Week 5 - Rayleigh Taylor Instability

What is Rayleigh-Taylor instability?

• Inversion in density develops gravitational instability at the interface between two fluids of contrasting densities, where the relatively low-density fluid pushes the overlying denser fluid.This kind of instability termed Rayleigh–Taylor (RT) instability .
• The classical Rayleigh–Taylor instability occurs when a heavy fluid overlies a lighter one, and the two fluids are separated by a horizontal interface, with a sharp interface separating them lie horizontally with the heavier fluid above the lighter one
• Gravity acts downward, and any small disturbance to the interface results in the unstable growth of fingers of the heavy fluid moving downward and bubbles of the lighter fluid penetrating upward.
• Rayleigh and Taylor considered planar flow and carried out linearized analyses.

we come across Multiphase:

• Multiphase flow is simultaneous parallel flow of Materials with different states or two phases.(liquid_gas phase),its is flow of materials with different chemical properties ,it also exist with same state_or same phase (liquid_liquid).

Types of phases of Multiphase:

• Separated_phase_one/more immisicible fluid in a continious phase separated by an interface.
• Mixed_phase___one/more micible (mixed ) together it contain both separated and dispersed phase.
• Dispersed_phase_it is defined as the phase in which the finite no of phases are spreaded in the continious phase

Types of flow Regimes in Multiphase:

• Bubble flow__
• slug flow
• churn flow
• annular flow
• disperse flow

Types of flows in Multiphase:

• Liquid_gas flow__water and water vapor in heat pipe
• Liquid _solid flow__Turbulent Binghamian slurries
• Gas_solid flow__Pnuematic transport.

Forces in Multiphase:

• Pressure forces__acts on to the surface of the component,the pressure force only enables fluid to flow in the downward direction due to pressure gradient.
• Inertia forecs____The inertia force is a volume force it is responsible for magnitude & direction of the flow ,if the inertia is high then the flow is turbulent,if its low then laminar.
• Viscous forces__acts on the surface of the component,the viscous force only to fluid flow by reducing in a uniform mannner by reducing velocity gradeint b/w them.
• Buoyancy force__is a volume force represent action of gravity in the fluid flow
• Surface_Tension___the surface Tension acts on the line or curve ,its minimizes the surface area of interface.

Solutions for Mutiphase simulations:

• Volume model
• Mixture model
• Eulerian Model
• Wet steam model

Volume of fluid Method:

• Volume of fluid method is a surface tracking method,it is designed to find out the interface b/w two or more immicible fluids,The volume fraction of each fluid in each of the cell throught the domain can be tracked by this method and in this the single set of Momentum eq is shared by the fluid domain.In this method the entire control volume must be filled with single phase or multiphase .This method is only applicable

let us consider

alpha1=phase1

alpha2=phase2

Equations of VOF:

• Volume fraction eq
• Momentum eq
• Energy Eq

Volume fraction eq:

Volume fraction is a "space" occupied by each phase, the tracking of interface b/w the phases is accomplished by the solution of continuity eq for the volume fraction of one or more phases.

Let be p=primary phase

q=secondary phase

For primary phase:

for secondary phase:

Implicit scheme:

The formula for Implicit scheme is as follows

Explicit Scheme:

The formula for Explicit scheme is as follows.

where

n+1=index for new current time step

n=index for previous time step

V=volume of the cell

Momemtum Eq:

In VOF method the single momentum equation is shared throught out the domain and the resulting velocity is shared among the phases.the momentum eq is dependent on the volume fraction of each phases by the properties of density & viscosity. If the viscosity exceeds the value of 1e^3,which leads to the convergence ..

The momentum equation is as follows

The Energy Eq:

• The Energy equation is also shared among the phases,the VOF treats E & T as mass averaged variables.
• The energy eq is as follows

Mixture Model:

• Mixture models are used to model multiphase flow where phases move with different velocities but assume local equillibriumover short spatial length,this method is applicable for liquid _solid flows & pressure based solver.it uses the concept slip velocities to solve different velocity phases.
• Equations for mixed model
• Volume fraction eq
• Continuity eq
• Momentum eq
• Energy eq

The VOF of equation represents space occupied by each phase

• The VOF is of Secondary phase is as follows

• Continuity eq

• Momentum eq

• Energy eq

Eulerian Model:

Eulerian model in ansys fluent for modelling multi phases,this phases can be liquids ,gases,solids,it can be any form of combinations,Many no of secondary phases can be used in this model but limited only by the memory requirements & convergence beheavior,here only single pressure is only shared by the all phases.

Equations of Eulerian model:

• volume fraction eqs
• Continuity eq
• Momentum eq
• Energy eq

Volume fraction eq

Continuity eq

Energy eq

Case1:0.05mm of mesh

Geometry_

geometry_sketch_rectangle_

geometry_pull

Mesh

• Mesh of 0.05mm _mesh _generate Mesh.
• Right click on mesh _generate mesh.

Open fluent:

• Physics_Multiphase_set to volume of fluid_Implicit_apply
• Physics_viscous_set to laminar_apply
• Physics_materials_air_fluid medium
• physics_materials_aluminium_solid medium
• physics_cellzones_select_fluidsie.,air,water.

Case2:

• 0.02mmof Mesh

Case3:

• 0.02mm of mesh with changing some properties
• Like density& viscosity.(density = 400 kg/m³, viscosity = 0.001 kg/m-s)

My observation:

• Whenever relate to the densities the higher density of fluids settle _down as deposits & lower densities float over the fluid,so we called Rayleigh instability of fluids
• eg.,
• A human being is higher denser than water,so during swimming we will be float on the water untill we swim with hands and legs if not, we dipped into the water
• A water bottle with closed cap will float on water,if its opened its dipped into water,due to the density of water.

Here,the density of water, is higher compared to the air, so due to the gravity, its gets distructed, the interface b/w two phases of fluid leads to the instability in water & air , causes water to be settle down & air float over water.. this is called as Rayleigh instability b/w two fluids.

Conclusion:

• Here we gone with mesh dependent study...I gone with 0.05 & 0.02 and also so by changing some parameters like density, viscosity with the finer mesh, we able to see perfectly what happening in the rayleigh instabilityie.,finer mesh increases the accuracy of solution ...more of cells leads more accuracy of solution..