Turbulence model definitions, strengths, weaknesses and best practices for your CFD simulation.

Turbulent CFD simulation of the air velocity around landing gear. (Image courtesy of CD-adapco/Siemens.)

Since the 19th century, finding a simulation model to describe turbulence perfectly has proven to be a bumpy ride.

Despite this, engineers need ways to simulate turbulent fluid flow to optimize their designs for the real world. Various empiric or semi-derived turbulence models have been created to help engineers to find the best model to fit their system of study, but this process could take a lot of trial, error and physical testing.

“To make the selection of a turbulence model easier for end users,” suggested David Corson, director of program management at Altair, “[here are] what are widely accepted to be the most accurate general-purpose models: Spalart-Allmaras, SST and *k*-omega. For the majority of engineering applications, these models provide a good trade-off between [computational] cost and accuracy.”

Unfortunately, engineers need more than just a short list to make a correct selection. MIT professor Emilio Baglietto noted the importance of understanding the fundamental challenges, myths, fallacies, successes and failures of computational fluid dynamics (CFD) to determine a model with accuracy.

### The Difficulty Defining Turbulent Fluid Flow

Baglietto explained that the mission to find a general solution to turbulence is known as the turbulence closure problem. The aim is to close the Navier-Stokes and Reynolds stress equations that describe turbulent flow. The solution has remained elusive, as averaging nonlinear occurrences of fluctuating quantities will only create new unknowns without governing equations.

Simulation of the fluid flow in a washing machine using SIMULIA. (Video courtesy of Dassault Systèmes.)

Turbulence models attempt to close the system of equations that describe turbulent flows by devising new equations through experimentation or derivations for specific applications.

Corson noted that in making a turbulent model, many assumptions are made to reduce the computational costs of the simulation. Based on the type of flow being modeled, different assumptions will be made.

This has created a ballooning number of available turbulent models. This can make choosing a CFD simulation software solution a considerable challenge for engineering teams because while more is not always an advantage, if your software has too few turbulence models then you might miss the one you need.

“When someone shops for a CFD code, they might think that it would be an advantage to have many turbulence models,” said Paul Malan, director of fluids applications for SIMULIA R&D. “Let’s say that they make the purchase of their perfect code with, say, 50 different models. They are thrilled, because surely at least one of these will give the right answer. But when he starts to solve a real problem, he has to choose one of the 50. Which should he choose? And once he has made the choice, how does he know it is giving the right answer?”

The key to choosing the right model is to understand its strengths, weaknesses and definitions. According to Corson, “Until there is a single model of turbulence developed, CFD engineers will always be faced with the challenge of selecting the right model for the right job.”

The following is a list of turbulent model families and how they compare.

### Reynolds-averaged Navier-Stokes Models

The family of Reynolds-averaged Navier-Stokes (RANS) models is the largest in the field of turbulence. These models attempt to close the turbulence equations using viscosity terms. A common variable calculated in these models is *k*, or the kinetic energy per unit mass of turbulent fluctuations.

Baglietto explained that there are numerous ways to perform these closures, but some are much more common and instructive than others. Typically, algebraic models have been used with either one or two equations.

“That loss of degrees of freedom bakes in an inherent assumption that the turbulence is isotropic and not stretched by the proximity of the wall, strong shear, or swirling flow,” said David Mann, product manager for STAR-CCM+ at CD-adapco. “We should look for extra treatments in RANS models to overcome these limitations, or they will perform badly for these flows.”

There are some limitations with RANS models as they are based on the definition of turbulent viscosity. These limitations are:

- Lack of physical description
- Turbulence-induced secondary flows
- Streamlined curvatures
- Swirling flows or flows with rotations
- Transitional flows between turbulent and laminar
- Unsteady flows like internal combustion engines
- Stagnant regions in flows

#### RANS Single-Equation Model: Spalart-Allmaras

Simulation of the turbulent flow around a NACA profile calculated using Spalart-Allmaras within COMSOL. (Image courtesy of COMSOL.)

“Spalart-Allmaras (SA) is a one-equation turbulence model that has been developed specifically for aerodynamic flows such as transonic flow over airfoils,” said Baglietto.

The model is based on kinematic eddy viscosity and mixing length. This mixing length defines the transport of the turbulent viscosity.

Baglietto noted that this popularity is in large part due to the model’s robustness and fast implementation when modeling specialized flows. Spalart-Allmaras is not memory-intensive and has good convergence but it has no wall functions. The model is also a popular addition to various CFD codes.

“When we look at the benefits and drawbacks, the Spalart-Allmaras model has historically been a strength … due to its speed and robustness,” said Corson.

“Because we are only solving a single equation for turbulence,” Corson added, “the non-linear convergence is outstanding and the model is very forgiving of poor quality mesh, particularly in the near wall region. The drawback is that it does have some limitations due to the single-equation formulation. The turbulence length and time scales are not as well defined as they are in other models such as SST.”

Limitations of Spalart-Allmaras include:

- Shear flows
- Under predicting separation
- Decaying turbulence

#### RANS Two-Equation Model: Standard k-epsilon, Realizable k-epsilon, RNG k-epsilon

Turbulent flow around a car-like model calculated in COMSOL using a k-epsilon model. (Image courtesy of COMSOL.)

“In [the standard *k*-epsilon model] we solve for two variables, the turbulent kinetic energy, *k*, and the rate of dissipation of kinetic energy, epsilon [*ε*],” said Valerio Marra, marketing director at COMSOL.

Marra explained that the model uses wall functions to analytically account for the fluid velocity in the viscous sublayer near the wall.

The technique offers good convergence and isn’t memory-intensive. Marra also explained that the model is typically used for external flows with complex geometry. However, it is also a good general-purpose model.

Baglietto noted that the equation for epsilon is postulated, so it isn’t perfect. Nonetheless, the model is used for the largest number of applications. This is partly because many of the model’s limitations are well-known.

Limitations of *k*-epsilon include:

- No-slip walls
- Adverse pressure gradients
- Strong curvatures
- Jet flows
- Difficulty solving for epsilon

Despite this, the model is reliable due to its predictability and numerous variants that aim to improve the model for several applications.

Perhaps the most famous variation of the model is the realizable *k*-epsilon model. This variation modifies the equation for epsilon and introduces the effect of the mean flow distortion on turbulent dissipation.

“[Realizable *k*-epsilon] is the default recommendation in mainstream commercial packages, therefore represents the most proven, well-quantified and widely-documented of all closures,” said Baglietto. “The model has improved performance for planar surfaces, round jets, rotation, recirculation and streamline curvature. It also improves the boundary layer under strong adverse pressure gradients or separation. But it cannot do magic as it’s still based on [turbulent] viscosity.”

Malan clarifies that *k*-epsilon has also become the “de facto” standard two-equation model because its two-layer formulation has improved its applicability to well-resolved boundary layers. It also has improved results for complex separated industrial flows.

Another popular modification is the renormalization group (RNG) *k*-epsilon model. The model was originally derived by attempting to solve for epsilon using the Navier-Stokes equation. The result was very much like the original equation. However, an update of the method added a term to the epsilon equation that accounts for the mean flow distortion of turbulence dissipation.

The result is that RNG produces lower turbulence levels and can underestimate the value of *k*. This produces a less viscous flow that creates more realistic flow features with complex geometry. Though the method is popular, Baglietto notes that it gets on the nerves of many modeling veterans as it is more accurate for the wrong reasons.

“It is the production of *k* that is overestimated by the EVM (eddy viscosity models) and not the level of epsilon,” explained Baglietto, “so the cure should be found in a better representation of anisotropy and essentially of the normal stresses.”

Though the standard, realizable and RNG variations of *k*-epsilon are all popular with CFD vendors, Baglietto is correct that the RNG model does have its detractors. This has caused at least one vendor to take action. “Although [SIMULIA] used to provide a version of the RNG *k*-epsilon model, we will not be supporting it for the R2017x release,” explained Malan. “We feel that it offers little or no advantage over the realizable *k*-epsilon model which has superseded it and we cannot convincingly articulate why one would select it.”

#### RANS Two-Equation Model: Standard k-omega and SST k-omega

Another popular two-equation model pairs *k* with the specific rate of dissipation of kinetic energy, or omega (*ω*). Baglietto explained that the aim of the standard *k*-omega model is to model near-wall interactions more accurately than *k*-epsilon models.

However, he noted that *k*-omega can over-predict shear stresses of adverse pressure gradient boundary layers and that the model has issues with free stream flows. The model is also very sensitive to inlet boundary conditions, which is a disadvantage not seen in *k*-epsilon.

“The most significant advantage of the *k*-omega model is that it may be applied throughout the boundary layer without further modification,” said Baglietto. “Furthermore, the standard *k*-omega model can be used in this mode without requiring the computation of wall distance.”

“[*k*-omega] is a popular model for turbomachinery simulations and for simulations where strong vortices are present such as those originating from wing tips,” said Mann. “[It] performs well for swirling flows and in the near wall region, but it over-predicts separation.”

Limitations of *k*-omega include:

- Difficulty of convergence compared to
*k*-epsilon - Sensitivity to initial conditions

One variant of *k*-omega that has gained popularity, especially in the aeronautics area, is the shear stress transport (SST) model. The model has gained this popularity based on its ability to predict separation and reattachment better when compared to *k*-epsilon and the standard *k*-omega.

“The SST *k*-omega model is an enhancement of the original *k*-omega model and addresses some specific flaws of the base model, such as the sensitivity to freestream turbulence levels,” explained Malan. “It has the advantage that it can be applied to the viscous-affected region without further modification, which is one reason it has become a popular choice in aerospace applications where the flow is deemed too complex for Spalart-Allmaras.”

The SST model accounts for cross-diffusion which better marries the *k*-epsilon and *k*-omega models. Using a blended function based on wall distance, engineers can include cross-diffusion when away from the wall but not near it. In other words, using the wall distance as a switch, SST works like *k*-epsilon in the far field and *k*-omega near the target geometry.

“Purists may object strongly that the blending function crossover location is arbitrary and could obscure some critical feature of the turbulence,” noted Baglietto. Clearly the model isn’t perfect; it also requires limiters to improve the prediction of stagnant regions of the flow. Additionally, it has issues predicting turbulence levels and complex internal flows and it doesn’t take buoyancy into account.

Malan added, “Some people claim that the model has superior performance to the *k*-epsilon model in simulating boundary layers with adverse pressure gradients. Ultimately, though, the performance of SST *k*-omega is not very different from the realizable *k*-epsilon two-layer model. The choice between the two will typically be made based on user preference.”

It seems that many engineers do prefer *k*-omega as all the CFD vendors interviewed have the SST model and most have the standard *k*-omega within their code.

### Large-Eddy Simulation and Detached Eddy Simulation Models

Simulation of a turbulent flow around a cylinder using Altair’s Acusolve LES turbulence model. (Image courtesy of Altair.)

RANS models simulate all scales of turbulence and resolve none. Large-eddy simulation (LES) and detached-eddy simulation (DES) models, on the other hand, resolve the largest scales of turbulence and model the rest by use of sub-grid turbulence models or by blending with a RANS model.

The LES model is used to predict large turbulent eddy structures when solving a CFD model system with a fine mesh. However, since turbulent scales are small near the wall, the model is unable to predict these regions with accuracy.

“LES and DES simulations are being carried out more and more often for applications such as aeroacoustics or combustion and again there are several variants of these models,” explained Mann. “DES is a hybrid RANS-LES method which combines the benefits of LES for resolving the large turbulent structures away from the wall, with the benefits of RANS near the wall where the turbulent eddies are too small to resolve. It is important to remember that the RANS portion of DES models is still responsible for the prediction of separation, heat transfer and other near-wall effects.”

The biggest limitations with both the LES and DES models are their high computational and programming costs. This likely explains why LES and DES models are not that popular with CFD software vendors. So if you need to use one for your application, choose your CFD software wisely.

“All RANS models [are] limited in accuracy for highly separated flows,” explained Corson. “For these types of applications, or those that require explicit resolution of turbulent structures, it is necessary to move towards a scale resolving simulation. DES models fulfil this requirement for users, but come at the expense of increased compute time.”

“When it comes to scale resolving simulations, Spalart-Allmaras-based DES—more specifically, delayed detached eddy simulation—is by far the most popular among our users,” said Corson. “This model is very stable and provides excellent accuracy for highly separated flows. For attached flows in which the smaller scales of turbulence are important, users typically choose the dynamic subgrid scale LES model. This model has excellent accuracy, and has little or no drawbacks in comparison to the fixed coefficient version.”

### Reynolds Stress Model

“Reynolds Stress Model (RSM) is the most complete physical representation of turbulent flows,” said Baglietto. “It is useful for new challenges and is able to capture complex strains like swirling flows and secondary flows. For swirling flows, such as cyclones, RSM is the only accurate closure.”

These models attempt to model the flow and terms directly in RANS equations. These models are based on the six equations that represent turbulent stresses. They represent the flow very well but at the cost of high computational work. They are typically reserved for flows that are extremely complex or have never been studied before.

Limitations of RSM include:

- Computational expense
- Sensitivity to initial conditions
- Amount of modeling required
- Requirement of high-quality mesh

Due to the difficulty in using these models, they are not that popular with CFD vendor software. Therefore, engineers looking to use RSM will need to do their research, or read this eBook.

### So, How Do I Choose My Turbulence Model Again?

Simulation of a 19.7-ft (6-m) ozone reactor calculated in COMSOL Multiphysics. Proper assessment of the turbulence allows for estimations of the residence times of each chemical species. (Image courtesy of COMSOL.)

Now that you’ve learned all this information about the turbulence model families, you might be asking yourself, “So, how do I choose my turbulence model again?”

“Before choosing a model we need to ask ourselves what question is it that I am looking for an answer to,” said Mann. “Then we need to understand the strengths and more importantly weakness of each model so that we can be sure that the strengths of the model we choose is aligned with the type of problem and that we are not asking a model to do something it is weak at.”

Mann explained this with a great example; let’s say you want to look at the air flow around an airplane. Spalart-Allmaras would be a great choice in this instance because it’s tested and well-known for this sort of application. However, if you want to dig into your design further and determine the angle of attack that will cause the airfoil to stall, then Spalart-Allmaras is no longer the model of choice.

“It will tell you the flow is still attached long after it has separated in reality,” explained Mann. “The reason being is that although the model was designed for attached aerospace flows it simply does not have enough degrees of freedom to predict stall adequately.”

Other factors affect the choice of model such as the mesh resolution near the wall. This is because turbulent flow near the wall is different from that in the bulk. Normal to the wall, the flow is constrained and eddies become anisotropic; near the wall, the flow becomes laminar at the viscous sub-layer. This doesn’t fit many turbulent models that assume the flow is completely turbulent and isotropic.

If the mesh is fine near the wall, the model will need to be compatible with near-wall turbulent flow. “Knowing how your chosen turbulence model deals with the anisotropy in the near-wall flow and in other features such as swirling flow is key to getting the best out of your model choice,” said Mann.

Marra agreed that certain models treat the viscous sublayer and buffer layer differently through the usage of wall functions. These models will differ based on the number of variables solved, what these variables represent and the velocity and pressure values.

“Each turbulence model has strength and weaknesses. Being aware of their range of applicability is of the essence in picking the right one,” noted Marra. “Some models are well-suited for internal flows, others for external flow around complex geometries. Some engineers might be interested in separated flows, jets, or need to compute lift, drag, heat fluxes [and more] with high accuracy. Once a model that meets the criteria for the job at hand is picked, the next step is to use a mesh able to capture all details of the flow.”

Corson explained that best practices at Altair include identifying the dominant feature of a turbulent flow and basing the choice of model on this feature. The engineer can then study how the model performs with situations and canonical flows where these features are dominant and compare the results to experimental data.

“Once you’ve identified which model performs best for the canonical flows of interest, you can apply that to your more complex application,” mentioned Corson. “We can’t guarantee that the models will provide the same level of accuracy on the complex case, but this approach provides a good starting point that should result in reasonable levels of accuracy.”

However, Marra also suggests that other factors can affect why an engineer would choose a certain turbulence model. These factors are:

- Accuracy of the model when used in their original scope
- Model’s ability to produce appropriate results in applications it isn’t intended for
- Computational cost of the model and its ability to produce quick preliminary results to rule out early design options

Sometimes, an engineer will need to still use a computationally expensive model with limited computational power. In these situations, Marra suggests a best practice of using boundary layer meshes at the wall and adaptive mesh refinements within the bulk of the fluid. This will help engineers to balance the accuracy they need with the computational power they have.

But in the end, choosing the right model comes down to practice. A seasoned CFD simulation expert might be able to look at an application and name off a few models of choice. They can then verify the correct model from the shortlist based on the convergence of the solution and mesh.

However, no matter who you are or what you are simulating, it is always a good practice to verify that the turbulence model is producing results in line with experimental data. Even the best of us can get it wrong and it’s best to find that error early in development cycles.

“For applications that demand highly accurate resolution of specific flow features, the only way to determine the best modeling approach is to rely on comparison to experimental data that is specific to that application,” noted Corson. “In that case, a turbulence model sensitivity study is necessary to identify which model produces the best results in comparison to experimental data. Once the best practice for that application is desired, similar applications in the future can rely on the same modeling guidelines.”

Once you have chosen a few potential turbulence models for your application, you will then need to ensure that these models are available in the CFD software you have access to. Otherwise, you might need to look to source a new CFD platform.

To see what models are available from various CFD vendor software, check out the eBook: Turbulence Models Offered by CFD Simulation Vendors.

To learn more about Simulation, read: Current Overview of Simulation Technology.