# Turbulence modeling

Turbulence modeling is the construction and use of a model to predict the effects of turbulence. Turbulent flows are commonplace in most real life scenarios, including the flow of blood through the cardiovascular system[1], the airflow over an aircraft wing[2], the re-entry of space vehicles[3], besides others. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows[4].

## 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, ${\displaystyle R_{ij}}$.[5] 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.

${\displaystyle -{\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}=\nu _{t}\left({\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {\upsilon }}_{j}}{\partial x_{i}}}\right)-{\frac {2}{3}}k\delta _{ij}}$
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 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.[6] 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

Joseph Smagorinsky (1964) was the first who proposed a formula for the eddy viscosity in numerical models, based on the local derivatives of the velocity field and the local grid size:

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

In the context of Large Eddy Simulation, turbulence modeling refers to the need to parameterize the subgrid scale stress in terms of features of the filtered velocity field. This field is called subgrid-scale modeling.

## 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–ε and k–ω models use two.

## Common models

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

The Spalart–Allmaras model[7] 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.[citation needed]

K-epsilon (k-ε) turbulence model [8]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[9] 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[10] 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.

The Reynolds stress equation model (RSM), also referred to as second moment closure model [11], is the most complete classical turbulence modelling approach. Popular eddy-viscosity based models like the k–ε (k–epsilon) model and the k–ω (k–omega) models have significant shortcomings in complex engineering flows. This arises due to the use of the eddy-viscosity hypothesis in their formulation. For instance, in flows with high degrees of anisotropy, significant streamline curvature, flow separation, zones of recirculating flow or flows influenced by rotational effects, the performance of such models is unsatisfactory.[12] In such flows, Reynolds stress equation models offer much better accuracy.[13]

Eddy viscosity based closures cannot account for the return to isotropy of turbulence,[14] observed in decaying turbulent flows. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid Distortion limit,[15] where the turbulent flow essentially behaves like an elastic medium [16].

## References

### Notes

1. ^ Sallam, Ahmed; Hwang, Ned (1984). "Human red blood cell hemolysis in a turbulent shear flow: contribution of Reynolds shear stresses". Biorheology.
2. ^ Rhie, C; Chow, Li (1983). "Numerical study of the turbulent flow past an airfoil with trailing edge separation". AIAA Journal.
3. ^ Reddy, K; Silva, D; Krishnendu, Sinha (1983). "Hypersonic turbulent flow simulation of Fire II reentry vehicle afterbody". AIAA Journal.
4. ^ Pope, Stephen (2000). Turbulent Flows.
5. ^ Andersson, Bengt; et al. (2012). Computational fluid dynamics for engineers. Cambridge: Cambridge University Press. p. 83. ISBN 978-1-107-01895-2.
6. ^ John J. Bertin; Jacques Periaux; Josef Ballmann, Advances in Hypersonics: Modeling hypersonic flows
7. ^ Spalart, P.; Allmaras, S. (1992). "A one-equation turbulence model for aerodynamic flows". 30th aerospace sciences meeting and exhibit, AIAA.
8. ^ Hanjalic, K.; Launder, B. (1972). "A Reynolds stress model of turbulence and its application to thin shear flows". Journal of Fluid Mechanics.
9. ^ Wilcox, D. C. (2008). "Formulation of the k-omega Turbulence Model Revisited". AIAA Journal.
10. ^ Menter, F. R. (1994). "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications". AIAA Journal.
11. ^ Hanjalić, Hanjalić; Launder, Brian (2011). Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure.
12. ^ Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures". Journal of Fluid Mechanics. 731: 639–681. Bibcode:2013JFM...731..639M. doi:10.1017/jfm.2013.343.
13. ^ Pope, Stephen. "Turbulent Flows". Cambridge University Press, 2000.
14. ^ Lumley, John; Newman, Gary (1977). "The return to isotropy of homogeneous turbulence". Journal of Fluid Mechanics. 82: 161–178. Bibcode:1977JFM....82..161L. doi:10.1017/s0022112077000585.
15. ^ Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures". Journal of Fluid Mechanics. 731: 639–681. Bibcode:2013JFM...731..639M. doi:10.1017/jfm.2013.343.
16. ^ Sagaut, Pierre; Cambon, Claude (2008). Homogeneous Turbulence Dynamics.

### 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