TURBULENCE MODELLING

For turbulent flow, a proper turbulence model should be incorporated into the boundary-layer solvers to predict the results accurately. Turbulence is a property of the flow rather than that of the fluid, hence, empirical correlation is employed to represent the variation of eddy viscosity inside a boundary layer. Turbulence modelling may be classified as follows (White (1991)).

1. Algebraic models 2. One-equation models 3. Two-equation models

Algebraic models

An algebraic turbulence model requires no additional differential equations to be solved. The eddy viscosity distribution is specified in two parts, the inner region and the outer region. Examples of this model are the Cebeci-Smith model and the Baldwin-Lomax model. They are used by many existing calculation methods and seem to be the most popular class of turbulence models (Humphreys and Lindhout (1988)). Lakshminarayana (1991) suggested that the algebraic eddy-viscosity model is strictly valid only for two-dimensional "simple shear flows". He also concluded that:

1. The model is adequate for two-dimensional compressible flows with a mild pressure gradient

2. The model is suitable for three-dimensional boundary layers with small cross flows.

3. The model is not valid for flows with curvature, rotation, or separation. 4. The model is not valid for pressure or turbulence-driven secondary flows and when abrupt changes in strain or shear rate are presented.

5. The model cannot accurately predict shock-induced separated flow.

One-equation models

In one-equation model, the turbulent eddy viscosity is evaluated by a model of turbulent kinetic energy (k). This model solves only one partial differential equation, therefore, it is less computationally intensive compared with two-equation model.

The use of one-equation models in the 1968 Stanford Conference showing that the results were satisfactory but, apparently, no better than the best algebraic methods that merely used a model for eddy viscosity (White (1991)). The implementation of one-equation model is extremely difficult for extending a

length-scale correlation to complex flows (Lakshminarayana (1991)). These are the reasons why a one-equation model is not presently popular.

Two-equation models

In the two-equation model, the turbulent kinetic energy (k) and dissipation (e) equations are solved to calculate turbulent eddy viscosity. Lakshminarayana (1991) summarised the model as follows.

1. The k-e model is much superior to algebraic models.

2. The k-e model is accurately predicted for most two-dimensional flows. 3. The k-e model is not good for three-dimensional flows with high cross flows, swirl, rotation, curvature and shock-induced separation. This is because the model assumes an isotropy and low Reynolds number formulation near the wall.

Sondak (1995) developed a method for the application of wall functions to generalised curvilinear co-ordinate systems with non-orthogonal grids. The method has made the use of wall functions and the k-e turbulent model to accurately predict the viscous effects near the wall in low-Reynolds number models. It has been tested on a flat plate with a non-orthogonal grid and a prolate hentispheroid with an orthogonal grid. The results are compared with experimental data and Baldwin-Lomax model. The k-e model with the wall function gave good results for both the fme-grid and coarse-grid cases.

Ekaterinaris and Menter (1994) tested one- and two-equation eddy- viscosity models for unsteady massively separated flow. An implicit numerical scheme is used for the integration of the compressible, Reynolds-averaged Navier- Stokes equations. These turbulent models are tested for steady separated flow and for unsteady flow over oscillating airfoils. The selected models include the Baldwin-Barth (B-B) model and the Spalart-Allmaras (S-A) model for one- equation models, the k-e, the k-co and the shear stress transport (SST) k-co models

for two-equation models. The results showed that for the light-stall case the S-A model did not yield sufficient separation and underpredicted the extreme values of the unsteady loads (lift coefficient Cp drag coefficient c^, and pitching moment coefficient c^). The B-B model overpredicted the values for the light-stall case and the attached flow cases. The standard k-e and the k-co models did not predict separation even for the deep-stall case. The SST k-co model gave good predictions for the attached and the light-stall cases.

Reynolds Stress Models (RSM)

RSM is potentially the superior model since it provides a more realistic physical simulation of turbulent flow. RSM is intended to account for complex turbulent effects, such as surface curvature and rotation, in three-dimensional layers. However, the modelled Reynolds stress equations are extremely complicated to solve for a three-dimensional flow since there are about 10-20 transport equations involved (Lakshminarayana (1991)). Zhang and Lakshminarayana (1990) modified a model in conjunction with the k-e equations. This model is called an algebraic Reynolds stress model (ARSM). ARSM simulates the turbulent stress more realistically by relating the properties to local conditions. This model is efficient and inexpensive and can predict important features of flow in a turbomachinery (e.g. rotation and curvature effects). Therefore ARSM may be used to illustrate the ability of boundary layer codes to predict complex three-dimensional boundary layers.