Literature review RANS derivation and analysis

Objective:

To Apply reynold's decomposition to the NS equations and come up with 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:

The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newton's Law of Motion to a fluid element and is also called the momentum equation

The equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using methods from calculus. But, in practice, these equations are too difficult to solve analytically. In the past, engineers made further approximations and simplifications to the equation set until they had a group of equations that they could solve. Recently, high speed computers have been used to solve approximations to the equations using a variety of techniques like finite difference, finite volume, finite element, and spectral methods. This area of study is called Computational Fluid Dynamics or CFD.

(Note: You will notice that the differential symbol is different than the usual "d /dt" or "d /dx" that you see for ordinary differential equations. The symbol "partial" is is used to indicate partial derivatives. The symbol indicates that we are to hold all of the independent variables fixed, except the variable next to symbol, when computing a derivative)

Turbulence and turbulence modelling:

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows.in parallel layers, with no disruption between those layers .And Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence.

Bascially we can solve any kind flow using this equations which is the best way of computation also known as DNS (Direct numerical simulation).But solving this can be computationally very expensive and also time consuming so industries use models to capture the physics ,and are also computationally feasible.There are total 3 models 

  • Reynolds Averaged navier-strokes equations (RANS)
  • Large Eddy simulation( LES)
  • Direct Numerical Simulation (DNS)

Solving RANS:

Now while solving a turbulent problems number of sizes of eddies are generated also a convergence graph to this kind of graph has a fluctuating  . RANS is amethod of solving such type of problems ,in which all the terms in the navier stokes equation are seperated between the mean value and the fluctuating value

For example 

       U      =                 μ           +            u

Instantaneous            mean           fluctuating

This is called reynold's decomposition

`barU` = `1/tintU(x,y,t)dt`

Integration of fluctuating component is always zero so we are left with

u'(x,y,z,t)=0(u' & v' are fluctuating component)

Apply decompostition to continuity equation we get 

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

`del (u + u' )/(delx) + del(v+v')/(dely)`

`1/tintdel/(delx)(U+u)dt + del/(dely)(V+v)dt=0`

after intregration 

`1/tint(delu)/(delx)dt` & `1/tint(delv)/(dely)dt` = 0

As integral of fluctuating component is zero

so on integration of remaining terms  

`(delu)/(delx)1/tint dt + (delv)/(dely)1/tint dt =0`

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

which means this satisfies the continuity equation and there is no change of turbulence in continuity equation

Now lets take the decomposition terms to the momentum equation 

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

adding the continuity equation as it is zero it does not make any diffrence 

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

applying product rule and simplifying the equation we get ,

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

Now appllying the time averaging reynolds decomposition

`1/tint(delu)/(delt)dt + (delu')/(delt)dt +(delu^2)/(delx)dt+del/(delx)2u u'dt +del/(dely)(uv)+del/(dely)(uv')+del/(dely)(u'v)dt+del/(dely)(u'v')dt`

`1/tint-1/(rho) (delp)/(delx)dt-1/rho(delp')/(delx)dt+v(del^2u)/(dely^2)dt+v(del^2u')/(dely^2)dt`

Taking out the fluctuating terms as integration of fluctuating term is always zero only the square term remains as integration of them wont be zero 

but u' velocity fluctuating along x axis is very small we can neglect it 

Also gradient of velocity component y>>>>u

`therefore 1/tint del/(delx)(u')^2dt= 0`

Now the equation becomes 

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

simplifying the RHS terms we get

`-1/rho(delp)/(delx) + 1/rhodel/(dely)(u(delu)/(dely)-rho/tint(u'v')dt)`

                                       viscous stress                           reynolds stress

`rho/tint(u'v')dt`= reynold's stress =`tauundersets`

Now as inertia is so small in viscous sub layer so eqn becomes

`-1/rho(delp)/(delx) + 1/rhodel/(dely)(u(delu)/(dely)-rho/tint(u'v')dt)`

As we consider the inertial terms to be zero also with negligible pressure gradient it becomes

`1/rhodel/(dely)(u(delu)/(dely)+rho/tint(u'v')dt)=0`

`(mu(delu)/(dely)-rho/tint(u'v')dt) = constant`

Now when the above eqn is applied to a region outside viscous sublayer turbulence term dominates

`rho/tint(u'v')dt = constant`

here the viscous term is so small it is taken as zero and this term derived itself is called the reynolds stress term 

rho/tint(u'v')dt =`tauundersets`

reynolds stress is the stress induced into the fluid by convective momentum transfer due to random motions caused by turbulence fluctuation of velocity

Now when the reynolds stress is written in the form of viscous stress we get

`tau=rhoepsilon(delu)/(delt)`

`epsilon`is turbulent viscosity

so the total stress experienced is

`tau=tauundersets``+tauundersetv`

`tau=mu(delU)/(dely)+rhoepsilon(delU)/(dely)`

on simplifying we get

`tau=rho(v+epsilon)(delU)/(dely)`

v= molecular viscosity 

`epsilon`= turbulent turbulent viscosity

Theory:

1 Turbulent viscosity:The turbulent transfer of momentum by eddies creates internal fluid friction. It is the fundamental idea of how we define viscosity in turbulent flow, i.e. internal fluid resistance. Eddy viscosity explains what causes the internal friction

2 molecular viscosity: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.

Now based on the derived equation earlier 

`taus= rhou*^2`

`u*^2 =`friction velocity 

`therefore u*=sqrttau/sqrtrho`

u+ = U/u*= `Usqrt(rho/tau)`

Now if U+ is very small then U is smaller than turbulent boundary layer  means it is inside viscous sublayer 

And 

y+ = y / Lchar 

y = absolute distance from the wall

Lchar = Charateristic length

Y+ is small then you are inside the boundary layer 

Selecting the turbulence model depends on this y+ value 

Generally

for k-epsilon 

     y+<5 

OR Y +>30 

is recomended because in the buffer layer the solution we obtain will not be correct as in this region solution does not follow the linear nor the loglaw profile of the curve

Conclusion :

  • Significance of reynolds stress term was determined 
  • Importance of models was determined 
  • Concept of y+ was understood 
  • A good idea about  correct use of models based on y+ was known

(Extra :As the assiged topic is really vast further research on model was done and I am attaching a link containing detailed explaination of wall functions & also thermal wall function ,how they differ and how to select a turbulence model based on y+ https://drive.google.com/file/d/10Asngnkre3RhFyI-1eANa0PytMKM2q34/view?usp=sharing )

 

 


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