Week 8: Literature review RANS derivation and analysis
OBJECTIVE:
Apply Reynold's decomposition to the NS equations and develop the expression for Reynold's stress.
- Explain your understanding of the terms Reynold's stress
- What is turbulent viscosity? How is it different from molecular viscosity?
THEORY:
RANS:
The Reynolds-averaged Navier-Stokes (RANS) equations are the time-averaged equations of motion for fluid flow. The equations are based on Osborne Reynolds' theory of Reynolds decomposition, which divides an instantaneous quantity into its time-averaged and fluctuating components. Turbulent flows are generally described using the RANS equations. The Navier-Stokes equations can be approximated with the help of these equations using approximations based on the understanding of the characteristics of flow turbulence.
Reynolds Decomposition:
Reynolds decomposition is a mathematical technique in fluid dynamics used to separate the actual value of a fluid property into time-averaged and fluctuating values.
For example, if `mu` is the quantity, then through Reynolds decomposition:
`mu(x,y,z,t) = bar mu(x,y,z) + muprime(x,y,z,t)`
where, `mu` = actual value,
`muprime` = fluctuating value,
`bar mu` = time-averaged value
Derivation:
Consider an incompressible, two-dimensional flow, the continuity equation can be written as:
`(delu)/(delx) + (delv)/(dely) = 0 - - - - - - - - - (a)`
Applying Reynolds decomposition on `u` and `v`, we get:
`u(x,y,t) = bar u(x,y) + uprime(x,y,t) - - - - - - - (i)`
`v(x,y,t) = bar v(x,y) + vprime(x,y,t)- - - - - - - (ii)`
Since, `bar u` is a time-averaged value of `u`, therefore:
`bar u(x,y) = 1/(t_("int")) int_0^(t_("int")) u(x,y,t) dt - - - - - - - - - (iii)`
Integrating, eq.`(i)`:
`1/(t_("int")) int_0^(t_("int")) u(x,y,t) dt = 1/(t_("int")) int_0^(t_("int")) bar u(x,y)dt + 1/(t_("int")) int_0^(t_("int")) uprime(x,y,t)dt`
`bar u(x,y) = bar u(x,y) + 1/(t_("int")) int_0^(t_("int")) uprime(x,y,t)dt` ( from `(iii)`)
`1/(t_("int")) int_0^(t_("int")) uprime(x,y,t)dt = 0 - - - - - - - (iv)`
Substituting `(i)` and `(ii)` in `(a)`:
`del/(delx) (baru + uprime)+ del/(dely)(barv + vprime) = 0`
Integrating w.r.t. `t` and time-averaging:
`1/(t_("int")) int_0^(t_("int")) del/(delx) (baru + uprime) dt + 1/(t_("int")) int_0^(t_("int")) del/(dely)(barv + vprime) dt = 0`
`1/(t_("int")) int_0^(t_("int")) ((delbaru)/(delx)) dt + 1/(t_("int")) int_0^(t_("int")) ((deluprime)/(delx)) dt + 1/(t_("int")) int_0^(t_("int")) ((delbarv)/(dely)) dt + 1/(t_("int")) int_0^(t_("int")) ((delvprime)/(dely)) dt = 0`
from `(iii)`:
`((delbaru)/(delx)) 1/(t_("int")) int_0^(t_("int")) dt + ((delbarv)/(dely)) 1/(t_("int")) int_0^(t_("int")) dt = 0`
`(delbaru)/(delx) + (delbarv)/(dely) = 0 - - - - - - - - - - (b)`
Consider the momentum equation:
`rho((delu)/(delt) + (udelu)/(delx) + (vdelu)/(dely)) = - (delP)/(delx) + mu(del^2u)/(dely^2)`
`(delu)/(delt) + (udelu)/(delx) + (vdelu)/(dely) = - 1/rho (delP)/(delx) + mu/rho (del^2u)/(dely^2)`
`(delu)/(delt) + (udelu)/(delx) + (vdelu)/(dely) = - (delP)/(rhodelx) + nu (del^2u)/(dely^2)`
Adding `(a)` to L.H.S:
`(delu)/(delt) + (udelu)/(delx) + (vdelu)/(dely) + u((delu)/(delx) + (delv)/(dely)) = - (delP)/(rhodelx) + nu (del^2u)/(dely^2)`
`(delu)/(delt) + 2(udelu)/(delx) + ((vdelu)/(dely) + (udelv)/(dely)) = - (delP)/(rhodelx) + nu (del^2u)/(dely^2)`
`(delu)/(delt) + (delu^2)/(delx) + del/(dely)(uv) = - (delP)/(rhodelx) + nu (del^2u)/(dely^2)`
Applying reynolds decomposition and time-averaging the equation:
`1/(t_("int")) int_0^(t_("int")) del/(delt)(baru + uprime) dt+ 1/(t_("int")) int_0^(t_("int")) del/(delx)(baru + uprime)^2 dt+ 1/(t_("int")) int_0^(t_("int")) del/(dely)(baru + uprime)(barv + vprime)dt = - 1/(t_("int")) int_0^(t_("int")) del(bar P + Pprime)/(rhodelx)dt + 1/(t_("int")) int_0^(t_("int")) nu del^2/(dely^2)(baru + uprime)dt`
From `(iii)` and `(iv)`:
`1/(t_("int")) int_0^(t_("int")) (delbaru)/(delt)dt + 1/(t_("int")) int_0^(t_("int")) (delbaru^2) /(delx)dt + 1/(t_("int")) int_0^(t_("int")) (deluprime^2)/(delx)dt + 1/(t_("int")) int_0^(t_("int")) del/(dely)(barubarv)dt + 1/(t_("int")) int_0^(t_("int")) del/(dely)(uprimevprime)dt = - 1/(t_("int")) int_0^(t_("int")) (delbarP)/(rhodelx)dt + 1/(t_("int")) int_0^(t_("int")) nu (del^2baru)/(dely^2)dt`
`(delbaru)/(delt) + (delbaru^2) /(delx)+ 1/(t_("int")) int_0^(t_("int")) (deluprime^2)/(delx) + del/(dely)(barubarv) + 1/(t_("int")) int_0^(t_("int")) del/(dely)(uprimevprime) = - (delbarP)/(rhodelx) + nu (del^2baru)/(dely^2)`
`(delbaru)/(delt) + (delbaru^2) /(delx)+ del/(dely)(barubarv) = - (delbarP)/(rhodelx) + nu (del^2baru)/(dely^2) - ( 1/(t_("int")) int_0^(t_("int")) (deluprime^2)/(delx)dt + 1/(t_("int")) int_0^(t_("int")) del/(dely)(uprimevprime)dt)`
Assuming that inside boundary layer, the velocity gradient is negligible along the x-axis when compared to the y-axis:
`1/(t_("int")) int_0^(t_("int")) (deluprime^2)/(delx) = 0`
`implies (delbaru)/(delt) + (delbaru^2) /(delx)+ del/(dely)(barubarv) = - (delbarP)/(rhodelx) + nu (del^2baru)/(dely^2) - 1/(t_("int")) int_0^(t_("int")) del/(dely)(uprimevprime)dt`
`(delbaru)/(delt) + (delbaru^2) /(delx)+ del/(dely)(barubarv) = - (delbarP)/(rhodelx) + 1/rho del/(dely)( (mudelbaru)/(dely) - 1/(t_("int")) int_0^(t_("int"))(uprimevprime)dt)`
These equations are called the Reynolds-Averaged Navier Stokes Equations.
Molecular Viscosity:
Molecular Viscosity is the same as the fluid's viscosity; it is a well-defined fluid property and a transport coefficient. It refers to the naturally occurring friction between layers of fluid.
Turbulent Viscosity:
Turbulent Viscosity is also known as Eddy Viscosity; it is an imaginary term introduced to make the study of turbulent force simpler. Physically, it is the internal resistance that results due to eddy formation and obstructs the natural flow in a turbulent problem.
Unlike Molecular Viscosity, it is not a well-defined constant or property of the fluid, as it depends on the flow field and its solution.
Reynolds Stress:
In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid derived when averaging is carried out on the Navier-Stokes Equation. It compensates for the turbulent fluctuations in a fluid flow and is a crucial part of the turbulence modelling
In the above RANS Equation:
Reynold Stress = `1/(t_("int")) int_0^(t_("int"))(uprimevprime)dt`
CONCLUSION:
We have successfully derived the RANS Equations and understood the role of Reynolds stress in turbulence modelling.