Rayleigh Taylor instability behaviour of two diffrent fluids

Title:  Rayleigh Taylor instability  behaviour of two diffrent fluids

Objective

1. To study  CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves

2.Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes.

3.Define the Atwood Number.

Theory 

Rayleigh instability mean

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

Example 

1.Behavior of water suspended above oil in the gravity earth.

2. Mushroom clouds like those from volcanics eruption and atmospheric nuclear explosion.

3. Supernova explosions in which expanding core gas is accelerated into denser shell gas.

4. Instabilities in plasma fusion reactors.

5. Inertial confinement fusion.

 Behavior of water suspended above oil in Rayleigh instability

              Water suspended a top of oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immisicble fluid, the more dense on top & less dense on bottom and both subject to the Earth\'s gravity.

                The equilbrium here is unstable to any 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 setup called as Lord Rayleigh instability.

1. k-L turbulence model,

The k-L turbulence model, where k is the turbulent kinetic energy and L represents the turbulent eddy scale length, is a two-equation turbulence model that has been proposed to simulate turbulence induced by Rayleigh-Taylor (RT) and Richtmyer Meshkov (RM) instabilities. k-L model is describing both shear-driven and buoyancy-driven instabilities.

2. Richtmyer–Meshkov instability Model

                    Richtmyer–Meshkov instability 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 small amplitude perturbations which initially grow linearly with time. The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers.

3. Kelvin–Helmholtz instability 

                       Kelvin–Helmholtz instability can occur when there is velocity shear in a single continious fluid, or where there is a velocity difference across the interface between twom fluids.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.

4. Lattice Boltzmann Method (LBE)

               It is a minimal form of Boltzmann kinetic equation which is meant to simulate the dynamic behaviour of fluid flows without directly solving the equations of continuum fluid mechanics. Instead, macroscopic fluid dynamics emerges from the underlying dynamics of a fictitious ensemble of particles, whose motion and interactions are confined to a regular space-time lattice.

5.Plateau–Rayleigh instability

      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 instability and is part of a greater branch of fluid dynamics concerned with fluid thred breakup.

Geometry made in Spaceclaim

Setup for simulation

1. Time   -Transient

2. Gravity- Y-Direction=-9.81m/s2

3.Viscous Mode-Laminar

4. Homogenous Model-Volume of Fluid

5.Volume Fraction parameter- Implicit

6. Phase

              1. Air- Primary phase

              2. Water-Secondary phase

7. Patch

       zone of patch-water surface=1

      zone of patch-air surface=0

8.  No.of Time step=0.005 sec, No.of time step=1000

Case I: Coarse Mesh

Coarse mesh use of

element size 0.5m, Node=3321, elemets=3200

Residual plot

Case II: Medium Mesh

Medium mesh use of

element size 0.3m, Node=9180, elemets=8978

Case III: Fine Mesh

fine mesh use of

element size 0.2 m, Node=20301, elemets=20000

Diffrence between three cases

Its found that air is pushes the water in upward direction & hence Rayleigh Taylor instability observed

Case I:   In this case at the intially there is create bubble & this bubble is travel in upward direction of water area. Also it is found that there is generation some vertices are created.

Case II:  In this case we can see more amoun of air travelled in upward direction , also observed that small air pockets area created in water area.

Case III:  In this case two fluids is completely seprate out. Air is reach out in upword diraction, while due to gravity force water is reach in bottom portion

Steady state Not suitable for this Challanges- 

      As this challange is related with study of  instability of an interface between two fluids. so ,here we are interested to study how water & air mixture is taken place, how air is behave with water. So, this can be identify by the only by Transient state. In case of steady state we are intersted for only final solution.

Atwood No.

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

where

= density of heavier fluid is water

= density of lighter fluid is Air

 so ,                         A= (998-1.225)/(998+1.225)

                               A=0.9975

At Atwood No=0 Rayleigh Taylor instability flow take place like fingers

At Atwood No=1 Rayleigh Taylor instability flow take place like bubble like plumes

As Atwood no. is increases the instability of fluid is increase & which result in compression & expansion of effect is identify . where 0<A<1,indicate that mixing becomes qualitatively different at high density ratios (the variable density case) compared to the case when the densities of the two fluids are commensurate  Thus, in an idealized triply periodic configuration, pure heavy fluid mixes more slowly than pure light fluid.

In our case we found that Atwood no is nearly equall to 1, so , heavier fluid takes bubble formation, it shows that simulation is right & it validate.