Week 5 - Rayleigh Taylor Instability

Aim :- To perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5mm.To  explain the results by showing the animations and to study the effect of mesh size on simulation. We use Water and Air in this simulation and in one more case we use Water and a User-defined material having density = 400kg/m^3 and viscosity = 0.001kg/m-s . In these simulations we have to suspend Water over Air and User-defined material and then we have to run the simulations and make an animation by calculating the Water Volume fraction.

Introduction and Theory :- 

Rayleigh Taylor Instability :- The Rayleigh Taylor instability or RT instability (named after Lord Rayleigh and G.I Taylor), is an 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 fluid on top of the less denser one and both subjecting to the Earth's gravity.This was the set-up as studied by Lord Rayleigh.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 as the denser material moves down under the gravitaional field. As the RT instability develops, the initial peturbations progress from a linear growth phase to a non-linear growth phase,eventually developing "plumes" flowing upwards (in the gravitational buyoyancy sense) and "spikes" falling downwards.
• The evolution of the Rayleigh Taylor instability involves 4 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.At the end of his first stage,when non-linear effects begin to appear,one observes the beginnings of the formation of the mushroom shaped spikes.The growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models.The spikes and bubbles then begin to interact with one another in the third stage.Bubble merging takes place,where spikes and bubbles of smaller wavelength 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 the final stage in the evolution.

Crab Nebula:-

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

Kelvin-Helmholtz instability (named after Lord Kelvin and Hermann von Helmholtz):- The Kelvin-Helmholtz instability can occur when there is a velocity shear in a single continuous 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 the waves on the water surface.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 introduced at the boundary connecting the fluids.For a continuously varying distribution of densities and velocities (with the lighter layers uppermost, so that the fluid is RT-stable) the dynamics of the KH instability is described by the Taylor-Goldstein equation and its onset is given by the Richardson number (Ri) 

Plateau Rayleigh Instability :- The Plateau 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 a part of a greater branch of fluid dynamics concerned with the fluid thread breakup.This fluid instability is exploted in the design of a particular type of ink jet technology where 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.

Buoyancy Drag model :- It is a model which uses ordinary differential equations to evolve the width of the mixing layer.These models can't model multiple mixing interfaces,also donot adress demixing known as counter gradient transport i.e, reduction of total fluid masses within the mixing zone.

K-L model :- The K-L model was proposed by Dimonte and Tipton for describing the turbulent self similar regime of RT and RM induced mixing.

Computational methodology :- 

• All the simulations were performed with SIMPLE algorithm using 2nd-order Upwind scheme.To account for the difference in accuracy,the grid spacing is refined.
• The steady state analysis is used when we are more concerned with the end result.But for Rayleigh-Taylor instability we are more interested in observing the real time changes of the interface and the fluid region.The real time observation is possible in Transient state analysis.In steady state simulation we can't see the small time variation of instability and it is used for the final state results.

Performing the Rayleigh Taylor instability simulation for 2 different mesh sizes(with the base mesh 0.5mm):  

Geometry :- Click and drag Ansys Fluent in Ansys workbench to create a new project and name it Rayleigh Taylor instability.

1. Double click on geometry to open the spaceclaim and now we have  to create two squares of dimensions 20mm X 20mm one upper and the other lower.
2. First create the lower volume and then extract it,then create the upper volume and then extract it.
3. Name the lower volume as 'Air' and the upper volume as 'Water'.
4. And then share the topology such that they share one interface edge.(Use 'shareprep' tool from workbench.)
5. Now close the geometry.

SOLVING AND MODELLING APPROACH:

1.First i open the workbench and then i drag the anasys fluent into the work space. then i open then geometry on the popup and i open the spaceclaim and import the geometry into the spaceclaim.

 

fig:setup window

2.then i made square with 20*20m for water and i convert into 3D by pull command option, then i created another one for air with same dimensions and i converted into 3D. then i used command share tropology.

fig:geometry

fig:gemetry setup.

3.then i went to the meshing part to make mesh on the geometry with the 0.5mm and 0.3mm.

fig:mesh setup.

case 1 mesh size with 0.5mm water and air.

fig:meshed geometry of 0.5mm.

fig: mesh quality.

case 2 mesh size with 0.3mm water and air.

fig:meshsize 0.3mm.

fig: mesh quality.

case 3 mesh size with 0.3mm water and user material i.e density = 400 kg/m3, viscosity = 0.001 kg/m-s.

fig:mesh size 0.3mm

fig:mesh quuality.

4.then i went to the solver setup to give the input conditions to run the simulation. here i have given the boundary condition like input fluids has air and water for case 1 and 2. for case 3 water and user define material that is properties density = 400 kg/m3, viscosity = 0.001 kg/m-s.

5.then i went to the multphase and here i have set the volume of fraction and i set the domains for air and water. then i initialized the simulation and i went for the patching here i set water for 1 and air for 0, for the case water for one and user define material of zero.

6.then i have created the report for the volume fraction, contour plot for the volume fraction and also i have made animations for each time steps.

OUTPUT:

 CASE 1

fig: Residual plot.

fig:volume fraction of the stable stage.

fig:volume of fraction at middle.

fig:volume of fraction of middle.

fig:time steps.

 

 

CASE 2

fig:Residual plot

fig:volume of fraction final stage.

fig:volume of fraction of middle stage.

fig: volume of fraction at middle stage.

fig: Time steps.

 

CASE 3

fig: residual plot.

fig: volume of fraction at stage one.

fig: volume of fraction at middle.

fig: volume of fraction at meddle.

fig: volume of fraction at stable.

fig:time steps.

 

Q2. Explain why a steady-state approach might not be suitable for these types of simulation?

Here in this study, the instabilities information is sensitive and completely time-dependent. At each and every time step, the instabilities grow exponentially. Thugh steady state simulation is capable t capture those instabilities but the result would not be time accurate. Anyways transient state calculation with reduced time-steps can capture these instabilities at each an every time-steps and it gives accurate time dependent results.

CALCULATIONS:

Atwood number=(ρ1-ρ2)/ρ1+ρ2)

ρ1= water density

ρ2=user define material density

ρ1=1000 kg/m3

ρ2=400 kg/m3

Atwood number=1000-400/1000+400=0.428571.

CONCLUSION:

here i conclued that the penetration speed is dependent on the density differenace between ligther fluid and heavier fluid.

the mesh size capture the accurate result in the peneration of the fluid motion. if the density different is small the motion of the penetration of the fluid motion is slow and if density different is high then the penetration motion is high.