“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

**Abstract**

**Abstract**

Three versions of the k-omega two-equation turbulence model will be presented. The first is the original David d. Wilcox model. In contrast to what we learned about the near wall dependency k-ε model the model of Wilcox is wall insensitive, tends to have on arbitrary freestream values. Then, Florian Menter Base-Line (BSL) model is introduced. designed to give results similar to those -The original k-omega The BSL model is identical to the Wilcox model in the inner 50 percent of the boundary-layer but changes gradually to the high Reynolds number Jones-Launder k-epsilon model (in a k-ω formulation) towards the boundary-layer edge. The new model is also virtually identical to the Jones-Lauder model for free shear layers. Then, the second version of the model called Shear-Stress Transport (SST) model is presented. It is based on the BSL model, but has the additional ability to account for the transport of the principal shear stress in adverse pressure gradient boundary-layers. The model is based on Bradshaw’s assumption that the principal shear stress is proportional to the turbulent kinetic energy, which is introduced into the definition of the eddy-viscosity. Both models were tested for a large number of different flow fields. The results of the BSL model are similar to those of the original k-ω model, but without the undesirable freestream dependency. The predictions of the SST model are also independent of the freestream values and show excellent agreement with experimental data for adverse pressure gradient boundary-layer flows.

The chapter ends with the introduction of The Generalized k-ω (GEKO) Turbulence Model (GEKO)which allows for a total there are six of these free parameters, which may be tuned to achieve desirable and specific flow attributes, and enter the final formulation as part of what are ultimately switching functions which are changing the formulation’s behavior with respect to different flow attributes.

Press play watch a little of beautiful art, then continue reading. It will loop…💕

*2. *__Two Equation Turbulence Transport Equation Turbulence Models__

__Two Equation Turbulence Transport Equation Turbulence Models__

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 whatsoever.

*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 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.

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

*Near wall cell size calculation*

*The above “Near wall cell size calculation” explanatory video *

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 * MANY* ad-hoc modeling assumptions (to avoid perturbative variables in the formulation).

## 2. Deriving the 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:

after defining the total kinetic energy:

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:

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):

The 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):

Now 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:

**Pressure work due to only turbulence.****Transport of turbulent kinetic energy due to fluctuations.****Diffusive transport of turbulence kinetic energy.****Turbulence production, or to be more precise the amplification of the Reynolds stress tensor by the mean strain.****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:

*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).

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 referred 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.

DNS of the turbulent flow around a square cylinder at Re=22,000

## 3. David D. Wilcox k- ω Turbulence model

There are many candidates for the other variable, the most popular, as we saw in the previous chapter, 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).

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.

Turbulence: The Legacy of A. N. Kolmogorov

(recommended download)

### 3.1 Deriving the Wilcox k-ω Turbulence model

As with the k-ε turbulence model, deriving exact ω-equation can be derived from the Navier-Stokes. As in the case for the derivation of the ε-equation from the Navier Stokes equation this derivation and the obtained equation are both very complex and contain many higher order terms. For this reason we do the same as we did with the ε-equation and formula in analogy with the turbulent kinetic energy turbulence transport equation (strong modeling assumption ), It is assumed that -equation has the same structure (same terms) as the k-equation Each of the terms is modeled as proportional to the corresponding terms in the k-equation. as applying multiplication of each term is multiplied by ε/k to get correct dimension, we set it now according to:

Coefficients are introduced to allow calibration of -equation.

**There are two ways to derive the ω****-equation:**

**Same derivation as ε****-equation – by analogy with k****-equation****By direct transformation of model based on analytical relation between ω, k***,*and ε

One of the shortcomings of Wilcox k-ω, is a strong dependence on free-stream values of ω specified outside the shear layer.

*4. Florian Menter Base Line (BSL) Turbulence Model*

Freestream values for k and omega outside boundary- and shear-layers should have very little effect on the flow inside the layer. With some k-ω models the freestream values can change the eddy-viscosity inside the layer by a factor of 2 or more. This is not acceptable as freestream values are not easy to control.

The k- ω models is a strong foundation for a generic two-equation model. Methods have to be found to avoid freestream sensitivity.

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 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.

The blending function blends the these two transport equations but does not solve an ε equation in the outer part of the boundary layer but an ω equation transformed from the that Jones-Launder k-ε. In other words since it’s undesirable to solve bot the ω-equation and the ε-equation and then blend the solution, the ε-equation is transformed mathematically to an **exact** equivalent ω-formulation and then blended with the original wilcox k-ω model.

We get the terms of the Wilcox model, but we get an additional term from the transformation, and that is a cross-diffusion term, so it has the gradient of the turbulent kinetic energy as part of the term multiplied by the gradient of ω divided by ω and the coefficient that comes from the ω equation. This cross-diffusion term is zero in most of the calibration cases, for example it would have a negligible impact of a flat plate, but near the boundary layer edge it has a tremendous effect in reducing the free stream sensitivity of the k-ω model and therefore plays the role that we would like to use it for.

The BSL k-ω model final formulation:

**5. k-ω Shear-Stress Transport (SST)**

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.

If we look at any standard 2-equation and insert the eddy viscosity that we have we find that the relation:

This leads to over prediction of turbulent stresses in adverse pressure gradient flows (Pk/ε)>1 leading to no or delayed separation. In equilibrium flows that ratio of production to dissipation is 1. In adverse pressure gradient it could lead to a shear stress which is even 20% higher in the middle of the boundary layer, and that’s already sufficient to prevent and delay separation.

The ingenious idea by Menter, 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:

If we switch the correlation obtained from the Bradshaw assumption by the standard one we get the idea behind the SST limiter such that turbulent viscosity is computed from:

- is nearly equal to 1 Inside the boundary layer and goes to zero far from the wall and free shear layers,
- The coefficient is used to fine tune the model (originally proposed as 0.31, increasing it will make the coefficient less aggressive to separation.

The end result is Menter’s k-ω SST, one of the most reliable RANS turbulence models:

*6. The Generalized k-ω (GEKO) Turbulence Model*

*6. The Generalized k-ω (GEKO) Turbulence Model*

**6.1** The differences between 2-equation models are not of a fundamental nature, but difference in calibration and especially on the second length scale equation seem impact the results profoundly.

As far as boundary layers are concerned, the models differ mostly on how well they estimate separation onset, and in the very near wall region, the effect of their different approach to wall-treatment is clearly apparent (drag calculation and heat transfer applications are just a few examples).

There are also noticeable differences in the way each model handles free shear flows.

There is no decisive answer for specific model superiority, as each of them may perform better in certain instances.

Most importantly however, each model features different sensitivity to calibration constants and limiters in the formulation. This might not affect baseline flows, but these same differences become enormously pronounced when flow complexity arises, as not only bad predictions are made at specific high complexity locations, but they also tend to contaminate the entire domain.

Choosing a specific model per application is certainly possible by following some rule of thumbs, but it certainly does not guarantee that the best choice of model for the application has been made. Even when it seems that such a choice has been made, slight variations might tilt the deciding factor in favor of a different turbulence model.

First, CFD practitioners will be much better off working in a consolidated framework of which the turbulence model presents one set of calibration constants and limiters without having to rethink basic definitions in the construction of such, each time the slightest variability in the application occurs. Second, and perhaps more important, there is a lack of flexibility in tuning the types of turbulence models’ constants, as they are specifically calibrated according to the law-of-the-wall. This does not mean a model’s calibrated constants can’t be tuned. For example, Thies and Tam proposed a specific set of new model constants for the standard k-ε model designed specifically for predicting jet flows (A. Thies, C.K.W. Tam, “Computation of Axisymmetric and Nonaxismmetric Jet Flows Using the k-ε model” – AIAA journal, 1996):

Yet, a common thread through most of such specific targeting of the model constants is the fact their range of validity is very limited, such that including a solid boundary for the above example will hamper the model’s predicting power as the calibration shall not linked to the law-of-the-wall anymore.

- To overcome these two limitations, that of choosing from a a multitude of turbulence models (along with their different conceptualized limiters and calibration constants), and especially to be able to tune a model without hampering its calibration according to the law-of-the-wall, ANSYS developed a consolidated infrastructure, a kind of an “all-in-one” branch of turbulence models. Although this branch is based upon the k-ω formulation it can be tuned to match a variety of flows.

The ingenuity is in a set of free parameters, such that they may be tuned without adversely affecting the regular law-of-the-wall calibration. In other words:.*The Generalized k-ω (GEKO) Turbulence Model*

In total there are six of these free parameters, which may be tuned to achieve desirable and specific flow attributes, and enter the final formulation as part of what are ultimately switching functions which are changing the formulation’s behavior with respect to different flow attributes (specific portion of the turbulent boundary layer or shear flows vs. boundary layer flows for example).

The details on the exact structure of these switching functions is propriety, but which of these free parameters contribute to which switching function is readily identifiable due these functions location in the k-ω formulation, and the specific intended impact of each of the free parameters.

This free parameter is the main contributor for the amount of “aggressiveness” of a model in the prediction of flow separation. It is also the most important free parameter due to its impact on what is in essence the most common source of prediction variability between such models.

Increasing this specific parameter reduces the eddy-viscosity, subsequently making the model more sensitive to adverse pressure gradient. Increasing it also affects the spreading rates of shear layers.

Ultimately, increasing this parameter will significantly reduce the ratio between the eddy viscosity to the dynamic viscosity (turbulent viscosity ratio), and it does so without affecting the expected logarithmic velocity profile (which is actually the point…).

Streamlines for plane diffuser flow for both models (k-ε and k-ω SST). The k-ε model failed to capture the separation as the flow is completely attached while the k-ω SST predicts a strong separation and a re-circulation zone and is in close agreement with data by K. Gersten et al.

As the former free parameter is intended by design to not impact the wall shear stress and heat transfer rate upon variation, this parameter, limited to affect the inner portion of the boundary layer (and to not have an effect on free shear flows) may be tuned for that exact objective, but for non-equilibrium flows situations (we wouldn’t like for it to have an effect in a flat-plate boundary layer flow of course – and indeed it is verified to not affect the the shear stress and heat transfer coefficient for such application).

**The two free parameters above are intended to affect boundary layers.** It’s important to understand that both may be tuned to achieve a desired effect (which of course must be experimentally validated) for certain flow features, but to not have any significant effect for basic flows such as flat plate boundary layer behavior (again, that’s the hwole point…)

While the former two free parameters are designed to affect the turbulent boundary layer, the next two presented are designed for free shear flows. I’ve shown an example of calibration constants variation for jet applications, but such a variation shall not be adequate whenever a solid boundary shall be introduced to the application somewhere in the domain.

This free parameters affects only free shear flows. This is ensured through the use of a blending function switching from 1 inside the boundary layer to 0 for free shear flows.

When free shear flows are encountered, increasing this free parameter value shall increase the spreading rates of free shear flows.**This is flexibility is actually very important** as far as RANS models predictions are concerned. It is well known that free shear flows are composed of a variety of three-dimensional turbulent structures. Nonetheless, the organization of those structures is related to the dominant instability modes, which differ between wakes, mixing layers , and jets.

A **wake** forms downstream of any object placed in a stream of fluid.

A **mixing layer** occurs between two parallel streams moving at different speeds.

A** jet** occurs when fluid is ejected from a nozzle or orifice.

These differences can be seen in experimental flow visualization and in solutions from an LES. This change in the way that the turbulence is organized presents a particular challenge for RANS models prediction, which model statistical averages rather than structure. It is among the reasons why most turbulence models cannot accurately predict both jets and mixing layers using the same set of coefficients.

This free parameter is actually a sub-model of the former, meaning it has no impact when the former is zero.

As explained above, due to the variability in the dependence of specific instability modes of the different types of shear flow (e.g. jets), this free parameter allows for further adjustment of the spreading rate of jets while maintaining a desirable and spreading rate for the mixing layer.

In applications containing rectangular corners (e.g. rectangular channel flows), secondary flows develop and evident in a plane normal to the mean flow direction. The Boussinesq hypothesis ties between the average velocity tensor of the flow and the Reynolds stresses in a linear stress-strain relation. therefore even in the equations for the kinetic energy enters the influence of the strain tensor which is the symmetric part of the velocity tensor after a decomposition to a symmetric an antisymmetric part.

The antisymmetric part is the rotation tensor defined as:

Doesn’t appear in the equation for the kinetic energy nor in the Boussinesq hypothesis. As a consequence the behavior of the Reynolds stresses does not take into account instances such as secondary flows (among other rotation related flows…). When such flows are computed applying the suggested linear stress-strain relation, phenomenon such as early separations can occur, and shall further impact and contaminate the prediction downstream in the domain.

This free parameters is in essence a non-linear stress-strain term to account for secondary flows in corners.

Square duct – normal to mean velocity plane cut

This parameter is actually not new and is already in use as an option in the k-ω SST Turbulence Model. Nevertheless, since my intended purpose for this post is to emphasize turbulence modeling attributes of eddy-viscosity based turbulence models through GEKO, it is quite valuable to dwell a little on the motivating features for this parameter.

In the description of the last free parameter I have elaborated about the inherent weakness of eddy-viscosity based turbulence models, and their inability to capture effects of streamline curvature and system rotation. This inability is a consequence of 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) which 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 as the improved physical fidelity that is worth the time and computational resources consumed in most cases, and not only that, they often do not converge.

The approach to alleviating this inability is actually based on an approach taken by Philippe Spalart and Michael Shur in the aim of accounting for rotation and curvature effect by offering a modification for the Spalart-Allmaras turbulence model formulation, based on empirical grounds.

The route for altering the transport equation (for the eddy-viscosity itself in the case of the Spalart-Allmaras turbulence model) kicks off with the identification of the effect of curvature and rotation in two types of extreme flows:

- Thin shear flows with weak rotation (compared with the shear rate) or weak curvature (compared with the inverse of the shear-layer thickness), highly impacting the level of the turbulent shear stress.
- Homogeneous rotating shear flow and free vortex cores of which strong rotation reduces the turbulent shear stress sharply.

Spalart and Shur offer an alteration to the original eddy viscosity transport equation, based on the first type of reasoning presented above (thin shear flows) and another empirical alteration added to the former to account for the second type of extreme (i.e. homogeneous rotating shear flow and free vortex cores).

The physical reasoning returns yet again to the chosen way to relate the eddy viscosity relation to the strain rate **and vorticity** in a way as to alleviate non-addressing the discrepancy between the principal axes of the Reynolds stress tensor and rate of strain tensor.

Subsequently to exploring a satisfactory relationship between strain-rate and vorticity to be accounted for by a scalar quantity to handle curvature and rotation for thin shear flows with weak rotation and also (albeit less reliably but could be improved by empirical additives) for homogeneous rotating shear flow and free vortex cores, the production term of the eddy viscosity transport equation is multiplied by a “rotation function”.

Basing on the Spalart and Shur Correction, Florian Menter and Pavel Smirnov sensitized the k-ω SST Turbulence Model to rotation and streamline curvature, with a slight modification of the Spalart-Shur correction to control the production terms of the original SST model, and by that increase its range of validity to comply adequately for a range of flows between stabilized flows with minimal turbulence production characteristic of strong convex curvature to those characterized by of enhanced turbulence production of strong concave curvature.

k-ω SST with and without the curvature correction. Both results show a strong recirculation region on the inner wall close to the end of the U-turn. SST with Curvature Correction (bottom) predicts a larger recirculation zone (geometry from Monson et al.)

GEKO free parameters should be set in the following range to obtain a desirable flow attribute dependent performance (subsequent validation is mandatory):

Ranges to tune the free parameters (Cmix is corelated by default to ensure that changes in Csep do not affect free mixing layers)

Then as a good storyline should end we go back to the enchanted moment of beginning

**REFERENCES**

[1] Improved Two Equation Turbulence Models for Aerodynamic Flows – F.R. Menter Ames research center, October 1992.

[2] Ten years of industrial experience with the SST turbulence model – F.R. Menter (ANSYS), RB Lantgry (the Boeing company), M. Kunz (ANSYS)

[3] Generalized k-ω Two-Equation Turbulence Model in ANSYS CFD (GEKO).- by ANSYS

[4] “On the Sensitization of Turbulence Models to Rotation and Curvature” – P. Spalart, M. Shur.

[5]“Sensitization of the SST Turbulence Model to Rotation and Curvature by Applying the Spalart-Shur Correction Term” – P. Smirnov, F. Menter

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

Turbulence Modeling – The Gist (by Tomer Avraham)

Very beautifully explained the evaluation of Turbulence modeling in a brief article. I really enjoyed the reading.

Thank you very much for the appreciation

This is a very useful article describing the brief evaluation of the RANS models. I strongly recommend you to keep writing such articles.

–Ender

Thank you very much for the support Ender. I’m glad it’s appreciated.

There are other brief evaluations such as that for Detached Eddy Simulation (DES):

https://cfdisraelblog.wordpress.com/2017/03/19/detached-eddy-simulation-an-attractive-methodology-to-rans-in-the-aid-of-les/

Large Eddy Simulation (LES) WALE:

https://cfdisraelblog.wordpress.com/2017/03/21/thats-a-big-whale/

Partially Averaged Navier-Stokes (PANS):

https://cfdisraelblog.wordpress.com/2017/03/29/pots-and-pans/

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Oh how I long for the day when we no longer have to assume that the turbulent tensor is exactly aligned with the mean flow gradient. Then, we can finally discard these two-equation models and honor these great scientists like Mentor, Jones, Launder, Wilcox, etc. by standing on their shoulders moving beyond.

i try to use sst model on my cfd simulation of smal axial hydraulic turbine but never converse even with using very fine mesh. on the contrary k-epsilon or k-omega works without any problem

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Yeah this is good, I’m still very new in this area, but I appreciate it

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