RANS Derivation and Analysis

The governing equations for the boundary layers are the Navier Stokes equation shown below:

1. `(delu) / (delx) + (delv)/(dely) = 0` - Continuity Equation

2. `rho((delu)/(delt) + u(delu)/(delx) + v(del u)/(dely))= -(drho)/(dx) + (mudel^2u)/(dely^2)` - Momentum Equation

3. `rhoc((delT)/(delt) + u(delT)/(delx) + v(delT)/(dely))= K(del^2T)/(delx^2)+mu((delu)/(dely))^2` - Energy Equation

Applying Reynold\'s Decomposition to the Navier Stokes equations to derive Reynold\'s Stress:

u(x,y,z,t) = `baru(x,y,z)+u\'(x,y,z,t)`

v(x,y,z,t) = `barv(x,y,z)+v\'(x,y,z,t)`

w(x,y,z,t) = `barw(x,y,z) + w\'(x,y,z,t)`

where,

u = instantaneous velocity

`baru(x,y,z)`= function of space

u\'(x,y,z,t) = function of time

u\' = fluctuating component

on integrating,

`baru(x,y)= 1/t int_0^tu(x,y,t)dt`

`barv(x,y)=1/tint_o^tv(x,y,z)dt`

Note: Integrating a fluctuating component = 0.

`1/tint_0^tu\'(x,y,z,t)dt=0`

on simplifying,

u(x,y,z,t) = `baru(x,y)+u\'(x,y,z,t)`

`1/tint_0^tu(x,y,z,t)dt=1/tint_0^tbaru(x,y)+1/tint_0^tu\'(x,y,z,t)`

`darr`

`cancel(baru(x,y,z)) = cancel(baru(x,y,z))+1/tint_0^tu\'(x,y,z,t)`

`1/tint_0^tu\'(x,y,z,t)dt=0`

Applying the Decomposition :

`(delu)/(delx)+(delv)/(dely)=0`

`del/(delx)(baru+u\')+del/(dely)(barv+v\')=0`

`1/tintdel/(delx)(baru+u\')dt+del/(dely)(barv+v\')dt=0`

`1/tint(delu)/(delx)dt+cancel(1/tint(delu\')/(delx)dt) +1/tint(delbarv)/(dely)+cancel(1/tint(delv\')/(dely))=0`

`(delu)/(delx)1/tintdt+(delbarv)/(dely)1/tintdt=0`

`(delbaru)/(delx)1/tint_0^tdt+(delbarv)/(dely)1/tint_0^tdt=0`

`(delbaru)/(delx)cancel((t-0)/t)+(delbarv)/(dely)cancel((t-0)/t)`

`(delbaru)/(delx)+(delbarv)/(dely)=0`- Continuity Equation , thus proved.

Momentum Equation:

`rho(delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)=-(drho)/(dx)+mu(del^2u)/(dely^2)`

`(delu)/(delt)+mu(delu)/(delx)+v(delu)/(dely)=1/rho-(delrho)/(delx)+mu/rho(del^2u)/(dely^2)`

Replace `mu/rho`with `delta`(kinematic viscosity)

`(delu)/(delt)+u(delu)/(delx)v(delu)/(dely)=-(delrho)/(rhodelx)+delta(del^2u)/(dely^2)`

Multiply `u(delu)/(delx)`with`(delu)/(delx)`

where,

`u((delu)/(delx)+(delv)/(dely))=0`

`(delu)/(delt)+delu(delu)/(delx)+v(delu)/(dely)+v(delv)/(dely)=-1/rho(delrho)/(delx)+delta(del^2u)/(dely)^2`

By Product Rule:

`del/(dely)(uv)=u(delv)/(dely)+v(delu)/(dely)`

`(delu)/(delt)+delu(delu)/(delx)+del/(dely)(uv)=-1/rho(delrho)/(delx)+nu(del^2u)/(dely^2)`

`(delu)/(delt)+(delu^2)/(delx)+del/(dely)(uv)=-1/rho(delrho)/(delx)+nu(del^2u)/(dely^2)`

`1/tint_o^tdel/(delt)(baru+u\')dt+del/(delx)(baru+u\')^2dt+del/(dely)[(baru+u\')(barv+v\')]dt=1/tint_0^t-1/rho[(del((barrho+rho\'))/(delx))]dt+nudel^2/(dely^2)(baru+u\')dt`

`1/tint_0^t(delbaru)/(delt)dt+cancel((delu\')/(delt)dt)+(delbaru^2)/(delx)dt+(delu\'^2)/(delx)dt+cancel((del/(delx)(delbaru u\')dt))+del/(dely)(barubarv)dt)+cancel(del/(dely)(baruv\')dt)+cancel(d/(dely) (u\'barv)dt))+del/(dely)(u\'v\')dt`

`1/tint_0^t-1/rho(delbarrho)/(delx)dt-1/rhocancel((delrho\')/(delx))dx+nu(del^2baru)/(dely^2)dt+nucancel((del^2u)/(dely^2))dt`

`(delbaru)/(delt)+(del(baru^2))/(delx)+(del(barubarv))/(dely)=-1/rho(delbarrho)/(delx)+nu(del^2baru)/(dely^2)-1/tint_0^t(del(u\')^2)/(delx)dt-1/tint_0^t(del(u\'v\'))/(dely)dt`

`(delbaru)/(delt)+(del(ubaru)^2)/(delx)+(del(barubarv))/(dely)=-1/rho(delrho)/(delx)+nu(del^2baru)/(dely^2)-1/tint_0^tdel/(dely)(u\'v\')dt`

=`ubrace((-1/rho (delbarrho)/(delx)))_(\"Pressure term\") 1/rhodel/(dely)[mu(delbaru)/(dely)+rho/tint_0^t(u\'v\')dt]`

where,

`mu(delbaru)/(dely) = Viscous Stress`

`rho/tint_0^t(u\'v\')dt`= Reynold\'s Stress

Reynold\'s Stress:

The velocity field of a flow can be split into a mean part and a fluctuating part using Reynolds Decomposition. We write:

with u(x,t) being the flow velocity vector having components in the coordinate direction (with denoting the components of the coordinate vector x). The mean velocities are determined by either time averaging, spatial averaging or ensemble averaging, depending on the flow under study. Further denotes the fluctuating (turbulence) part of the velocity.

Therefore, Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier Stokes Equation to account for turbulent fluctuations in fluid momentum.

Turbulent Viscosity:

The turbulent transfer of momentum by eddies giving rise to an internal fluid friction, in a manner analogous to the action of molecular viscosity in laminar flow, but taking place on a much larger scale, is known as turbulent viscosity.

Molecular viscosity is the transport of mass motion momentum solely by the random motions of individual molecules not moving together in coherent groups. Molecular viscosity is analogous in laminar flow to eddy viscosity in turbulent flow.

The primary difference between turbulent and molecular viscosity is that in turbulent viscosity, molecular action takes place in a much larger scale. Also, the friction produced by the former is higher due to its turbulent nature.

Result:

The Reynold\'s Stress can be derived by applying the Reynold\'s Decomposition formula to the Navier Stokes Equations that govern the boundary layers. The molecular and turbulent viscosities can also be obtained from this process.