Turbulence models in CFD

Turbulence models in Computational Fluid Dynamics (CFD) are methods to include the effect of turbulence in the simulation of fluid flows. The majority of simulations require a turbulence model as turbulent flows are prevalent in nature and in industrial flows and occur in most engineering applications.

What is a turbulent flow?

Turbulence is the apparent chaotic motion of fluid flows. Fluid flows can be laminar, when they are regular and flow in an orderly manner. When the speed or characteristic length of the flow is increased, the convective forces in the flow overcome the viscous forces of the fluid and the laminar flow transitions into a turbulent one. The ratio between convective and viscous forces is called the Reynolds number. This number can be used to classify the type of flows, the higher the number the more turbulent the flow is.

Laminar to turbulence transition
Transition from laminar to turbulent flow

Turbulent flows are characterised by a large range of vortical structures at different scales, both in time and space, which interact with each other and exchange energy. The largest scales contain most of the kinetic energy for the flow. As larger structures are broken into smaller ones, this energy is transferred to progressively smaller scales. The process of energy transfer from large to small scale is called direct energy cascade and it continues until the viscous dissipation can convert the kinetic energy into thermal energy. The scale at which this occurs is referred to as Kolmogorov length scale,  after the mathematician Andrey Kolmogorov who worked on the energy cascade in turbulence.

Turbulence models in CFD

As engineering flows are mostly of turbulent nature when dealing with CFD simulations, most of the time we need to solve turbulent flows. The modelling of turbulence constitutes one of the most important aspects of CFD modelling and correctly modelling turbulence is key in obtaining correct and reliable CFD results.

Why do we need to model turbulence? Can it be resolved? It is possible to solve directly the governing equations of fluid flows, the Navier-Stokes equations, without the use of any modelling assumption. This approach is called Direct Numerical Simulation, or DNS in short, and it requires to solve the extensive range of temporal and spatial scales of a turbulent flow, from very large to very small, down to the Kolmogorov length scale. It can be estimated that the mesh resolution and time steps required to correctly solve the complexity of the fluid structures scales approximately with the cube of Reynolds number. This makes the DNS approach virtually impossible for engineering applications. DNS is indeed almost exclusively used in academia and research institutions to model simple flows and, along with experiments, it is used to improve the understanding of turbulence and to develop simplified turbulence models that are less expensive to calculate but still useful to predict the main contribution of turbulence on the flow.

Most of the time, in engineering applications we interested in mean or integral quantities like forces on a body or mass flow rate through a passage. In order to obtain such quantities, solving turbulent flows with a turbulence model is not only sufficient, but recommend too, as in this way it is possible to obtain reliable solutions in a more efficient and cost effective way. The figure below shows a summary of the most common approaches to solve turbulent flows. The computational cost a CFD simulation increases from RANS to DNS as the number of degrees of freedom required to solve the flow increases. As a consequence of the computational cost, scale resolving approaches like DNS and LES are generally applied to simple geometries and academic configurations, while hybrid RANS-LES, URANS and RANS can be applied to complex industrial problems

Turbulence models in CFD
Turbulence models in CFD from RANS to DNS

RANS models

Let’s take as an example a flow around a body. We can investigate the flow with an experiment. If the Reynolds number is large, the flow around the body will be turbulent. Taking a snapshot of the flow we would see that the initially organised and ordered flow would develop a dynamic chaotic motion with a large range of vortices. As we zoom in on a region downstream of the body, we would continue to see more and more intricate and small structures that vary in space and time. If we repeat the experiment with the same conditions several times, each time we would see a different evolution of the flow. However, if we over impose the different solutions and calculate an ensemble average we would start to recognise coherent structures in both space and time. If the flow has no time variations or periodicity, we could also average the results over time to increase the sampling of the mean.
To calculate engineering quantities of interests, these mean flows are sufficient. For example force coefficients acting on a plane at a certain flight condition, or on a car at a specific attitude.

An averaging operation can be applied to the Navier-Stokes equations to obtain the mean equations of fluid flows called Reynolds Averaged Navier-Stokes (RANS) equations. These are very similar to the original equations but contains some additional terms in the momentum equations called Reynolds stress terms that are unknown and need to be modelled.

Turbulence models aim to represent the effect of turbulence via the closure of unknown Reynolds stress terms. Turbulence models are generally classified based on the number of additional equations that are required in order to model the effect of turbulent on the flow. Models range from very simple algebraic relations and increase in fidelity and complexity as the number of equations used is increased. RANS modelling is the most common and widespread approach in industrial applications. If the flow of interest is characterised by moving parts or periodic flow feature, these can be resolved using unsteady RANS (URANS). 

All RANS models have some limitations due to the modelling assumptions used to derive the mathematical formulation of the model. For certain applications, it might be required to use more detailed approaches that instead of modelling all of the turbulence scales, try to resolve the most energy containing structures of the flow. This approach is called Large Eddy Simulations (LES).

Resolved scales in turbulence models
Energy distribution and scales in turbulence modeling

LES models

In LES, the smallest scales of turbulence are spatially filtered out while the largest, most energy containing scales are resolved directly. 

Due to the nature of turbulence, at a very small scale, the flow structures tend to be similar to each other even in different applications. This allows the use of simpler turbulence models that tend to be more universal and can be applied to several applications with a reduced requirement of model tuning.

Similarly to RANS modelling, in LES turbulence models aim at resolving the unknown terms in the filtered Navier-Stokes equations, called the Sub-grid Scale stresses. The term comes from the fact that in most LES models, the filtering of the equations is obtained at mesh size level, relegating the modelling to flow scales smaller than the grid size.

LES modelling offers increased range of applicability and increased fidelity of the solution but all of this comes with an increased computational cost due to the time step requirements, as we can no longer consider the flow steady, and increased mesh resolution required to capture more details of the flow.

Hybrid RANS-LES models

There is a family of turbulence models where the benefits of LES modelling are being exploited while retaining the efficiency of RANS models where LES would be too expensive. These go under the name of hybrid RANS-LES models.

The majority of the mesh resolution tends to be in proximity of surfaces and bodies of interest as in LES the flow needs to be fully resolved near the wall. The cost of LES can therefore be greatly reduced if the constraints of the increased resolution near the wall can be relaxed. One of the most common hybrid approaches uses LES modelling away from the walls and RANS modelling near the wall. This approach is called Detached Eddy Simulation (DES). The DES approach is becoming very popular in industrial applications as it helps overcoming some of the limitations of the RANS models as well as offering increased insight in the solution as the simulation is always run as unsteady flow, even for cases that have a steady state solution, and the finer spatial resolution allows to study detailed behaviour of the flow of interest. All of it at a reduced cost compared to a fully fledged LES approach.

 

CFD of a turbulent jet
CFD modelling of a turbulent jet using different approaches

Conclusions

All turbulence models have strengths and limitations due to the nature of the modelling assumptions. There is no single model that can be readily applied to all turbulent flows without a degree of adjustment and experience in selecting the most appropriate model. In addition, tuning the model coefficients is of utmost importance to obtain accurate, consistent and robust simulations.

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