Turbulence modeling
Turbulent flows are commonplace in most real-life scenarios.
In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows.
[1] The Navier–Stokes equations govern the velocity and pressure of a fluid flow.
[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
as a function of the mean flow, removing any reference to the fluctuating part of the velocity.
Joseph Valentin Boussinesq was the first to attack the closure problem,[3] 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.
, the (kinematic) turbulence eddy viscosity, has been introduced.
[4] 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).
The Boussinesq hypothesis – although not explicitly stated by Boussinesq at the time – effectively consists of the assumption that the Reynolds stress tensor is aligned with the strain tensor of the mean flow (i.e.: that the shear stresses due to turbulence act in the same direction as the shear stresses produced by the averaged flow).
It has since been found to be significantly less accurate than most practitioners would assume.
[5] Still, turbulence models which employ the Boussinesq hypothesis have demonstrated significant practical value.
In cases with well-defined shear layers, this is likely due the dominance of streamwise shear components, so that considerable relative errors in flow-normal components are still negligible in absolute terms.
Beyond this, most eddy viscosity turbulence models contain coefficients which are calibrated against measurements, and thus produce reasonably accurate overall outcomes for flow fields of similar type as used for calibration.
Later, Ludwig Prandtl introduced the additional concept of the mixing length,[6] 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:
where 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.
Joseph Smagorinsky was the first who proposed a formula for the eddy viscosity in Large Eddy Simulation models,[7] based on the local derivatives of the velocity field and the local grid size: 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.
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
The following is a brief overview of commonly employed models in modern engineering applications.
This arises due to the use of the eddy-viscosity hypothesis in their formulation.
[13] In such flows, Reynolds stress equation models offer much better accuracy.
