Literature review RANS derivation and analysis

Literature review RANS derivation and analysis:

1.1 INTRODUCTION:

The Reynolds avergaed Naviers stokes equation. They are used to create turbulence models or studying engineering simulations. In Research and development they are used for flow based analysis. We solve equations for the Ensemble Averaged Velocity fields i.e Time averaged velocity fields and magnitude of turbulent fluctations.

Derivation of RANS equation:

The basic tool of derivation for the RANS equation can be done from the Naviers stokes equation. This is called the Reynolds decomposition. Reynolds decomposition refers to seperation of flow variable (like velocity u) into the mean (time-averaged) component (`bar u`) and the fluctiating component (`u^\'`) . Because the mean operator is the Reynolds operator, It has a set of properties. One of these properties is that the mean of fluctuating quantity is equal to zero ( `bar u = 0`).

Hence `u(x,t) = baru(x,t) + u^\'(x,t) ,`where `x = (x,y,z)`is the position vector . Some authors prefer using U instead f `bar u`for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term `u^\'`is represented instead of u. This is possible beacuse of two terms do ot simultaneously in the same equation. To avoid confusion, the notation u, `bar u`and `u^\'`will be used to represent the instantaneous and fluctuating term respectively.

The properties of the Reynolds operator are re useful in the dervation f the RANS equation. Using these properties he Naviers stokes equation of motion expressed in tensor notation re ( for an incompressible Newtonian fluid).

`(delu_i)/(delx_i) =0`

`(delu_i)/(delx_i) + u_j(delu_i)/(delx_i) = f_i - 1/ rho (delrho)/(delx_i) + nu(del^2u_i)/(delx_idelx_j)`

where `f_i` is the vector representing external forces.

Next, each instantaneous quantity an be split into time averaged nd fluctuating components and the resulting equation time averaged to yeild:

`(delu_i)/(delx_i) =0``(delbaru)/(delt) + bar u_j( delbaru_i)/(delu_j) + bar (u_j^\'(delu_j^\')/(delx_j))= f_i-1/rho (delbarrho)/(delx_i) + nu(del^2baru_i)/(delx_idelx_j)`

The momentum equation can also be written as

`(delbaru)/(delt) + bar u_j( delbaru_i)/(delu_j) =f_i-1/rho (delbarrho)/(delx_i) + nu(del^2baru_i)/(delx_idelx_j) +(delu_i^\'delu_j^\')/(delx_j)`

On further manipulation this yeilds,

`rho(delbaru)/(delt) + rhobar u_j( delbaru_i)/(delu_j) =rhof_i+ (del)/(delx_j) (-barpdelta_(ij) + 2mubarS_(ij)-rhobar(u_i^\'u_j^\'))`

where, `barS_(ij) = 1/2 ((delbaru_i)/(delx_j) +(delbaru_j)/(delx_i))` is the mean rate of stress tensor.

Finally since the integraton in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:

`rhobar u_j( delbaru_i)/(delu_j) =rhof_i+ (del)/(delx_j) (-barpdelta_(ij) + 2mubarS_(ij)-rhobar(u_i^\'u_j^\'))`

Example: