Introduction
The K-epsilon model is one of the most common turbulence models. It is a two equation model, that means, it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy.
The first transported variable is turbulent kinetic energy, . The second transported variable in this case is the turbulent dissipation, . It is the variable that determines the scale of the turbulence, whereas the first variable, , determines the energy in the turbulence.
There are two major formulations of K-epsilon models (see References 2 and 3). That of Launder and Sharma is typically called the "Standard" K-epsilon Model. The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows. As described in Reference 1, the K-epsilon model has been shown to be useful for free-shear layer flows with relatively small pressure gradients. Similarly, for wall-bounded and internal flows, the model gives good results only in cases where mean pressure gradients are small; accuracy has been shown experimentally to be reduced for flows containing large adverse pressure gradients. One might infer then, that the K-epsilon model would be an inappropriate choice for problems such as inlets and compressors.
To calculate boundary conditions for these models see turbulence free-stream boundary conditions.
Standard k-epsilon model
Transport equations for standard k-epsilon model
For turbulent kinetic energy
For dissipation
Turbulent viscosity is modelled as:
Production of k
Where is the modulus of the mean rate-of-strain tensor, defined as :
Effect of buoyancy
where Prt is the turbulent Prandtl number for energy and gi is the component of the gravitational
vector in the ith direction. For the standard and realizable - models, the default value of Prt is
0.85.
The coefficient of thermal expansion, , is defined as
Realisable k-epsilon model
Transport Equations
Where
In these equations, represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model. is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model.
Modelling Turbulent Viscosity
where
; ;
where is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity . The model constants and are given by:
Model Constants
RNG k-epsilon model
The RNG model was developed using Re-Normalisation Group (RNG) methods by Yakhot et al
to renormalise the Navier-Stokes equations, to account for the effects of smaller scales of motion. In the standard k-epsilon model the eddy viscosity is determined from a single turbulence length scale, so the calculated turbulent diffusion is that which occurs only at the specified scale, whereas in reality all scales of motion will contribute to the turbulent diffusion. The RNG approach, which is a mathematical technique that can be used to derive a turbulence model similar to the k-epsilon, results in a modified form of the epsilon equation which attempts to account for the different scales of motion through changes to the production term.
Contents
[hide]
• 1 Transport Equations
• 2 Constants
• 3 Applicability and Use
• 4 References
Transport Equations
There are a number of ways to write the transport equations for k and , a simple interpretation where bouyancy is neglected is
where
and and
With the turbulent viscosity being calculated in the same manner as with the standard k-epsilon model.
Constants
It is interesting to note that the values of all of the constants (except ) are derived explicitly in the RNG procedure. They are given below with the commonly used values in the standard k-epsilon equation in brackets for comparison:
(0.09) (1.0) (1.30) (1.44) (1.92)
(derived from experiment)
Applicability and Use
Although the technique for deriving the RNG equations was quite revolutionary at the time, it's use has been more low key. Some workers claim it offers improved accuracy in rotating flows, although there are mixed results in this regard: It has shown improved results for modelling rotating cavities, but shown no improvements over the standard model for predicting vortex evolution (both these examples from individual experience). It is favoured for indoor air simulations.
k-ε model
The k-ε model is the most commonly used of all the turbulence models. It is classified as a two equation model. This denotes the fact that the transport equation is solved for two turbulent quantities k and ε. Within the model the properties k and ε are defined through two transport equations (1) and (2).
(1)
(2)
k-ω model
The k-ω model is the second most widely used turbulence model and is also classified as a two equation model. This model is very similar to the k-ε model and uses the same definition for k as previously outlined (1). However the k-ω model differs in its selection of a second variable for characterising turbulent behaviour. The equation for ω is defined by the following equation (3).
(3)
Reynolds Stress
Reynolds Stress models solve for individual Reynolds stresses and for the turbulent dissipation ε. The Reynolds Stress model assumes that the factors within it contain no dependence on the Reynolds number of the flow. However it should be noted that this is not always true especially at moderate Reynolds numbers. The central concept of the Reynolds stress model is that the stress tensor Rij is determined locally within the cell. This assumption provides the main
weakness to the Reynolds stress model in that it often disregards long range effects caused by walls and other objects within the flow.
Strengths and Weaknesses
Strengths Weaknesses
k-ε model and k-ω model are computationally
cheap k-ε model overestimates turbulence k-ω model more accurate at boundary level
Reynolds stress generally more accurate Reynolds Stress is computationally expensive Reynolds Stress underestimated long range effects
Choice
k-ε model because k-ω model offers no advantage and Reynolds Stress would be too computationally demanding for a large 3-D case.
