Rayleigh Taylor Instability Challenge

Objectives:

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:

The Rayleigh–Taylor instability:

The Rayleigh–Taylor instability, or RT instability, is the instability of an interface between two fluids of different densities.

If a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.

The Plateau–Rayleigh instability:

The Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh-Taylor Instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. 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.

The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.

The Saffman–Taylor instability:

The Saffman–Taylor instability, also known as viscous fingering, is the formation of patterns in a morphologically unstable interface between two fluids in a porous medium. This situation is most often encountered during drainage processes through media such as soils. It occurs when a less viscous fluid is injected, displacing a more viscous fluid. (In the inverse situation, with the more viscous displacing the other, the interface is stable and no patterns form.) Essentially the same effect occurs driven by gravity (without injection) if the interface is horizontal and separates two fluids of different densities, the heavier one being above the other: this is known as the Rayleigh-Taylor Instability. In the rectangular configuration the system evolves until a single finger (the Saffman–Taylor finger) forms. In the radial configuration the pattern grows forming fingers by successive tip-splitting.

Richtmyer-Meshkov instability (RMI)

A wide variety of fluid motions can be generated, following the interaction of a shock wave with an interface separating two fluids of different properties. Any perturbation initially present on the interface will, in most cases, be amplified following the refraction of the shock. This class of problems is generally referred to as the Richtmyer-Meshkov instability (RMI). The basic mechanism for the amplification of perturbations at the interface is baroclinic vorticity generation resulting from the misalignment of the pressure gradient of the shock and the local density gradient across the interface. As the interface between the two fluids becomes more distorted, secondary instabilities, such as the Kelvin-Helmholtz shearing instability, develop and a region of turbulence and mixing ultimately results.

The Kelvin–Helmholtz instability

The Kelvin–Helmholtz instability typically occurs when there is in a single, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen with wind blowing over water, the instability constant is able to manifest itself through waves on a water surface. The Kelvin-Helmholtz instability is not only restricted to a water surface as clouds, but is evident through other natural phenomena as the ocean, Saturn's bands, and the sun's corona.

Mushroom clouds:

Mushroom clouds result from the sudden formation of a large volume of lower-density gases at any altitude, causing a Rayleigh-Taylor Instability. The buoyant mass of gas rises rapidly, resulting in terbulent vortices curling downward around its edges, forming a temporary vortex ring that draws up a central column, possibly with smoke, debris, and/or condensed water vapor to form the "mushroom stem". The mass of gas plus entrained moist air eventually reaches an altitude where it is no longer of lower density than the surrounding air; at this point, it disperses. The stabilization altitude depends strongly on the profiles of the temperature, dew point, and wind shear in the air at and above the starting altitude.

Salt fingering :

Salt fingering is a mixing process that occurs when relatively warm, salty water overlies relatively colder, fresher water. It is driven by the fact that heated water diffuses more readily than salty water. A small parcel of warm, salty water sinking downwards into a colder, fresher region will lose its heat before losing its salt, making the parcel of water increasingly denser than the water around it and sinking further. Likewise, a small parcel of colder, fresher water will be displaced upwards and gain heat by diffusion from surrounding water, which will then make it lighter than the surrounding waters, and cause it to rise further. Paradoxically, the fact that salinity diffuses less readily than temperature means that salinity mixes more efficiently than temperature due to the turbulence caused by salt fingers.

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Steps in Simulation:

Open Ansys Workbench. Drag and drop Fluid Flow (Fluent) onto Project Schematic.

Here, we will see Steps we need to follow to solve a particular engineering system from a CFD standpoint.

1. Geometry: Spaceclaim, where we import the geometry.
2. Mesh: Ansys Mesh, where Boundary naming, element sizing and mesh generation with required grid size will be done.
3. Setup: Ansys Fluent, where actual case setup will be done.
4. Solution: Ansys Fluent, where runtime results visualization done.
5. Results: CFD-Post, where Post Processing of Results will be done.

CAD Cleanup:

Import Exhaust Port geometry into Spaceclaim, Spaceclaim is CAD cleanup software that can be used to prepare CAD models for simulation setup. It could be for CFD or FEA.

For this Simulation, we have to create to components of Air and Water respectively

• Create two rectangles of size 20*20mm, one for water and one for Air.
• Create Conformal Mesh to have common interface between two interface and to share the information.

Mesh Generation

Baseline Mesh:

• Create a Baseline mesh of 0.5mm

Refined Mesh:

Create two more refined mesh of 0.3mm

and 0.2mm

Fluent Setup:

Setting up physics: Laminar Model.

Solver: Pressure-based.

Assign Time as Transient(Non-steady State). Because:

Material Selection: 1)Water and Air. 2)Air and Custom material

Properties of Material:

For Air: Density = 1.22 kg/m^3

Viscosity = 0.0000181 kg/m.s

For Water: Density = 997 kg/m^3

Viscosity = 0.0000890 kg/m.s

Setting Boundary Condition:

For Inlet:

Velocity = 5m/s

• Enable Gravity as -9.81 m/s^2.
• Select Volume of fluid method and number of Eulerian phases are 2 one for water and one for air.
• Formulation Implicit Method
• Rename phases as air and water.
• Standard Initialization.
• Create Phase contour
• Create Animation.
• Assign Time Step size 0.005s and Number of time steps 1000.

Animation:

For Mesh element size 0.5mm (Air and Water)

https://youtu.be/JLydb9UcUtQ

For Mesh element size 0.3mm (Air and Water)

https://www.youtube.com/watch?v=NjKrB_uuOZk

For Mesh element size 0.2mm (Air and Water)

https://youtu.be/nFdcZ-haEUU

For Mesh element size 0.2mm (Air and user-material having density=400kg/m^3 & dynamic viscosity=0.001kg/m-s)

https://youtu.be/Vp5-DKIC9Rs

Comparison between Baseline(0.5mm) and refined Mesh(0.3mm & 0.2mm) :

Element Size(mm)	Cell counts	Nodes
0.5	2739	2849
0.3	7637	7840
0.2	17100	17381

Why a steady-state approach might not be suitable for these types of simulation

• There is perturbation(unsettlenes) of interface generates wrt time due to penetration of heavier and lighter fluid into one another.

ATWOOD NUMBER:

The Atwood number (A) 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

A = (rho1-rho2)/(rho1+rho2)

where, rho1 = density of a heavier fluid

rho2= density of lighter fluid.

Field of Application:

Atwood number is an important parameter in the study of Rayleigh-Taylor Instability and Richmyer-Mechkow instability. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, where g is the gravitational acceleration and t is the time.

For Air and Water, Atwood Number is 0.9975.

For Air and User-material (density=400kg/m^3 & dynamic viscosity=0.001kg/m-s), Atwood Number is 0.9939.

Conclusion:

• As the Mesh becomes finer, the mixing becomes much clear and visible.
• Hence accuracy of solution also increases as there are mode number of nodes.
• However, the computation time also increases with increses in the quality of Mesh