Week 5 - Rayleigh Taylor Instability

Rayleigh Taylor Instability

INTRODUCTION
The Rayleigh–Taylor instability, is the instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid with the effect of gravity.

Here we have prepared a model with two surfaces on over the above and considered one with light fluid a bottom phase and heavy fluid at top phases. here the potential energy of the configuration is lower than the initial state due to the movement of fluid due to gravity difference. Thus the disturbance will grow and lead to a further release of potential energy, as the denser material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.

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.

Figure (1): The evolution of the RTI follows four main stages.

In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth. In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. However, after the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid).

In the second stage, the growth of the mushroom structures continues and can be modeled using buoyancy drag models, resulting in a growth rate that is approximately constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored. The spikes and bubbles then begin to interact with one another in the third stage.

Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Also, bubble competition takes place, where spikes and bubbles of smaller wavelengths that have become saturated are enveloped by larger ones that have not yet saturated.

This eventually develops into a region of turbulent mixing, which is the fourth and final stage in the evolution. It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the Reynolds number is sufficiently large.

OBJECTIVE

1. 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.
2. Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behavior of the instability. How can we use the Atwood number to validate our simulation result?

PART-I
Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations( Attach them from your google drive). Also, explain why a steady-state approach might not be suitable for these types of simulation.

PART-II
Perform the analysis for a refined mesh with water and oil. Find Atwood number for this case and discuss the effect of Atwood number & viscosity by observing the behavior of instabilities.
Oil Properties:- Density= 400 kg/m , Dynamic Viscosity = 1.06 kg/m-s.

Geometry Preparation
Step1: Open the workbench & double click open Flow Fluent
Step2: Double click on geometry to open space claim> open the file from saved location.

prepare the surface by
Part 1
water-side (top) = (20 x 20) mm
air-side (bottom) = (20 x 20) mm
Part 2
water-side (top) = (20 x 20) mm
oil-side (bottom) = (20 x 20) mm
as shown below.

Figure (3): The evolution of the RTI follows four main stages.

Step4: Go to prepare the tab and select the surface generation tool and select the air and waterside one by one.
select both prepared surfaces one above another and use share >share topology> tick click and shared top done

Step5: Close space claim workbench and save files if required.

Preprocessing
Step6: Double click on Mesh from workbench>Generate Mesh by right-clicking the mesh icon and click generate mesh option and mesh will be generated; here you can change mesh size from element size option in the mesh tree.
Step7: Give a name to required parts by face section/edge selection>click update by right-clicking on the mesh>check the mesh metric matrix by select elementquality should be not less than 5%.

CASE-1
Element Size = 0.5mm
Nodes = 4753
Elements = 4608
Mesh Type: Baseline mesh coarse
Element Quality

Figure (6): shows different parameters for 0.5mm mesh size.

Step8: close mesh module.

Setup/Processing
Step9: Double click on setup to open the workbench

Step10: select for Time -Transient solution, Solver Type -pressure-based solvers, Velocity Formulation - Absolute and Physics-none from the model.>select the laminar model for turbulent flow based upon the condition.

Properties:
Solver: Transient Pressure-based
Gravity: X = 0 m/s^2
Y = -9.81 m/s^2
Flow type: Laminar
Model type: Multiphase
Method: Implicit
Material: Air and Water-Liquid

Step11: setup the boundary condition as per the given data.

In the multiphase model select VOF i.e volume of fluid it is a widely used method. for simulation of multiphase flow>implicit >volume. fraction cut off.

Material

select material air and water >specify primary phase =water>secondary phase=air for the first case.
select material oil and water >specify primary phase =water>secondary phase=oil for the second case.

Oil Properties:- Density= 400 kg/m , Dynamic Viscosity = 1.06 kg/m-s.

perform standard initialization >patch >1 for water surface >0 for air surface for the first case.
perform standard initialization >patch >1 for water surface >0 for oil surface for the first case.

Create Animation

Time Step Size = 0.025
No. of Time Steps =1500 Iterations
Max. no. iterations per time steps=20

Post-processing/Results
Step12: Not required /the results. are the animation. file extract from the solution animation.
and the residual plot which gets generated along the simulation run and completes after settling of high-density fluid at the bottom.

base line simulation

RESULT AND DISCUSSIONS
The results of various Simulation tests conducted Rayleigh-Taylor instability as well as are calculated and explained with the help of different graphs and charts which are plotted by using Ansys Fluent & MS-Excel software. Comparisons of results are also done Case I and Case II. The compared results are summarized as the following:-
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.

Models used for Rayleigh-Taylor instability[2]

Plateau–Rayleigh

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 Ryleigh- 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, they tend to minimize their surface area.

Kelvin–Helmholtz instability:
The theory predicts the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds. This instability occurs where there is a velocity shear or velocity difference between the interface of the multiphase fluids.

Richtmyer–Meshkov instability:
Instability occurs when there is an impulsive acceleration between the two fluids. This type of instability can occur when there is a shock at greater speeds. 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 Ryleigh- Taylor instability.

Rayleigh- taylor instability :
1. Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behavior of the instability. How can we use the Atwood number to validate our simulation result?

Atwood Number:
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 water /air multiphase fluid
A = (1000 - 1.25) / (1000 + 1.25) = 0.9975

Atwood number is an important parameter in the study of Rayleigh-Taylor instability. For Atwood number close to 0, RT instability flows take the form asymmetric “finger” of fluid; for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

The calculated Atwood number is close to 1 and from the simulation results, it is found that when high dense fluid i.e. water poured upon low dense fluid i.e. air under gravity, the formation of air bubble-like plumes takes place which travels towards the upward region in the form of waves and some gets trapped at the lower regions during the initial stages, which afterward try to move towards the upper region and at the end, both phases gets separated with some diffusivity left at the middle portion between them. Thus, the calculated Atwood number is validated for our simulation results.

PART-II
Perform the analysis for a refined mesh with water and oil. Find Atwood number for this case and discuss the effect of Atwood number & viscosity by
observing the behavior of instabilities.
Oil Properties:- Density= 400 kg/m , Dynamic Viscosity = 1.06 kg/m-s.

Everything is same as above, just change the material oil instead Water

Conclusion

The numerical analysis for the RTI phenomena with Air and Water as phases in one domain and user-defined material and Water as phases in another domain under transient conditions using the laminar viscous and VOF Multiphase model in ANSYS Fluent 2020 R1 was well understood.

It was realised that while the first stage of the RTI phenomena due to its relatively small velocity field or amplitudes smaller to its wavelengths could be computed using linear form of dynamical equations, the underlying influence of the density ratio or Atwood number governed the formation of different structures in the latter stages of fluid flow development which subsequently determined the interaction between them that would eventually deciding whether the regime of chaotic mixing would be arrived to earlier or later.

The analysis of the RTI phenomena carried out in this numerical analysis does not consider the role of other real world scenarios such as surface tension, viscosity, etc., and that the accuracy of the obtained solution could improve with further reduction in the time-step size, however, the
overall instability behaviour with the formation of structures like spikes, bubbles, mushrooms, etc., was well observed using the CFD model configured for the given setup across its different stages.