Week 8: Literature review - RANS derivation and analysis

OBJECTIVE: Reynolds-averaged Navier-Stokes derivation and analysis

THEORY: The RANS equations are time-averaged equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, wherein an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities. 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 equations.

Reynolds stress

The Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-stroke equation to account for turbulent fluctuations in the fluid moments.

They arise from the nonlinear advection term when the Navier Strock equation is Reynolds decomposed and Reynolds averaged. The general form of the Reynolds stress tensor is `tauij = -rho overlineui overlineuj`

Reynolds stress equation model (RSM), also referred to as second-moment closures are the most complete classical turbulent model. In these models, the eddy-viscosity hypothesis is avoided and the individual components of the Reynolds stress tensor are directly computed

Turbulent viscosity and Molecular 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 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

DERIVATION:

From Governing equation

Continuity

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

Momentum

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

Energy

`rhoc((delT)/(delt) + u(delT)/(delx) + v(delT)/(dely) )= K(del^2u)/(delx^2) + mu((delu)/(dely))^2`

Time averaged velocity

`U(x,y) = (1)/(tINT) int_0^(tINT) u(x,y,z)dt`

`overline v(x,y) = (1)/(tINT) int v(x,y,z)dt`

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

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

Inserting the values we get

`(1)/(tINT) int_0^(tINT) u'(x,y,z,t) = 0`

Continuity Equation

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

`(del(overlineu + u'))/(delx) + (del(overlinev +v'))/(dely) = 0`

`1/(tINT) int (del(overlineu + u'))/(delx) dt+ (del(overlinev +v'))/(dely)dt = 0`

expanding and resolving this we get

`((deloverlineu)/(delx))((TINT - 0))/(tINT) + ((deloverlinev)/(delx))((TINT - 0))/(tINT) = 0`

on further simplification we get

`(deloverlineu)/(delx) + (deloverlinev)/(dely) = 0`

Momentum Equation

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

dividing by `rho`on both side we get

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

Now,

`(1)/(tINT) int_0^(tINT ) (del(overlineu + u'))/(delt)dt + (del(overlineu + u'))/(delx)dt + (del[(overlineu +u')(overlinev +v')])/(dely)dt``= -1/(tINT) int_0^(tINT) -1/(rho) (del(overlineP + P'))/(delx) + v(del^2(overlineu + u'))/(dely^2)`

On simplifying further

`(deloverlineu)/(delt) + (delu^2)/(delx) + (deloverlineuoverlinev)/(dely) = -1/(rho)(deloverlineP)/(delx)+ 1/(rho)(del)/(dely) [ (mu(deloverlineu))/(dely)``+ 1/(tINT) int_0^(tINT)(u'v')dt ]`

Above mathematical equation in words can be expressed as,

Inertial term = Pressure term + Molecular viscosity (viscous stress) + momentum diffusivity because of turbulence (Reynolds stress)