Reynolds averaged Navier Stokes

This report will answer the question of what is Reynolds stress, turbulent viscosity, and molecular viscosity while deriving Reynolds averaged Navier stokes equation by applying Reynolds decomposition to Navier stokes equation.

Reynolds decomposition

Consider a sensor measuring velocity in turbulent flow the plot of velocity over time will be very fluctuating and noisy which is difficult to solve. Thus, Reynolds averaging is used to take time average. The mean of this fluctuating plot is denoted as `(:U:)` and fluctuating part is denoted by u(t) as fluctuation depends on time. the mean vector than can be decomposed as follows:

`ul U(ulx,t) = (:ul U(ul x):) + ulu (ulx,t)`--------------(1)

the mean value (:U:) is the function of space, the fluctuating value is function of space & time.

This is Reynolds decomposition.

Rules for averaging

`(:(delU)/(delx):) = (del(:U:))/(delx)`

`(:(:U:):)=(:U:)`

`(:u:)=0`

`(:U*Q:)=(:U:)*(:Q:)*(:u*q:)` (u &q are the fluctuating part of mean vector U & Q respectively)

`(:u*q:) ne 0`

Navier Stokes Equation 

continuity equation 

`grad * ulU = 0`

momentum equation

`(DulU)/(Dt) = -1/rho gradp + nu grad^2 ul U`

Reynolds decomposition on continuity equation 

`grad * ulU = (del U_i ) /(del x_i) = (del U_1)/(delx_1) + (del U_2)/(delx_2) + (del U_3)/(delx_3)= 0`

If we consider  2D and apply Reynolds decomposition we get 

`(del ( (: U_1:) + u_1))/(del x_1) + (del ( (: U_2:) + u_2))/(del x_2) =0`

`(del (:U_1:))/(del x_1) +(del u_1)/(del x_1) + (del (:U_2:))/(del x_2) +(del u_2)/(del x_2)=0`

averaging both sides

`(:(del (:U_i:))/(del x_i) + (del u_i)/(del x_i):) = (:0:)` - - - - - - - - (a)

`(:(del(:U_i:))/(delx_i):) + (:(delu_i)/(delx_i):) = 0`  - - - - - - - - - (b)

`(del(:U_i:))/(delx_i) = 0 because` average of a fluctuating part is zero by the averaging rule`

the above equation is Reynolds averaged continuity equation 

(a) - (b) => `(del u_i)/(del x_i) = 0`

the continuity equation is valid on mean and fluctuating part seperately.

Reynolds decomposition on the momentum equation 

`(DulU)/(Dt) = -1/rho gradp + nu grad^2 ul U`

`(delu_i)/(delt)+U_j (delu_i)/(delx_j) = -1/rho* (delp)/(delx_i)+ nu *(del^2 U_i)/(del x_j ^2`

`(del ((:U_i:)+u_i))/(delt)+((:U_j:)+u_j)*(del((:U_i:)+u_i))/(delx_j)=1/rho*(del((:p:)+p\'))/(delx_i)+nu*(del^2((:U_i:)+u_i))/(delx_j^2)`

averaging both sides

`(del(:U_i:))/(delt)+(:U_j:)*(del(:U_i:))/(delx_j)+(:(delu_iu_j)/(delx_j):)=1/rho*(del(:p:))/(delx_i)+nu*(del^2(:U_i:))/(delx_j^2)`

rewriting the equation above

`(D(:U:))/(Dt)=-1/rho (del(:p:))/(delx_i)+del/(delx_j)[nu(del(:U_i:))/(delx_j)-(:u_iu_j:)]`

The above equation is the Reynolds Averaged Navier Stokes

Expressing viscous terms as shear stresses.

`nu(del^2U_i)/(delx_j^2)=1/rho(del tau_(ij))/(delx_j)`

`tau_(ij)` is the viscous stress (stress tensor)

substituting the above equation in RANS

`(D(:U:))/(Dt)=-1/rho (del(:p:))/(delx_i)+1/rhodel/(delx_j)[tau_(ij)-rho(:u_iu_j:)]`

The term `rho (:u_iu_j:)`is the Reynolds stress.

This is the additional term we get in the Navier Stokes equation after the Reynolds averaging

the Reynolds stress tensor for 3D is shown below

`[[u_1^2,u_1u_2,u_1u_3],[u_2u_1,u_2^2,u_2u_3],[u_3u_1,u_3u_2,u_3^2]]`

the diagonal u_i^2 represents normal stresses and the others are shear stresses.

to solve the Navier Stokes equation we must have the transport equation to solve for these terms which doesn\'t exist.

This is where turbulence models are used which can describe these quantities.

Turbulent Viscosity

A computer performs a fluid dynamics simulation by numerical discretizing the Navier-Stokes equation over a domain that is split into grid boxes. The size of the grid boxes (also called resolution) determines the smallest scale of the eddies that the machine can resolve. Which means that the net effective momentum transfer due to small eddies occurring within that box, is unknown to the machine. In order to estimate the net effective mixing at each of these grid boxes, we specify an effective viscosity and multiply it with the shear (or velocity gradient) of the fluid surrounding the grid box. This effective viscosity is called turbulent viscosity. The product of this turbulent viscosity and the shear gives us the net diffusive flux of fluid across the grid box. For momentum, that would be a diffusive momentum flux, and for temperature/density it would be a scalar flux.

In simple terms - turbulent viscosity is a representation of a mixing rate that helps us to estimate the net effective mixing between fluids ‘at a scale that can’t be resolved by a computer simulation’.

Molecular Viscosity vs Turbulent Viscosity

Molecular Viscosity

it is a fluid property defined as the ratio of the shearing stress to the shear of the motion.

It is independent of the velocity distribution and the dimensions of the system.

Turbulent Viscocity

it is a flow property and is a representation of mixing rate

it is a function of velocity fluctuations and depends on velocity and Reynolds number 

Conclusion 

The assignment gives a clear understanding of time-averaging of the Navier Stokes equation and also increases our knowledge on Reynolds stress, turbulent and molecular viscosity.