Reynolds Stress
Two different types of viscous fluid flows are encountered in general: laminar and turbulent. The solution of the navier-stokes equation does not raise any fundamental difficulties in the case of laminar flow. However, the simulation of turbulent flows still constitutes a significant challenge and it can be found out by reynolds euation governed by the navier-stokes equation.Reynolds stress is a part of total stress in a fluid which gives out results for turbulant fluctuations.
The average shear stress and rate of velocity gradient in the turbulent flow is called turbulent viscosity
Molecular viscosity is the tangential force per unit area required to move one horizontal plane with respect to an other plane.It is also known as Dynamic viscosity or Absolute viscosity.
Reynolds equation is governed by navier-stokes equation
The following are the boundary layers eqaution:
`(del u )/ (del x)+(del v)/(del y)= 0` This is the continuity equation......(1`
The continuity equation says that if charge is moving out of a differential volume then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative.
`rho[(del v)/(del t)+u(delv)/(del x)+v( delu)/(del y)]=-(delp)/(del x) +mu(del^2u)/(del y^2)`This is the momentum equation.........(2`
This equation is derived from the newton\'s second law of eqation which states that The momentum equation is a statement of Newton\'s Second Law and relates the sum of the forces acting on an element of fluid to its acceleration or rate of change of momentum.
`rhoc((del T)/(del t)+ u(delT)/(delx)+v(delT)/(dely)=K(del^2T)/(del x^2)+mu((del u)/(dely))^2`This is the energy eqaution...........(3)
The reynolds decomposition goes as follows
`u(x,y,z,t)=baru(x,y,z)+u\'(x,y,z,t)` ..........[4]
`v(x,y,z,t)=barv (x,y,z)+v\'(x,y,z,t)` ..........[5]
`w(x,y,z,t)=barw(x,y,z)+w\'(x,y,z,t)` .........[6]
`baru(x,y)=1/t int_0^t u(x,y,t) dt`....[7]
`barv(x,y)=1/t int_0^tv(x,y,z)dt`....[8]
we know that
`u(x,y,z,t)=baru(x,y)+u\'(x,y,z,t)`........[9]`
Time averaging the equation
`1/t int_0^t u(x,y,z,t) dt=1/t int_0^t baru(x,y)+1/t int_0^t u\'(x,y,z,t) dt`
we get
`baru(x,y,z)=baru(x,y,z)+1/t int_0^t u\'(x,y,z,t) dt`
cancelling out the like equations we get
`1/t int_0^t u\'(x,y,z,t) dt=0`......[10]
This shows that the when fluctuating componenets are intergrated over longer period of time they tend towards zero
Now applying the intergral rule to the continuity equation
`(delu)/(delx)+(delv)/(dely)=0`
applying decomposotion
`(del(baru+v\'))/(delx)+(del(barv+v\'))/(dely)=0`
intergrate w.r.t time and preform time averaging
`1/t int[(delbaru)/(delx)+v\']dt + [(del (barv+v\'))/(dely)]dt = 0`
`1/t int(delbaru)/(delx)dt + cancel[1/t int(delu\')/(delx)dt] + 1/t int[(delbarv)/(dely)dt + cancel[1/t int[(delv\')/(dely)dt] = 0`
the cancelation takes place, as we know the fluctuation tends to zero.
further simplying
`(delbaru)/(delx) 1/tintdt + (del barv)/(dely) 1/t (delbarv)/(dely) = 0`
Applying limits
`(delbaru)/(delx) 1/tint_0^tdt + (del barv)/(dely) 1/tint_o^t dt= 0`
`(delbaru)/(delx) cancel([t-0]/t) +(delbarv)/(dely) cancel([t-0]/t)`
`(delbaru)/(delx) +(delbarv)/(dely) = 0`...........[11]
equation 11 is the continuity equation interms of time average quantities.
Now taking the momentum equation
`rho[(delu)/(delt)+u(delu)/(delx) +v(delu)/(dely)] = -(dP)/(dx) + mu(del^2u)/(dely^2)`
`[(delu)/(delt)+u(delu)/(delx) +v(delu)/(dely)] = 1/rho-(dP)/(dx) + mu/rho(del^2u)/(dely^2)`
`[(delu)/(delt)+u(delu)/(delx) +v(delu)/(dely)] = -(dP)/(rhodx) + vartheta(del^2u)/(dely^2)`...........`vartheta = mu/rho`
On the left hand side adding a zero term from continuty equation [11]
`[(delu)/(delt)+u(delu)/(delx) +v(delu)/(dely)] +u((delu)/(delx)+(delv)/(dely))= -1/rho(dP)/(dx) + vartheta(del^2u)/(dely^2)` ..........[12]
`(delu)/(delt)+delu(delu)/(delx) +v(delu)/(dely)+u(delv)/(dely)= -1/rho(dP)/(dx) + vartheta(del^2u)/(dely^2)`............[13]
further simplifying using product rule
`(del(u,v))/(dely)=u((delv)/dely)+v((delu)/(dely))`.....[14]
substituting equation 14 in 13
`(delu)/(delt)+delu(delu)/(delx) +((del(u,v))/(dely)) = -1/rho(dP)/(dx) + vartheta(del^2u)/(dely^2)`....[15]
`(delu)/(delt)+(del(u^2))/(delx) +((del(u,v))/(dely)) = -1/rho(dP)/(dx) + vartheta(del^2u)/(dely^2)`....[16]
Now applying time intergration and reynolds decomposition
`1/tint_0^t(del(baru+u\'))/(delt)delt+(del((baru+u\')^2))/(delx)delt +((del(baru+u\'),(barv+v\'))/(dely)) =1/tint_0^t -1/rho(d(barP+P\'))/(dx) delt+ vartheta(del^2(baru+u\'))/(dely^2) delt`
expanding these terms we get
`1/tint_0^t(delbaru)/(delt)delt+cancel((delu\')/(delt)delt)+(del(baru)^2)/(delx)delt +(del(u\')^2)/(delx)delt +cancel((del delbaru u\')/(delx)dt)+(del(baru barv))/(dely)dt + cancel((del(baru v\'))/(dely)dt)+cancel((del(baru v\'))/(dely)dt)+(del(u\' v\'))/(dely)dt =1/tint_0^t -1/rho(d(barP))/(dx) delt -cancel(1/rho(d(P\'))/(dx) delt)+ vartheta(del^2(baru))/(dely^2) delt +cancel(vartheta(del^2(u\'))/(dely^2) delt)`
we get
`1/tint_0^t(delbaru)/(delt)delt+(del(baru)^2)/(delx)delt +(del(u\')^2)/(delx)delt +(del(baru barv))/(dely)dt +(del(u\' v\'))/(dely)dt =1/tint_0^t -1/rho(d(barP))/(dx) delt + vartheta(del^2(baru))/(dely^2) delt`
further simplyfing
`(delbaru)/(delt)delt+(del(baru)^2)/(delx)delt +(del(baru barv))/(dely)dt =-1/rho(d(barP))/(dx) delt + vartheta(del^2(baru))/(dely^2) delt-1/tint_0^t(del(u\')^2)/(delx)delt-1/tint_0^t(del(u\' v\'))/(dely)dt`
1/tint_0^t(del(u\')^2)/(delx)delt-->This term is considered to to be zero as change in velocity is less in x-axis when compared to change in velocity in y-axis this is only true inside the boundary layer.In order to solve this equation we have to do Order of magnitude analysis to determine its validity. So now neglecting this term we get
`(delbaru)/(delt)delt+(del(baru)^2)/(delx)delt +(del(baru barv))/(dely)dt =-1/rho(d(barP))/(dx) delt + vartheta(del^2(baru))/(dely^2) delt-1/tint_0^t(del(u\' v\'))/(dely)dt`
simplifying the eqation
`(delbaru)/(delt)+(del(baru)^2)/(delx) +(del(baru barv))/(dely) =-1/rho(d(barP))/(dx) +1/rho del/(dely)[mu(delbaru)/(dely)+rho/t int_0^t(u\'v\')dt]`.......[17]
(delbaru)/(delt)+(del(baru)^2)/(delx) +(del(baru barv))/(dely) .... These are the initial terms
1/rho(d(barP))/(dx) .................................................................This is the pressure gradient
mu(delbaru)/(dely)...................................................................Molecular viscosity/Viscous stress
rho/t int_0^t(u\'v\')dt.................................................................Momentum diffusivity due to turbulence/Reynolds stress