“Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people” – Andrey Kolmogorov
Most of nowadays CFD simulations are conducted with the Reynolds Averaging approach. Reynolds-Averaged Navier-Stokes (RANS) simulation 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.
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 maybe of 1-equations transport for the eddy viscosity itself such as Spalart-Allmaras (SA) 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.
As 2-equations are the most popular non-algebraic turbulence models, they usually take the form of one equation for the turbulent kinetic energy easily obtained by taking a dot product of the Navier-Stokes equation by its velocity solution vector, invoking the Reynolds decomposition and performing some manipulation and modeling assumptions until the final result takes the form of the transport equation below:
Turbulent kinetic energy transport equation
There are many candidates for the other variable, the most popular and well-known is the turbulence kinetic energy dissipation rate – ε (or at least was in the past while today it has a true battle in quantity of use with the following presented model…). The k-ε model (Jones and Launder) and its variants proved to be very successful in a variety of flows to which it has been calibrated for, but found to suffer from some major drawbacks already noticed in the early days of its use.
One major k-ε model shortcoming stems from the need for a careful near wall treatment, consequence of the fact that the equation for ε does not go to zero at the wall, hence mandating additional highly non-linear damping functions for its integration through the laminar sublayer (y+<5). This produces numerical stiffness and in case is problematic to handle in view of linear numerical algorithms.
Another major drawback is the model lack of sensitivity to adverse pressure-gradient as it is observed that under such conditions the model tends to overestimate the shear stress and by that delay separation (I shall refer to the reason for that in the following paragraphs).
k-ω Shear-Stress Transport (SST)
One of the most successful models alleviating some of the shortcomings presented above, was presented by the k-ω model as devised by D. Wilcox. It should be noted though, that the first k-ω model devised on a purely dimensional analysis grounds and a rare understanding of the physics of turbulent flows was that of Andrey Kolmogorov, where ω, coined as the specific turbulent dissipation rate, also referred to as turbulent frequency (which is an amazing concept). Nevertheless, Kolmogorov’s model suffered from many deficiencies and was essentially never used for CFD calculations.
Andrey Nikolaevich Kolmogorov
As far as the first major drawback of the Jones-Launder k-ε model presented above, it could be shown that the ω equation in Wilcox’s k-ω takes on what might be refered to as an “elliptic” near wall behavior (partial differential wise), meaning that it has an inherent nature of being able to “communicate” with the wall and actually has Dirchlet (as in no-slip in this case) boundary conditions.
The term “elliptic” as coined here refers to a concept introduced as the basis for the v2-f 4-equation turbulence model, that of elliptic relaxation and could actually shown to be an inherent feature of the k-ω turbulence model solely by inspecting the ω-equation in the near wall region when combined with the specified ω values at the wall :
The implication of such behavior is the straightforward possibility of integrating through the laminar sublayer without additional numerically destabilizing damping functions or two additional transport equation.
Blending function concept
One of the shortcomings of Wilcox k-ω, is a strong dependence on free-stream values of ω specified outside the shear layer. To alleviate both shortcomings, that Jones-Launder k-ε near wall behavior and that of Wilcox k-ω ambiguity to freestream values of ω in one formulation, Menter (fan alert 😉 ) decided to blend continuously the Wilcox k-ω such that its formulation shall be applied in the near-wall region with the Jones-Launder k-ε applied towards the end of the boundary layer through reformulation of the last in a k-ω form, thereby getting the best out of both (Menter will use the blending function concept yet again in devising the Stress-Blended Eddy Simulation (SBES) – a modular approach for hybrid RANS-LES simulation).
A second major drawback is evident in almost all eddy-viscosity models relating the Reynolds stress to the mean flow strain and is one of the salient differences between such a modeling approach and a full Reynolds-stress model (RSM) as the RSM approach accounts for the important effect principal turbulent shear-stress transport (although generally found to have stabilization issues due to reciprocity between the many variables). The alleviation of the above drawback comes about through the introduction of the Shear Stress Transport (SST) concept. The ingenious idea by Menter, apparent already in the revised k-ω model (termed the Baseline (BSL) model) is related to an observed success in implementing the Bradshaw’s assumption, according to which the shear-stress in the boundary layer is proportional to the turbulent kinetic energy.
To make this desirable feature apply in the appropriate portion of the boundary layer, yet another switching function is introduced .
The end result is Menter’s k-ω SST, one of the most reliable RANS turbulence models:
A review of different approaches to turbulence modeling could be found in the presentation below.