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9.1. Introduction

PHOENICS provides the standard high-Reynolds-number form of the two- equation eddy-viscosity KE-EP turbulence model. For wall-bounded flows the model requires the application of so-called wall functions, by which the wall boundary conditions are transferred to the near-wall grid points located in the fully-turbulent fluid.

The advantage of this approach is its computational economy; for it eliminates the need to introduce many grid points in the near-wall layer so as to resolve the steep gradients that prevail there.

Despite the not-inconsiderable success of the wall-function approach, there are many flow situations where its use may be unsuitable. Examples are:-

The alternative approach is the low-Reynolds-number treatment, which involves integrating the equations through the viscous sublayer with the aid of a turbulence model which allows for viscous effects.

The simplest example of a near-wall modification to a turbulence model is the Van Driest [1956] damping function for the Prandtl mixing length (see the "Encyclopaedia": VAN-Driest damping model).

For the two-equation KE-EP model, several low-Reynolds-number extensions have been suggested in which wall damping and viscous effects are incorporated by making several of the model coefficients functions of a local "turbulence Reynolds number". Also, in some models, extra source terms are added to the turbulence-transport equations in order to represent the near-wall behaviour better.

A fairly comprehensive review of the available low-Reynolds-number extensions has been given by Patel et al [1984].

The review concludes that the models of:

The LB model employs a transport equation for the total dissipation rate, while the other two models solve for a modified dissipation rate which permits the 'dissipation variable' to be set to zero at the wall. It has the advantage that it requires no additional source terms; but a disadvantage is that one of the damping functions requires the calculation of the local distance to the nearest wall.

The Chien formulation requires not only additional source terms and the wall distance, but also the friction velocity associated with the nearest wall.

At present only the LB model is available for use in PHOENICS.

Provision has also been made in PHOENICS 2.0 for the LB model to be used as a low-Reynolds-number extension of the modified KE-EP model of Chen and Kim (hereafter denoted CK) [1987, 1990], which performs better than the standard KE-EP model for separated and reattaching flows.

Several workers have observed that the original form of the EP equation returns progressively too high length scales as the near- wall value of turbulence production/dissipation falls to zero. The consequences of this are that boundary-layer separation will be predicted too late and, in separated flow, too large heat-transfer coefficients are predicted.

The effects become more severe if the numerical integration is carried to the wall, because the low-Re model allows the excessive length scales to reach the wall itself. For example, the wall heat- transfer coefficients are typically six times higher than measured in separated and reattaching flows.

A modification of the high-Re form of the EP equation is essential for such applications, and the CK modification goes some way towards providing what is needed.

Alternatively, PHOENICS provides for the standard KE-EP to be modified by way of the so-called Yap correction term to the EP equation. This correction is reported to provide much-improved predictions in separated and reattaching flows (see Yap [1987]).

As an economical, and often more accurate, alternative to the low-Re KE-EP models, PHOENICS is also equipped with a two-layer k-e model, which uses the high-Re k-e model only away from the wall in the fully-turbulent region. The near-wall viscosity-affected layer is resolved with a one-equation model involving a length-scale prescription.

A complete account of the model is provided under the Encyclopaedia entry 'TWO-Layer KE-EP turbulence model'.

Finally, a low-Reynolds-number version of Wilcox's [1988] KE-OMEG model has been provided in PHOENICS.

This model produces converged solutions much more rapidly than the LB and CK KE-EP models, and in this respect is very competitive with the two-layer model.

Another advantage of the model is that it does not require the calculation of wall distances for use in low-Re damping functions.

A complete description of this model is provided under the Encyclopaedia entry 'K-Omega turbulence model'.