Turbulence flow exhibits randomness but it is possible to approach them via statistical methods. It fluctuates in time and space. It is possible to do spatial and time averaging. In RANS approach, we time average the turbulent quantities to extract the mean flow properties from the fluctuating ones.
Now, we have to find some way to put information back into the momentum equation for example strong mixing process that turbulence motion has on the mean flow.
That’s where Turbulence Modelling comes into the picture.
It is very important to highlight that, It is a very strong simplification because all of the turbulence information is lost by averaging.
Reynolds Average Navier Stokes Equation
These equations are known as Reynolds Average Navier Stokes Equation. We have an additional term in the right-most corner. Its unit is same as the unit of stress hence it is known as Reynolds Stress. We can see the expanded form of the Reynolds Stress term.
We can write Reynolds Stress term in a Matrix Form.
Diagonal Terms are known as Normal Stresses.
Off Diagonal Terms are known as Shear Stresses.
Originally we have 9 unknowns – 3 diagonal and 6 off-diagonal elements. But, The Reynolds stress tensor matric is symmetric. This means, Three terms that appear above the diagonal elements are the same as the three terms that appear below the diagonal element. Hence, The total number of unknowns is actually six i.e. 3 along the diagonal and 3 off-diagonal.
This is another way of writing the RANS equation as shown above. On LHS we have total acceleration written in averaged quantity. On RHS we have pressure gradient term, the laminar contribution of stress, and turbulence contribution of stress. The main task in Turbulence Modelling is to model these additional stress terms. Turbulence flow exhibits random fluctuation both in time and space. So it is difficult to predict those fluctuations. But, a Statistical way of dealing is possible.
Replacing the Reynolds stresses with an appropriate quantity that can be modeled mathematically is known as Turbulence Modelling.
II term Mean Viscous Stress Tensor – is due to laminar viscosity/dynamic viscosity. It is a mean viscous stress tensor because mean velocity quantities are used.
III term Reynolds Stress Tensor – is due to averaging procedure that we apply on the Convection Term.
We have 3 Momentum and 1 Pressure Equation. When the flow is laminar we have 4 variables x, y, z velocity, and pressure. Hence, the problem is closed. In the Turbulence flow problem, We have additional unknown stress terms in our momentum equation. But, the number of equations is still 4. Hence no. of variables is not equal to the number of variables hence Problem remains unclosed.
Now to close the problem either one has to get an additional equation or replace the unknown variables with suitable known variables. We can derive the additional equations to model our Reynolds Stresses but it will end up adding more unknowns. The alternative way is to replace the unknown terms with known variables. This process is known as Turbulence Modelling. One of the famous Modelling methods is Eddy Viscosity Method. It is based on the Boussinesq hypothesis where Reynolds Stress is related to mean velocity gradient.
In 1887, Boussinesq proposed that the Reynolds Stresses (τij) can be related to the mean velocity gradients by a turbulent (eddy) viscosity, µt. Here, Turbulence is characterized by isotropic eddy viscosity μt which enhance mixing between various constituent of the flow.
Based on the dimensional argument, Eddy Viscosity is proportional to the product of density, eddy velocity scale (Vt) and eddy length scale (lt) as shown above. Between Eddy length, time and velocity scale, If any two quantities are known then the third quantity could be computed. Here eddy viscosity μt is a flow property, not a fluid property.
Momentum exchange through turbulence eddies. The largest eddies are responsible for mixing. They have a time scale (T – turn-over time) and a length scale L. Now we need two additional equations to describe L and T so that we can compute eddy viscosity. Once the eddy viscosity is computed, we can put it back into the Boussinesq formula and calculate the unknown Reynolds Stress. In this way, we can close the RANS equation.
The most natural framework to compute the two independent scales required for the eddy-viscosity μt ~ L2∕T ~ k2∕ε ~ k∕ω. One turbulence scale is computed from the turbulent kinetic energy, k, which is provided from the solution of its transport equation. The second scale is typically estimated from the dissipation rate of turbulence, ε, or its turn-over frequency, ω.
From turbulent kinetic energy, we can get velocity scale. Dissipation rate will give a length scale or specific dissipation will give a time scale. Now using k and ε or ω, We can calculate the eddy viscosity.
Derivation of Turbulent Kinetic Energy Equation
At first, We will start with obtaining the equation of Reynolds Stresses.
To do that, We can take the difference between the Instantaneous Navier-Stokes equation and Reynolds Averaged Navier-Stokes Equation.
By taking the difference, We can obtain the Equation of Reynolds Stresses as shown below:-
In order to reduce the unknowns, We will follow one step of tensor algebra which is known as contraction. Contraction is nothing but taking the trace or the isotropic part of the Reynolds Stress Tensor Matrix. The isotropic Part is nothing but the Diagonal Element of the Reynolds Stress Tensor Matrix. If we make i = j, We can obtain the Isotropic Part.
Turbulence Kinetic Energy (k) is the mean kinetic energy per unit mass associated with eddies in a turbulent flow. Physically, the turbulence kinetic energy (k) is characterized by root-mean-square (RMS) of velocity fluctuations. Turbulence Kinetic Energy (k) only contains the normal/isotropic part of the Reynolds Stress Tensor. The shear component or anisotropic part of the Reynolds Stress (Which is very important in strong rotating flows) is not considered in the 2 Equation Model Turbulence Modelling Approach.
After adding all the equations of normal stresses we will get an Equation for Turbulent Kinetic Energy, k as shown below.
Interpretation of Terms in Turbulent Kinetic Energy Equation
Unsteady Term: How the TKE at a certain point changes with time.
Convection Term: How the TKE is convected from one point to another by the mean flow velocity.
Production Term: Most important one, has an interaction between turbulent fluctuation which is Reynolds Stress times the mean velocity gradient, So the turbulence extracts the energy from the mean flow. So if there is no mean flow gradient, then there would be no turbulence. Meaning there would be no production of turbulence.
Dissipation Term: Here we have molecular viscosity times velocity gradient square. This term takes out the energy. It dissipates energy into heat.
Turbulent Diffusion Term: It is one of the most complex terms because it has the triple product of fluctuating velocities and a product of velocity and pressure. But the advantage here is this term is under divergence, What that means is, if we integrate this term across the mixing layer, The integration of this term outside the mixing layer is zero as turbulence variables are zero outside the shear layer. So this term does not produce or destroy anything. It just distributes the energy slightly inside the layer. For example: If we have a mixing layer or jet flow, This term will take some energy from someplace and diffuse it to another place that’s why it is called a turbulent diffusion term.
Molecular Diffusion Term: This term is the classical gradient diffusion or laminar diffusion term which is the viscosity time gradient of turbulent kinetic energy.
Modelling of Terms in Turbulent Kinetic Energy Equation
This led to the famous 2 Equations Turbulence Model, K-Epsilon or K-Omega Turbulence Model.
Reynolds averaging is one of the approaches used to eliminate the turbulence scales. The application of this approach leads to the Reynolds Averaged Navier-Stokes (RANS) equations.
The Reynolds stress terms in the RANS equation require modeling in order to obtain a closed system of equations. The Boussinesq hypothesis is one of the key elements of turbulence modeling. The eddy-viscosity can be computed if two independent scales are available.
This leads naturally to two-equation turbulence models. Turbulence Kinetic Energy (k) is the mean kinetic energy per unit mass associated with eddies in a turbulent flow.
Turbulence Kinetic Energy (k) only contains the normal/isotropic part of the Reynolds Stress Tensor.
The shear component or anisotropic part of the Reynolds Stress (Which is very important in strong rotating flows) is not considered in the 2 Equation Model Turbulence Modelling Approach.