Turbulence modeling

Turbulence modeling is the construction and use of a model to predict the effects of turbulence. Averaging is often used to simplify the solution of the governing equations of turbulence, but models are needed to represent scales of the flow that are not resolved.

Closure problem

A closure problem arises in the Reynolds-averaged Navier-Stokes (RANS) equation because of the non-linear term ${\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}$ from the convective acceleration, known as the Reynolds stress,

$R_{ij}={\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}$

Closing the RANS equation requires modeling the Reynold's stress $R_{ij}$.

Eddy viscosity

where $S_{ij}$ HOW ARE YOU EVERY BODY.....BOOOOOOO mean rate of strain tensor

$\nu _{t}$ is the turbulence eddy viscosity

$K={\frac {1}{2}}{\overline {\upsilon _{i}'\upsilon _{i}'}}$ FFFFFFFFFF turbulence kinetic energy

and $\delta _{ij}$ is the Kronecker delta.

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:

$\nu _{t}=\left|{\frac {\partial u}{\partial y}}\right|l_{m}^{2}$

where:

${\frac {\partial u}{\partial y}}$ is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction (y);

$l_{m}$ 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 , 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:

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

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 $\nu _{t}$. 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 list of commonly employed models in modern engineering applications.

• Spalart–Allmaras (S–A)
• k–ε (k–epsilon)
• k–ω (k–omega)
• SST (Menter’s Shear Stress Transport)
• Reynolds stress equation model

References

Other

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