eddyviscosity

Eddy viscosity, also called turbulent viscosity, is a modeling concept used to represent turbulent momentum transport in fluids. It treats the effect of chaotic eddies as an enhanced, effective viscosity that adds to the molecular viscosity, allowing the Reynolds-averaged equations to be closed with a diffusion-like term.

Under the Boussinesq hypothesis, the turbulent shear stresses are assumed proportional to the mean rate of strain, with the proportionality constant nu_t, called the eddy viscosity: tau_ij = -rho nu_t (du_i/dx_j + du_j/dx_i) + (2/3) rho k delta_ij. nu_t is not a physical viscosity but a model parameter dependent on the flow.

In practice, nu_t is estimated by turbulence models such as the k-ε model, the k-ω model, or their variants, which relate nu_t to turbulence kinetic energy k and its dissipation ε or to vorticity-based quantities. For large-eddy simulation, subgrid-scale eddy viscosity is computed using models like Smagorinsky, with nu_t = (C_s Δ)^2 |S|.

Eddy viscosity has limitations: it assumes isotropy of the turbulent stresses and is less reliable near walls or in strongly anisotropic flows. It also cannot capture backscatter or non-diffusive transport. Despite these issues, eddy viscosity remains a foundational concept in computational fluid dynamics and is used in Reynolds-averaged and large-eddy simulations, often with wall treatments and more advanced subgrid models. The concept originated with Boussinesq in the 1870s and was refined by Prandtl and others.