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One proposal suggests that turbulence consists of small eddies which are continuously forming and dissipating, and in which the Reynolds stresses are assumed to be proportional to mean velocity gradients. This defines an ‘eddy viscosity model.'
The eddy viscosity hypothesis assumes that the Reynolds stresses can be related to the mean velocity gradients and Eddy (turbulent) Viscosity by the gradient diffusion hypothesis, in a manner analogous to the relationship between the stress and strain tensors in laminar Newtonian flow:
where is the Eddy Viscosity or Turbulent Viscosity. This has to be modeled.
Analogous to the eddy viscosity hypothesis is the eddy diffusivity hypothesis, which states that the Reynolds fluxes of a scalar are linearly related to the mean scalar gradient:
where is the Eddy Diffusivity, and this has to be prescribed. The Eddy Diffusivity can be written as:
where is the turbulent Prandtl number. Eddy diffusivities are then prescribed using the turbulent Prandtl number.
The above equations can only express the turbulent fluctuation terms of functions of the mean variables if the turbulent viscosity, , is known. Both the - and - two-equation turbulence models provide this variable.
Subject to these hypotheses, the Reynolds averaged momentum and scalar transport equations become:
where is the sum of the body forces, and is the Effective Viscosity defined by:
and is a modified pressure, defined by:
By default, the solver actually assumes that , but the contribution can be activated by setting the expert parameter “pressure value option = 1“. In Equation 12 above, there is a term which although included in the fundamental form of the equation (For details, see .) is neglected in the solver and thus not included here.
The Reynolds averaged energy equation becomes:
Note that although the transformation of the molecular diffusion term may be inexact if enthalpy depends on variables other than temperature, the turbulent diffusion term is correct, subject to the eddy diffusivity hypothesis. Moreover, as turbulent diffusion is usually much larger than molecular diffusion, small errors in the latter can be ignored.
Similarly, the Reynolds averaged transport equation for Additional Variables (non-reacting scalars) becomes:
Eddy viscosity models are distinguished by the manner in which they prescribe the eddy viscosity and eddy diffusivity.