Week 8: Literature review RANS derivation and analysis
Objective:
Apply reynold's decomposition to the NS equations and come up with the expression for reynold's stress.
- Explain your understanding of the terms reynold's stress
- What is turbulent viscosity? How is it different from molecular viscosity?
Answer:
Part 1: Deriving RANS equation
To derive Reynolds Averaged Navier Stokes equations, first Reynolds decomposition tool needs to be applied on the Navier Stokes equation. Reynolds decompostion is the processs of seperating mean (time averaged) & fluctuating component of the quantity. To decompose velocity, u; the two components, mean(`bar u`) & fluctuating (`u'`) will be seperated. The mean component is Reynolds operator.
Reynolds decomposition,
Using these properties, the motion eqaution of Navier Stokes equation is expressed in tensor notation as follows:
where, `f_i` is the vector represting external forces.
Now splitting above equation in two components, mean & flutuating:
The momentum eqaution can be written as :
On further manipulation of above equation:
where
Finally, since integration in time removes the time dependence of resultant terms, the time derivative must be eliminated, leaving:
The time evolution equation of Reynolds stress is given by:
In above equation, if `bar (u_i' u_j')` is traced, then turbulent kinetic energy can be determined.
The last term,
is turbulent dissipation rate.
All RANS models are based on the above equation.
Part 2: Explain Reynolds Stress term
In fluid dynamics, the Reynolds Stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-Stokes equations to account for turbulent fluctuations in fluid momentum.
Part 3: What is turbulent viscosity? How is it different from molecular viscosity?
Turbulent viscosity is the turbulent transfer of momentum by eddies giving rise to an internal fluid fiction, in a manner analogous to the action of molecular viscosity in laminar flow, but taking place in a much larger scale.