Week 5 - Rayleigh Taylor Instability

AIM:

To perform Rayleigh Taylor Instability simulation for the following cases:

1. Fluid 1: Air, Fluid 2: water

Mesh size(base) : 0.5mm

Refined case of mesh

2. Fluid 1: Air, Fluid 2: User-defined fluid of density: 400 kg/m³, viscosity = 0.001 kg/m-s.

Refined case of mesh.

Introduction:

The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is the instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.

Examples include the behaviour of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modelled by two completely plane-parallel layers of immiscible fluid, the denser on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh. The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated, with the less dense fluid accelerating into the more dense fluid. This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.

As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equations, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe fluid motions. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here).

The difference in the fluid densities divided by their sum is defined as the Atwood number, A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.

Atwood number for Case 1, 2, and 4 is:

A = (998.2-1.225)/(998.2+1.225) = 0.99755

Atwood number for Case 3 is:

A = (998.2-400)/(998.2+400) = 0.427836

Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:

1. Kelvin–Helmholtz instability

After Lord Kelvin and Hermann von Helmholtz; typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen when the wind blows across a water surface; the instability constant manifests as waves. Kelvin-Helmholtz instabilities are also visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter, and the atmospheres (coronas) of stars, including the Sun.

2. Richtmyer–Meshkov instability (RMI)

It occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and spikes appearing in the case of a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleigh–Taylor instability.

3. Plateau–Rayleigh instability

often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh–Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of inkjet technology whereby a jet of liquid is perturbed into a steady stream of droplets. The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.

SIMULATION OF Rayleigh–Taylor instability on ANSYS FLUENT:

Modelling and Preparation:

1. Created geometry in Spaceclaim
2. Pulled the geometry to make it into a 2D object
3. Then used share topology to create a relation between the two phases.
4. Created separate components for Air and Water

Air column:

Water column:

Pulled geometry:

Case 1: Base case

Base mesh: 0.5 mm

Cell count: 6521

Fluent Set up:

• Viscous model: Laminar
• Phases:
• Pase 1: Air
• Phase 2: Water
• Time step size: 0.005

Results of case 1:

Case 2: Refined case

Base mesh: 0.16 mm

Cell count: 62,876

Fluent set up is the same as case 1.

Results of case 2:

Case 3: Refined case, user-defined fluid

Base mesh: 0.16 mm

Cell count: 62,876

Fluent set up is the same as case 1.

Results of case 3:

Case 4: Steady case

Base mesh: 0.5 mm

Cell count: 6521

Fluent set up is similar to case 1 except that we have chosen a steady-state rather than transient analysis.

Results of case 4:

Observations:

• The simulation of the base mesh size of 0.5 mm is coarse, lacks detail and doesn't depict the Rayleigh Taylor instability properly.
• Whereas in the refined case of the mesh of 0.16 mm size, the simulation is smooth, fine, filled with detail and depicts the Rayleigh Taylor instability properly.
• Therefore, a finer mesh size gives a more appropriate result of the simulation.
• Case 4, which is solved using a steady-state solver gives a wrong interpretation of the Rayleigh Taylor instability and the phases are separated distinctly