RANS

Let us consider the Navier Stokes equation for a 2D, incompressible flow. The continuity equation for this can be written as below.

`(delu)/(delx)+(delv)/(dely)=0rightarrowEqn(1)`
Applying Reynold's decomposition by substituting `u=(baru+u')and v=(barv+v')`, Eqn(1) becomes

`(delbaru)/(delx)+(delu')/(delx)+(delbarv)/(dely)+(delv')/(dely)=0rightarrowEqn(2)`

Integrating with respect to time 't' from 0 to t,

`1/tint_0^t(delbaru)/(delx)dt+cancel(1/tint_0^t(delu')/(delx)dt)+1/tint_0^t(delbarv)/(dely)dt+cancel(1/tint_0^t(delv')/(dely)dt)=0rightarrowEqn(3)`

Since time integral of fluctuating component is zero, `int_0^t(delu')/(delx)dt=0 and int_0^t(delv')/(dely)dt=0`

`thereforeEqn(3)Rightarrow1/t(delbaru)/(delx)int_0^tdt+1/t(delbarv)/(dely)int_0^tdt=0rightarrowEqn(4)`

`(delbaru)/(delx)(cancel((t-0)/t))+(delbarv)/(dely)(cancel((t-0)/t))=0rightarrowEqn(5)`

`bb((delbaru)/(delx)+(delbarv)/(dely)=0)rightarrowEqn(6)`

Eqn(6) is the Reynold's Averaged continuity equation.

The momentum equation in the x- direction can be written as follows.

`(delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)=-1/rho(delP)/(delx)+nu(del^2u)/(dely^2)rightarrowEqn(7)`

In order to simplify, let us add `u((delu)/(delx)+(delv)/(dely))` to the LHS. From Eqn(1) this term is zero

`(delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)+u underbrace(((delu)/(delx)+(delv)/(dely)))_(0)=-1/rho(delP)/(delx)+nu(del^2u)/(dely^2)rightarrowEqn(8)`

`(delu)/(delt)+2u(delu)/(delx)+v(delu)/(dely)+u(delv)/(dely)=-1/rho(delP)/(delx)+nu(del^2u)/(dely^2)rightarrowEqn(9)`

Consider `(del(uv))/(dely)=u(delv)/(dely)+v(delu)/(dely)
and (del(u^2))/(delx) = 2u(delu)/(delx)`, substituting these in Eqn(9)

`(delu)/(delt)+(del(u^2))/(delx)+(del(uv))/(dely)=-1/rho(delP)/(delx)+nu(del^2u)/(dely^2)rightarrowEqn(10)`

Applying reynold's decomposition and integrating with respect to time from 0 to t,

`1/tint_0^t(delbaru)/(delt)dt+1/tcancel(int_0^t(delu')/(delt)dt)+1/tint_0^t(del(baru)^2)/(delx)dt+1/tint_0^t(del(u')^2)/(delx)dt+1/tint_0^tcancel((del(2baru u'))/(delx)dt)+1/tint_0^t(del(barubarv))/(dely)dt+1/tcancel(int_0^t(del(baruv'))/(dely)dt)+1/tcancel(int_0^t(del(u'barv))/(dely)dt)+1/tint_0^t(del(u'v'))/(dely)dt=-1/rho(1/tint_0^t(delbarP)/(delx)+1/tcancel(int_0^t(delP')/(delx)))+nu(1/tint_0^t(del^2baru)/(dely^2)+1/tcancel(int_0^t(del^2u')/(dely^2)))rightarrowEqn(11)`

Rearranging the terms

`(delbaru)/(delt)+(del(baru)^2)/(delx)+(del(barubarv))/(dely)=-1/rho(delbarP)/(delx)+nu(del^2baru)/(dely^2)-1/tint_0^t(del(u')^2)/(delx)dt-1/tint_0^t(del(u'v'))/(dely)dtrightarrowEqn(12)`

It can be proven by order of magnitude analysis that `int_0^t(del(u')^2)/(delx)dt` is negligibly small, and can be equated to zero.

`bb(thereforeunderbrace((delbaru)/(delt)+(del(baru)^2)/(delx)+(del(barubarv))/(dely))_("Inertial Terms")=underbrace(-1/rho(delbarP)/(delx))_("Pressure Term")+underbrace(1/rho(del)/(dely)(underbrace(mu(delbaru)/(dely))_("Viscous Stress")-underbrace(rho/tint_0^t(u'v')dt)_("Reynold's Stress")))_("Diffusion Term"))rightarrowEqn(13)`

Eqn(13)is the Reynold's Averaged momentum equation in the X direction. Applying Reynold's decomposition for Y direction would result in similar equation.

Comparing Eqn(10) and Eqn(13), it can be observed that there are some extra terms which are dependent on the fluctuating components of velocity. The time integral `rho/tint_0^t(del(u'v'))/(dely)dt` is called as Reynold's Stress. This term gives the value of momentum diffusivity due to turbulence. Reynold's stress is the term which simplifies the Navier-Stokes equation and represents the For a 3D flow, there would be another term, `rho/tint_0^t(del(u'w'))/(delz)dt`. It can be observed that in the RANS momentum equation, the derivative of fluctuating components only in the perpedicaular directions .