Let's Talk CFD #1: RANS Turbulence Models for Design Engineers

Big whorls have little whorls

That feed on their velocity,

And little whorls have lesser whorls

And so on to viscosity.

- Lewis F. Richardson

1. Introduction

Since time immemorial, mankind has always struggled with turbulences. To tame the whirling beast, Computational Fluid Dynamics has come a long way in simulating turbulences in more ways than one. And choosing a simulation model to accurately determine turbulence in a system of fluid flow, has been daunting of a task for engineers. On the flip side, the need for simulating turbulence to optimize the design for the real world applications in making products more reliable and efficient is now more needed than ever. The commercially available solvers offer several different formulations for solving turbulent flow problems. Some of these formulations that are widely used are SST, k-epsilon and k-omega turbulence models and their insidious variations.

Choosing the right model that fits the problem then becomes crucial to arrive at the desired results. This process can be quite challenging and could take several instances of trial and error and experimental testing to determine the right model.

2. Turbulence and Boundary Layer 

To begin with, let’s start by considering flow over a flat plate. In Figure 1 below, the flow with uniform velocity hits the leading edge of the flat plate and a laminar boundary layer develops. This flow regime is easily understood, predictable and modelled. However, after certain length into the flow, small disturbances or oscillations begin to develop at the boundary layer and a transition into turbulence begins which eventually develops as fully turbulent flow.

Figure 1: Flow over a flat plate and boundary layer formation

The Reynolds Number (Re) is a key dimensionless quantity in predicting the transition of flow patterns. At low Re numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Re numbers turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (eddy currents).

In the laminar region, the flow can be completely solved using Navier-Stokes equations. The solution provides velocity and pressure fields for the given boundary conditions. For the ease of understanding, let us assume that velocity is not time-variant. As the flow transitions into turbulence, eddies appear. Despite the inlet velocity being time-invariant and steady, it is no longer possible to assume that flow in the turbulent region is not time-invariant. Time-dependent NS equations would then need to be solved along with a finely refined mesh to capture the eddie resolution in that region.

3. Reynolds-Averaged Navier-Stokes (RANS) Formulation

In the turbulent region, the oscillations due to the eddies become so minute that using NS equations to resolve them become computationally unfeasible. Enter, Reynolds-Averaged Navier-Stokes (RANS) formulation. RANS is based on the observation that the flow field (u) in Figure 2, has small local fluctuations and can be treated as a time-averaged entity (ū).

Figure 2: Velocity fluctuations in a turbulent flow.

NS equations being non-linear, averaging the equations means that the velocity fluctuations will still appear in the RANS equations. This nonlinear term is known as Reynolds Stress. To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the Reynolds stress term as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is called the closure problem.

The objective of the turbulence models for the RANS equations is to compute the Reynolds Stresses, which can be done by three main categories of RANS-Based turbulence models. Below I have consolidated popular RANS-based models:

1)    Linear Eddy Viscosity Models

a.     Algebraic models (Zero equation models)

                                                   i.    Baldwin-Lomax model

                                                  ii.    Cebeci-Smith model

b.     One Equation Models

                                                   i.    Baldwin-Barth model

                                                  ii.    Spalart-Allmaras model

                                                 iii.    Rahman-Agarwal-Siikonen model

c.     Two Equation Models

                                                   i.    k-epsilon model (and its variants)

                                                  ii.    k-omega model (and its variants)

2)    Nonlinear Eddy Viscosity Models

3)    Reynolds Stress Model (RSM)

Of course, there are many more models under the classifications but for now, we will limit to those turbulence models that are widely accepted in the industry which is k-epsilon and k-omega models. You can read more about the several models being benchmarked by NASA here - https://turbmodels.larc.nasa.gov/

3.1 Two Equation Models

Models like the k-epsilon model and the k-omega model have become industry-standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.

The basis for all two-equation models is the Boussinesq eddy viscosity assumption, which postulates that the Reynolds stress tensor, is proportional to the mean strain rate tensor. The Boussinesq assumption is both the strength and the weakness of two-equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like turbulence intensity and turbulence length scale.

The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decelerated flows the Boussinesq assumption is simply not valid. This gives two-equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decelerated flows like stagnation flows.

By definition, two-equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two-equation model to account for history effects like convection and diffusion of turbulent energy. Most often one of the transported variables is the turbulent kinetic energy, k. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent dissipation, epsilon, or the Specific turbulence dissipation rate, omega. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or timescale), whereas the first variable, k, determines the energy in the turbulence.

The turbulence dissipation usually dominates over viscous dissipation everywhere, except for in the viscous sublayer close to the solid walls shown in Figure 3. Here the turbulence model has to continuously reduce the turbulence level, such as in Low Reynolds Number (LRN) models. Or, new boundary conditions have to be computed using Wall-Functions.

Figure 3: Near Wall Treatment

3.2 Low Reynolds Number (LRN) Models

It seems preposterous at first glance to term a turbulence model “low Reynolds number” since the turbulence region is known to have much high Re. The idea of “low Reynolds number” here does not refer to the flow on a global scale, but to the region close to the wall where viscous effects dominate at the viscous sublayer.

Most omega-based models are low Reynolds number models by construction. But the standard k-epsilon model and other commonly encountered k-epsilon models are NOT low Reynolds number models. Some of them can, however, be supplemented with so-called damping functions that give the correct limiting behaviour. They are then known as low Reynolds number k-epsilon models.

LRN integrates every equation up to the viscous sublayer. This demands very fine meshes close to the layer behaviour which often makes the equations stiff and further increases computation time. This is why alternative methods to model the flow close to the walls are often employed for industrial applications.

3.4 Wall Functions - High Reynolds Number (HRN) Model

The turbulent flow near a flat wall can be divided into four regions as shown in Figure 3.

1)    Viscous/Laminar Sublayer: At the wall, the fluid velocity is zero. In a thin layer above this, the flow velocity is linear with the distance from the all.

2)    Buffer Layer: Turbulence stresses begin to dominate over viscous stresses.

3)    Log-Law layer: Flow is fully turbulent, and the average flow velocity is related to the log of the distance to the wall.

4)    Free-Stream: Further from Log-law layer.

The viscous and buffer layers are very thin. If the height of the buffer layer from the wall is ‘y’, then the Log-Law layer extend to 100y from the wall.

It is possible to use a RANS model to compute the flow field in all four of these regions. However, since the thickness of the buffer layer is so small, it can be advantageous to use an approximation in this region. Wall functions ignore the flow field in the buffer region and analytically compute a nonzero fluid velocity at the wall. By using a wall function formulation, you assume an analytic solution for the flow in the viscous layer and the resultant models will have significantly lower computational requirements. This is a very useful approach for many practical engineering applications.

4. The million-dollar question - How to decide what model to use?

You are now equipped to answer the above question.

If you need a level of accuracy beyond what the wall function formulations provide, then you will want to consider a turbulence model that solves the entire flow regime as described for the low Reynolds number models above. For example, you may want to compute lift and drag on an object or compute the heat transfer between the fluid and the wall.

4.1 k-epsilon Model

The k-ε model solves for two variables: k, the turbulence kinetic energy; and epsilon, the rate of dissipation of turbulence kinetic energy. Wall functions are used in this model, so the flow in the buffer region is not simulated. The k-epsilon model has historically been very popular for industrial applications due to its good convergence rate and relatively low memory requirements. It does not very accurately compute flow fields that exhibit adverse pressure gradients, strong curvature to the flow, or jet flow. It does perform well for external flow problems around complex geometries. For example, the k-epsilon model can be used to solve for the airflow around a bluff body.

The turbulence models listed below are all more nonlinear than the k-epsilon model and they can often be difficult to converge unless a good initial guess is provided. The k-epsilon model can be used to provide a good initial guess.

4.2 k-omega Model

The k-omega model is similar to the k-epsilon model, but it solves for omega — the specific rate of dissipation of kinetic energy. It is a low Reynolds number model, but it can also be used in conjunction with wall functions. It is more nonlinear, and thereby more difficult to converge than the k-epsilon model, and it is quite sensitive to the initial guess of the solution. The k-omega model is useful in many cases where the k-epsilon model is not accurate, such as internal flows, flows that exhibit strong curvature, separated flows, and jets. A good example of internal flow is - flow through a pipe bend.

4.3 Low Reynolds Number k-epsilon

The low Reynolds number k-epsilon model is similar to the k-epsilon model but does not need wall functions: it can solve for the flow everywhere. It is a logical extension of the k-epsilon model and shares many of its advantages, but generally requires a denser mesh; not only at walls but everywhere its low Reynolds number properties kick in and dampen the turbulence. It can sometimes be useful to use the k-epsilon model to first compute a good initial condition for solving the low Reynolds number k-epsilon model. An alternative way is to use the automatic wall treatment and start with a coarse boundary layer mesh to get wall functions and then refine the boundary layer at the interesting walls to get the low Reynolds number models.

The low Reynolds number k-epsilon model can compute lift and drag forces and heat fluxes can be modelled with higher accuracy compared to the k-epsilon model. It has also shown to predict separation and reattachment quite well for several cases.

4.4 SST k-omega model

The SST k-omega turbulence model has become very popular. The shear stress transport (SST) formulation combines the best of two worlds. The use of a k-omega formulation in the inner parts of the boundary layer makes the model directly usable down to the wall through the viscous sub-layer, hence the SST k-omega model can be used as a Low-Re turbulence model without any extra damping functions. The SST formulation also switches to a k-epsilon behaviour in the free-stream and thereby avoids the common k-omega problem that the model is too sensitive to the inlet free-stream turbulence properties. Engineers often merit it for its good behaviour in adverse pressure gradients and separating flow. The SST k-omega model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This tendency is much less pronounced than with a normal k-epsilon model.

5. Conclusion

Now you have a good understanding of the nature of turbulence modelling and the various models available at your disposal that are widely expected by engineers. You may now incorporate the understanding in your thought process when choosing a turbulence model. However, choosing a model is only half the story. A well-defined mesh is equally important in getting the best out of any chosen model. We will discuss more on meshing in my article.