Theory of Turbulent Flows

1. Notations and units

Notation Quantity Unit

ρ

fluid density

kg⋅m−3

\boldsymbol{u}

fluid velocity

m \cdot s^{-1}

\boldsymbol{\sigma}

fluid stress tensor

N \cdot m^{-2}

\boldsymbol{f}^t

source term

kg \cdot m^{-3} \cdot s^{-1}

p

pressure fields

kg \cdot m^{-1} \cdot s^{-2}

\mu

dynamic viscosity

kg \cdot m^{-1} \cdot s^{-1}

\bar{U}

characteristic inflow velocity

m \cdot s^{-1}

\nu

kinematic viscosity

m^2 \cdot s^{-1}

L

characteristic length

m

2. Reynolds-averaged Navier–Stokes equations

The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged equations of motion for fluid flow. The Reynolds averaging consists to decompose the solution variables of Navier-Stokes equations (like velocity \boldsymbol{u}) into the mean component (\overline{\boldsymbol{u}}) and the fluctuating component (\boldsymbol{u}^{\prime}), which can be written in the following form

\boldsymbol{u} = \overline{\boldsymbol{u}} + \boldsymbol{u}^{\prime}.

2.1. Incompressible case

The incompressible Navier-Stokes equations required to find the velocity \boldsymbol{u} and the pressure p which verify following equations

\begin{eqnarray*} \rho \left. \frac{\partial\mathbf{u}}{\partial t} \right|_\mathrm{x} + \rho \left( \boldsymbol{u} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u} - \nabla_{\mathrm{x}} \cdot \boldsymbol{\sigma}(\boldsymbol{u},p) = \boldsymbol{f}^t , \quad \text{ in } \Omega^t \times \left[t_i,t_f \right] \\ \nabla_{\mathrm{x}} \cdot \boldsymbol{u} = 0, \quad \text{ in } \Omega^t \times \left[t_i,t_f \right] \end{eqnarray*}

with the stress tensor \boldsymbol{\sigma}

\boldsymbol{\sigma}(\boldsymbol{u},p) = -p \boldsymbol{I} + 2\mu D(\boldsymbol{u})

and strain tensor D(\boldsymbol{u}) :

D(\boldsymbol{u}) = \frac{1}{2} (\nabla_\mathrm{x} \boldsymbol{u} + (\nabla_\mathrm{x} \boldsymbol{u})^T)

By substituting the Reynolds decomposition on the velocity and pressure, i.e

\begin{eqnarray*} \boldsymbol{u} = \overline{\boldsymbol{u}} + \boldsymbol{u}^{\prime} \quad \text{and} \quad p = \overline{p} + p^{\prime} \end{eqnarray*}

into the incompressible Navier-Stokes equations and in computing the Reynolds average of these equations, we can derived Reynolds-averaged Navier–Stokes in the incompressible case. The RANS probleme corresponds to find \overline{\boldsymbol{u}} and \overline{p} such that

\begin{eqnarray*} \rho \left. \frac{\partial\overline{\boldsymbol{u}}}{\partial t} \right|_\mathrm{x} + \rho \left( \overline{\boldsymbol{u}} \cdot \nabla_{\mathrm{x}} \right) \overline{\boldsymbol{u}} + \rho\ \overline{\left( \boldsymbol{u}^{\prime} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u}^{\prime} } - \nabla_{\mathrm{x}} \cdot \boldsymbol{\sigma}(\overline{\boldsymbol{u}},\overline{p}) = \boldsymbol{f}^t , \quad \text{ in } \Omega^t \times \left[t_i,t_f \right] \\ \nabla_{\mathrm{x}} \cdot \overline{\boldsymbol{u}} = 0, \quad \text{ in } \Omega^t \times \left[t_i,t_f \right] \end{eqnarray*}

These system is very close to the incompressible Navier-Stokes equations exept that we have an additional term \rho\ \overline{\left( \boldsymbol{u}^{\prime} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u}^{\prime}}. Then, this extra term can be rewritten (by using the continuity equation of the fluctuation) as

\begin{eqnarray*} \rho\ \overline{\left( \boldsymbol{u}^{\prime} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u}^{\prime}} = \nabla \cdot \left( \rho \overline{ \boldsymbol{u}^{\prime} (\boldsymbol{u}^{\prime})^T } \right) \end{eqnarray*}

2.2. The Boussinesq approximation

\begin{eqnarray*} - \rho \overline{ \boldsymbol{u}^{\prime} (\boldsymbol{u}^{\prime})^T } = \mu_t D(\boldsymbol{u}) - \frac{2}{3} \rho k \boldsymbol{I} \end{eqnarray*}

3. Turbulence Models

3.1. Spalart Allmaras Model