Literature review RANS derivation and analysis

Objective :

To apply Reynolds decomposition to the Navier-Stokes equations and come up with the expression for Reynolds stress.

Intruduction:

Reynolds Number:- The Reynolds number is the ratio of inertia force to viscous force withinin a fluid which is being subjected to relative internal movement due to different fluid velocities, which is known as a boundry layer in case of bounding surface such as the interior of a pipe.

Reynolds Decomposition:- Reynold decomposition is a mathematical techinue to spilt the flow variables into main componet and flucuating component.

Types of fluid flow:-

Fluid flow can be split into three types-

• Laminar flow
• Transient flow
• Turbulance flow

The type of flow can be determined using the Reynolds Number. A low Reynolds number corresponds to laminar flow where the flow is in an orderly manner and parameters like pressure and velocity remain steady. A high Reynolds number corresponds to turbulent flow where the flow is chaotic and the parameters are flucuating.

Turbulance modeling:- Turbulance modeling is used to calculate Reynolds stress term and Turbulent viscosity. The effect of turbulence can be modelled using a method called as Direct Numerical Simulation(DNS) which involes in solving Navier-Stokes equation over turbulent time step. But the turbulent time step is very small thus computational cell size is required to solve. Therefore turbulence model is used to capture the effect of turbulence in coarse size.

Three different turbulence models used are:-

• Reynold Averaged Navier Stokes equation (RANS)
• Large eddy simulation(LES0
• Direct Numerical Simulation(DNS)

Solving Rans:

Rans is the most used models amongest the because of its requires less computational resoure than the other two methos. RANS is used to make turbulence models for studying engineering simulations.

Applying time integration and Reynolds decomption fluctuating componet becomes zero

From the above eqn the olny thing we do not know is time integral. The turbulence modeling is desiged to capture that

Reynolds shear stress:- Reynolds shear stress is the function of produt of turbulent viscosity and velocity gradient.

Reynolds stress :- Reynolds stress is a componet of stress tensor in fluid which is obtained by averaging operation over the Navier Stokes eqn used to account for turbulent fluctuation in the fluid momentum.

-`-rho(du\'.v\' div dy)`

Turbulent viscosity:-

• Turbulent viscosity is a function of fluctuations in velocity term and is influenced by randomness or disorderlines inflow.
• It is non-linear and difficult to estimate.
• It is dependent on flow domains such velocity , Reynolds Number and is not a fluid property.
• It is dominant in turbulent flow regimes with high Reynolds Number.
• The flow velocity is high and turbulent viscosity depend on the velocity & flow conditions of the fluid.

Molecular viscosity:-

• Is a resistance to the movement of one layer of fluid over an adjecent of fluid.
• It si linear and is easy to measure.
• It is a fluid property and is independant of geometry.
• it is dominat in laminar flow regims where the Reynolds number is low.
• The velosity of flow is generally & the molecular viscosity depends on the thickness of the fluid.