Reynolds-Averaged Navier-Stokes (RANS) Simulations

For incompressible turbulent flow the instantaneous velocity field can be decomposed into a time averaged velocity and its corresponding fluctuation.

This is commonly called the Reynolds’ decomposition, which is shown in Figure 1. (1)

u_{i} = \bar{u_{i}} + u_{i}^{'}

where

$u_{i}$ : instantaneous velocity,
$\bar{u_{i}} = \frac{1}{Δ t} \int_{t}^{t + Δ t} u_{i} d t$ : time averaged velocity over $Δ t$ ,
$u_{i}^{'}$ velocity fluctuation.

Figure 1. Turbulent Velocity Signals as a Function of Time

Similarly, instantaneous pressure can be decomposed into the time averaged and fluctuation terms. (2)

p = \bar{p} + p^{'}

where $p$ is the instantaneous pressure.

Once this concept is substituted into the instantaneous Navier-Stokes equations and then time averaging is performed the RANS equations are obtained. The following equations are, respectively, the Reynolds-averaged continuity and Reynolds-averaged momentum equations: (3)

\frac{\partial ρ}{\partial t} + \frac{\partial (ρ {\bar{u}}_{j})}{\partial x_{j}} = 0

(4)

\frac{\partial (ρ {\bar{u}}_{i})}{\partial t} + \frac{\partial (ρ {\bar{u}}_{i} {\bar{u}}_{j})}{\partial x_{j}} = - \frac{\partial (\bar{p} δ_{i j})}{\partial x_{j}} + 2 μ \frac{\partial \bar{S_{i j}}}{\partial x_{j}} + \frac{\partial τ_{i j}^{R}}{\partial x_{j}}

where

$\bar{S_{i j}} = \frac{1}{2} (\frac{\partial \bar{u_{i}}}{\partial x_{j}} + \frac{\partial \bar{u_{j}}}{\partial x_{i}})$ is the mean strain rate tensor,
$τ_{i j}^{R} = - ρ \bar{u_{i}^{'} u_{j}^{'}}$ is the Reynolds stress tensor.

It is noted that the momentum equation has an unsteady term (the first term in the left hand side of the momentum equation). Wegner et al. (2004) argued that the RANS equations with the time term (unsteady RANS) can be applied to simulate unsteady flows when a time scale (T2) for large scale motions is larger than a time averaging period for RANS (T1) (see Figure 2). It is worth mentioning that RANS time averaging period (T1) should be considerably larger than the largest turbulence (integral) time scale, as URANS can only give a time averaged mean value for the velocity field, not solving turbulence. Thus, a representative URANS solution is demonstrated by the smoothed red line in the image below. Whereas the blue line represents the velocity profile with instantaneous fluctuations.

Steady RANS can be obtained by removing the time term when there is no large scale unsteadiness observed in turbulent flow.

Figure 2. Time Scales for T2 for Large Unsteady Oscillation vs T1 for Turbulent Time Scale

Although Reynolds-averaging eliminates a need to compute the instantaneous flow field, it introduces a new unknown term in comparison to the Navier-Stokes equations. This unknown term is referred to as the Reynolds stress tensor, which represents an added stress due to the turbulent motions. This term arises from the decomposition of the convective term using the Reynolds decomposition. Since your goal is to eliminate any dependence on the instantaneous flow field in these simplified equations, this term should be modeled in terms of the mean flow. This challenge is known as the closure problem of turbulence.

One method that is available for computing the Reynolds Stresses that appear in the RANS equations is to use the Boussinesq approximation. This approach assumes a linear relationship between the turbulent Reynolds stresses and the mean rate of strain tensor: (5)

τ_{i j}^{R} = - ρ \bar{u_{i}^{'} u_{j}^{'}} = μ_{t} [\frac{\partial \bar{u_{i}}}{\partial x_{j}} + \frac{\partial \bar{u_{j}}}{\partial x_{i}} - \frac{2}{3} \frac{\partial \bar{u_{k}}}{\partial x_{k}} δ_{i j}] - \frac{2}{3} ρ k δ_{i j}

where

$μ_{t}$ : the eddy viscosity,
$k = \frac{1}{2} \bar{u_{i}^{'} u_{j}^{'}}$ : the turbulent kinetic energy per unit mass,
$δ_{i j}$ : the Kronecker delta.

Since

\frac{\partial \bar{u_{k}}}{\partial x_{k}}

= 0 for incompressible flow the isotropic eddy viscosity becomes (6)

τ_{i j}^{R} = μ_{t} [\frac{\partial \bar{u_{i}}}{\partial x_{j}} + \frac{\partial \bar{u_{j}}}{\partial x_{i}}] - \frac{2}{3} ρ k δ_{i j}

The only unknown in this equation is the eddy viscosity, which must be determined by the turbulence model. There are hundreds of available turbulence models that have been developed to provide this eddy viscosity. Some common Boussinesq models include

Mixing length model
Spalart-Allmaras model
$k - ε$ model, RNG (Renormalized Group) $k - ε$ model, Realizable $k - ε$ model
$k - ω$ model, SST (Shear Stress Transport) model
$v 2 f$ model, zeta-f model

This approach is valid for many turbulent flows, for example, boundary layer, channel flow, round jets, turbulent shear flow and mixing layer, but not for all, for example, impinging flow and swirling flow. Under these circumstances, a model that is capable of predicting anisotropic Reynolds stresses is more appropriate. This can include a full Reynolds stress model (RSM), which determines the turbulent stresses by solving a transport equation for each stress component, or an eddy viscosity model that uses a non linear stress strain relationship to compute each Reynolds stress component separately. Although the improvement in accuracy achieved from this type of modeling is appealing, the introduction of anisotropic stresses is often accompanied by increased compute cost and a decrease in robustness.

The following section provides a review of some commonly used RANS models, ranging from algebraic models (zero equation models) up to the seven equation Reynolds Stress Model.