The k-tau RANS Model

NekRS's - RANS model solves the incompressible Navier-Stokes equations with two additional PDEs for and . This model includes equations governing conservation of mass, momentum, and energy,

(1)

(2)

(3)

where is the velocity, is the fluid density, is the pressure, is the laminar dynamic viscosity, is the turbulent dynamic viscosity, is a general momentum source, is the fluid isobaric specific heat capacity, is the fluid temperature, is the laminar fluid thermal conductivity, is the turbulent thermal conductivity, and is a general energy source. Assuming similarity between turbulent momentum and energy transfer, is related to through the turbulent Prandtl number ,

(4)

where is the turbulent kinematic viscosity and is the turbulent thermal diffusivity. The - model is a modification of the standard - turbulence model that bases the second transport equation on , the inverse of the specific dissipation rate ,

(5)

The equation for the turbulent kinetic energy is a model equation based on a gradient diffusion approximation for the turbulent transport and pressure diffusion terms in the true equation, which itself is derived by taking the trace of the Reynolds stress equation. The equation in NekRS is

(6)

where and are constants and is the production of turbulent kinetic energy by velocity shear. The equation is obtained by inserting Eq. (5) into the model equation of the - equation, giving

(7)

where , , and are constants. The turbulent dynamic viscosity is then related in terms of and as

(8)