2. The k-ε Turbulence Model
The k-ε turbulence model still remains among the most popular, most known is the standard (Jones-Launder) k-ε
turbulence model.
As in all eddy-viscosity turbulence model derivation initializes with the Boussinesq hypothesis:
The second steps is to devise (on purely dimensional grounds) a relation between the eddy-viscosity and the two
chosen characteristics:
Obtaining a transport equation for the total kinetic energy is a simple mathematical step of forming a dot product of
NSE with the velocity vector:
A transport equation for the total kinetic energy could be written as:
Decomposing the velocity vector according to Reynolds decomposition
and defining the turbulent kinetic energy as:
3. The k-ε Turbulence Model
The construction of an energy transport equation for the mean flow by the same procedure as the total kinetic energy
transport equation was constructed (i.e. dot product of the mean velocity with RANS equations):
The next steps consider time (or ensemble) averaging the total kinetic energy transport equation and
the subtraction of the mean flow energy transport equation:
1. Pressure work due to only turbulence.
2. Transport of turbulent kinetic energy due to fluctuations.
3. Diffusive transport of turbulence kinetic energy.
4. Turbulence production, or to be more precise the amplification of the Reynolds stress tensor by the mean strain.
5. Dissipation rate of turbulence kinetic energy.
4. The k-ε Turbulence Model
Another important step evident to all such models is ad-hoc simplification relying on (some) physical justification for
each of the above terms to achieve a final transport equation:
In the construction of an equation for the turbulence dissipation ε. To do so we first invoke local isotropy for the
dissipation.
as far as the Reynolds decomposition is concerned it is somewhat harder to justify local isotropy as the fluctuating
term in RANS is not an actual representation of high wave-number (small spatial scale) behaviour in general (for LES it
does!).
It is possible to derive the equation following the same route as for the turbulent kinetic energy to finaly achieve a
transport equation for the turbulence dissipation ε:
5. Final form of the k-ε Turbulence Model
After constructing both equations and defining the relations between the transported variables to the eddy viscosity
the final form of the standard k-ε turbulence model may be presented:
The last important step is to calibrate the model constants. In turbulence modeling calibration of the model is at least
as important as the derivation of the model itself. Calibration is achieved with the help of experimental and numerical
results of the type of flow that should be modeled. The calibration process is also the first step in which the range of
validity of the model would be revealed to close inspection and not just postulated from physical reasoning.
For the standard k-ε turbulence model the calibrated closure constants are:
6. Shortcomings of the k-ε Turbulence Model
the model is essentially a high Reynolds model, meaning the law of the wall must be employed and provide velocity
"boundary conditions" away from solid boundaries (what is termed "wall-functions").
From a mathematical standpoint, even if one could impose Dirichlet conditions for ε on solid boundary, after meshing
it would still be difficult to numerically approach the problem due to what is termed in numerical analysis as
stiffness of the numerical problem, partially related to the high gradients.
In order to integrate the equations through the viscous/laminar sublayer a "Low Reynolds" approach must be
employed. This is achieved as additional highly non-linear damping functions are needed to be added to low-
Reynolds formulations (low as in entering the viscous/laminar sublayer) to be able to integrate through the laminar
sublayer (y+<5). This again produces numerical stiffness and in case is problematic to handle in view of linear
numerical algorithms.
The model suffers from lack of sensitivity to adverse pressure-gradient. It was observed that under such conditions it
overestimates the shear stress and by that delays separation.
See link for a thorough description of the k-ε Turbulence Model
7. The Realizable k-ε Turbulence Model
A straightforward option to circumvent some of the standard drawbacks still in the framework of the same variables as
transport transport equations was made by invoking realizability constraints.
There are a number of such constraints, the usual ones are that all normal stresses should remain positive and the
correlation coefficients for the shear stress should not exceed one:
Remembering that the velocity gradient tensor may be decomposed to a symertic part (strain rate tensor) and an anti-
symmetric part (rotation tensor):
8. The Realizable k-ε Turbulence Model
If it is assumed that the flow that on one axis the flow approaches a wall, the Boussinesq Hypothesis for the normal
stress becomes:
It could be seen that if s11 is too large then: , hence nonphysical, i.e. non-realizable.
It is customary at this stage to introduce the concept of "invariant", meaning something that is independent on a
coordinate system. For the above case this relates to rotation.
The Invariants are calculated via solving for the eigenvalues of the strain rate tensor. The eigenvalues of S correspond to
the strains in the principal axis, since we have applied the equation on the principle axis, S11 is replaced by the largest
eigenvalue such that:
This simple modification to an eddy viscosity model ensures that the normal stresses stay positive.
9. The Realizable k-ε Turbulence Model
Another realizability constraint appears when we require that if: , This shall be done smoothly.
This is ensured by demanding:
We also impose a requirement that when v1’2 approaches zero, the transport equation for it shall do so to.
This is one of these cases where the normal stress goes to zero faster (O)(x4) than the parallel one (O(x2)) and creates
the state of turbulence called the two component limit.
See link for a thorough description of the realizable k-ε Turbulence Model
10. Near Wall Treatment for ε-equation
Turbulence Models
Standard wall functions:
Wall functions are applicable for a suitable range of y* and by that to the flow's Reynolds. As the purpose is to
allow not integrating through the viscous sublayer, the lower limit of y* is 11 hence standard wall functions
should not be used below that limit as the solution's accuracy might deteriorate in an uncontrolled manner.
Scalable wall functions:
To avoid deterioration of the solution by standard wall functions in situations where it's unavoidable for the first
grid point to be located at y*<11, ANSYS Fluent proposes wall functions that produce consistent results for
arbitrary grid refinement by forcing the usage of the log law in conjunction with the standard wall functions
approach:
Non-equilibrium wall functions:
sensitizing the log-law for mean velocity to pressure gradient effects and by the use of the two-layer-based
concept to compute the reciprocal relations between turbulence kinetic energy production, Gk, and its
dissipation ε, by non-equilibrium means.
Therefore to some extent the non-equilibrium formulation for the wall functions takes into account the effect of
pressure gradients on the distortion of the velocity profiles and by that account for some non-equilibrium effects.
11. Near Wall Treatment for ε-equation
Turbulence Models
Enhanced Wall Treatment ε-Equation (EWT-ε):
by incorporating the two-layer model with enhanced wall functions a separation of the two regions is conducted via a
wall-distance-based, turbulent Reynolds number:
For regions where the above defined Reynolds number is above 200 the original standard k-ε is employed.
By the same token if the above Reynolds number is below 200, a one equation for the transport of turbulence kinetic
energy is employed (Wolfstein's k-equation).
Menter-Lechner ε-Equation (ML-ε):
To avoid EWT-ε shortcomings such as regions with small turbulence kinetic energy that might also have a turbulent
Reynolds number below 200 and as such treated with a near-wall formulation even though they are actually away
from the wall.
The approach offers a y+ insensitive which is a not based on a two layer model assuming a sufficient resolution of the
boundary layer, predicts in a y+ independent manner the wall shear stress and wall heat flux. Such a
formulation switches gradually from wall functions to a low-Reynolds formulation when the mesh is refined. The
concept is achieved by adding a a source term to the transport equation of the turbulence kinetic energy that
accounts for near-wall effects.
See link for a thorough description of k-ε near wall treatement
12. The law of the wall –
y+ according to Fluent turbulence models
The flow demands solving the viscous sublayer – The first element away
from solid boundary must be located at 𝑦+
< 1.
Includes cases of which wall bounded effects are very important. Common
examples are aerodynamic drag calculation, heat transfer, flow
reattachment prediction, and flows with adverse pressure gradient.
The flow does NOT demand solving the viscous sublayer – The first
element away from the wall should be located such that 𝑦+ = 30 − 300
(note that the extent of the log layer or in other words its upper bound is
flow dependent. Low Reynolds flows for example might have a very low
𝑦+
cuttof for the log-layer)
Includes cases for which wall-bounded effects are of secondary
importance or when the flow undergoes geometry induced separation.
Common examples are aerodynamic lift calculations, and flow
surrounding a bluff body.
See link for a thorough description for The Law of The Wall
