Week 5 - Rayleigh Taylor Instability

Aim: Study and Simulation of Rayleigh Taylor Instability using VOF method.

Objective:

• To research the significance of the Rayleigh Taylor instability

• ANSYS FLUENT multiphase flow analysis.

• To research the fluid volume approach and its significance.

• Examining how mesh refinement affects the Rayleigh Taylor instability

• To research and compute the Atwood number for the modelling of the difference in density.

• To describe the Atwood number-based behaviour of instability.

Introduction: 

When applied to pipe flow and other conveying media, single phase fluid flows predominate the majority of the time. Multiphase flows have recently been introduced in industrial development applications and are growing in popularity. Multiphase flow refers to the idea of doing a task using two or more distinct liquids, whether or not they are miscible. Depending on the application, the multiphase flow may consist of two liquids that are identical but have different densities, or it may be a gas-liquid flow. Multiphase flows are used in the chemical industry, nuclear industry, manufacturing facilities, and product agitator mixing.

Rayleigh Taylor instability is one of the sector of multiphase flow which is based on the density stratification in which heavier fluid is on top of lighter fluid which is inherently unstable. A small perturbation can be providing to the interface which will allow fluid to displace in the field of gravity. The cause of instability is mostly depending on the density of the two liquids. The governing equation which resembles the Rayleigh Taylor instability in 2 dimension is mostly assumed on the basis of two immiscible liquids i.e. viscosity of liquids is not consider. In such instability state the fluid always tries to come towards equilibrium position hence can be tracked with respect to time. The equation of Rayleigh Taylor instability is both linear and non-linear in nature. Linear in nature when the perturbation or disturbance are small and non-linear in nature when perturbation are high. Rayleigh Taylor instability are common in real life application in which one of the best example is oil floating over water. From which we know that oil having higher density and water has lower henceforth oil always float over water. The other application of the Rayleigh Taylor instability can be seen in the plasma fusion reactors and confinement fusion where inertial effects are predominant. In space also we can see Rayleigh Taylor instability in supernova explosion also where single star can be collapse when the matter at the core is greatly increase where gravitational pull is increased cause to collapse because of difference in matter and densities. Mushroom clouds are another example of Rayleigh Taylor instability in which sudden formation of large low density gases at any particular altitude.

Rayleigh Taylor instability has been used for many other CFD codes to run simulation. As we know that depending upon the density of liquid variation of fluid flow can be different also depending upon the velocity of flow and turbulence, generation of plateau can be different henceforth different model are developed. Those models are:

• Saffman-Taylor Instability

Viscosity fingering, a morphologically imbalanced interface between the two fluids that occurred in the porous media, is another name for this instability. This phenomena is primarily seen through soil-based mediums' drainage processes. The more viscous fluid is further displaced downward to produce the fingering effect when the less viscous fluid is injected. This instability is also produced when no forced displacement is offered, hence it also affects the gravitational force as an RT instability.

• Richtmyer-Meshkov Instability  

Richtmyer-Meshkov instability, in contrast to Rayleigh Taylor instability, is caused by an external force and is governed by fluid acceleration. When two fluids with different viscosities and densities are impulsively accelerated, this instability develops. This instability begins as a little perturbation, increases linearly over time, and then enters a nonlinear phase during which bubbles occur in the lighter liquid as it penetrates the heavier fluid and spikes develop in the higher density fluid as it penetrates the lower density fluid. After a tumultuous period, the two fluids are combined. Examples of the Richtmyer-Meshkov instability include inertial confinement fusion, where the instability is produced by accelerating the hot shell material to the fuel layer. Another example of Richtmyer-Meshkov Instability in Scramjet engines

• Plateau-Rayleigh Instability

Plateau-Rayleigh In more specific aspects, instability is more like the Rayleigh Taylor instability. The mixing of two liquids might vary depending on the type of liquid utilised and the disturbance applied. The fluid thread breakdown, a component of fluid dynamics, is the major focus of this instability. The process of a fluid bubble breaking into multiple pieces as a result of surface tension and contact area is known as fluid thread breakdown. the simulations that concentrate more on the Plateau-Rayleigh Instability is the movement of an ink-jet through air or another liquid.This simulation allows us to watch an ink bubble burst as it passes through a fluid with a decreased density. Water leaking from faucets and inkjet printers are two frequent examples of this instability.

• The Atwood number (A)It 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

   = density of lighter fluid

Modelling Approach:

The approach for simulation is very simple and it is a 2D analysis in which two surface are connected to each with an interface between them. The solution methodology is created based on the objective defined for simulation. Simulation will be carried in 4 different cases which are:

• Case 1: - Mesh size 0.5mm for Water and Air as default material.
• Case 2: - Mesh size 0.25mm for Water and Air as default material.
• Case 3: - Mesh size 0.25mm for Water as default material and one user defined material.

As long as there are two fluids present, multiphase simulation will be carried out. We will employ the Volume of Fluid approach, which is well known for sloshing simulations, for the multiphase. We previously talked about the existence of a phase interface between two liquids, which may or may not be immiscible. It is crucial to develop a mechanism that accurately anticipates the kinematic motion of the fluid interface in order to monitor it.The moving grid method, marker particle methods, and VOF method are the commonly utilised methods in FLUENT to track the interface. Small free surface deformation is typically treated with the moving grid or adaptive grid approach. Large deformation problems cannot be handled by this method, which necessitates changing the mesh topology. Using polynomials and nearby marker particles, the interface is traced in the case of the marker technique. However, it can only be used for brief 2D simulations. The Volume of Fluid technique offers a 3D interface with excellent volumetric conservation.We cannot have two fluids with the same density in the VOF model because they will interact on a molecular level. The interface of a VOF model cannot be smaller than a volumetric mesh, which is its single restriction.

Geometry :

• Geometry is created in spaceclaim with a dimension of 20x20.
• Two squares has been created in such a way that it is placed one over the other.
• After that pull tool is used to convert a sketch into surfaces.
• Then share the topology between two surfaces for the effective transfer of information and to connect  a mesh between two interfaces.

Mesh :

Case 1 :

Details :

Mesh	Coarse
Element size	0.5 mm
Number of elements	3200
Density of primary fluid	Air (1.225 kg/$m^3$)
Density of secondary fluid	Water (998.2 kg/$m^3$)
Atwood number	0.9975

 

 

Mesh quality :

 

STEP 3 : SETUP

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Unsteady
Models	Viscous model : laminar Multiphase      : Volume of fluid                      : Implicit formulation                      : 2 phases
Material	Fluid type   : Air                   : water liquid  Solid-type   : Aluminium
Cell zones	Fluid type  : Air                  : water
Boundaries	Standard
Reference values	Standard
Initialization	Standard
Patch	Water-surface (1 value of volume fraction of water)  Air-surface (0 value of volume fraction of water)

 

Residual plot :

Animation :

 https://youtu.be/OB-_Ujjj-Ho

 

Case 2 :

Details :

Mesh	Refined
Element size	0.25 mm
Number of elements	8000
Density of primary fluid	Air (1.225 kg/$m^3$)
Density of secondary fluid	Water (998.2 kg/$m^3$)
Atwood number	0.9975

 

Mesh quality :

STEP 3 : SETUP

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Unsteady
Models	Viscous model : laminar Multiphase      : Volume of fluid                      : Implicit formulation                      : 2 phases
Material	Fluid type   : Air                   : water liquid  Solid-type   : Aluminium
Cell zones	Fluid type  : Air                  : water
Boundaries	Standard
Reference values	Standard
Initialization	Standard
Patch	Water-surface (1 value of volume fraction of water)  Air-surface (0 value of volume fraction of water)

Residuals :

 

Animation :

 https://youtu.be/2btwEyPjcmw

 

Case 3 :

Details :

Mesh	Refined
Element size	0.25 mm
Number of elements	24642
Density of primary fluid	User-defined material               (400 kg/$m^3$)
Density of secondary fluid	Water (998.2 kg/$m^3$)
Atwood number	0.4278

 

STEP 3 :

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Un-Steady
Models	Viscous model : laminar Multiphase      : Volume of fluid                      : Implicit formulation                      : 2 phases
Material	Fluid type      : user-defined                      : water liquid  Solid-type      : Aluminium
Cell zones	Fluid type      : Air                      : water
Boundaries	Standard
Reference values	Standard
Initialization	Standard
Patch	Water-surface (1 value of volume fraction of water)  Air-surface (0 value of volume fraction of water)

 

Residual :

 

Animation :

https://youtu.be/HFATcD2E9mc

 

Conclusion :  

• Why not steady state solver:

Although the steady state simulation can also be solved, the VOF model is typically employed to handle time-dependent problems. Only when our solution is independent of the beginning conditions and specific inflow and outflow boundary conditions are present for each phase can a steady state simulation of the VOF model make sense. Only when results are needed at the conclusion of a time step and no intermediate phase results are needed, is a steady state solution employed. It is crucial to ensure that the volume fraction is a function of time in the event of instability since it will alter with regard to time.

• Effect of Mesh on Results

The area weighted average of the volume fraction for water is dropping earlier in the baseline mesh configuration, and as the mesh size is decreasing, the drop is going forward in time. This indicates that as the number of elements are increased, we are better capturing flow phenomena. Thus, we can conclude that mesh size influences how hydrodynamic instability convergences. Additionally, it was discovered that the number of simulation iterations rose by a factor of 2.5 to 3 as the mesh size was reduced.

• Comparison of Atwood number

In cases 1 and 2, the Atwood number is getting closer to 1, while in the case of user-defined content, it has nearly been cut in half. The stratification effect is more pronounced when the Atwood number is close to 1, which indicates that mixing is symmetrical in nature. As we raise material density in the case of user-defined material, mixing effects are decreased and just a minor plateau is visible in the simulation.

• Comparison of Animation

Case 1 made it very clear that a lack of mesh elements causes very rapid mixing and a rapid gain in equilibrium position for both liquids. In the second instance, we can observe that the mixing is slower and that there are more iterations required to finish the simulation. In the third scenario, mixing happens very slowly and we can more clearly see how water is moving downward owing to gravity. The user defined content has been introduced in the fourth example. In this instance, water interacting with user-defined material results in the production of a mushroom shape.