Week 5 - Rayleigh Taylor Instability

Objective:

• To understand practical CFD models which are based on the mathematical analysis of Rayleigh Taylor waves.
• To perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm.
• To Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
• Define Atwood number and calculate Atwood number for above cases and its effect on instability.

Introduction:

Rayleigh-Taylor instability:

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.

Examples include Water suspended atop oil, water suspended above the gravity of earth.

What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations.

Ans: 

CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves are as follow:

• Kelvin–Helmholtz instability 
• Richtmyer–Meshkov instability
• Plateau–Rayleigh instability

Kelvin–Helmholtz instability 

The Kelvin–Helmholtz instability  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 wind blows across a water surface; the instability constant manifests as waves. Kelvin-Helmholtz instabilities are also visible in the atmosphere of planets and moons, such as in cloud formation on earth or the red spot on Jupiter, and the atmospheres of stars, including the Sun's.

Richtmyer–Meshkov instability

The Richtmyer–Meshkov instability  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 with 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.

Plateau–Rayleigh instability

The 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 ink jet 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 tension, 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 in Ansys Fluent:

Geometry:

• Create a square shaped geometry with dimension  20*20 mm and convert it into surface using pull command.
• now create new component and named it as water and in that create square surface of dimension 20*20 mm.
• To share the topology between two geometries, click on shareprep tool under workbench.

Solver:

Viscous Model: laminar

Solver : pressure based transient solver.

Check on gravity box and add value of acceleration due to gravity -9.81 m/s.

Select Volume of fluid under multiphase tab with implicit scheme.Ensure Air as primary phase and water as secondary phase.

After clicking on initialize, click on patch to show water phase in entire cell zone.

Result:

Air-Water instability:

1. mesh size: 0.5 mm

No of elements = 3200.

Residue:

Simulation is run for timestep 0.025 and 500 iterations.

Animation:

2. mesh size: 0.35 mm

No of elements = 6498.

Residue:

Simulation is run for timestep 0.005 and 500 iterations.

Animation:

3. mesh size: 0.2 mm

No of elements = 20000.

Residue:

Simulation is run for timestep 0.005 and 1000 iterations.

Animation:

water and user defined material:

Properties of user definedd materials are as follow:

density = 400 kg/m³, viscosity = 0.001 kg/m-s

Mesh size = 0.35 mm

simulation is run for timesteps 0.005 and 800 iterations.

Residue:

simulation is run for timesteps 0.005 and 800 iterations.

Animation:

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

We prefer to run steady state simulation approch when we are only interested to know what is happening on initial and final state of simulation and uninterrested at intermediate state. In the Rayleigh taylor instability we are interested to know how two fluids are mixed together and how shock wave propagate and this cann't be monitored at only final state hence we have to monitor each and evry timestep in the simulation to understand the phenomenon.

Atwood number(A):

The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as

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.

For Air-Water instability,

ρ1 = 998.2 kg/m3  (Water)

ρ2 = 1.225 kg/m3   (Air)

Atwood Number (A) = (998.2-1.225)/(998.2+1.225)

                                 = 0.9975

 Atwood Number (A) = 0.9975

As A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.

For Water-User defined material instability,

ρ1 = 998.2 kg/m3  (Water)

ρ2 = 400 kg/m3   (Air)

Atwood Number (A) = (998.2-400)/(998.2+400)

                              = 0.4278     

Atwood Number (A) = 0.4278