Week 5 - Rayleigh Taylor Instability

AIM - 

To understand different CFD models that are based on the mathematical analysis of Rayleigh-Taylor waves

To perform Rayleigh-Taylor instability simulation for a 2D model using air and water as the material, then simulate again for a refine mesh and finally change the materials to water and a user defined material (density = 400 kg/m^3 , viscosity = 0.001 kg/m-s) and perform simualtion of the refined mesh.

Define Artwood number and explain how the variation of atrwood number for the above two cases affects the behavior of the instability.

OBJECTIVE -

1. To perform Rayleigh Taylor instabilty simulation for water and air as the material
2. To perform a simulation on a refiend mesh.
3. To change the materials of the refined mesh as given above in the aim and perform simulation again 

RAYLEIGH TAYLOR INSTABILITY -

Rayleigh taylor instability is the instability at the interface between two fluids of different densities which occur when the lighter fluid is pushing the heavier fluid. For example , behavior of water suspended above oil in the gravity of earth.

It is modeled as two completely plane-parallel layer of immiscible fluid, the more dense one is on the top of the less densed one and both are subjected to the earths gravity. The equillibrium here is unstable any preturbutations or distribunces of the interface. if the parcel of the heavier fluid is displaced downward and same volume of the less densed fluid is moved upward and this result in less potential energy than the initial state. this disturbance grow further and realases more potential energy as the more densed fluid is displaced downwards due to gravity and the less densed fuild moves upward.

There are more such models based on the mathematical analysis of Rayleigh-Taylor instability as given below

1. Richtmyer-Meshkov instability 
2. Plateau_Rayleigh instability
3. Saffman-Taylor instability
4. Kelvin-Helmholtz instability

Richtmeyer-Meshkov Instability :

Richtmyer-Meshkov instability occur when two fluid of different densities are impulsively accelerated . This instability begins with small preturbation which grows linearly with time. This is followed by bubbles if light fluid penetrates the heavy fluid and with spikes if heavy fluid enters light fluid. the both fluid mix eventually forming a choatic regime. This instability is considered impulsive-acceleration of Raylegh-Taylor instability.

Plateau-Rayleigh Instibility :

Plateau-Rayleigh instability often known as Rayleigh instability. This instability explain how and why the falling stream of fluid breas up into small packects of same volume but less surface area. This was first exploited in the ink jet technology when a jet of liquid is pretubated into a steady stream of droplets. Liquids tend to minimize their surface area by the virtue of their surface tension.

A special case of this is the formation of small droplets when water is dripping from a faucet/tap. When a segment of water begins to separate from the faucet, a neck is formed and then stretched. If the diameter of the faucet is big enough, the neck doesn't get sucked back in, and it undergoes a Plateau–Rayleigh instability and collapses into a small droplet. 

Saffman-Taylor Instability :

Saffman-Taylor instability also know as viscous fingering is the formation of patterns in a morphologically unstable interface between two fluid in a porous medium. It occurs when a less viscous is injected , displacing a more viscous fluid. Viscous fingering can be seen in the experiment on a Hele-Shaw cell which consist of two closely spaced parallel sheets of glasses containing a viscous fluid . Instabilities similar to viscous fingering can also be seen in selfgenerated biological systems.

Kelvin-Helmholtz Instability :

Kelvin-Helmholtz instability occur when there is a velocity shear in a single continuous fluid, or when there is a velocity difference across the interface between two fluids a common exapmle is when wind flows over a water surface , the instablity generated in the form of waves. the kelvin-helmholtz instabilities are also visible in the atmosphere of planets and moons such as cloud formation in the earth's atmosphere .

RAYLEIGH-TAYLOR INSTABILITY SIMULATION -

BASELINE MODEL :

Now here the RT instability simulation for a 2D model is performed using the air and water as the two fluids. We have used a laminar model with volume of Fluid method to simulate the calculation for the two different fluid phases.

for the geometry we have used two square plates of length 20 mm each sharing a common edge between them as shown in the following image. Also we have turned on the share topology to have a conformal mesh.  

For the Meshing , element size of 0.5 mm was used for the baseline case. The elements we by default quadratic which is good and we can use this mesh for our fluent setup.

 the mesh has about 3200 elements

For the Fluent setup ,

We will setup the solver type to pressure based. where we will set the time to transient and apply gravity in the negatve Y direction . 

here we have two materials forst is the  air and another is water , both having all the default properties set by fluent

we will set the viscous model to lanimar and for the calculation in mutliple phases we will be using volume of fluid model, with implicit formulation for the volume of fluid parameters.

for setting up the phases we have set our primary phase to air and the secondary phase to water

to initilize the simulation we will be hitting the initilize button and then patch the solution . while patching we select the phase to water and the variable as volume fraction and use value '0' for the air surface zone and use value '1' for teh water sueface zone as shown below.

now a countour will be set to view the initial patching of the phase for the volume fraction of water on all the surfaces.

the contour shown 1 for the water in red and 0 for the air in blue colour so we can see that water is above the air initally.

Now we will create a solution animation for the above contour so that we can see the distribution of phase over the time until is reaches a steady state.

To run the calculation we will use a time step value of 0.005 s and run the calculation for about 300 time steps and then increase the time step to 0.05 s and run for another 50 time steps so that our solution reaches steady state faster and also the solution wont blow up.

Then we have our final solution where the calculation are complete the hase contour looks like as shown below

the transient simulation animation is given in  the link

Animation Link - Rayleigh-Taylor instability using baseline mesh

REFINED MESH :

Now we will perform the same simulation with a more refined mesh having an element size of 0.1 mm and achieve this meshing we have to insert face meshing on both the faces and set the method as quadrilaterals the total element count for this mesh is 80000

after this we will setup the simulation as we did with the baseline mesh and calculate the results here we will need to start with a smaller time step value of 0.001 s and run for about 3000 time steps the we havent increased the timestep here beause then the solition animation will be showing smooth animation for the first value of time step but the when we increase the timestep the jump is the succeccive contour will be high and we dont see proper animations 

the transient animtion for this simulation is given in the link

Animation Link - Rayleigh_taylor instability refined mesh 

from the above results we can see that as we refine the mesh the instabilities are growing more and have smaller sizes that as compared to the baseline mesh . also it takes a smaller timestep value to compute the calculation or else the solution blows up.

CHANGING THE MATERIALS : 

Now we will be changing the materials used in the simulation to water and a user defined material (density = 400 kg/m^3 ,  viscosity = 0.001 kg/m-s) for this simulation we will be using the refined mesh and set the materials in fluent as shown below

Next is setting up all the other condition the same as earlier and run the simulation with a time step value of 0.05 s and for about 200 and then speed up the solution using a timestep value of 0.1 s for another 50 timesteps

the transient animation for this simulation is given in the link

Animation Link- Rayleigh-Taylor Instability using a used defined material

here we can see as there is less difference in the densitites of the materails as compared to the air and water case the it required less number of timesteps and also we can use a bigger time step value. 

ARTWOOD NUMBER:

Artwood number is a dimensionless number that gives the hydrodynamic instability in density stratified flows, it is the dimensionless density ratio mathematically defined as

`A =(rhounderset(1)() - rhounderset(2)())/(rhounderset(1)() + rhounderset(2)())`

where `rho underset(1)()` = density of heavier fluid

and `rho underset(2)()` = density of lighter fluid

 Artood number is an important parameter in the study of Rayleigh-Taylor instability and Richtmyer-Meshkov instability . In RT instability , the penetration distance of the heavy fluid bubbles into the light fluid is a function of acceleration time scale , given by `Agt^2` , where g is the gravitational acceleration and t is the time.

For the first case with air and water as materials the artood number is given by `A underset(1)()` 

`A underset(1)() = (rho underset(water)()- rho underset(air)()) /(rho underset(water)() + rho underset(air)()) = (998.2-1.225)/(998.2+1.225) =0.998`

Similarlty the artwood number for the second case with water and the user defined material is given by `A underset(2)()` 

`A underset(2)() = (rho underset(water)()- rho underset(air)()) /(rho underset(water)() + rho underset(air)()) = (998.2-400)/(998.2+400) =0.428`

we can clearly see that for second case the artwood number is small so the bubbe formed in this case are small as compared to the bubbles formed for the first case . and it take less time to seperate after the instability.

CONCLUSION :

1. The Rayleigh Taylor instability shows how the instability are formed when a heavier fluid is palced above the lighter fluid under the influence of gravity 
2. When we redice the mesh size it take smaller value of timestpe for the calculation or else the solution blows 
3. after some calculations we can increase the timestep value to speed up our calculations amd attain steady state earlier. the residuals may jump when the timestep is increased but enventually they should come down . if the residuals grows by a huge order then the timestep the we have used should be reduced.
4. Artwood number help to determine the hydrodynamic instability that occur in a density stratified flows.
5. For the water and air we see that the bubbles formed are larger due to higher difference between the densities of both the fluids 
6. For the water and user defined material the bubbles formed are smaller due to lesser difference between the densities of both the fluids