Week 5 - Rayleigh Taylor Instability

AIM:

To conduct the Rayleigh Taylor CFD simulation.

OBJECTIVE:

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.
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]
3. Run one more simulation with water and user-defined Material (density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
4. 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.

PROCEDURE:

Rayleigh Taylor Instability:

• Rayleigh Taylor Instability is an instability of the interface betweeen the two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.
• It is clear from Newtons law of viscosity,fluid cannot resist the slightest shear force and Density is one of the main property of the fluid to incorporate the shear force.
• Based on the density, when lighter and heavier fluids are mixed together, denser fluid comes to top of the lighter fluid, due to the gravity and the density variation at the interface of the liquid is not stable.
• This instability of the interface between the different densities of fluid when the lighter fluid is accelerated towards the heavier fluid is called RAYLEIGH TAYLOR INSTABILITY.

Practical CFD models:

SPH method(Smoother particle hydrodynamics)

• Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows.
• Smoothed-particle hydrodynamics is being increasingly used to model fluid motion as well. This is due to several benefits over traditional grid-based techniques.
• First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass.
• Second, SPH computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations.
• Finally, unlike grid-based techniques, which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly since the particles represent the denser fluid and space represents the lighter fluid.
  

Single-fluid model

• A typical approach used for the analysis of two-phase flows is a mixture model, i.e. the individual fluid phases are assumed to behave as a flowing mixture described in terms of the mixture properties.
• The applied single-fluid model is a five-equation model consisting of the mass, momentum, and energy equations for a vapor/liquid mixture, and two equations describing the formation and growth of the liquid phase.

Two-fluid model

• In the two-fluid model, separate sets of the governing equation for the vapor and liquid phases have been used.
• The interaction between the droplets and the heat exchange between the liquid phase and the solid boundary is not modeled here as well. Additionally, the velocity slip between the vapor and the liquid phase is in this model taken into account.

Turbulence model

• Turbulence models are needed to predict the average mixing behavior inflows that are on average one- or two-dimensional.
• The approach to the construction of the tile turbulence model is guided by experimental behavior. The equations governing turbulent flows can only be solved directly for simple cases of flow.
• For most real-life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

SOLUTION:

In this project, we consider 3 different cases in which 2 cases are based on the type of MESHING and the other is with user-defined materials.

We consider 2 materials air and water placing one above the other and see the effect of gravity acting on them.

1. GEOMETRY:

• Create a square of 20mm*20mm.Use Pull tool in order to create a surface.
• Move the generated component to the New Component. Name it as Air. Now click on design and create a new component --> Water. Repeat sketch just above the Air with the similar dimensions as mentioned.
• Check if both the components i.e Air and Water are created.
• Workbench > click on Share > Ok

2. MESH:

OPEN MESH --> Generate the mesh --> baseline mesh element size (0.5mm).

3. SETUP:

CASE 1:

Overall Element size: 0.5mm

• DOUBLE CLICK SETUP - > FLUENT LAUNCHER ->DOUBLE PRECISION.
• Ansys Fluent is opened.

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Transient
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
Initialization	Patch: Water-surface (1 value of volume fraction of water) : Air-surface(0 value of volume fraction of water)

4. Solution:

• SOLVING -> CONTROLS, REPORTS, INITIALIZATION, RUN CALCULATION.
• CREATE -> select REPORT FILE and REPORT PLOT.
• INITIALIZATION -> select METHOD = HYBRID -> click t=0 INITIALIZE
• RUN CALCULATION -> NO. OF ITERATIONS =150 and press CALCULATE.

RESIDUAL PLOT:

VOLUME FRACTION:

ANIMATION VIDEO: https://drive.google.com/file/d/19F4alfI4R5SGeRDIE4w-xdAkodKI1FcX/view?usp=sharing

CASE 1:

Overall Element size: 0.4mm

MESHING:

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Transient
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
Initialization	Patch: Water-surface (1 value of volume fraction of water) : Air-surface(0 value of volume fraction of water)

RESIDUAL PLOT:

VOLUME FRACTION:

ANIMATION VIDEO: https://drive.google.com/file/d/1-E0XXmGljBMj18hvjN2iZaMP5jkuEAbM/view?usp=sharing

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-fluid(Density =400 kg/m3, viscosity = 0.001 kg/m-s) : water liquid Solid-type: Aluminium
Cell zones	Fluid type: Air : water
Boundaries	Standard
Initialization	Patch: Water-surface (1 value of volume fraction of water) : Air-surface(0 value of volume fraction of water)

RESIDUAL PLOT:

VOLUME FRACTION:

ANIMATION VIDEO: https://drive.google.com/file/d/1SBmKRkE4Qo0EDeQXw_UGzj45dRmhIrn0/view?usp=sharing

It is clear from the results that instabilities that are occuring at the interface ie, formation of bubbles and vortices are explicitly seen as we refine the mesh quality or finer mesh.

Why not Steady State Approach: It is used when we are only interested in end result and the intermittent behaviour is not important. Whereas in this case we are interested in the real-time changes at the interface as well as to see the changes in the fluid region. This real time observations can only be studied in Transient state Analysis. We are interested in the behaviour at interface when the fluids of various densities collapses as a function of time and this time step is too small to capture explicit behaviour the Instabilitty effect known as Rayleigh Taylor Instability. Thus Transient approach was selected.

Atwood Number: 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:

The calculated Atwood number is close to 1 and from the simulation results, it is found that when high dense fluid i.e. water poured upon low dense fluid i.e. air under gravity, the formation of air bubble-like plumes takes place which travels towards upward region in the form of waves and some gets trapped at the lower regions during the initial stages, which afterward try to move towards upper region and at the end, both phases gets separated with some diffusivity left at the middle portion between them. Thus, the calculated Atwood number is validated for our simulation results.

For above cases

case 1:

heavier fluid(water)

density 1=998.2 kg/m^3

lighter fluid(air)

density 2=1.225 kg/m^3

Atwood number(A) =(density 1-density2)/(density 1+density2)

Atwood number(A)=0.996

case 3:

heavier fluid(water)

density 1=998.2 kg/m^3

lighter fluid(user-defined material)

density 2=400 kg/m^3

Atwood number(A) =(density 1-density2)/(density 1+density2)

Atwood number(A)=0.427

variation in Atwood number in the above two cases affects the behavior of the instability.

For Atwood number close to 0, RT instability flows take the form of symmetric fingurs of fluid.

When Atwood number close to 1, the much lighter fluid below the heavier fluid takes the form of larger bubble like plumes.

Therefore, from the value of Atwood number it is seen from the simulations that air forms larger bubbles when water is poured down the air pushes through and forms large bubbles. Therefore, the results can be validated by obtaining the value of Atwood number.

Atwood number is an important parameter in the body of Rayleigh taylor instability and Richtmyer-Meshkov instability. In Rayleigh-Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale.

CONCLUSION:

• From the above cases, we observe that finer mesh captures the instability better than the coarse mesh, but it takes more time for the heavier fluid to settle down. As computational time is long but produces results with better accuracy.
• Since the simulation involves capturing instabilities that are changing in an unpredictable manner as a function of time, the transient approach is preferred over the steady-state approach.
• It is observed from the simulations that air forms larger bubbles when water is burst down the air and forms large bubbles. The Atwood number calculated in first case is almost 1 other case is 0.42
• Both cases have larger Atwood number showing the Rayleigh Taylor Instability.