# What is the Boussinesq approach?

I want to simulate a turbine in Fluent. For this reason, I studied some papers. In these papers, Turbulence is modeled using the realizable k–e model and Non-equilibrium wall functions are used in the near wall flow modeling. There are the following sentences in these papers:

• The Boussinesq approach is applied to relate the Reynolds stresses to the mean velocity gradient.

• The Reynolds stress is related to the mean velocity gradient by means of the Boussinesq approach.

What is the exact meaning of above sentences? And is there a direct option in Fluent?

Is there any adjustment that must be made in Fluent before starting the calculation for this approach?

The basic assumption of the RANS approach (see the answer by @Algo) is that there is an overall steady solution to the problem. This overall steady solution features small scale oscillations around the mean value.

• In a first step the Navier Stokes equations (NS) are rewritten with this assumption $u_i = \bar{u_i} + u_i'$
• The second step is to average the NS again
• By doing this only on term does not cancel out except: $\frac{\partial }{\partial x_j}(\overline{\rho {u_i}'{u_i}'})$ (this is a 3x3 matrix and it's called the Reynolds Stress Tensor)

Since there is no analytical solution for the mean of the product of the fluctuations an additional assumption was needed to solve the equation system. There were a lot of ideas on how to model or solve this problem. One turned out to be very useful because it predicted the reality very well.

The idea is based on the observation that turbulence basically increases mixing of the fluid. To put it in other words: velocity gradients mix out a lot faster in turbulent flows, as if the viscosity (momentum diffusion) is increased.

So the first step in the solution was to model the unknown term as a kind of additional viscosity term as shown in the answer from @Algo. Here the mean velocity gradient is known and only one unknown remains (after the RANS approach we were faced with a 3x3 matrix).

Turbulence models are then used to calculate $\mu_t$. So the only adjustment you can make is to choose a turbulence model and in some cases tweak its parameters.

To take into account the effect of turbulent fluctuations on a flow field, the Navier-Stokes equations are modified to include such effects. The obtained equations are called Reynolds-averaged Navier-Stokes (RANS) equations.

As an example, the steady incompressible momentum equation can be written as follows (in Einstein tensor notation):

$$\rho \frac{\partial u_i}{\partial t} + \rho u_i \frac{\partial u_i}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \frac{\partial t_{ij}}{\partial xj}$$

Where $t_{ij}$ is the viscous stress tensor defined as:

$$t_{ij} = 2\mu s_{ij}$$ where $\mu$ is viscosity and $s_{ij}$ is the strain-rate tensor:

$$s_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$$ After time-averaging, one can obtain the following equation:

$$\rho \frac{\partial U_i}{\partial t} + \rho U_i \frac{\partial U_i}{\partial x_j} = -\frac{\partial P}{\partial x_i} + \frac{\partial }{\partial xj}(2\mu s_{ij} - \overline{\rho {u_j}{u_i}})$$

The resulted term $- \overline{\rho {u_j}{u_i}}$ is called the Reynolds-stress tensor (or turbulent shear stress), which is the basis of all turbulence models.

The Boussinesq approximation is merely assuming that turbulent shear stress is analogous to viscous shear stress (by introducing a new term called eddy viscosity ${\mu}_t$). Thus, one can write: $$- \overline{\rho {u_j}{u_i}} = {\mu}_t \frac{\partial U_i}{\partial x_j}$$

The approximation didn't solve the closure problem of turbulence, but it was used to come up with turbulent models that can model that eddy viscosity. By definition it's not a complete model, so there is no direct option for this in Fluent.