Week 8: Literature review - RANS derivation and analysis

                                                                LITERATURE REVIEW – REYNOLDS AVERAGED NAVIER STOKES DERIVATION AND ANALYSIS

                                                                                                                       (WEEK-8 CHALLENGE)

1. AIM

Our aim is to derive Reynolds Averaged Navier Stokes (RANS) equation by applying Reynolds decomposition to the incompressible Navier Stokes equation and analyze the Reynolds stress term.  

2. THEORY/EQUATIONS/FORMULAE USED

Incompressible Navier Stokes Equation

 `grad.vec u = 0`                                                        ...............................................Continuity Equation

  `vec u_t + (vec u.grad) vec u = - grad p + 1/(Re) grad^2 vec u`...........................................Momentum Equation 

We are going to apply Reynolds Averaging process to the Navier stokes equation. Lets take the flow field variable u and decompose it into average and fluctuating component. 

Reynolds Decomposition

                                                                                     `U(x,t) = bar U(x) + U prime (x,t)`    

                                                               where `bar U(x)` is average component and `U prime (x,t)` is fluctuating component`

                                                               Also `vec U(x,t) = [[u],[v],[w]] = [[bar u + u prime],[bar v + v prime],[bar w + w prime]]` 

we are doing this to model the partial differential equation. 

                                                                          `bar U(x) = lim_(T->oo) int_0^T U(x,t)dt`     

Important Laws used in Reynolds averaging process

1. `bar (Uprime) = 0`

2. `bar (U+V) = bar U + bar V`

3. `bar(barU) = bar U`

4. `bar (bar a.b) = bar a.bar b`

5. `bar (a+b) != bar a + bar b`

6. `(bar ((dela)/(dels))) = (del bar a)/(del s)`

7. `bar (U prime^2) != 0`

Continuity Equation 

`grad.vecU = 0 => (del barU)/(del x) + (del barV)/(del y) + (del barW)/(del z) +(del bar(uprime))/(del x) + (del bar(vprime))/(del y) + (del bar(wprime))/(del z) = 0`

  using law 1 we can say that `(del bar(uprime))/(del x) + (del bar(vprime))/(del y) + (del bar(wprime))/(del z) = 0`                      So,  `(del barU)/(del x) + (del barV)/(del y) + (del barW)/(del z) = 0`

                                                                               We can say that `grad.bar(barU) = 0` , `grad.bar(Uprime) = 0`

Momentum Equation

`U_t + (u(del)/(delx)+v(del)/(dely)+w(del)/(delz))u = -(delp)/(delx)+1/(Re)grad^2u`

`bar (U_t) + uprime_t + barUbar(U_x) +barUuprime_x+uprimebar(U_x)+uprimeuprime_x+barVbar(U_y) +barVuprime_y+vprimebar(U_y)+vprimeuprime_y+barWbar(U_z) +barWuprime_z+wprimebar(U_z)+wprimeuprime_z = -barP_x- Pprime_x+1/(Re)grad^2barU+1/(Re)grad^2Uprime`

`barUbar(U_x) +bar(uprimeuprime_x)+barVbar(U_y)+bar(vprimeuprime_y)+barWbar(U_z)+bar(wprimeuprime_z) = -barP_x+1/(Re)grad^2barU`

Above equation shows that after time averaging there are no fluctuating terms, all are average quantities.

Using Chain Rule, `bar(uprimeuprime_x) = (delbar(uprimeuprime))/(delx) - bar(uprime(deluprime)/(delx)), bar(vprimeuprime_y) = (delbar(vprimeuprime))/(dely) - bar(uprime(delvprime)/(dely)), bar(wprimeuprime_z) =(delbar(wprimeuprime))/(delz) - bar(uprime(delwprime)/(delz))`

                                                                                                But  `uprime grad.vecu = 0`

                                       So,  `barUbar(U_x)+barVbar(U_y)+barWbar(U_z)+(delbar(uprimeuprime))/(delx)+(delbar(vprimeuprime))/(dely)+(delbar(wprimeuprime))/(delz)= -barP_x+1/(Re)grad^2barU`

                                                   Reynolds Stress Term = `(delbar(uprimeuprime))/(delx)+(delbar(vprimeuprime))/(dely)+(delbar(wprimeuprime))/(delz)`

Reynolds Stress Term

The whole strategy of turbulence modelling is to approximate the fluctuating components in terms of mean flow then there is a chance of solving mean flow qualities (bulk properties) of the problem which is much needed from the engineering simulation perspective. 

In general the Navier Stokes equation is solved for fluid physics but we dont really solve for fluctuating components. So we cannot solve the Navier Stokes for flow quantities.

So the reynolds stress term needs to be modelled. Solving N-S equation by modelling the Reynolds Stress term is called as Turbulence Closure problem. 

Eddy Viscosity Models are class of turbulence model used to calculate Reynolds Stress. Shear stress from the mean flow is the viscous stress. Molecular Viscosity dominates in viscous sub layer. 

                                                                                                     `tau = mu (delU)/(dely)`

Also we assume that the flow of particles follows the brownian motion. So for example in the flow over flat plate the momentum is transferred downwards which is opposite to the velocity gradient. 

                                                                            So the reynolds stress is proportional to `(delU)/(dely)`

Molecular Viscosity

Viscosity is often referred to as the thickness of a fluid. You can think of water (low viscosity) and honey (high viscosity). However, this definition can be confusing when we are looking at fluids with different densities.

At a molecular level, viscosity is a result the interaction between the different molecules in a fluid. This can be also understood as friction between the molecules in the fluid. Just like in the case of friction between moving solids, viscosity will determine the energy required to make a fluid flow.  It depend on the fluid properties and it is linear so its easy to measure.

Turbulent Viscosity

                  According to Boussinesq Approximation,                       `bar(uprimevprime) = mu_t(delU)/(dely)`

Shear stress from the turbulence/eddies is the Reynolds stress. Turbulent Viscosity stress dominates in log law region. 

                                                                                                        `tau = - bar (uprimevprime)`

The proportionality constant, mu_t is called the turbulent viscosity. mu_t is artificial and controls the strength of diffusion. Further modelling is needed to calculate mu_t. So now the whole calculation boils down to turbulent viscosity to calculate the flow quantities. It depends on the fluid flow.