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

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

3. Turbulence Models

3.1. Spalart Allmaras Model