Week 8: Literature review - RANS derivation and analysis

Aim:

• 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?

Turbulence:

Turbulence is an irregular motion of the air resulting from eddies and vertical currents. It may be as insignificant as a few annoying bumps or severe enough to momentarily throw an airplane out of control or to cause structural damage. Turbulence is associated with fronts, wind shear, thunderstorms, etc.

Reynolds-averaged Navier–Stokes equations: 

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 equations.

The basic tool required for the derivation of the RANS equations from the instantaneous Navier –Stokes equation is the Reynolds Decomposition. Reynolds decomposition refers to the separation of the flow variable (like velocity u)into the mean (time averaged) component(¯u) and the fluctuating component (u'). Because the mean operator is a Reynolds Operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero (u'=0).

The velocity at any component at can be divided into the mean component and the fluctuating component as shown  below:

What is turbulent Viscosity?

The Coefficient that relates the average shear stress and the rate of velocity gradient in the turbulent flow is called the turbulent viscosity. The viscosity in the turbulent flow is non homogenous and it varies in space.

The turbulent transfer of momentum by eddies giving rise to an internal field fraction, in a manner analogous to the action of molecular viscosity in the laminar flow but taking place on a much larger scale Eddy viscosity is the function of the flow, not of the fluid. It is greater for flows with more turbulence.

Molecular viscosity is the same as viscosity. The co efficient of molecular viscosity is the same value as the dynamic viscosity. Molecular viscosity is the transport of mass motion momentum solve by the random motions of individual molecules not moving together in Coherent groups. Molecular viscosity is analogous in laminar flow to eddy viscosity in the turbulent flow.

The molecular viscosity is the microscopic property of the fluid and the eddy viscosity is the flow property emerging from the increased momentum transport through the flow fluid due to the turbulent velocity fluctuations.