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. 

Introduction:

                      In Multiphase flow, the dynamic variables like the velocity, viscosity, pressure and density are generally used to explain the movement of fluids under the gravitational field. Multiphase flows takes place in nuclear geothermal power plants, food processing industries and many others. One of the basic example is Rayleigh Taylor Instability. The Rayleigh Taylor Instability is a hydrodynamic instability arises when a high density fluid is placed over a low density fluid in a gravitational field. The RTI phenomenon was initially introduced by Rayleigh 1883 and later was observed for all accelerated fluids by Taylor 1950. When the density interface is disturbed, the hydrostatic pressure is generated, so the heavier fluid moves downwards and mixes with the lighter one, with the amplitude of perturbation initially growing exponentially but after sometime saturates and forms characteristic mushroom shape. When a heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, it results in decrease of potential energy than it’s initial state. This RTI has been applied to a wide range of problems including the mushroom clouds like volcanic eruptions, nuclear explosions and plasma fusion reactors instability.

 

 Rayleigh–Taylor instability:

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.

 

1.Richtmyer-Meshkov instability:

 The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers. This review focuses on the basic physical processes underlying the onset and development of the Richtmyer-Meshkov instability in simple geometries. It examines the principal theoretical results along with their experimental and numerical validation. It also discusses the different experimental approaches and techniques and how they can be used to resolve outstanding issues in this field.

2.Plateau–Rayleigh instability:

The Plateau-Rayleigh Instability describes the phenomenon where a falling jet or cylinder of fluid (such as water) at one point ceases to be a jet and breaks into multiple droplets of smaller total surface area. The short explanation for this is that the surface energy is minimized by this change, but the Plateau-Rayleigh Instability can be analyzed more quantitatively and in greater detail.

The phenomenon is primarily one of surface tension. For this reason gravity and viscous forces are not significant and hence are ignored.

 

3.Kelvin–Helmholtz instability:

Kelvin-Helmholtz (KH) instability typically occurs between two immiscible fluids with a velocity difference across their interface.

Kelvin–Helmholtz instability can occur when there is velocity shear in a single continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is a wind blowing over water: The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn's bands, and the sun's corona show this 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.

 

Modeling Approach:

Pre-Processing:

Geometry setup:

• Choose Fluent from the workbench setup, edit Space Claim to create geometry.
• Choose the XY plane to sketch 2Dgeometry, create a square of 20mm, and select pull tool to make it a surface of that.
•  Move the geometry to a new component and rename it as Air.
•  Now create new geometry under the main design as water and create a square of 20mm right on top of the air square which creates an interface between the 2 boxes.
• Select the entire geometry for share prep under the workbench and select share so that the interface can face information and conformal meshing can be achieved between the 2 phases. 

 

MESHING:

After creating the geometry open the mesh and generate the mesh,

Mesh properties:

Element size = 0.5 mm / 0.0005 m

 

Mesh Quality:

1. Element Quality:

2. Aspect Ratio:

 

Meshing Statistics: 

Elements: 3200

Nodes: 3321

 

Solving using Fluent:

After updating the mesh open the fluent,

 

• Solver Type   =  Pressure-Based

 

• Velocity Formulation =  Absolute

 

• Time = Un-Steady

 

•  Models = Multiphase= [  Volume of fluid ] 

 

• Material = Fluid type =  Air , water liquid

 

• Cell zones =  Fluid type: Air , water

 

• Boundaries = Standard

 

• Reference values = Standard

 

• Initialization = Patch: Water-surface (1 value of volume fraction of water) , Air-surface(0 value of volume fraction of water)

 

• Under physics Use laminar in viscous flow and in the multiphase select volume of the fluid model because here we  designed to find out the position of the interface between the two or more immiscible fluids

 

 

• Add material water using fluent database and then assign the air as primary and water as secondary phase,

 

 

 

• Initialize using standard initialization, then use the patch option to give the alpha value for water as 1 and 0 for air.

• create a contour for the phases and also the animation for the volume fraction of water
• Now run the simulation with a time step size of 0.025 and 100no. of time steps if we encounter any error while running the simulation try reduce the momentum value in the under relaxation factors.

 

CFD post-Processing:

Results:

Time step size = 0.005

No. of Time steps = 500

 

 Residual plot:

 

 

contours of volume fraction:

 

 

 

 

 

 

Animation Video:

https://youtu.be/LLJOnmexaLw

 

 

CASE 2: refined mesh with an elemental size of 0.2 mm and having the fluid materials air and water.

MESHING 

Overall Element size: 0.0002m

 

 

Meshing Matrics:

Meshing Statistics: 

Elements: 20000

Nodes: 20301

Results:

 

Residual plot:

 

 

contours of volume fraction:

 

 

Animation Video:

https://youtu.be/5IfKRoDJ4Mg

 

CASE 3: refined mesh with an elemental size of 0.2 mm and having the user material and water.

MESHING 

Overall Element size: 0.0002m

 

Results:

 

Residual plot:

 

contours of volume fraction:

 

 

 

 

Animation Video:

https://youtu.be/inVwDYFiins

4. Atwood number:

The Atwood number is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio related to the density of heavier fluid and the density of lighter fluid. (The difference in the fluid densities divided by their sum). Atwood number is an important parameter in the study of Rayleigh–Taylor instability. For Atwood number close to 0, RT instability flows take the form of symmetric “fingers” of fluid; for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

 

Case 1,2 :For fluid materials air and water

Case 3:For fluid materials user defined and water

 

The difference in the fluid densities divided by their sum is defined as the Atwood number A.

For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.

 

 

Comparison or Difference Between Low Atwood number and large Atwood Number:

low Atwood number (0.427), bubble and mushroom head shape is relatively symmetrical, the disturbances or perturbations are linear for a larger time: find below image

 

 

large Atwood number(0.997), bubble and mushroom head shape is less symmetrical: find below image:

 

 

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.