Reynolds-Averaged Navier-Stokes (RANS) Turbulence Models

RANS turbulence models provide closure relations for the Reynolds-Averaged Navier-Stokes equations, that govern the transport of the mean flow quantities.

To obtain the Reynolds-Averaged Navier-Stokes equations, each solution variable $ϕ$ in the instantaneous Navier-Stokes equations is decomposed into its mean, or averaged, value $\bar{ϕ}$ and its fluctuating component $ϕ'$ :

Figure 1. EQUATION_DISPLAY

ϕ = \bar{ϕ} + ϕ'

(1142)

where $ϕ$ represents velocity components, pressure, energy, or species concentration.

The averaging process may be thought of as time-averaging for steady-state situations and ensemble averaging for repeatable transient situations. Inserting the decomposed solution variables into the Navier-Stokes equations results in equations for the mean quantities.

The mean mass, momentum, and energy transport equations can be written as:

Figure 2. EQUATION_DISPLAY

\frac{\partial ρ}{\partial t} + \nabla\cdot (ρ \bar{v}) = 0

(1143)

Figure 3. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \bar{v}) + \nabla\cdot (ρ \bar{v} \otimes \bar{v}) = - \nabla\cdot {\bar{p}}_{\mod} I + \nabla\cdot (\bar{T} + T_{RANS}) + f_{b}

(1144)

Figure 4. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \bar{E}) + \nabla\cdot (ρ \bar{E} \bar{v}) = - \nabla\cdot {\bar{p}}_{\mod} \bar{v} + \nabla\cdot (\bar{T} + T_{RANS}) \bar{v} - \nabla\cdot \bar{q} + f_{b} \bar{v}

(1145)

where:

$ρ$ is the density.
$\bar{v}$ is the mean velocity.
${\bar{p}}_{\mod} = \bar{p} + \frac{2}{3} ρ k$ is the modified mean pressure, where $\bar{p}$ is the mean pressure and $k$ is the turbulent kinetic energy.
$I$ is the identity tensor.
$\bar{T}$ is the mean viscous stress tensor.
$f_{b}$ is the resultant of the body forces (such as gravity and centrifugal forces).
$\bar{E}$ is the mean total energy per unit mass.
$\bar{q}$ is the mean heat flux.

These equations are essentially identical to the original Navier-Stokes equations (see Eqn. (654), Eqn. (656), and Eqn. (658)), except that an additional term now appears in the momentum and energy transport equations. This additional term is the stress tensor, which has the following definition:

Figure 5. EQUATION_DISPLAY

T_{RANS} = - ρ (\begin{matrix} \bar{u' u'} & \bar{u' v'} & \bar{u' w'} \\ \bar{u' v'} & \bar{v' v'} & \bar{v' w'} \\ \bar{u' w'} & \bar{v' w'} & \bar{w' w'} \end{matrix}) + \frac{2}{3} ρ k I

(1146)

The challenge is thus to model $T_{RANS}$ in terms of the mean flow quantities, and hence provide closure of the governing equations. Two basic approaches are used in Simcenter STAR-CCM+:

Eddy viscosity models
Reynolds stress transport models