Fundamentals of Turbulence Modeling

The transmission dynamics of COVID-19 has again exposed the challenges of modeling turbulent multi-phase flows. For effective modeling, it is very crucial to understand the fluid dynamics associated with coughing / sneezing and the motion of the turbulent gas cloud with cluster of evaporating droplets entrained with it. A turbulent flow of this nature is very complex and might not be accurately modeled using the traditional RANS techniques. In this article we will review the various turbulence models and the challenges associated with them 

Attempts to understand turbulence dates back to the days of Leonardo Da Vinci when he tried to visualize it through his drawings. But, even today, after years of considerable research, turbulence still remains an immature science with many complex physics yet to be deciphered.

What is turbulence: In 1937, Taylor and von Karman proposed the following definition of turbulence:

"Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another"

Turbulence is a chaotic flow with velocity fluctuations in all directions. These velocity fluctuations arise due to flow instabilities. At low Reynolds number, any instabilities (velocity & pressure fluctuations) are damped out at molecular level under the influence of viscosity and the flow remains laminar. However, with increasing Reynolds number, the damping is inadequate to control the instabilities and hence eddies (Eddies are circular current of fluid) larger than the molecular scale are formed and the flow becomes turbulent.

What causes turbulence: Turbulence is caused due to uncontrolled flow instabilities. These instabilities arise either from contact of the flowing stream with solid boundaries (wall bounded flows) or from contact between two layers of fluid moving at different velocities (Free shear flows). Due to the resulting shear, larger eddies are continuously formed. Thereafter, they break down into smaller and smaller eddies and finally disappears

Eddies and its effects: The movement of eddies results in velocity fluctuations and enhanced transport of mass, momentum and energy by diffusion. Eddies are continuously created and broken down and hence its length scale varies considerably from the larger ones (Integral length scale) to the smallest one (Kolmogorov length scale) where it is finally dissipated into heat by viscosity. The ratio of largest to smallest length scale is a function of Reynolds number and is given by

L (Integral length scale)/ η (Kolmogorov length scale) ~ (ReL)3/4. where ReL is the Reynolds number associated with the large eddies.

Hence, higher the Reynolds number, the larger is the variation of eddy sizes, and hence more difficult to resolve

Transition to Turbulence: Onset of turbulence is mainly predicted by Reynolds number and the type of flow as mentioned below

External Flows

Rex ≥ 500,000 along a surface

Red ≥ 20,000 around an obstacle

Internal Flows

Red ≥ 2300

Natural Convection

Ra/Pr ≥ 10^9

Modeling turbulence:

Turbulence is characterized as eddies superimposed on the laminar flow. Hence, we need to either resolve the eddies or model them so that their effect on flow can be captured. Considering the complexity, there are various approaches to turbulence modeling as mentioned below

DNS (Direct Numerical Simulation): Resolves all the length and time scales of the eddies. However, this is not practically feasible due to huge requirement of computational resources

LES (Large Eddy Simulation): Resolves only the larger eddies, while the effect of smaller eddies (near the walls) are modeled (Sub grid scale models). This is less expensive than DNS, but the amount of computational resources and efforts are still too large for most practical applications

RANS (Reynolds Averaged Navier Stokes): Model all the eddies using suitable turbulence models. The random fluctuation and its effect is estimated by solving the ensemble averaged Navier Stokes Equation. This is the most widely used approach in industry and will be discussed here.

Averaging the N-S equations with the fluctuations results in additional stresses (Reynolds stresses) that needs to be modeled. This is done via eddy viscosity models (Spalart Allmaras, Standard k–ε, RNG k–ε, Realizable k–ε, Standard k–ω, SST k–ω) or Reynolds Stress Models (Via transport equations for Reynolds stresses).

In Eddy viscosity models, the Reynolds stresses are modeled using an eddy (or turbulent) viscosity μ_t . This hypothesis is reasonable for simple turbulent shear flows with boundary layers, round jets, mixing layers, channel flows, etc.

There are a number of eddy viscosity models and they can be used either with or without wall functions

Eddy viscosity models:

One Equation Model:

Spalart-Allmaras: Can be used for wall attached flows or flows with mild separation and recirculation. But it performs poorly in massively separated flows, free shear flows and decaying turbulence.

Two Equation Model:

k-e standard: Easy to implement, leads to stable calculations and gives reasonable prediction for many flows. However, it is not suitable for swirling and rotating flows, flows with strong separation and axisymmetric jets

k-e RNG: Improvement to k-e standard and gives improved prediction for high streamline curvature and strain rate

k-e realizable: Improved model that gives good prediction for planar and round jets, adverse pressure gradient, rotation / recirculation and strong streamline curvature

k-omega and k-omega SST: The k–ω model has many good attributes and performs much better than k–ε models for boundary layer flows. However, Wilcox original k–ω model is overly sensitive to the free stream value of ω and inlet boundary conditions, while the k–ε model is not prone to such problem. k-omega model has gained popularity (Turbo machinery and aerospace applications), as it can be integrated to the wall without using any damping functions and gives accurate and robust prediction for a wide range of boundary layer flows with pressure gradient.

Another variant of k-omega that has gained popularity, especially in the aeronautics area, is the shear stress transport (SST) model. The reason being its ability to predict separation and reattachment better when compared to k-epsilon and the standard k-omega models. SST model removes some the deficiencies of the original k-omega model and accounts for cross diffusion by merging k-epsilon and k-omega models. However, it is also not a perfect model. It uses limiters for predicting turbulence in stagnation regions and it doesn’t take buoyancy into account. Some studies have shown that the model has superior performance compared to the k-epsilon model in simulating boundary layers with adverse pressure gradients.

Reynolds Stress Model: It is more accurate seven equation model but is more costly and difficult to converge than the 2-equation models. This model is particularly suitable for complex 3-D flows with strong streamline curvature, swirl and rotation

Which Turbulence model to use?

Due to the complexities associated with ?uid turbulence, there are no turbulence model that seems to be universally applicable under a wide variety of applications and therefore it is important to choose a suitable turbulence model depending on the situation. Generally, two equation models are good enough and they perform well for a wide range of engineering applications. However, the most important assumptions for these models are that the flow does not depart far from local equilibrium and Reynolds number is high enough so that local isotropy is approximately satisfied. Both k-epsilon realizable and k-omega SST is good enough to predict free shear flows produced by jets, wakes and mixing layers. For flows with adverse pressure gradient, k-omega SST is somewhat better than k-epsilon realizable model (Two layer).

Normally, the performance of SST k-omega is not very much different from the realizable k-epsilon model. The choice between the two will typically be made based on user preference.

However, neither of these model gives good results for more complicated flows such as flows with sudden changes in mean strain rate, curved surfaces, secondary motions, rotation and separation. In this case RSM or more advanced models needs to be used

Wall Functions: All the above-mentioned turbulence models can be used with or without wall functions. Wall functions are approximations that are used to reduce computational effort of modeling wall bounded turbulence. Very close to the wall and in the laminar sublayer, the effect of turbulence dies out very quickly. This leads to very high gradients of turbulence and velocity and hence very fine grid is required to capture the effects.

Moreover, neither the k-epsilon nor the k-omega equation is applicable in viscosity affected zone. However, k-ω model does come close enough to allow the viscous sublayer to be resolved without significant error. Resolving the viscous sublayer using the standard k-ε model cannot be done without corrections and hence damping functions are added to modify the equation in the viscosity affected zone. If the mesh is fine enough, then we can go ahead with Low Reynolds model, or else we have to approximate with wall functions.

Low Reynolds Modeling: A very fine grid is required throughout (yplus ~ 1) and Low Reynolds model is used in the viscosity affected zone. If required, the turbulence equations are modified with damping functions and all the equations are solved to the wall.

High Reynolds Modeling with Wall Functions: A coarse grid (yplus ~ 30-100) is used so that the first cell always remains in the log layer. In this case, the near wall velocity and its gradient are not calculated, and the first cell velocity is analytically fitted to obey the log law relationship. Effectively this approach uses the law of the wall as a constitutive equation for the wall shear stress in terms of the velocity at the first grid point. Using this approach, the value of the velocity at first grid point is used to determine wall shear stress which in turn is used to set the values of k and ε or ω. However, in this case, problem will arise if the first cell yplus falls in laminar or buffer zone. This results in improper use of log law wall function which is not valid in this zone. This is a strict constraint and needs to be taken care of. In most of the commercial solvers, this is taken care of by using enhanced wall treatment

High / Low Reynolds Modeling with Enhanced Wall Treatment: Enhanced wall treatment is a two layer model (Low Reynolds turbulence model with damping functions is used in viscosity affected zones) with enhanced wall functions. Enhanced wall functions use a single equation to formulate the law of the wall and hence the first grid point can be placed either in laminar sub-layer, buffer or logarithmic zone. This is achieved by blending the linear and logarithmic wall law using a function.

Summary: To summarize, a brief description of turbulence and its modeling approach has been explained. The workflow below gives an idea about the whole discussion

References:

Georgi Kalitzin ; Gorazd Medic; Gianluca Iaccarino; Paul Durbin. Near-wall behavior of rans turbulence models and implications for wall functions. Computational physics, 204(205):265–291, 2004

Introductory Turbulence Modeling : Lectures Notes by Ismail B. Celik

J.R. Edwards and S. Chandra, “Comparison of eddy viscosity-transport turbulence models for three dimensional, shock separated ?ow?elds”, AIAA Journal, 34, 756–763 (1996).