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.

THEORY:

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. It is a dynamic process wherein the system tries to reduce its combined potential energy. Initially, the system is in a state of hydrostatic equilibrium, in which the heavier fluid(ex: water) sits atop a lighter fluid(ex: air) in a constant gravitational field. As the RTI develops, the initial perturbations progress from a linear growth phase, eventually developing "plumes" flowing upwards(in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, equations can be linearized and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, the perturbation amplitude is too large for the non-linear terms to be neglected. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). Examples include supernova, water suspended atop oil, mushroom clouds formed due to volcanic eruptions.

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 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: 0.0005m

Step 3: Setup

Solver Type Pressure-Based

Velocity Formulation Absolute

Time Un-Steady

Viscous model: laminar

Models

Multiphase: Volume of fluid

: Implicit formulation

: 2 phases

Material

Fluid type

Phase 1 : Air

Phase 2 : water liquid

Cell zones Standard

Boundaries 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

Link of Animation 1 :- https://drive.google.com/file/d/1QUsba361CaJd1uHBSsSM9U2Doc-WWhFo/view?usp=sharing

CASE 2:

Overall Element size: 0.0003m

MESHING

Setup:-

Solver Type Pressure-Based

Velocity Formulation Absolute

Time Un-Steady

Viscous model: laminar

Models

Multiphase: Volume of fluid

: Implicit formulation

: 2 phases

Material

Fluid type

Phase 1 : Air

Phase 2 : water liquid

Cell zones Standard

Boundaries Standard

Initialization Standard

Patch : Water-surface (1 value of volume fraction of water)

: Air-surface(0 value of volume fraction of water)

Results:-

Residual plot

Change in the Counters phase-vise

Link of Animation 2 :- https://drive.google.com/file/d/1VgQBRvKZDLYvMz-MDlkjO-0WoVM74Jnc/view?usp=sharing

CASE 3

MESHING:-

Overall Element size: 0.0002m

Setup:-

Solver Type Pressure-Based

Velocity Formulation Absolute

Time Un-Steady

Viscous model: laminar

Models

Multiphase: Volume of fluid

: Implicit formulation

: 2 phases

Material

Fluid type

Phase 1 : Air

Phase 2 : water liquid

Cell zones Standard

Boundaries Standard

Initialization Standard

Patch : Water-surface (1 value of volume fraction of water)

: Air-surface(0 value of volume fraction of water)

Results:-

Residual plot

Change in the Counters phase-vise

Link of Animation 3 :- https://drive.google.com/file/d/1SEiOYg2vjcXyUqM6ypUTRCcfnxjsrii6/view?usp=sharing

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,

`Ag t^2`

where g is the gravitational acceleration and t is the time.

The behavior of RT Instability on the variation of Atwood number:

RTI evolves in three distinct stages: Linear stability, mushroom head(falling) or bubble(rising) formation, long term evolution due to bubble merging and mixing. For low Atwood number, bubble and mushroom head shape is relatively symmetrical, the disturbances or perturbations are linear for a larger time. In contrast, for large Atwood number, bubble and mushroom head shape is less symmetrical, also it occurs very rapidly at an accelerating rate which leads to a nonlinear growth rate.

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.