Week 8: Literature review - RANS derivation and analysis

Aim:

• Derive the Reynolds Averaged Navier Stokes equation(RANS equation) and find an expression for Reynolds stress.
• Determine the terms Reynolds stress, turbulent viscosity, and molecular viscosity.

Introduction:

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 is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds. The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier-Stokes equation.

The Governing equations for the boundary layer are:

Continuity equation:

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

Momentum equation:

`rho((delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely)) = -(delP)/(delx)+mu(del^2u)/(dely^2)`

Energy equation:

`rhoc*((delT)/(delt)+(udelT)/(delx)+(vdelT)/(dely)) = (kdel^2T)/(delx^2)+mu((delu)/(dely))^2`

Reynolds Decomposition:

Reynolds decomposition allows the simplification of the Navier–Stokes equations by substituting in the sum of the steady component and perturbations to the velocity profile and taking the mean value. The resulting equation contains a nonlinear term known as the Reynolds stress which gives rise to turbulence.

In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations. For example, for a quantity, u,v, and w the decomposition would be

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

`v(x,y,z,t) = barv(x,y,z) + v'(x,y,z,t)`

`z(x,y,z,t) = barz(x,y,z) + z'(x,y,z,t)`

where denotes the expectation value of u, (often called the steady component/time, spatial or ensemble average), and , are the deviations from the expectation value (or fluctuations). The fluctuations are defined as the expectation value subtracted from quantity u such that their time average equals zero.

Time-averaged velocity in x and y-direction.

`baru(x,y) = 1/t_sf"int"*intu(x,y,z)dt`

`barv(x,y) = 1/t_sf"int"intv(x,y,z)dt`

Also the time averaged value of the fluctuating component is always zero.

`1/t_sf"int" int_0^(t_sf"int")u'(x,y,z,t)dt = 0`

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

Integrating on both sides

`1/t_sf"int" int_0^(t_sf"int")u(x,y,z,t)dt =1/t_sf"int" int_0^(t_sf"int")baru(x,y,z)dt +1/t_sf"int" int_0^(t_sf"int")u'(x,y,z,t)dt`

we know that the time averaging of a time averaged quantity results in the same value. so

`1/t_sf"int" int_0^(t_sf"int")baru(x,y,z,t)dt = baru`

`cancel(baru(x,y,z)) = cancel(baru(x,y,z))+1/t_sf"int" int_0^(t_sf"int")u'(x,y,z,t)dt`

So `1/t_sf"int" int_0^(t_sf"int")u'(x,y,z,t)dt = 0`

Reynolds Averaged NS Equation Derivation:

Firstly apply Renolyds decomposition in continuity equation:

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

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

Now, integrate the total equation with time averaging.

`1/(t_(sf"int"))int(del(baru+u'))/(delx)dt+(del(barv+v'))/(dely)dt = 0`

By splitting the integral

`1/(t_(sf"int"))int(delbaru)/(delx)dt+1/(t_(sf"int"))int(delu')/(delx)dt+1/(t_(sf"int"))int(delbarv)/(dely)dt+1/(t_(sf"int"))int(delv')/(dely)dt = 0`

Time integral of a fluctuating component is equal to 0. And coming to the `baru and barv`is not a function of time.

So `1/(t_(sf"int"))int(delu')/(delx)dt = 1/(t_(sf"int"))int(delv')/(dely)dt = 0`

`1/(t_(sf"int"))int(delbaru)/(delx)dt+cancel(1/(t_(sf"int"))int(delu')/(delx)dt)+1/(t_(sf"int"))int(delbarv)/(dely)dt+cancel(1/(t_(sf"int"))int(delv')/(dely)dt) = 0`

`(delbaru)/(delx)1/(t_(sf"int"))intdt + (delbarv)/(dely)1/(t_(sf"int"))intdt = 0`

Apply the limits

`(delbaru)/(delx)1/t_sf"int" int_0^(t_sf"int")dt + (delbarv)/(dely)1/t_sf"int" int_0^(t_sf"int")dt = 0`

`(delbaru)/(delx)(cancel((t_(sf"int")-0)/(t_(sf"int"))))+(delbarv)/(dely)(cancel((t_(sf"int")-0)/(t_(sf"int")))) =0`

Finally `(delbaru)/(delx) + (delbarv)/(dely) = 0` --------------- Continuty eq interms of time averaged variables`

Now apply Renoylds decomposition in Momemtum equation:

`rho((delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely)) = -(delP)/(delx)+mu(del^2u)/(dely^2)`

Move density to RHS

`((delu)/(delt)+(udelu)/(delx)+(vdelu)/(dely)) = -1/rho(delP)/(delx)+mu/rho(del^2u)/(dely^2)`

We know that Kinematic viscosity(v) = `mu/rho`

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

From continuity eq `(delu)/(delx) + (delv)/(dely) = 0`

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

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

From the product rule `(del(uv))/(dely) = v(delu)/(dely)+u(delv)/(dely)`

substitute the product rule in eq-1

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

substitute

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

Applying integration on both sides

`1/t_sf"int" int_0^(t_sf"int")(del(baru+u'))/(delt)dt+(del(baru+u')^2)/(delx)dt+(del[(baru+u')(barv+v')])/(dely)dt = 1/t_sf"int" int_0^(t_sf"int")-1/rho(del(barp+p'))/(delx)dt + (vdel^2)/(dely)(baru+u')dt`

`1/t_sf"int" int_0^(t_sf"int")(delbaru)/(delt)dt+(delu')/(delt)dt+(delbaru^2)/(delx)dt+(delu'^2)/(delx)dt+(2delbaru*u')/(delx)dt+(del(barubarv))/(dely)dt+(del(baruv'))/(dely)dt+(del(u'barv))/(dely)dt+(del(u'v'))/(dely)dt = 1/t_sf"int" int_0^(t_sf"int")-1/rho(delbarp)/(delx)dt-1/rho(delp')/(delx)dt + (vdel^2baru)/(dely)dt+(vdel^2u')/(dely)dt`

Time averaging fluctuating quanties(`u' and v'`) are equal to 0.

`1/t_sf"int" int_0^(t_sf"int")(delbaru)/(delt)dt+cancel((delu')/(delt)dt)+(delbaru^2)/(delx)dt+(delu'^2)/(delx)dt+cancel((2delbaru*u')/(delx)dt)+(del(barubarv))/(dely)dt+cancel((del(baruv'))/(dely)dt)+cancel((del(u'barv))/(dely)dt)+(del(u'v'))/(dely)dt = 1/t_sf"int" int_0^(t_sf"int")-1/rho(delbarp)/(delx)dt-cancel(1/rho(delp')/(delx)dt) + (vdel^2baru)/(dely)dt+cancel((vdel^2u')/(dely)dt)`

By simplifying the above eq, we get

LHS:- `1/t_sf"int" int_0^(t_sf"int")[(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely) +(delu'^2)/(delx)+(delu'v')/(dely)]dt`

RHS:- `1/t_sf"int" int_0^(t_sf"int")[-1/rho(delbarp)/(delx)+(vdel^2u)/(dely^2)]dt`

By simplifying the LHS and RHS

`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely)=-1/rho(delbarp)/(delx)+(vdel^2baru)/(dely^2)-cancel(1/t_sf"int" int_0^(t_sf"int")(del(u')^2)/(delx)dt)-1/t_sf"int" int_0^(t_sf"int")(delu'v')/(dely)dt`

`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely)=-1/rho(delbarp)/(delx)+(vdel^2baru)/(dely^2)-1/t_sf"int" int_0^(t_sf"int")(delu'v')/(dely)dt`

The LHS terms are known as Inertial terms.

Simplyfing the RHS

`-1/rho(delbarp)/(delx)-1/rho(del)/(dely)[(mudelbaru)/(dely)+rho/t_sf"int" int_0^(t_sf"int")(u'v')dt]`

`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbarubarv)/(dely) = -1/rho(delbarp)/(delx)-1/rho(del)/(dely)[(mudelbaru)/(dely)+rho/t_sf"int" int_0^(t_sf"int")(u'v')dt]` is known as RANS equation.

`1/rho(delbarp)/(delx)` is known as the Pressure gradient term.

`(mudelbaru)/(dely)` is known as Molecular viscosity.

`rho/t_sf"int" int_0^(t_sf"int")(u'v')dt` is a momentum diffusivity due to the turbulence motion.

`rho/t_sf"int" int_0^(t_sf"int")(u'v')dt = rho(u'v')` is known as Reynolds stress.

Explain your understanding of terms Reynold's stress:

The 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.

Turbulent viscosity and how it differs from molecular viscosity:

Viscosity is often referred to as the thickness of the fluid. The turbulent viscosity is also known as eddy viscosity. This is due to Reynold's number is greater than 4000 then the turbulent viscosity occurs. For turbulent flow, Reynold's number is high due to the high velocities.

Molecular viscosity(`mu`) is the same as dynamic viscosity. At a molecular level, viscosity is a result of the interaction between the different molecules in a fluid. This can be also understood as friction between the molecules in the fluid. Just like in the case of friction between moving solids, the viscosity will determine the energy required to make a fluid flow. So the molecular viscosity depends on the fluid type and the temperature of the fluid whereas the turbulent viscosity depends always on the flow.