Literature review RANS derivation and analysis

Reyonlds-Averaged Navier-Strokes (RANS) equations are time-averaged equations of motion used to describe turbulent fluid flow.

To derive the equation, Reynolds decomposition must be used. Reynolds decomposition separates the flow variable into the mean and the fluctuating component as shown below.

`u = obrace(baru)^(\"mean\") + obrace(u\')^(\"fluctuating\")`

Using Reyondls operators and tensor notation, Navier-Stokes equations of motion are:

`(partialu_i)/(partialx_i) = 0`

`(partialu_i)/(partialt)+u_j(partialu_i)/(partialx_j) = vec(f_i) - 1/rho(partialrho)/(partialx_i)+nu(partial^2u_i)/(partialx_jpartialx_j)`

where `vec(f_i)`is a external forces.

Applying Reyonlds decomposition yield:

`(partialbaru_i)/(partialx_i)=0`

`(partialbaru_i)/(partialt)+baru_j(partialbaru_i)/(partialx_j)+ bar(u\'_j(partialu\'_i)/(partialx_j)) = bar(vec(f_i)) - 1/rho(partialbarrho)/(partialx_i)+nu(partial^2baru_i)/(partialx_jpartialx_j)`

Applying Reyonlds decomposition to momentum yield:

`(partialbaru_i)/(partialt) + baru_j(partialbaru_i)/(partialx_j) = bar(vecf_i)-1/rho(partialbarrho)/(partialx_i)+nu(partial^2baru_i)/(partialx_jpartialx_j)-partialbar(u\'_iu\'_j)/(partialx_j)`

Simplifying the equation, and integration the equation:

`rhobaru_j(partialbaru_i)/(partialx_j) = rhobar(vecf_i)+partial/(partialx_j)[-barpDelta_(ij)+2mubarPhi_(ij)-obrace(rhobar(u\'_iu\'_j))^(\"Reyonld Stress\") ]`

Reynolds Stress
Reynolds Stress is a component of the total stress tensor on an object obtained from time-averaging the Navier-Stokes equations to define the momentum of turbulent fluid flow.

Viscosity
In turbulent flow, the diffusion mechanism is not only performed by molecular motions but also by eddies motion. To account this, when modeling, turbulence viscosity is used to intensify diffusion coefficient for momentum and heat transfer. This is different than molecular viscosity, which is same as viscosity.