Week 8: Literature review - RANS derivation and analysis

the Navier-Stokes equations are equations, that describe the three-dimensional motion of viscous fluid substances.The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation.

An averaging operation can be applied to the Navier-Stokes equations to obtain the mean equations of fluid flows called Reynolds Averaged Navier-Stokes (RANS) equations. These are very similar to the original equations but contains some additional terms in the momentum equations called Reynolds stress terms that are unknown and need to be modelled.in this study we will derive RANS equation, the procedure as follows -

Continuity Equation -

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

Decomposing into instantaneous and mean velocity component

`u = baru+u'`

`baru`=Mean component

`u'`=fluctuating component

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

Integrating and time averaging -

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

`t_i`=integration time

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

Time average of instantaneous component will be zero

`(1/t_i)*int(delu')/(delx)*dt =0`

`(1/t_i)*int(delv')/(dely)*dt =0`

Final equation -

since u and v are independent of time so, we can take these term out from integral term -

`(delbaru)/(delx)(1/t_i)intdt+(delbarv)/(dely)(1/t_i)intdt=0`

applying limit`0->t_i`

`(delbaru)/(delx)*[(t_i-0)/t_i]+(delbarv)/(dely)*[(t_i-0)/t_i]=0`

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

So, from the above written equation we can say continuity equation satisfies the time average Reynolds decomposition

Momentum Equation -

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

Kinematic viscocity(`nu`) =`mu/rho`

and adding a term `u((delu)/(delx)+(delv)/(dely))` to both the sides, this term is euqal to zero, so in LHS we will write this term and in RHS we will right 0

`(delu)/(delt)+u*(delu)/(delx)+v*(delu)/(dely)+u((delu)/(delx)+(delv)/(dely))=-(1/rho)*(delp)/(delx)+nu*((del^2u)/(dely^2))+0`

Rearranging -

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

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

after replacing term 

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

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

Decomposing and time averaging -

`(1/t_i)int((del(baru+u'))/(delt)+(del(baru+u')^2)/(delx)+del/(dely)*(baru+u')(barv+v'))dt=(1/t_i)*int(-(1/rho)*(del(barp+p'))/(delx)+nu*((del^2(baru+u'))/(dely^2)))dt`

Now, expand and make fluctuating term zero(dashed terms) except squared terms, remaning terms are -

`(1/t_i)*int(delbaru)/(delt)*dt+(1/t_i)*int(delbaru^2)/(delx)*dt+(1/t_i)*int(del(u')^2)/(delx)*dt+(1/t_i)*int(delu'v')/(dely)*dt+(1/t_i)*int(delbarubarv)/(dely)*dt=-(1/t_i)*int(1/rho*(del(barp))/(delx))*dt+(nu/t_i)*int(del^2baru)/(dely^2)dt`

time averaging of average terms will result into same term, so -

`(delbaru)/(delt)+(delbaru^2)/(delx)+(1/t_i)*int(del(u')^2)/(delx)*dt+(1/t_i)*int(delu'v')/(dely)*dt+(delbarubarv)/(dely)=-(1/rho*(delbarp)/(delx))+nu*(del^2baru)/(dely^2)`

rewriting -

`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely)=-(1/rho*(delbarp)/(delx))+nu*(del^2baru)/(dely^2)-(1/t_i)*int(del(u')^2)/(delx)*dt-(1/t_i)*int(delu'v')/(delt)*dt`

in boundary layer

`(1/t_i)*int(del(u')^2)/(delx)*dt =0`

so equation becomes -

`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely)=-(1/rho*(delbarp)/(delx))+nu*(del^2baru)/(dely^2)-(1/t_i)*int(delu'v')/(delt)*dt`

rewriting -

`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely)=-(1/rho*(delbarp)/(delx))+1/rho*del/(dely)[mu*(delbaru)/(dely)+rho/t_i*int(u'v')dt]`

This is RANS equation.

here -

term `u*(delbaru)/(dely`  is called as molecular viscosity and term  `rho/t_i*int(u'v')dt` is called as turbulent viscosity.

if we take derivative of molecular viscosity, we will get viscous stress and if we take derivative of tubulent viscosity, we will get Reynolds stress.

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. Reynolds stress is produced when fluid particles decelerate or accelerate while changing direction.

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. In a turbulent flow, the mechanism of transverse momentum transport is dominated by bulk mixing at scales mostly well above the mean free path. So, the effective viscosity in a region such as a turbulent boundary layer is much higher than in laminar flow and this is turbulent viscosity. So molecular viscosity is laymen term but turbulent viscosity arises in turbulent region only.