Understanding The k-ε Turbulence Model

“To mechanical progress there is apparently no end: for as in the past so in the future, each step in any direction will remove limits and bring in past barriers which have till then blocked the way in other directions; and so what for the time may appear to be a visible or practical limit will turn out to be but a bend in the road…” – Osborne Reynolds

osborne reynolds

Turbulence modelling is considered by many as witchcraft, by others as the art of producing physics out of chaos. The above quote by Osborne Reynolds, generally to remind the endless potential in “last unsolved problem” of classic physics, that of Turbulence, ever so affected by and coupled with our chaotic surrounding, incorporating circular cause and effect reciprocity, acting so unexpected that it seems one may only put his trust in god or the dice… Such is turbulence!

A full description of the phenomena is entangled in a seemingly simple set equation, the Navier-Stokes equations, their nature is such that analytic solutions to even the most simple turbulent flows can not be obtained and resorting to numerical solutions seems like the only hope.

But the resourcefulness of the plea to a direct numerical description of the equations is a mixed blessing as it seems the availability of such a description is directly matched to the power of a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body – The Reynolds Number.
It is found that the computational effort in Direct Numerical Simulation (DNS) of the Navier-Stokes equations rises as Reynolds number in the power of 9/4 which renders such calculations as prohibitive for most engineering applications of practical interest and it shall remain so for the foreseeable future, its use confined to simple geometries and a limited range of Reynolds numbers  in the aim of supplying significant insight into turbulence physics that can not be attained in the laboratory.

Turbulent Boundary Layer (P. Schlatter and D. Henningson of KTH)


Having said all that, engineering applications could not have been left out and simplified methodologies to capture flow features of interest were developed their complexity and range of applicability dictated by the simplifying assumption, a direct consequence of computational effort limitations and  generally predicted by “Moore’s Law”. 

On the ladder of CFD one may find many stages. Lifting-Surface Methods that model only the camber lines of lifting surfaces, not the thickness, vortex wakes that must of course be paneled. Linear Panel Methods that solve either the incompressible potential-flow equation or one of the versions applicable to compressible flow with small disturbances. Nonlinear Potential Methods where the velocity is represented as the gradient of a potential, as it is in incompressible potential flow, nonlinearity through effectively incorporating an entropic relation for the density as a function of the local Mach number. Euler Methods, solving the Navier-Stokes equations with the viscus and heat-conduction terms omitted. Coupled Viscous/Inviscid Methods solving the boundary-layer equations in the inner near wall region and matched to an outer region inviscid flow calculations.

One huge leap forward was achieved through the ability to simulate Navier-Stokes Methods Such as Reynolds-Averged Navier-Stokes (RANS).

Most of nowadays CFD simulations are conducted with the Reynolds Averaging approach. Reynolds Averaged Navier-Stokes (RANS) simulation, the “working horse” of industrial CFD is based on the Reynolds decomposition according to which a flow variable is decomposed into mean and fluctuating quantities. When the decomposition is applied to Navier-Stokes equation an extra term known as the Reynolds Stress Tensor arises and a modelling methodology is needed to close the equations. The “closure problem” is apparent as higher and higher moments of the set of equations may be taken, more unknown terms arise and the number of equations never suffices. This is of course an obvious consequence to the fact that taking these higher moments is simply a mathematical endeavor and has no physical contribution what so ever.

Reynolds Stress
Reynolds-stress tensor

Levels of modeling are related to the number of differential equations added to Reynolds Averaged Navier-Stokes equations in order to “close” them.
0-equation (algebraic) models are the simplest form of turbulence models, a turbulence length scale is specified in advance through experimenting. 0-equations models are very limited in applications as they fail to take into account history effects, assuming turbulence is dissipated where it’s generated, a direct consequence of their algebraic nature.
1-equation and 2-equations models, incorporate a differential transport equation for the turbulent velocity scale (or the related the turbulent kinetic energy) and in the case of 2-equation models another transport equation for the length scale (or time scale), subsequently invoking the “Boussinesq Hypothesis” relating an eddy-viscosity analog to its kinetic gasses theory derived counterpart (albeit flow dependent and not a flow property) and relating it to the Reynolds stress through the mean strain.
In this sense 2-equation models can be viewed as “closed” because unlike 0-equation and 1-equation models (with exception of 1-equations transport for the eddy viscosity itself as described in a former post: Understanding The Spalart-Allmaras Turbulence Model) these models possess sufficient equations for constructing the eddy viscosity with no direct use for experimental results.

2-equations models do however contain many assumptions along the way for achieving the final form of the transport equations and as such are calibrated to work well only according to well-known features of the applications they are designed to solve. Nonetheless although their inherent limitations, today industry need for rapid answers dictates CFD simulations to be mainly conducted by 2-equations models whose strength has proven itself for wall bounded attached flows at high Reynolds number (thin boundary layers) due to calibration according to the law-of-the-wall.

IMG_0646
The turbulent boundary-layer and the “law of the wall”

y+_Calculation
Near wall cell size calculation

A drawback evident in almost all eddy-viscosity models is the inability to inherently account for rotation and curvature. This drawback is resulted from relating the Reynolds stress to the mean flow strain and in fact is the major difference between such a modeling approach and a full Reynolds-stress model (RSM). The RSM approach accounts for the important effect of the transport of the principal turbulent shear-stress. On the other hand, RSM simulations are not computationally cost-effective, in as much that one does get an improved physical fidelity that is worth the time and computational resources consumed, not only that, they often do not converge.


The k-ε Turbulence Model

The k-ε turbulence model still remains among the most popular, most known is the Jones-Laun

der k-ε turbulence model.

To kick-off the brief description I shall remind the above paragraphs explanation, meaning 2-equation closure models carry two additional transport equations, in the case k-ε turbulence model they are the turbulence kinetic energy k and the turbulence dissipation ε:  eddy vviscosity in k-eThis eddy viscosity dimensional grounds based relation relating the Reynolds Stresses to the mean strain rate:

my 1-eq 2


Turbulence energy transport equation

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:

K-eq Mcdonough

after defining the total kinetic energy:

K-eq Mcdonough

A transport equation for the total kinetic energy could be written as:

K-eq McdonoughDecomposing the velocity vector according to Reynolds decomposition and defining the turbulent kinetic energy as:

K-eq Mcdonough 2

allows 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):

K-eq Mcdonough 2The next steps consider time (or ensemble) averaging the total kinetic energy transport equation and the subtraction of the mean flow energy transport equation. Then after tedious manipulations on the result (which shall not be presented as this is a blog and not a textbook… 😉 ), a transport equation for the turbulence kinetic energy transport equation is achieved (Tennekes and Lumley form):

k-e initialNow enters the most important (and fun?… 🙂 ) part subsequently following the mathematical endeavor in each and every construction of closure transport equations, the surgical identification and simplification by physical reasoning of the terms in the initial transport equation. For the turbulence kinetic energy equation (as the left hand side of the above equation is the advection of turbulent energy) they are identified as follows:

  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.

The acknowledgement that each of the terms has been identified shall allow me (again…) to not go into the surgical simplification of each of the initial turbulence kinetic energy transport equation terms, but just to add that it is the part where the witchcraft comes into play in turbulence modeling. As the final transport equation for the kinetic turbulence kinetic energy shall soon be presented, one should ask as to why should we expect so many simplifying assumptions to so many terms in the initial equation to even satisfy a transport equation in the first place… well, here goes:

K-eq Mcdonough final


Turbulence dissipation transport equation

Now to the construction of an equation for the turbulence dissipation ε…
To do so we first invoke local isotropy for the dissipation. As it seems reasonable (maybe for LES it actually somewhat is), 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.
Before presenting the final turbulence dissipation ε transport equation it should be added that its derivation is much more difficult, so as to simplify the view of the surgically simplified equation one should note that it takes a similar form as the kinetic energy equation by many aspects, albeit with some important subtleties:

E-eq Mcdonough


Final form of the standard k-ε turbulence model

After “constructing” (lazy-wise) 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:

final-form-k-e.pngSubsequently to performing the surgical identification of the different terms in the transport equation, it should be remembered that we are still left out with some added constants to be calibrated.
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:

Calibrated constants k-e


Shortcomings of the Standard k-ε Turbulence Model

Although being perhaps the most popular and known turbulence model (its alterations such as Renormalization group and realizable variants shall be described in future posts), the standard k-ε turbulence model carries along some harsh shortcomings which are important to acknowledge.
First, it is important to note that 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+<10). This again produces numerical stiffness and in case is problematic to handle in view of linear numerical algorithms.
Furthermore the low Reynolds methodology should not be confused with transitional Reynolds modeling, as sometimes many practitioners rely on low Reynolds models to achieve a measure of transition from laminar to turbulence prediction. As the low Reynolds methodology is devised to handle the near wall viscous/laminar sublayer, there is no reason to expect it also satisfactory transition predictions, especially noting that there are many mechanisms for transition onset. Perhaps the only predictions which shall be close to satisfactory by low Reynolds model (maybe “pseudo-transition” behavior) are the ones related to bypass transition of which high levels of turbulence in the free stream occur and transition is dominated by diffusion effects (a brief description of the mechanism in my former post: A Forest of Hairpins – on the quest for turbulence coherent structures )

Another major drawback is the model lack of sensitivity to adverse pressure-gradient. It was observed that under such conditions it overestimates the shear stress and by that delays separation. Menter’s k-ω SST alleviates this drawback through the Shear Stress Transport (SST) concept. This drawback is evident in almost all eddy-viscosity models, relating the Reynolds stress to the mean flow strain and in fact is the major difference between such a modeling approach and a full Reynolds-stress model (RSM). The RSM approach accounts for the important effect of the transport of the principal turbulent shear-stress. The ingenious idea of Menter to include it in the revised k-ω model (termed the Baseline (BSL) model) is related observed success in implementing what is termed as the Bradshaw’s assumption, that the shear-stress in the boundary layer is proportional to the turbulent kinetic energy (a brief evaluation of the model in former post: Understanding The k-ω SST Model ).

Nonetheless the k-ε turbulence model still remains among the most popular, possibly since the drawbacks in the above paragraph are not as important in most simulations or hard to overcome anyway, meaning Transition modeling still remains problematic in the RANS domain of simulations and integrating through the viscous sublayer is rarely of great importance as far as high Reynolds flows are concerned.

k-e

A review of ANSYS Fluent different approaches to turbulence modeling could be found in the presentation below.

turbulence-the-gist-2
Turbulence Modeling – The Gist (by Tomer Avraham)

5 thoughts on “Understanding The k-ε Turbulence Model

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