RANS ​

• RANS is an abbreviation of Reynolds Averaged Navier-Stokes.
• It uses time-averaged values of Navier-Stokes equation.
• When we draw velocity profile, we draw smooth profile of the flow.
• However, in reality, there are so many fluctuations around that smooth profile.
• It's convenient to look at smooth profile to get the sense of physical meaning of the flow.
• This is exactly what we're doing in RANS.
• Ease out fluctuations and look at bulk properties like lift or drag.
  
  
  

It all starts from continuity equation & momentum equation.

This is Newtonian, incompressible fluid.

The concept is, we decompose velocity vector into time averaged velocity and fluctuation velocity. (it's just rest of time averaged velocity)
We plug it into N-S equation.
We time-average the N-S equation.
The purpose of time-averaging is to get rid of tricky fluctuation terms.
A lot of terms will go away, and we get an equation with 'Reynolds Stress' term.
This 'Reynolds Stress' term still has fluctuation term.
So here comes 'Turbulence Closure Models'
It's all about expressing these fluctuation terms into time-averaged terms so that it is easy to compute.

we have a velocity vector which can be decomposed into time averaged value and extra term which is fluctuation term.

• velocity & are all vectors. I just omitted the vector sign for convenience.

We plug it in to continuity & N-S equation, and we time average two equations. We get

from continuity equation.

Next we plug into N-S equation, and time average the whole equation. Some fluctuation terms will go away, but some will not go away. The result is,

And this is the RANS equation.

But this is not our final equaion cause out objective was to get rid of fluctuation terms.
But they are still there.

So here comes the 'Turbulence Closure Problem'
Actually, RANS is all about how to express fluctuation terms with averaged terms.
From now on, we will take a look at model, model, etc, which are part of 'Turbulence Closure Problem'

Details can be seen at

• S. L. Brunton (2021, April 2), Turbulence: Reynolds Averaged Navier-Stokes (Part 1, Mass Continuity Equation), DOI: https://doi.org/10.52843/cassyni.tcxvxy
• S. L. Brunton (2021, April 16), Turbulence: Reynolds Averaged Navier Stokes (RANS) Equations (Part 2, Momentum Equation), DOI: https://doi.org/10.52843/cassyni.1xkvn0
• Fluid Mechanics 101 (2021, Feb 24), [CFD] Eddy Viscosity Models for RANS and LES, https://www.youtube.com/watch?v=SVYXNICeNWA&list=PLnJ8lIgfDbkrNyps1_36tNRRQ7hLzPFhV

Eventhough the improvement of computational power, simulating fluid is still a heavy computation.
Instead of simulating all the scales in N-S equation, aka DNS, we're gonna make approximation to make simulation more efficient and make turbulence models.
This is Tubulence Closure Models.

There are several models like , Smagorinsky which are part of Eddy Viscosity Model, and cubic , Full sub-grid scale, etc.

First, we are going to look at old, but simple approach which is Eddy Viscosity Model which is proposed by Boussinesq in 1877.

Eddy Viscosity Model is proposed by Boussinesq in 1877, in analogy to Newton’s law of friction. In 3D, Eddy Viscosity Model is

where is eddy viscosity,
and & where
is a Turbulent Kinetic Energy (TKE)
is a delta function.
will be used in the famous model.
Deriving this equation can be seen at Fluid Mechanics 101, [CFD] Eddy Viscosity Models for RANS and LES

What this equation mean is that, normal & shear reynolds stress is proportianal to some velocity gradient by a factor of eddy viscosity .

Actually, this formula is really similar to relationship between viscosity and shear stress.

So it means, turbulent scales that generates reynolds stress acts like viscosity.

Also, when you look at velocity gradient, motion of particles transfer the momentum downwards which is the direction of velocity gradient.

Eddy Viscosity Model is all about modeling .
There is a model proposed 'mixing length hypothesis' proposed by Prandtl, but thanks to the computatoinal power, we do not use this model.

And here comes the famous model.
I will cover it at next section.

Reference & Resources

• S. L. Brunton (2021, April 23), Turbulence Closure Models: Reynolds Averaged Navier Stokes (RANS) & Large Eddy Simulations (LES), https://doi.org/10.52843/cassyni.cjkr7f
• Fluid Mechanics 101 (2021, Feb 24), [CFD] Eddy Viscosity Models for RANS and LES, https://www.youtube.com/watch?v=SVYXNICeNWA&list=PLnJ8lIgfDbkrNyps1_36tNRRQ7hLzPFhV
• Schlichting, H., & Gersten, K. (2017). Boundary-Layer theory. In Springer eBooks. https://doi.org/10.1007/978-3-662-52919-5

• model is one of eddy visosity models that uses &
• & is obtained from transport equation and transport equation.
• It uses damping function to compensate the viscosity effect near the wall.

From Eddy Viscosity Model, - model is defined as

where is density
is adjustable coefficient,
is a damping function,
is TKE (Turbulence Kinetic Energy),
is Turbulence Dissipation Rate.

And and is obtained by transport equation.
transport equation is,

where is velocity component in corresponding direction,
is Laminar Viscosity,
is Turbulence Viscosity,
is adjustable coefficient,
is Favre-averaged turbulent stresses
and is strain-rate tensor

transport equation is,

where
is adjustable coefficient,
are adjustable coefficient,
are damping functions,
and is explicit wall term.