RAYLEIGH TAYLOR INSTABILITY

RAYLEIGH TAYLOR INSTABILITY

1)What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor Waves? Explain how these mathematical models have been adapted for CFD calculations.

Saffman-Taylor Instability:

Flow displacements in porous media can result in the development of an interfacial instability between the two fluids involved in the displacement process. This instability, known as the Saffman-Taylor instability, manifests itself in the form of finger-shaped intrusions of the displacing fluid into the displaced one and can have a dramatic impact on the displacement process. When the driving factor behind the instability is the viscous mismatch between the two fluids, the instability is referred to as the viscous fingering instability. The viscous fingering instability is observed in a variety of processes including secondary and tertiary oil recovery, fixed bed regeneration in chemical processing, hydrology, soil remediation and filtration.

Kelvin Hemholtz Instabilities:

Kelvin Hemholtz instabilities arise when two fluids are flowing next to each other with different fluid velocities. The most common place that they are seen, are in wind driven water waves. L. Gramer drives the instability condition for this as:

g(ρ₂² - ρ₁²) < kρ₁ρ₂ (u₂-u₁)²

so when this condition is satisfied, the fluid is unstable. His derivation is for the two fluid case where ρ₁ < ρ₂ with a discontinuous boundary between the fluids. Notice that this instability cannot arise if u₁ = u₂. This reinforces the fact that the fluid velocity difference is what supplies the free energy that drives this instability. Having the lower fluid be much more dense helps to minimize this instability. The larger the density difference, the more work it takes to raise the lower fluid and mix the two along the boundary. Also having a large difference in the fluid velocities will make the system more unstable as larger velocity difference supply more kinetic energy to the system, making mixing more probable.

Rayleigh- Bénard Convection:

Rayleigh- Bénard Convection occurs when a fluid is heated from beneath. The heating provides a source of free energy. As the fluid expands, it decreases in density and rises, creating convection cells. This is the instability that drives lava lamps. It is governed by the Rayleigh number:

`Ra_L = (gbeta)/(valpha) (T_t - T_b)L^3`

where `beta` = thermal expansion coefficient

`alpha`= thermal diffusivity

L = characteristic length

v = kinematic viscosity.

The fluid becomes unstable when `Ra_L > 1708`

Taylor-Couette instability:

The Taylor-Couette instability arises when one cylinder rotates inside another cyliner with a layer of fluid between them. When the rotating cylinder rotates quickly enough, there is enough free energy in the system to create a stable flow pattern that has vortices. That condition is met when the Taylor number is greater than some critical value. For this system, that means :

`Ta_c < (Ω^2 R_1 (R_2- R_1)^3)/v^2`

Where Ω is the relative angular velocity of the two cylinders.

2) Rayleigh Taylor instability simulation.

The Rayleigh-Taylor instability occurs when a denser fluid is on top of a less dense fluid. The denser fluid will then fall to decrease the potential energy of the system, mixing the fluids. In this simulation there are two surfaces placed over each other as shown in the image below.

The dimensions of the surfaces are 20mm x 20mm. The upper surface represents water whereas the lower one represents air. In this setup the gravity is enabled in the negative Y-axis, so when the simulation is carried out, the water du to gravity comes down and air moves upwards.

Three cases of different mesh setups are being used for this simulation inorder to study the instability effect at the interface.

CASE 1:-

In this case the element size is kept coarse (baseline), the mesh size is 0.5mm and the number of cells are 3200.

Results-

CASE 2:-

In this case the element size is kept medium, the mesh size is 0.25mm and the number of cells are 12800.

Results-

CASE 3:-

In this case the element size is kept fine, the mesh size is 0.15mm and the number of cells are 35378.

Results

Explain why a steady state approach might not be suitable for these types of simulation?

As we know that transient state analysis is time dependent whereas steady state analysis is independent of time and is used to study the end results. But in this simulation we already know the end result viz. the denser fluid (water) will settle down and the lighter one (air) will rise above. Here we are interested in studying the instability at the interface from the instant the instability occurs. Hence a steady state approach might not be suitable for these types of simulation.

3) Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behaviour of the instability. How can we use the Atwood number to validate our simulation result?

Atwood Number:-

Atwood number (abbreviated as A), is a dimensionless number in fluid dynamics, that describes density difference between two adjacent fluids with a common interface.

Mathematically, `A = (rho_1-rho_2)/(rho_1+rho_2)`

Where, A= Atwood number

ρ₁ = Density of heavier fluid

ρ2 = Density of lighter fluid

Therefore, `A= (997-1.25)/(997+1.25) =995.75/998.25 = 0.9975`

For Atwood number close to zero (o), down welling (spikes) disturbances are formed.

For Atwood number close to One (1), up-welling (bubbles) disturbances are formed.

Therefore comparing the above value of Atwood number and observation from the simulations i.e formation of bubbles when water comes down and air rises above; we can say that the simulation result is validated.