Week 8: Literature review - RANS derivation and analysis
Objective: The Reynolds Average Navier Stokes equation is used to solve when the flow is no longer laminar. When the flow is turned into the turbulent we can't use the standard Navier-Stokes equation to predict the flow. For this, the RANS model is derived by averaging the N-S equations to calculate the turbulence fluctuating components in the flow domain. This is obtained by applying Reynold's decomposition to the NS equations by including the Reynolds stress.
What is Turbulence:
Turbulence is commonly observed in everyday phenomena such as surf, fast-flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent.
Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids.
In general terms, in turbulent flow, unsteady vortices appear of many sizes that interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe.
Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.
The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Richard Feynman has described turbulence as the most important unsolved problem in classical physics.
`Re=(rho*v*d_h)/(mu)`
Derivation of Reynolds Averaged Navier-Stokes Equation: (RANS)
The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity (average + fluctuating components) is decomposed into its time-averaged and fluctuating quantities.
Reynolds decomposition:
Instantaneous quantity (velocity) = Time-averaged quantity + Fluctuating component.
Whereas it looks like:
`u(x,y,z,t) =baru(x,y,z,t)+uprime(x,y,z,t) -> Eq-1`
Velocity components can change in terms of it's direction (u, v, w)
Time-Average integration applying to the Eq-1:
`u(x,y,z,t)=1/Tint_0^Tu(x,y,z,t) -> Eq-2`
Applying time integration to the instantaneous quantity:
Instantaneous quantity:
`1/Tint_0^Tu(x,y,z,t)dt = 1/Tint_0^Tbaru(x,y,z,t)dt + 1/Tint_0^Tuprime(x,y,z,t)dt -> Eq-3`
Time-Averaged quantity:
`1/Tint_0^Tbaru(x,y,z,t)dt`
Fluctuating quantity:
`1/Tint_0^Tuprime(x,y,z,t)dt`
Note-1:
When we furthur integrates the fluctuating term of the velocity in longer time period it becomes zero.
`1/Tint_0^Tuprime(x,y,z,t)dt = 0`
Continuity Equation:
`(delu)/(delx)+(delv)/(dely) = 0 -> Eq-4`
When we substitute the instantaneous quantities in the equation, it looks like:
`(del(baru+uprime))/(delx)+(del(barv+vprime))/(dely) = 0 -> Eq-5`
Time-Average integration applying to the Eq-5:
`1/Tint_0^T(del(baru+uprime))/(delx)*dt+1/Tint_0^T(del(barv+vprime))/(dely)*dt = 0`
`1/Tint_0^T(delbaru)/(delx)*dt+1/Tint_0^T(deluprime)/(delx)*dt+1/Tint_0^T(delbarv)/(dely)*dt+1/Tint_0^T(delvprime)/(dely)*dt = 0 -> Eq-6`
As we discuss in the Note-1, the integration of the fluctuating velocity terms becomes zero. So,
`1/T*int_0^T(delbaru)/(delx)*dt+1/T*int_0^T(delbarv)/(dely)*dt = 0`
`(delbaru)/(delx)*1/Tint_0^Tdt+(delbarv)/(dely)*1/Tint_0^Tdt = 0`
`(delbaru)/(delx)*1/T T_0^T+(delbarv)/(dely)*1/T T_0^T = 0`
`(delbaru)/(delx)*1/(cancel(T))(cancel(T) - 0)+(delbarv)/(dely)*1/(cancel(T))(cancel(T) - 0) = 0`
`(delbaru)/(delx)+(delbarv)/(dely) = 0 -> Eq-7`
Eq - 7, represents the Reynold's Averaged Navier-Stokes (RANS modified) Continuity equation.
Momentum Equation:
The Momentum equation in the X-direction is:
`(delu)/(delt)+(del(u)(u))/(delx)+(deluv)/(dely)+(deluw)/(delz) = -(1/rho)*(delP)/(delx)+nu[(del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2)] -> Eq -8`
Now we need to insert the instantaneous quantities in the velocity, pressure components:
`(delbaru)/(delt)+(delbarubaru)/(delx)+(deluprimeuprime)/(delx)+(delbarubarv)/(dely)+(deluprimevprime)/(dely)+(delbarubarw)/(dely)+(deluprimewprime)/(delz) = -(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)+(del^2uprime)/(delx^2)+(del^2uprime)/(dely^2)+(del^2uprime)/(delz^2)] -> Eq - 9`
As the fluctuating velocity components gets canceled because they become zero after the integration, the equation is looks like:
`(del^2uprime)/(delx^2)+(del^2uprime)/(dely^2)+(del^2uprime)/(delz^2) = 0`
`(delbaru)/(delt)+(delbarubaru)/(delx)+(deluprimeuprime)/(delx)+(delbarubarv)/(dely)+(deluprimevprime)/(dely)+(delbarubarw)/(dely)+(deluprimewprime)/(delz) = -(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]`
Taking the velocity fluctuating terms from left hand side to right side.
`(delbaru)/(delt)+(delbarubaru)/(delx)+(delbarubarv)/(dely)+(delbarubarw)/(dely) = -(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]-(deluprimeuprime)/(delx)-(deluprimevprime)/(dely)-(deluprimewprime)/(delz) -> Eq - 10`
Analysing the Reynolds Stress tensors in a matrix form is:
`tau_(i,j) = -rho*[[bar(uprime^2)bar(uprimevprime)bar(uprimewprime)] , [bar(vprimeuprime)bar(vprime^2)bar(vprimewprime)] , [bar(wprimeuprime)bar(wprimevprime)bar(wprime^2)]]`
Then we convert the Reynolds stress tensors as:
`tau_(x,x) = -rho[uprimeuprime]`
`uprimeuprime = - (tau_(x,x))/(rho) -> Eq - a`
`tau_(x,x) = -rho[uprimevprime]`
`uprimevprime = - (tau_(x,y))/(rho) -> Eq - b`
`tau_(x,x) = -rho[uprimewprime]`
`uprimewprime = - (tau_(x,z))/(rho) -> Eq - c`
Dump or substitute all the above a,b and c equatuins in the Eq. no: 10
`(delbaru)/(delt)+(delbarubaru)/(delx)+(delbarubarv)/(dely)+(delbarubarw)/(dely) = -(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]+1/rho[(del(tau_(x,x)))/(delx)+(del(tau_(x,y)))/(dely)+(del(tau_(x,z)))/(delz)] -> Eq - 11`
The above equation represents the Reynolds Averaged Navier-Stokes equation.
Reynolds Stress:
In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-Stokes equation to account for turbulent fluctuations in fluid momentum.
The velocity field of a flow can be split into a meaningful part and a fluctuating part using Reynolds decomposition.
`u_i = baru_i + uprime_i`
with `u(x,t)`being the flow velocity vector having components `u_i` in the `x_i` coordinate direction. The mean velocities `baru_i` is determined by either time averaging or spatial averaging and `uprime_i` denotes the fluctuating quantity of the velocity.
We consider a homogeneous fluid, whose density ρ is taken to be a constant. For such a fluid, the components τ'ij of the Reynolds stress tensor are defined as:
`tau_(i,j) = -rho*bar(uprime)*bar(vprime)`
Turbulent Viscosity:
The coefficient that relates the average shear stress and rate of velocity gradient in the turbulent flow is called turbulent viscosity. The viscosity in the turbulent flow is non-homogeneous, and it varies in space.
The relation between the turbulent stress and turbulent viscosity is:
Turbulent stress:
`tau_(turb) = -rho*bar(uprime)*bar(vprime)`
Turbulent viscosity:
`mu_(turb) = mu*(du)/(dy)`
Here in above equations:
`uprime`and `vprime` are the fluctuating velocity components.
`rho` is the density of the fluid
`mu`is the turbulent viscosity
`baru`is a mean velocity
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 internal friction.
Molecular viscosity is the same as viscosity. The Coefficient of Molecular Viscosity is the same value as dynamic 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 a turbulent flow.
The general shear stress of the turbulent flow is written as:
`tau = tau_(lam)+tau_(turb)`
`tau = mu*(du)/(dy)+(-rho*bar(uprime)*bar(vprime))`
From the above equation, we must say that the first term in the RHS represents the viscous effect and the second term represents turbulence viscosity.
If the flow is considered as the laminar, the velocity fluctuations in the above equation (`baruprime and barvprime`) wille be zero.