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.

Note	In this context, unless otherwise stated, it should be understood that the term “turbulence model” refers to RANS closure models.

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'

(251)

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. The resulting equations for the mean quantities are essentially identical to the original equations, except that an additional term now appears in the momentum transport equation. This additional term is the stress tensor, which has the following definition:

Figure 2. 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

(252)

where:

$ρ$ is the density.
$u$ , $v$ , and $w$ are the velocity components.
$k$ is the turbulent kinetic energy.
$I$ is the identity tensor.

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

Eddy Viscosity Models

Eddy viscosity models use the concept of a turbulent eddy viscosity $μ_{t}$ to model the stress tensor as a function of mean flow quantities.

The most common model is known as the Boussinesq approximation:

Figure 3. EQUATION_DISPLAY

T_{RANS} = 2 μ_{t} S - \frac{2}{3} (μ_{t} \nabla\cdot \bar{v}) I

(253)

where:

$S$ is the mean strain rate tensor given by Eqn. (1130).
$\bar{v}$ is the mean velocity.

While some simpler models rely on the concept of mixing length to model the turbulent viscosity in terms of mean flow quantities (similar to the Smagorinsky Subgrid Scale model used in Large Eddy Simulation (LES)), the eddy viscosity models in Simcenter STAR-CCM+ solve additional transport equations for scalar quantities that enable the turbulent viscosity $μ_{t}$ to be derived. These include the following turbulence models:

Spalart-Allmaras models
K-Epsilon models
K-Omega models

The assumption that the stress tensor is linearly proportional to the mean strain rate does not consider anisotropy of turbulence. In order to account for this turbulence anisotropy, some of the models are provided with an option to extend the linear approximation to include non-linear constitutive relations.

For K-Omega and K-Epsilon models, a Scale Resolving Hybrid (SRH) model is available that allows the RANS model to continuously switch to LES mode to resolve unsteady information of large-scale turbulent structures.

Reynolds Stress Transport (RST) Models

Reynolds Stress Transport (RST) models, also known as second-moment closure models, approximate the stress tensor as:

Figure 4. EQUATION_DISPLAY

T_{RANS} = - ρ R + \frac{2}{3} tr (R) I

(254)

where $R$ is the Reynolds stress tensor defined as:

Figure 5. EQUATION_DISPLAY

R = (\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})

(255)

RST models solve transport equations for each component of $R$ .