Description of Turbulence
Turbulence is a difficult property to describe, general words used to describe it are chaotic, random, and swirling. Turbulent flows must also obey the laws of thermodynamics, the conservation of mass, and the conservation of momentum. Currently turbulence remains unsolved, several tools exist for approximating the effects of turbulent phenomena but there is no simple way to fully model turbulence.
Transition from laminar flow to turbulent flow typically occurs at a transition Reynold’s number, largely dependent on flow geometry.
Reynold’s Averaged Navier-Stokes (RANS)
Since turbulent flow is so chaotic it is difficult to solve the differential equations outright. Velocity can be broken into a time averaged component and a fluctuating component.
u=\overline{u}+u^\prime
The following averaging rules apply:
\overline{\overline{a}}=\overline{a}
\overline{\overline{a}\overline{b}}=\overline{a}\overline{b}
\overline{a^\prime}=0
\overline{\overline{a}b^\prime}=0
Using the component definition for velocity in the continuity equation:
\nabla\cdot\overline{\bm{V}}=0
By using the component definition for velocity in the Navier-Stokes equation and then averaging both sides we arrive at the RANS equation:
\rho\dfrac{D\overline{\bm{V}}}{Dt}=-\nabla\overline{p}+\nabla\cdot\bm{\sigma}_{total}+\rho\bm{f}
The stress term can be expressed as the sum of both viscous stresses and stresses caused by turbulent fluctuations.
\bm{\sigma}_{total}=\bm{\sigma}_{viscous}+\bm{\sigma}_{turbulent}
The viscous term has been previously resolved using Stoke’s hypothesis. The turbulent term is defined as the following:
\bm{\sigma}_{turbulent}=\begin{pmatrix}-\rho\overline{u^\prime u^\prime}&-\rho\overline{u^\prime v^\prime}&-\rho\overline{u^\prime w^\prime}\\-\rho\overline{v^\prime u^\prime}&-\rho\overline{v^\prime v^\prime}&-\rho\overline{v^\prime w^\prime}\\-\rho\overline{w^\prime u^\prime}&-\rho\overline{w^\prime v^\prime}&-\rho\overline{w^\prime w^\prime}\end{pmatrix}
The turbulent stress can be resolved using Boussinesq’s assumption, relating the turbulent stress to the strain rate just as was done for the viscous stress term. We define a turbulent viscosity, termed eddy viscosity \mu_t, for this purpose. Unlike viscosity, eddy viscosity is usually very sensitive to location, as such eddy viscosity can be expressed as a function of location.
\mu_t=f(x,y,z)
Therefore the RANS equation can be summarized as:
\rho\dfrac{D\overline{\bm{V}}}{Dt}=-\nabla\overline{p}+\nabla\cdot[(\mu+\mu_t)\nabla\overline{\bm{V}}]+\rho\bm{f}