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# Eddy Viscosity

Eddy viscosity is a model viscosity. It is used to account for the effects lost in averaging the turbulent effects in a CFD simulation. More specifically, it models the transport and dissipation of energy that was neglected as a result of turbulence modeling. These turbulence models are abstract fundamental equations that predict the mathematical development of turbulent flows $$^1$$.

Eddy viscosity is the proportionality factor describing the turbulent transfer of energy as a result of moving eddies, giving rise to tangential stresses.$$^2$$

It is also referred to as turbulent viscosity and doesn’t have any physical existence.

One can, thus, express the effective viscosity of the fluid flow as:

$$\nu_{eff} = \nu +\nu_{t} \tag{1}$$

where,

• $$\nu_{eff}$$ = Effective kinematic viscosity of the fluid.
• $$\nu_{t}$$ = Turbulent kinematic viscosity or eddy (kinematic) viscosity.
• $$\nu$$ = Kinematic viscosity without turbulent effects.

Hence, it wouldn’t be inaccurate to say that the difference between different turbulence models is the difference in calculating turbulent viscosity.

## Terminology

In Computational Fluid Dynamics, turbulence models where eddies of all scales are completely modeled, for e.g. RANS (Reynolds Averaged Navier Stokes) models like $$k-\epsilon$$ and $$k-\omega\ SST$$, use the term eddy viscosity. For turbulence models that use sub-grid scaling to resolve the larger eddies but model the smaller ones, for e.g. LES models like Smagorinsky and Spalart Allmaras, use the term sub-grid scale viscosity.

Hence, for sub-grid scale models equation (1) becomes

$$\nu_{eff} = \nu +\nu_{sgs} \tag{2}$$

Where,

$$\nu_{sgs}$$ = Sub-grid scale kinematic viscosity.

## Defining in the Workbench

SimScale allows its users to define dynamic turbulent viscosity $$(\mu_{t}$$) or $$(\mu_{sgs}$$) in units of $$[\frac{kg}{ms}]$$ as an initial condition and also as part of custom boundary condition.

Since,

$$\nu = \frac{\mu}{ \rho}\tag{3}$$

the value for $$\nu_{t}$$ or $$\nu_{sgs}$$ is automatically calculated.

References

• Pope, Stephen (2000). Turbulent Flows.
• P. Jarvis, B. Jefferson, J. Gregory and S. Parsons, “A review of floc strength and breakage”, Water Research, vol. 39, no. 14, pp. 3121-3137, 2005. Available: 10.1016/j.watres.2005.05.022.

Last updated: February 13th, 2023