Reynolds-averaged Navier–Stokes(RANS) derivation and analysis

Reynolds-averaged Navier–Stokes equations

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?

The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged equations of motion for fluid flow. The idea behind the equations is Reynold’s decomposition, whereby 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.

Derivation: https://drive.google.com/file/d/1EBcicp3ET7koyojnEmzGmLXHgv5bnXXn/view?usp=sharing

Turbulence viscosity: 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’.

Scenario A: The fluids share a distinct boundary in the form of a straight line.

Scenario B: The fluids share a boundary shaped in the form of small meandering structures, with a small radius of curvature.

Scenario C: The fluids share a boundary with the shape of a large meander with a large radius of curvature.

Also, consider that the volume of each colored fluid is the same in all the 3 scenarios. Now when you lift the boundary, the two fluids in all the 3 scenarios mix up to form a final fluid with an intermediate density (in this case shown in magenta) in case M. However, which among the 3 cases would take the shortest time to transform to the fluid M?

In case A, a linear boundary between the two fluids implies that the surface area between the fluids is the shortest among the 3 cases. Given that molecular mixing between the two fluids occurs at the interface between the fluids, the rate of the net effective transfer of density (or buoyancy) would be the least in scenario A. Now in case B, the interface between the fluids has a larger surface area, and hence would allow more net mixing between them, thus taking less time to transform to M. Considering case C, the surface area at the interface is a largest, which would allow maximum net mixing to occur between the two fluids. Hence, case C would take the least time to reach M. Thus even though the molecular mixing rate is the same in all cases, one case takes less time to completely homogenize than the other case, which is attributed to the pattern of distribution of the fluids. Thus, turbulence is a property of the flow, not the fluid.

In a turbulent fluid, a linear interface between two different fluids breaks apart to form small-scale structures, also known as eddies. As these eddies grow and diminish in size (by energy cascades), they effectively alter the surface area of the interface between the fluids with different properties, thus altering the net transfer of momentum and scalar properties through the interfaces.

A computer performs a fluid dynamics simulation by numerical discretizing the Navier-Stokes equation over a domain that is split into grid boxes, as shown below.

The size of the grid boxes (also called resolution) determines the smallest scale of the eddies that the machine can resolve. In other words, if the grid size of your domain is 1 km, your machine won’t be able to capture the eddies smaller than 1 km. 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.

Molecular viscosity: Molecular viscosity is a bit like friction. Viscosity is property of the fluid that quantifies how much the shearing motion of the fluid sliding past other types/part of the fluid results in shearing stress that dissipates the energy in the flow like friction.