RANS derivation and analysis

Reynolds-averaged Navier–Stokes equations are time-averaged equations of motion for fluid flow used to describe turbulent flows. They are obtained by Reynolds decomposition whereby instantaneous quantities are split into time averaged terms and fluctuating terms. Reynolds stresses are nonlinear stresses imposed on fluid flow by turbulent fluctuations and are required to be solved by RANS equations with additional modelling.The expression for Reynolds stress is also derived from Navier Stokes equation, by applying Reynolds decomposition.

Reynolds decomposition:

Reynolds decomposition is the separation of an instantaneous quantity into its ensemble average value and fluctuations. If u is instantaneous value of velocity at any point inside a fluid in motion, the velocity can be separated as shown below;

`u(x,y,z,t) = bar u(x,y,z) + u'(x,y,z,t)`

where, `bar u` is the time average component of velocity which is a function of space alone and `u'` is the fluctuating component which is a function of time and space. If the above equation is time integrated, it can be found that time integral of fluctuating component is always zero.

`(1/t)int_0^tu(x,y,z,t) dt= (1/t)int_0^tbaru(x,y,z)dt + (1/t)int_0^tu'(x,y,z,t)dt`

Now, the time average of velocity u is equal to`baru` at LHS. At RHS, `baru` is still unchanged as an existing time averaged value is being integrated by time. Hence, the equation now becomes;

`(1/t)int_0^tu'(x,y,z,t)dt = 0`

Which means that if the fluctuating component is integrated for a period of time, it will be zero.

Continuity Equation:

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

Applying the Reynold's decomposition to Continuity equation;

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

Integrating the above equation with respect to time;

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

`1/tint(del(baru))/(delx)dt+1/tint(del(u'))/(delx)dt+1/tint(del(barv))/(delx)dt+1/tint(del(v'))/(delx)dt`

As the time average of fluctuating components are zero and time average of time averaged components is unchanged, equation becomes;

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

Thus, the time averaged velocity satisfies the continuity equation.

Momentum Equation:

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

If density, `rho` is shifted to the right hand side of equation and multiplying left hand side with `u*((delu)/(delx)+(delv)/(dely))` which is zero`

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

Then the equation can be simplified as below;

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

Applying time integration and Reynold's decomposition to the above equation;

`1/tint_o^t(del(baru+u'))/(delt)dt+1/tint_o^t(del(baru+u')^2)/(delx)dt+1/tint_o^t(del(baru+u')(barv+v'))/(dely)dt=-1/tint_o^t(1/rho(del(barP+P'))/(delx)+nu(del^2(baru+u'))/(dely^2))dt`

Expanding this equation we get;

`1/tint_0^t((delbaru)/(delt)+(delu')/(delt)+(delbaru^2 )/(delx)+(delu'^2)/(delx)+(del(2baru.u'))/(delx)+(del(baru.barv))/(dely)+(del(baru.v'))/(dely)+(del(u'.barv))/(dely)+(del(u'v'))/(dely))dt`

`=-1/tint_0^t1/rho(delbarP)/(delx)dt-1/tint_0^ti/rho(delP')/(delx)dt+1/tint_0^tnu(del^2baru)/(dely^2)dt+1/tint_0^tnu(del^2u')/(dely^2)dt`

The above equation can be now simplified as;

`(delbaru)/(delt)+(del(baru)^2)/(delx)+(del(baru.barv))/(dely)+(del(u'v'))/(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)dt`

It can be observed that the above equation is quite similar to the original momentum equation if the last two integral terms at the right hand side are ignored. These two time integral terms may not be zero unlike the time average of a single fluctuating component. But out of the two additional terms, the time average of the square of velocity fluctuating component (u) can be assumed to be zero estimated by the order of magnitude analysis. Hence, the equation can be re-written as;

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

The different terms in the above equation can be defined as follows;

• The entire left hand side of the equation are inertial terms calculated upon average quantity
• At the right hand side, `-1/rho(delbarP)/(delx)` is the kinematic pressure gradient.
• The remaining terms of the equation are momentum diffusion terms. The momentum is diffused out of the boundary layer in two ways,

1) Molecular Viscosity: `mu(delbaru)/(dely)`

2) Turbulent/ Eddy viscosity: `rho/tint_0^t(u'v')dt`

Molecular viscosity is simply the friction generated by interaction between different molecules in a fluid. When the fluid flow is laminar, the momentum diffusion is only due to molecular motion and resulting molecular collisions. But in turbulent flow, apart from molecular collisions, eddies are present which alter the surface area of the interface between the fluids with different properties, thus altering the net transfer of momentum and scalar properties through the interfaces. So the effective viscosity in a region such as a turbulent boundary layer is much higher than in laminar flow. Hence, a virtual viscosity called Turbulent viscosity is introduced. Turbulent viscosity increases the shear rate and affects the velocity and heat transfer. It is not a physical concept and is used to estimate the net effective diffusive momentum flux or scalar flux in turbulent flow along with molecular viscosity.

Reynold's Stress:

From the above derivation, Reynolds stress is obtained as the derivative of turbulent viscosity on the right hand side of the final equation.

`rho/tint_0^tdel/(dely)(u'v')dt`

Reynold's stress or Eddy stress is a component of total stress tensor in a fluid which accounts for the turbulent fluctuations in fluid momentum. In turbulent flow eddies are formed. These eddies have a resultant motion which is not in the direction of the flow which can result in a variable volumetric flow, tangential loads (stresses) and cavitation. Reynold's stress accounts for these fluctuating changes and is conventionally written as;

`R_{ij} equiv rho overline {u'_{i}u'_{j}}`