Week 5 - Rayleigh Taylor Instability
AIM = Rayleigh Taylor Instability
Introduction=he Rayleigh-Taylor instability, or RT instability (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 behavior 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.
theory of the topic water suspended atop oil is an everyday example of Rayleigh-Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the more dense on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: 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 realization 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 (assuming other variables such as surface tension and viscosity are negligible here). 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.
Solving & Modeling approach
gemotry sections
In geometry section we have to create geometry part of solid as show in above figure.
After create geometry we have to use pull tool option as we know this geometry is 2D,and upper block is called as water block and lower block is called air block and both block is create in new component due to this we can use share tool
Then we use share option this option will help to create a fine mesh in between the two block after using share option the edge where the share tool is applied is become pink and after click ok this pink turn in to blue it means share tool is applied successfully in geometry.
Mesh section
In meshing section we have to create fine mesh in geometry here we use 0.13mm size of a element on meshing and then generate the mesh and then click on statistics option then we saw that number of element is 47895 and number of nodes is 47432 and as we know in student version maximum number of node is 50000 so this is fine mesh.
Setup section
Here we use pressure based solver because further we use fraction of fluid in multiphase. And transient state of fluid because ANSYS Fluent can solve the conservation equations in a time-dependent manner, to simulate a wide variety of time-dependent phenomena, such as
- vortex shedding and other time-periodic phenomena
- compressible filling and emptying problems
- transient heat conduction
and enable gravitational force but we know our geometry is 2D so z component is disable here.
And our geometry is in x,y coordinate so here y is vertical component and water is fall from upward to downwards so here we have to put y value that is – 9.81
Here we use laminar viscous model because here we use two immiscible liquid.
Creating material which is used as a fluid and for more material we can add material from fluent data base and here we also able to define the material property
From database we have to add water as a material and copy this material
Finally here we have two immiscible Fluid one is air and another is water.
After create material we have to move toward multiphase model here we choose volume of fluid because out fluid is immiscible and we use implicit formulation.
Then we have setup the primary and secondary phase here primary phase is air.
Secondary phase is water.
Then we have to patch the phase with material but before patching we must have to hybrid initialize the condition
Then patching option is occur
As we seen in above figure water is selected for phase and variable is volume fraction and value is 1
Here value indicate that water is different fluid and air is different fluid and value of air is 0.
After patching create contour this will help us to analysis the flow of fluids.
Then create animation on the bases of contours.
And finally hi the calculate option
After complete of calculation we get the result in the foam of contours and animation
Some slide is shown below
Starting phase of contours
After 50 iteration
After 120 iteration
After 175 iteration
ATWOOD NUMBER
998.2 - 1.225
-------------------- = 0.997
998.2 + 1.225
Here Atwood number is 0.997
QUESTIONS
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.
Practical CFD models that are based on mathematical analysis of rayleigh taylor waves:
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 two fluids.
Richtmyer-Meshkov instability:This type of instability occurs when there is an impulsive accelaration between 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.
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 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 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 tensions, 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.
2) Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations( Attach them from your google drive). Observe the differences in the animation and explain the effect of mesh size on simulation. Also, explain why a steady-state approach might not be suitable for these types of simulation.[ Choose Air and Water for this simulation]
Here we use two mesh size one is 0.5mm and other is 0.13mm
Case A
Element size 0.5mm
Elements and nodes number.
Mesh at 0.5mm element size
Case B
Element size 0.13mm
Element and nude number.
MESH
|
Element size |
Number of elements |
Number of nodes |
|
|
Case A |
0.05mm |
3321 |
3200 |
|
Case B |
0.13mm |
47895 |
47432 |
Animation of case A and case B
Case A
Starting contour of case A
Starting contour of case B
When we observe the contour of both case A and case B then we realize water start fall downward direction due to gravity and density of water but the behavior of falling down is different in case A initial it not foam mushroom structure and in case B it form mushroom structure. This happen because difference in number of element and number of nodes present in mesh.
Case A
Contour graph at 25 iterations of case A
Case B
Contour graph at 25 iterations of case B
In the above two figure in case A and case B observe after 25 iteration there are huge difference in contours graph this is because different in number of nodes and number of elements in case A less number of element are present so the contours graph are not fine and result are not prefect but in case B number of element and nodes are more so contours graph are more fine and perfect.
Case A
Final contours of case A
Case B
Final contours of case B
As shown in above figure in both cases A and B volume of fluid of water layer area is too large in case A and in case B the area is very small because number of element of case A is less and due to lass number of element it don’t not get fine result.
But in case B the area of volume of fluid of water is area because more number of element and nodes.
animation of case A
animation of case B
2) Run one more simulation with water and user-defined material(density = 400 kg/m3, viscosity = 0.001 kg/m-s) for refined mesh.
Geometry
Mesh
Setup
All steps are same as we shown in above setup condition here we only change the property of fluid material of air
Here we change the material properties like density and viscosity as given condition and finally we simulate.
Stimulation
As we show in above figure of contour the changing rate of volume of fluid of water is very slow this happen due to increase in viscosity and density it means viscosity and density is effect the rate of cahnve of volume of fluid.
simulation with water and user-defined material
ATWOOD NUMBER
998.2-400
---------------- = 0.42
998.2+400
So here Atwood number (A) is 0.42
3) Define the Atwood Number. Find out the Atwood number for both the cases and explain how the variation in Atwood number in the above two cases affects the behavior of the instability.
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
RESULT
When Atwood number is increase then rate of movement of volume of fluid is ALSO increase.