# Turbulence modeling

Turbulence modeling is the construction and use of a model to predict the effects of turbulence. A turbulent fluid flow has features on many different length scales, which all interact with each other. A common approach is to average the governing equations of the flow, in order to focus on large-scale and non-fluctuating features of the flow. However, the effects of the small scales and fluctuating parts must be modelled.[1]

## Closure problem

The Navier–Stokes equations govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS) equations, which govern the mean flow. However, the nonlinearity of the Navier–Stokes equations means that the velocity fluctuations still appear in the RANS equations, in the nonlinear term ${\displaystyle -\rho {\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}}$ from the convective acceleration. This term is known as the Reynolds stress, ${\displaystyle R_{ij}}$.[2] Its effect on the mean flow is like that of a stress term, such as from pressure or viscosity.

To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the Reynolds stress term ${\displaystyle R_{ij}}$ as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the closure problem.

## Eddy viscosity

Joseph Valentin Boussinesq was the first to attack the closure problem, by introducing the concept of eddy viscosity. In 1877 Boussinesq proposed relating the turbulence stresses to the mean flow to close the system of equations. Here the Boussinesq hypothesis is applied to model the Reynolds stress term. Note that a new proportionality constant ${\displaystyle \nu _{t}>0}$, the turbulence eddy viscosity, has been introduced. Models of this type are known as eddy viscosity models or EVM's.

${\displaystyle -{\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}=\nu _{t}\left({\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {\upsilon }}_{j}}{\partial x_{i}}}-{\frac {2}{3}}{\frac {\partial {\bar {\upsilon }}_{k}}{\partial x_{k}}}\delta _{ij}\right)-{\frac {2}{3}}K\delta _{ij}}$
Which can be written in shorthand as
${\displaystyle -{\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}=2\nu _{t}S_{ij}^{*}-{\frac {2}{3}}K\delta _{ij}}$
where ${\displaystyle S_{ij}^{*}}$ is the traceless mean rate of strain tensor
${\displaystyle \nu _{t}}$ is the turbulence eddy viscosity
${\displaystyle K={\frac {1}{2}}{\overline {\upsilon _{i}'\upsilon _{i}'}}}$ is the turbulence kinetic energy
and ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

In this model, the additional turbulence stresses are given by augmenting the molecular viscosity with an eddy viscosity.[3] This can be a simple constant eddy viscosity (which works well for some free shear flows such as axisymmetric jets, 2-D jets, and mixing layers).

## Prandtl's mixing-length concept

Later, Ludwig Prandtl introduced the additional concept of the mixing length, along with the idea of a boundary layer. For wall-bounded turbulent flows, the eddy viscosity must vary with distance from the wall, hence the addition of the concept of a 'mixing length'. In the simplest wall-bounded flow model, the eddy viscosity is given by the equation:

${\displaystyle \nu _{t}=\left|{\frac {\partial u}{\partial y}}\right|l_{m}^{2}}$
where:
${\displaystyle {\frac {\partial u}{\partial y}}}$ is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction (y);
${\displaystyle l_{m}}$ is the mixing length.

This simple model is the basis for the "law of the wall", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small pressure gradients.

More general turbulence models have evolved over time, with most modern turbulence models given by field equations similar to the Navier-Stokes equations.

## Smagorinsky model for the sub-grid scale eddy viscosity

Among many others[who?], Joseph Smagorinsky (1964) proposed a useful formula for the eddy viscosity in numerical models, based on the local derivatives of the velocity field and the local grid size:

${\displaystyle \nu _{t}=\Delta x\Delta y{\sqrt {\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial v}{\partial y}}\right)^{2}+{\frac {1}{2}}\left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)^{2}}}}$

## Spalart–Allmaras, k–ε and k–ω models

The Boussinesq hypothesis is employed in the Spalart–Allmaras (S–A), k–ε (k–epsilon), and k–ω (k–omega) models and offers a relatively low cost computation for the turbulence viscosity ${\displaystyle \nu _{t}}$. The S–A model uses only one additional equation to model turbulence viscosity transport, while the k models use two.

## Common models

The following is a brief overview of commonly employed models in modern engineering applications.

The Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart-Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.

K-epsilon (k-ε) turbulence model is the most common model used in Computational Fluid Dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two equation model which gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.

In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model, that is used as a closure for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).

SST (Menter’s Shear Stress Transport) turbulence model is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.

Reynolds stress equation model (RSM), also referred to as second order or second moment closure model is the most complete classical turbulence model. Due to the disparate character of complex engineering flows, turbulence models must be robust so as to be applicable for most cases, yet possessing a high degree of fidelity in each. Furthermore, as the processes of analysis and engineering design involve repeated iterations, the predictive method must be computationally economical. In this light, Reynolds Averaged Navier Stokes (RANS)-based models represent the pragmatic approach for complex engineering flows as opposed to computationally intensive methods like Large Eddy Simulations or Direct Numerical Simulations. However, popular RANS-based modeling paradigms like one or two-equation models have significant shortcomings in all but the simplest turbulent flows. For instance, in flows with streamline curvature or a high preponderance of mean rotational effects, the performance of such models is highly unsatisfactory. In such flows, Reynolds stress based models can offer much better predictive fidelity. In summary, the second moment closure approach offers better accuracy than one or two equation turbulence models and yet is not as computationally demanding as Direct Numerical Simulations.

## References

### Notes

1. ^ Ching Jen Chen; Shenq-Yuh Jaw (1998), Fundamentals of turbulence modeling, Taylor & Francis
2. ^ Andersson, Bengt; et al. (2012). Computational fluid dynamics for engineers. Cambridge: Cambridge University Press. p. 83. ISBN 978-1-107-01895-2.
3. ^ John J. Bertin; Jacques Periaux; Josef Ballmann, Advances in Hypersonics: Modeling hypersonic flows

### Other

• Townsend, A.A. (1980) "The Structure of Turbulent Shear Flow" 2nd Edition (Cambridge Monographs on Mechanics), ISBN 0521298199
• Bradshaw, P. (1971) "An introduction to turbulence and its measurement" (Pergamon Press), ISBN 0080166210
• Wilcox C. D., (1998), "Turbulence Modeling for CFD" 2nd Ed., (DCW Industries, La Cañada), ISBN 0963605100