Week 5 - Rayleigh Taylor Instability

* What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves?

The rayleigh taylor in stability is an instability of an interfance between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. examples include the behavior of water suspended above oil in the gravity of earth mushroom clouds like those from volcanic eruption and atmospheric nuclear instability in plasma fusion reactors and inertial confinement fusion.

There are some pratical instability CFD models that have been based on the mathematicaly analysis of rayleigh taylor waves:

* kelvin-helmholtz instability:

These can occur when there is velocity shear in a single continous fluid or where there is a velocity difference across the interface between two fluids. an example is wind blowing over water the instability manifests in wave on the water surface.more generally clouds, the ocean, saturn's bands , and sun's corona show this instability.

The theory predicts the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds. helmholtz studied the dynamics of two fluids of different densities when a small disturbance such as a wave was introduce at the boundary connecting the fluids. For a continously varying distribution of density and velocity the dynamics of the KH instability is described by the taylor-goldstein equation and its onset is given by the richardson number Ri. typically the layer is unstable for Ri<0.25. These effects are common in cloud layers. the study of this instability is applicable in palsma physics for example in inertial confinement fusion and the plasma berryllium interface. numerically the KH instability is simulated in a temoral or a spatial approch. the temporal approach experimental consider the flow in a periodic box moving at mean speed (absolute instability). in the spatial approach experimenters simulate a lab experiment with natural inlet and outlet conditions (convective instability).

* richtmyer-meshkov instability:

 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 samll amplitude perturbations which initialy grow lineraly 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 fluid mix. this instability can be considered the impulsive acceleration limit of the rayleigh taylor instability

* Plateau-rayleigh instability:

Its 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 fluid is perturbed into a steady stream of dropkets.

the driving force of the plateau rayleigh instability is that liquids by virtue of their surface tensions tend to minimize their surface area. considering the amount of work has been done recently on the final pinching profile by attacking it with self similar solutions.

* Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes.

GEOMETRY:

Opened workbench the fluid flow fluent was dragged into the main operating window then space claim was opened and a sketch was created on X-Y plane .the rectangle was created on the sketching module with 20*20mm dimension and surface was created from that sketch and using the pull command it is created into surface which is moved into new component air. again new component is created under this new sketch is creted same as air, one above the other water with same dimension and using pull command it is also coonverted to surface.now for these two surface we have to share topology so that we can generate a conformal mesh onnmeshing module .only single edge was shared from the surface which is connecting both the components.

A portion of water is suspende above a portion of air and is allowed to mix under the influence of gravity.

Due to the difference in the densities of water and air there will be an instability developing at the interface which will settle finally when the higher density water settles below the lower density air.

SETUP:

Physics

Solver: pressure-based

Time:Transient

Gravity: x=0 m/s^2

            Y=-9.81m/s^2

Flow type : laminar

Multiphase model type: volume of fluid

Formulation: Implicit

Material AIR and WATER LIQUID.

with the above shown geometry and setup options rayleigh taylor instability simulation is carried out for 2 different mesh sizes-

CASE 1= with element size 0.5mm

CASE 2= with element size 0.3mm

for both the cases geometry and setup setting are same as described above

MESHING 

CASE 1:

element size:0.5mm

no of nodes:3321

no of elements:3200

with the above mesh and given setup:

simulation is run with-Time step size=0.005 & no of time steps=1000

Results

scaled residuals

ANIMATION:

https://drive.google.com/file/d/1Gr7_CcZe2nT9fL60gu2-23DQAPGM1Xcl/view?usp=sharing

CASE 2:

Element size:0.3mm

no of nodes:9180

no of element:8978

simulation is run with-Time step size=0.005 & no of time steps=1500

Results

scaled residuals

ANIMATION:

 https://drive.google.com/file/d/1PuFvYzlilE0ukjB8OE4jgwQayFM0egT4/view?usp=sharing

CASE 3:

 one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.

Element size:0.2mm

 ANIMATION:

 https://drive.google.com/file/d/1TRu8URU817mZnZFD2cEUg8AWbZAYhjpz/view?usp=sharing

CONCLUSION:

* Rayleigh taylor instability was observed at the interface of the fluid with different density under the influence of gravity.

* the lower density pushes the higher density fluid above it. the heavier fluid which is water in this case comes at the bottom and replace the air from the bottom.

* the formation of air bubbles takes place which moves upward.

* the mesh refinment leads to better simulation of the rayleigh taylor instability all the instability and the irregularities that take place at the interface are better visualized as we refined the mesh.

*the lower dens fluid pierces into the high density fluid as it seems to be the propagation of shock wave.

*in case 1 at the interface we can see that some distributaion are formed as time passed the large distribution is formed its forces into the high dense fluid like a shock wave. 

*in case 2 at the interface we can see the distribution which are more than case 1 pushing through the high dense fluid mushroom like structure is formed and the formulation of bubbles can be seen around the mushroom structure.

*in case 3 we can observe that the final contour is much more detailed and accurate. this result can be considered to be highly accurated.however timer required to perform this simulation is also much higher than the previous meshes.hence only if an extremenly high accuracy is required from the solution such a high degree of mesh refinement is suggested.

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

The steady state analysis is used when we are more concreted with the end result. But for RT instability we are more intrested in observing the real time changes of the interface and the fluid region. the real time observation is possible is transient state analysis.

the different between the steady and transient is that you can see small time variation of instability the steady state simulation is performed if we are concerned more about the final state results or the equilibrium state. in RT instability CFD models we are more concerned to learn about the transient of thr irregularities that starts developing when we pour high dense fluid upon low dense fluid under gravity effects so by using transient solver along with refined mesh of the model, we can compute the smooth transient of irregularities that takes place at the interface of the fluids. the final state results for both the steady state and transient state will be the same.

ATWOOD NUMBER:

the atwood number is a dimensioneless number which is used in the field of fluid dynamic to study the hydrodynamic instabilities.

Atwood number for the instability involving water and air:

Atwood number for the instability involving water and air:

Based on the atwood number for A is closer to 0 then the instability begins with a finger formation whereas if the atwood number is closer to 1 it follows a bubble formation.