coupling of the implicitly discretised time-dependent fluid flow equations is described. The method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations. The main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step, with the split sets of equations being amenable to solution by standard techniques. At each time-step, the procedure yields solutions which approximate the exact solution of the difference equations. The accuracy of this splitting procedure is assessed for a linearised form of the discretised equations, and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time-step size. By virtue of this, it is possible to dispense with iteration, thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods. This is verified in a companion paper which presents results of computations carried out using the method.

Abstract: A finite volume numerical method is presented for the solution of the two-dimensional incompressible, steady Navier-Stokes equations in general curvilinear coordinates. This method is appied to the turbulent flows over airfoils with and without trailing edge separation. The k-e model is utilized to describe the turbulent flow process. Body-fitted coordinates are generated for the computation. Instead of the staggered grid, an ordinary grid system is employed for the computation and a specific scheme is developed to suppress the pressure oscillations. The results of calculations are compared with the available experimental data.