Turbulence

In fluid dynamics, flow can either be considered laminar or turbulent. In the first case, the fluid particles follow a smooth path with almost no mixing or disruption between adjacent paths. On the other hand, in turbulent flow, the flow becomes very irregular and the fluid particles follow a chaotic path, full of eddies, swirls and flow instabilities.

Turbulent flows are common in real life, think for example of the blood flow in your veins, the air flow over the wing of a plane, breaking waves at the beach, ….

In the \(19^{th}\) century, Reynolds was one of the first scientists to perform a set of experiments and show the transition between laminar and turbulent flow regimes and link this transition to the ratio of the inertial and the viscous forces. \[Re = \frac{inertia forces}{viscous forces} = \frac{\rho \cdot u \cdot l}{\mu} \] With \(\rho\) the density of the fluid, \(u\) the velocity of the flow, \(l\) a characteristic length scale and \(\mu\) the viscosity of the fluid.

This Reynolds number is still widely used today to characterize flow regimes. When the Reynolds number is smaller than 2300 in circular tubes, a laminar flow regime will occur. If the Reynolds number is higher than 4000, the flow will be turbulent. Everything in between is called the transitional regime.

When running simulations with turbulent flow, the turbulence needs to be resolved or an appropriate model needs to be selected. Turbulence models are simplified constitutive equations that predict the evolution of the turbulent flow. Even though a lot of research has been done over the past decades, turbulence modelling is still not a trivial task.

Three main approaches can be distinguished: DNS, LES and RANS to predict the evolution of the turbulent flow.

Turbulence can be fully described and resolved by the complete unsteady Navier-Stokes equations without the use of any modelling assumption regarding turbulence. This is called DNS, which means a Direct Numerical Simulation is performed. However, this requires extreme computational resources as the mesh resolution and the time steps need to be extremely small to capture the full range of temporal and spatial scales. This makes the DNS approach impractical in industrial applications and mainly considered as a research tool.

In the Large Eddy Simulation approach, or LES in short, the most energy containing structures of the flow, i.e. the large eddies are still resolved. Smaller scales of turbulence are treated by turbulence models instead.

The RANS, or the Reynolds-Averaged Navier-Stokes approach is the most commonly used approach in industrial applications. In most engineering applications, we are only interested in mean or integral quantities and to obtain these, solving turbulent flow with a turbulent model is the most cost-effective way and sufficient to obtain good results. In this approach, the Navier-Stokes equations are averaged to obtain the mean equations of the fluid flow. These averaged equations are similar to the full Navier-Stokes equations but contain some additional terms in the momentum equations called the Reynolds stresses due to the nonlinearity of the equations. These Reynolds stresses are unknown and need to be modelled with a turbulence model.

There are a lot of different RANS models available, only the most commonly used are elaborated on in more detail. Each of these models has their one limitations due to the specific modelling assumptions.

The \(k-\epsilon\) model is a two equation model. This means that in addition to the conservation equations, two additional transport equations are solved. One for the turbulent kinetic energy \(k\) which determines the energy in the turbulence and \(\epsilon\) one for which represents the turbulent dissipation. The \(k-\epsilon\) model also requires wall functions to obtain a value of near the wall. The \(k-\epsilon\) model has proven to provide reliable results for flows with relatively small pressure gradients. When a problem involves large adverse pressure gradients, large separations or complex flows with strong curvatures, the \(k-\epsilon\) model is not the most suitable turbulence model. A lot of different variations exist of the \(k-\epsilon\) model, a few examples are the RNG \(k-\epsilon\) model or the LaunderSharma \(k-\epsilon\) model. All of these variations have their own modifications to perform better under specific conditions.

The RNG \(k-\epsilon\) model was developed using the Re-Normalization Group (RNG) methods by Yakhot [1] to renormalise the Navier-Stokes equations to account for smaller scales of motions. This model is more suitable than the standard \(k-\epsilon\) model for swirling flows, separation or shear flows. It is also known to underestimate the value of the turbulent kinetic energy and thus creating a less viscous and more realistic flow with complex geometries. It is however not as widely used as the standard \(k-\epsilon\) model.

The LaunderSharma \(k-\epsilon\) model [2] is a low-Reynolds \(k-\epsilon\) model for incompressible, compressible and combusting flows including a rapid distortion theory (RDT) based compression term.

The \(k-\omega\) is also a two equation model, meaning two additional transport equations are solved. One for the turbulent kinetic energy \(k\) and one for the specific turbulent dissipation rate \(\omega \). The model used in the software, is based on the formulation by Wilcox [3]. \(k-\omega\) models are also commonly used. The standard formulation is a low Reynolds model, which means it can be used for flows with low Reynolds numbers where the boundary layer is relatively thick and the viscous sublayer can be resolved, without the need for wall functions. The \(k-\omega\) has shown to perform better in complex boundary layer flows under adverse pressure gradients and separations with respect to the \(k-\epsilon\) models. However this model is also known to overpredict separation or shear stresses of adverse pressure gradient boundary layers and is known to be sensitive to inlet free stream turbulence properties.

The \(k-\omega\) SST model is also a popular choice. This model, developed by Menter [4], employs a shear stress transport (SST) formulation which combines the \(k-\epsilon\) and the \(k-\omega\) model . In the outer region and in the outside of the boundary layer , the \(k-\epsilon\) model is used. In the inner boundary layer, the \(k-\omega\) model is used. By combining the two models, the disadvantages of both models are eliminated. The \(k-\omega\) SST model model can be used all the way down to the wall and the sensitivity to the inlet free stream turbulence is eliminated by using the \(k-\epsilon\) model in the outer region.

The selection of the turbulence model is important. Based on our experience, we can make the following recommendations.

First, the Reynolds number of the flow should be estimated. This Reynolds number just needs to be estimated to determine whether or not your case is turbulent. It is acceptable to make assumptions and simplifications regarding the geometry or parameters.

Let’s for example calculate the Reynolds number for water flowing through a pipe with a diameter \(d\) of 1.0 m and a speed of 1.5 m/s. In case you have a pipe or duct, the characteristic length scale in the Reynolds formula is equal to the hydraulic diameter, which can be calculated as follows, with \(A\) the area section of the duct or pipe in \([m^2]\) , \(P\) the perimeter of the duct or pipe in [m]: \[l = d_h = \frac{4A}{P} = \frac{4(\pi d^2/4)}{\pi d} = d \] So for a simple pipe or tube, the hydraulic diameter is equal to the diameter of the duct or pipe. So filling in the Reynolds equation as described before, gives: \[ Re=\frac{(1000 kg/m^{3}) (1.5 m/s) (1m)}{0.00959 kg/(ms)} = 156 413\] From which can be determined, the flow can be considered turbulent. If you don’t know what the Reynolds number would be for your case and find it difficult to calculate, it is probably turbulent (Re>4000). If you have a case with very small parallel channels and a low velocity, chances are high that your case will be laminar.

With the Reynolds number determined, an appropriate turbulence model can be selected. If this Reynolds number is very low (Re < 2400) a laminar model should be selected.

If you expect your flow to be in the transitional regime, i.e. a Reynolds number around 3000, the best choice according to our experience is the LaunderSharma model as this model is appropriate for low Reynolds simulations.

If turbulent flow is to be expected and you are simulating a liquid flow, the \(k-\omega\) SST model should be selected. In the case of air flow, \(k-\omega\) model is the best model for internal forced air flow in our opinion. For the case of external forced air flow, the \(k-\epsilon\) model should be selected.

Still in doubt about which model to choose after reading this? We suggest you take the \(k-\omega\) SST model as a starting point and see if the obtained results match your expectations.

1. V. Yakhot, S.A. Orszag, S. Thangam, and C.G. Speziale. Development of Turbulence Models for Shear Flows by a Double Expansion technique. Physics of Fluids A Fluid Dynamics, 4(7), 1992.
2. B.E. Launder and B.I. Sharma. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in heat and mass transfer, 1(2), 131-137, 1974.
3. D.C. Wilcox. Turbulence modelling for CFD, 2, 103-217. 1998.
4. F.R. Menter and T. Esch. Elements of Industrial Heat Transfer Prediction. 16th Brazilian Congress of Mechanical Engineering (COBEM), 2001.