Week 8: Literature review - RANS derivation and analysis

Reynolds-averaged Navier Stokes Equation

The Reynolds-averaged Navier–Stokes equations or RANS equations are a 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.

Reynolds Decomposition

`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)`

Contiunity Equation

The Contiunity Equation is as follows

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

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

`1/(ti nt) int del/(delx)(baru+u')dt+del/(dely)(barv+v')dt=0`

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

we can assume that `1/(ti nt)int(delu')/(delx)dt & 1/(ti nt)int(delv')/(dely)dt =0`

Therefore we have

`1/(ti nt)int(delbaru)/(delx)dt+1/(ti nt)int(delbarv)/(dely)dt=0`

Applying the time integral

`1/(ti nt)int_0^(ti nt) (delbaru)/(delx)dt+1/(ti nt)int_0^(ti nt)(delbarv)/(dely)dt=0`

Hence we get the final equation

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

Momentum Equation

The momentum equation is given as follows

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

Taking rho to the otherside we get

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

we know that mu/rho is gamma

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

Multiplying the Xcomponent of velocity to LHS of the equation

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

On simplifing

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

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

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

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

Applying Time Intergration

`1/(ti nt) int_o^(ti nt)del/(delt)(baru+u')dt+del/(delx)(baru+u')^2dt+del/(dely)(baru+u')(barv+v')dt= 1/(ti nt) int_o^(ti nt)-1/rhodel/(delx)(barp+p')dt+gamma del^2/(dely^2)(baru+u')dt`

`1/(ti nt) int_o^(ti nt)(delbaru)/(delt)dt+(delu')/(delt)dt+(delbaru^2)/(delx)dt+(delu' ^2)/(delx)dt+(del/(delx)(2baru u'))dt+(del/(dely)(baru barv))dt+(del/(dely)(baru v'))dt+(del/(dely)(u' barv))dt+(del/(dely)(u' v'))dt= 1/(ti nt) int_o^(ti nt)(-1/rho (delp)/(delx)dt-1/rho (delp')/(delx)dt+gamma (del^2 baru)/(dely^2)dt+gamma(del^2 u')/(dely^2)dt`

from the above equation some terms are zero

`(delu')/(delt)dt & (del/(delx)(2baru u'))dt & (del/(dely)(baru v'))dt & (del/(dely)(u' barv))dt & -1/rho (delp')/(delx)dt & gamma(del^2 u')/(dely^2)dt`

Therfore we get the following equation

`(delbaru)/(delt)+(delbaru^2)/(delx)+del/(dely)(baru barv)= -1/rho (delp')/(delx)+gamma(del^2 u')/(dely^2) - 1/(ti nt) int_o^(ti nt)(del/(delx))baru^2 dt-1/(ti nt) int_o^(ti nt)(del/(dely))barubarv dt`

On simplifying we

`(delbaru)/(delt)+(delbaru^2)/(delx)+del/(dely)(baru barv)= -1/rho (delp')/(delx)+gamma(del^2 u')/(dely^2) -1/(ti nt) int_o^(ti nt)(del/(dely))barubarv dt`

On simplifying the RHS we get

`(delbaru)/(delt)+(delbaru^2)/(delx)+del/(dely)(baru barv)= -1/rho (delp')/(delx)1/rho del/(dely)(mu(del u')/(dely) + rho/(ti nt) int_o^(ti nt) barubarv dt)`

Reynolds Stress

It is the total stress tensor in fluid obtained from averaging the navier stokes equation in the account for turbulent fluctuation in a momentum of the fluid.

Turbulent Viscosity

In a turbulent fluid, a linear interface between two different fluids breaks apart to form small-scale structures, also known as eddies and this viscosity is called turbulent viscosity.A coefficient that relates the average shear stress within a turbulent flow of water or air to the vertical gradient of velocity.

Molecular Viscosity

The Coefficient of Molecular Viscosity is the same value as dynamic 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.