# Theory of Turbulent Flows

## 1. Notations and units

Notation Quantity Unit

$\rho$

fluid density

$kg \cdot m^{-3}$

$\boldsymbol{u}$

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

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