Week 5 - Rayleigh Taylor Instability

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.

What is Rayleigh taylor Waves?

• Rayleigh Taylor Instability (RTI) is a phenomenon that occurs at the interface of fluids of different densities wherein the lighter fluid pushes the heavier fluid due to the density difference.
• 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.
• Example: Behaviour of water that is suspended above oil in gravity of Earth.

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

The analysis of two-phase flows (mixture model) is done using single fluid model. In this model individual fluid phases are assumed to behave as a flowing mixture described in terms of the mixture properties. It 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.

Plateau–Rayleigh instability

It is also known as Rayleigh instability, and explains why the falling stream of fluid breaks up into smaller packets with the same volume but less surface area. 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.

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.

SOLUTION AND RESULTS

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.

STEP 1: GEOMETRY

For our project, we consider a simple 2D geometry of which

• We draw 2 squares one above the other with a length of 0.02 m.
• Then naming the below square as air and above square as water in 2 different components.
• And share the topology for a perfect intersection between both surfaces.

STEP 2: MESHING

CASE 1:

There is not much change to the fundamental mesh as it is a 2D object

Overall Element size: 0.005m

STEP 3: SETUP

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: 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)

STEP 4: Results

Residual plot

Change in the Counters phase-vise

ONE

TWO

THREE

FOUR

FIVE

MESHING

CASE 2:

Overall Element size: 0.002m

STEP 5: Results

Residual plot

Change in the Counters phase-vise

ONE

TWO

THREE

FOUR

FIVE

CASE 3

STEP 3: MESH

Overall Element size: 0.002m

STEP 4: SETUP

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
Reference values	Standard
Initialization	Standard Patch: Water-surface (1 value of volume fraction of water) : Air-surface(0 value of volume fraction of water)

STEP 5: Results

Residual plot

Change in the Counters phase-vise

ONE

TWO

THREE

FOUR

FIVE

Atwood Number

The Atwood number (A) is dimensionless 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 the lighter fluid

In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the lighter fluid is a function of the acceleration time scale, where g is the gravitational acceleration and t is the time.

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.
• The Atwood no validates the CFD simulation as it is observed. In the case of water and air mixing, bubble formation takes place, whereas in the other case no bubble takes place.