A computational procedure has been developed for predicting separated turbulent flows in complex two-dimensional and three-dimensional geometries. The procedure is based on the fully conservative, structured finite volume framework, within which the volume are non-orthogonal and collocated such that all discrete fields of flow variables are stored at one and the same set of nodes. To ease the task of discretisation and to enhance the conservative property of the scheme, a Cartesian decomposition of the velocity field has been used, rather than a contravariant, coordinate directed decomposition. As an alternative, an adaptive decomposition has also been included, in which one velocity component is allowed to adapt itself to a pre defined, essentially arbitrary datum line, with the other two components normal to that adapted. This latter decomposition has been provided with the view to the procedure being preferentially applied to curved three-dimensional duct of variable cross-sectional area. The solution algorithm is iterative in nature, approaching the steady-state solution by a sequential solution for velocity components and pressure with the aid of a pressure-correction algorithm. Convection is approximated with a range of schemes, among them first-order, second-order and third-order upstream-weighted schemes, and with a Total Variation Diminishing MUSCL scheme, the last applied specifically to transport equations governing turbulence properties.

The procedure has been formulated so as to apply to both incompressible and compressible flows. The latter may contain shocks and highly supersonic regions. To achieve this range of applicability, the scheme solves, for compressible flows, equations for flux variables (i.e. density-weighted velocity components) and incorporates a density-retardation techniques which improves the procedure’s shock-capturing capabilities.

Computational economy is enhanced by use of a Full Multigrid Scheme which is shown to yield speed-up factor of order 5 - 100, depending on the nature of the flow and grid density. This multigrid scheme has been applied both for laminar and turbulent flows, in conjunction with all turbulent models investigated.

A range of turbulence models has been incorporated into the numerical algorithm, including high- and low-Reynolds-number k — e eddy-viscosity variants and a full Reynolds-stress-transport closure which computes all components of the Reynolds-stress tensor from related transport equations. Among these models, the low-Reynolds-number variants are original, having been developed by reference to constraints implicit in the length-scale prescription of existing one-equation turbulence models.

A broad range of flows, totally some 20 configurations, have been computed within an extensive validation exercise, encompassing inviscid, laminar and turbulent flows, both compressible and incompressible, some two-dimensional and others threedimensional. A number of geometries have been computed principally in order to verify or examine the adequacy of numerical practices or the performance of the multigrid convergence acceleration scheme. Many, however, involve the investigation of turbulence-model performance by reference to detailed experimental results. Among the most challenging flows computed are:​ shock-induced separation over bumps, computed with second-moment closure;​ separated flow through a sinusoidal constriction, also computed with second-moment closure;​ and three-dimensional flow through a S-shaped diffuser calculated with low-Reynolds-number models.

The study demonstrates that complex flows can be computed in a stable and fairly economical manner using advanced turbulence models. In particular, it has been shown how second-moment closure can be incorporated into a demanding non-orthogonal environment in which convection is approximated with numerically nondiffusive schemes, and that this type of closure offers some advantage in complex geometries, similar to those established for simpler configurations.