Week 8: Literature review - RANS derivation and analysis

AIM

RANS derivation and analysis

OBJECTIVE

Apply reynold's decomposition to the NS equations and come up with the expression for reynold's stress

THEORY

Reynolds-averaged Navier–Stokes equations

The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds. The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier–Stokes equations.

Derivation:

Reynold's decomposition

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity, `u`) into the mean (time-averaged) component (`baru`) and the fluctuating component (`u'`). Because the mean operator is a Reynolds operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero (`baru'`=0). Thus,

`u(x,y,z,t)`= `baru(x,y,z)+u'(x,y,z,t)`, where x = (x,y,z) is the position vector.

Governing equations for the boundary layer

Continuity equation

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

X-momentum equation

`rho((delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely))=-(delrho)/(delx)+(mudel^2u)/(dely^2)`

Energy equation

`rhoc((delT)/(delt)+(udelT)/(delx)+(vdelT)/(dely))=k((del^2T)/(delx^2))+mu((delu)/(dely))^2`

Substituting Reynold's decomposition in continuity equation

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

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

Applying time averaging, we get

`1/tint[(del(baru+u'))/(delx)dt+(del(barv+v'))/(dely)dt]=0`

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

Time integral of fluctuating component is equal to 0

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

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

`(delbaru)/(delx)+(delbarv)/(dely)=0`

This is the RANS continuity equation.

Substituting Reynold's decomposition in momentum equation

`rho((delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely))=-(delrho)/(delx)+(mudel^2u)/(dely^2)`

Dividing by `rho` on both sides

`(delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely)= -(delrho)/(rhodelx)+(mu)/(rho)(del^2u)/(dely^2)`

`(delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely)= -(delrho)/(rhodelx)+(nudel^2u)/(dely^2)`

`(delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely)+u((delu)/(delx)+(delv)/(dely))= -(delrho)/(rhodelx)+(nudel^2u)/(dely^2)`

`(delu)/(delt)+(2udelu)/(delx)+(vdelu)/(dely)+u(delv)/(dely)= -(delrho)/(rhodelx)+(nudel^2u)/(dely^2)`

`(delu)/(delt)+(2udelu)/(delx)+(del(uv))/(dely)= -(delrho)/(rhodelx)+(nudel^2u)/(dely^2)`

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

Applying time integration and Reynolds decomposition

`1/t_(Int)int_0^(t_(Int))[(del(baru+u'))/(delt)delt+(del(baru+u')^2)/(delx)delt+(del(baru+u')(barv+v'))/(dely)dt]=1/t_(Int)int_0^(t_(Int))[-1/rho(del(barrho+rho'))/(delx)delt+nu(del^2((baru+u'))/(dely^2)dt)]`

`1/t_(Int)int_0^(t_(Int))[(delbaru)/(delt)dt+(delu')/(delt)dt+(delbaru^2)/(delx)dt+(delu'^2)/(delx)dt+(del2(baru)u')/(delx)dt+(delbarubarv)/(dely)dt+(del(baru)v')/(dely)dt+(del(u')barv)/(dely)dt+(del(u')v')/(dely)dt]=1/t_(Int)int_0^(t_(Int))[-1/rho(delbarrho)/(delx)dt-1/rho(delrho')/(delx)dx+(nudel^2baru)/(dely^2)dt+(nudel^2u')/(dely^2)dt`

Time integral of fluctuating component is equal to 0

`(delbaru)/(delt)+(del(baru^2))/(delx)+(delbarubarv)/(dely)=-1/rho(delbarrho)/(delx)+(nudel^2baru)/(dely^2)-1/t_(Int)int_0^(t_(Int))(delu'^2)/(delx)dt-1/t_(Int)int_0^(t_(Int))(del(u')v')/(dely)dt`

On simplifying and ignoring negligible terms, we get

`ubrace((delbaru)/(delt)+(del(baru^2))/(delx)+(delbarubarv)/(dely))_("inertial terms")=ubrace(-1/rho(delbarrho)/(delx))_("pressure term")+1/rho(del)/(dely)[ubrace(mu(delbaru)/(dely))_("molecular viscosity")+ubrace(rho/t_(Int)int_0^(t_(Int))(u'v')dt)_("turbulent viscosity")]`

This turbulent viscosity term is the Reynold's stress term.

Reynold's stress

Turbulent flows are highly chaotic and unstable. The high fluctuation in the turbulent flows creates highly varying velocities. These velocities create additional stresses called Reynolds stress. In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.

Turbulent viscosity

Turbulent viscosity is also known as eddy viscosity. when flow is laminar, diffusion mechanism within the flow is only performed by molecular motions. Molecular viscosity is a fluid property and it depends on the type of fluid. It is the same as dynamic viscosity. When the flow becomes turbulent, diffusion mechanism is not only performed by molecular motions, but also it is going to be highly intensified by eddies motions. therefore when modeling the turbulence using eddy viscosity models, we need to intensify diffusion coefficient by adding turbulent viscosity to the molecular one.