Rayleigh Taylor Instability Challenge

Objective:

• To find out some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves and explain how these mathematical models have been adapted for CFD calculations.
• To Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm and compare the results by showing the animations.
• To explain, why a steady-state approach might not be suitable for the above types of simulation.
• To find out the Atwood number for the above case and explain how the variation in Atwood number affects the behavior of the instability and can we use it to validate our simulation result.

Rayleigh Taylor Instability:

The Rayleigh–Taylor instability, or RT instability, is instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

• Practical CFD model based on Rayleigh Taylor waves:

1. 1D RAYLEIGH-TAYLOR MIXING:

The turbulence models have been applied to the simulation of simple 1D RT mixing cases with a 3:1 density ratio (ρ1 = 3 g/cm3 and ρ2 = 1 g/cm3). The computational domain extends [−8, 20] cm with the heavy fluid placed on the left side of the domain and the initial interface at x = 0. Unless otherwise stated, the computational grid consists of 100 cells and the adiabatic exponent is γ = 5/3 for both fluids. The following relation is satisfied in all cases A0g = 1; thus, for the 3:1 density ratio the gravitational acceleration is g = 2 cm/s^2.

1. TILTED-RIG RAYLEIGH-TAYLOR MIXING:

The tilted-rig test case originates from a series of experiments performed at the Atomic Weapons Establishment in the United Kingdom in the late 1980s to study the mixing between two variable density fluids induced by the Rayleigh-Taylor instability. In the experiment, a tank containing two fluids of different densities, a heavy fluid placed in a tank below a lighter fluid (a stable configuration), is accelerated downwards between two parallel guide rods by firing rocket motors. The downward acceleration caused to the tank effectively changes the direction of “gravity” (external body force) so that the system becomes RT unstable, causing the two fluids to mix. The acceleration from the rocket motors was not constant but averaged approximately 35 times normal gravity. It eventually attains a roughly constant value, but the time during which the acceleration varied is significant and still needs to be considered. One approach is to directly incorporate the measured values into the simulation [i.e., a variable gravity g(t)]. This works well for incompressible Navier-Stokes solvers; however, it can create problems for compressible codes. An alternative approach is to make use of a constant acceleration for which a nondimensional time, τ, is used for comparison with experimental results:

3. SPRAY BREAKUP MODELING:

• CHU breakup model:

The Rayleigh–Taylor instabilities form the basis of the model formulated by the Chu. The theoretical correlation was constructed in order to predict droplet sizes based on an exponential function which was used in fuel–coolant interaction models. The model implemented in this study represents a modified version of Chu’s original model in that it includes a Taylor-series expansion of the exponential term to first order, and subsequently an explicit discretization in time in which drop diameter correlation is written as:

The default values for the constants are C₁ = 0.772, C₂ = 0.246.

• Wave Model:

The wave model (KH model) was proposed by Reitz. The model is based on a first-order linear analysis of a Kelvin–Helmholtz instability growing on the surface of a cylindrical liquid jet with initial diameter r₀ that is penetrating a stationary incompressible gas with a relative velocity u_rel as shown in fig.

The instability at the surface is a small axisymmetric displacement η.

where η₀ is an infinitesimal initial perturbation, k is the wavenumber, ω is the growth rate. Wavelength λ is related to the wavenumber, as follows:

• KHRT Model:

It is known that liquid droplets in diesel sprays have very high initial velocities and decelerate rapidly due to drag forces. In this case, Rayleigh–Taylor instability may also play an important role in droplet breakup mechanisms in addition to the Kelvin–Helmholtz instability as shown in fig.

In the KHRT model, the aerodynamic force on the drop flattens it into the shape of a liquid sheet, and the decelerating sheet breaks into large-scale fragments by means of RT instability. KH waves with a much shorter wavelength originate at the edges of the fragments, and these waves break up into micrometer-sized drops as illustrated in fig.

Performance of Rayleigh Taylor Instability Simulation:

The idea is, the dense fluid will be at the top and lighter fluid will be at the bottom and due to gravity, there will be instability at their interface, which is going to be calculated by using VOF multiphase method in ANSYS Fluent.

Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it can be modeled by two completely plane-parallel layers of immiscible fluid, the denser on top of the less dense one and both subject to the Earth\'s gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: 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. This was the set-up as studied by Lord Rayleigh.

• Geometry Model:

water-side (top) = (20 x 20) mm

air-side (bottom) = (20 x 20) mm

• Properties:

Solver: Transient Pressure-based

Gravity: X = 0 m/s^2

Y = -9.81 m/s^2

Flow type: Laminar

Model type: Multiphase

Method: Implicit

Material: Air and Water-Liquid

Time Step Size = 0.005

No. of Time Steps =1500 Iterations

• CASE-1

Element Size = 0.5mm

Nodes = 3321

Elements = 3200

Mesh Type: Coarse

Relevance = 0

Meshing Model:

Result:

Link: https://www.youtube.com/watch?v=0tz1Dmz0_lY&feature=youtu.be

• CASE-2:

Element Size = 0.3mm

Nodes = 9180

Elements = 8978

Mesh type: Medium

Relevance = 50

Result:

Link: https://www.youtube.com/watch?v=bslQ0NJv5mg&feature=youtu.be

• CASE-3

Element Size = 0.2mm

Nodes = 20301

Elements = 20000

Mesh Type: Fine

Relevance = 100

Meshing Model:

Result:

Link: https://www.youtube.com/watch?v=xg7zaUcmHfg&feature=youtu.be

• Observation:

In the above cases, the simulation starts with the state of hydrostatic equilibrium and the Rayleigh Taylor instability is observed at the interference when a lower density fluid pushes a higher density fluid due to which formation of shock waves at the interface takes place. Formation of air bubbles starts taking place, which compresses the heavy fluid around it, due to which shock waves of multidimensional fashion generates and it gets more stronger as it moves upward.

It is observed that the more we refine the mesh, the more the simulation results get smoother about the irregularities that take place at the interface of the two fluids. As higher density fluid replaces lower density fluid, the formation of air bubbles and vortex takes place, which travels towards the upward region with time. In case-3, we can see the more detail results about the formation of waves and the formation of some air bubbles that get trapped at the lower region during the initial stage and then they travel towards the upper region, generating shock waves. At the end of the simulation, it is to be observed that the two phases get separated from each other though, we can see some diffusivity between them which is a volume fraction of air and water.

why a steady-state approach might not be suitable for the above types of simulation?

The difference between the steady and transient is that you can\'t see the small-time variation of instability. The steady-state simulation is performed if we are concerned more about the final state results or the equilibrium state. In RT-Instability CFD models, we are more concerned to learn about the transition of the irregularities that starts developing when we pour high dense fluid upon low dense fluid under gravity effect so, by using transient solver along with refined mesh of the model, we can compute the smooth transition of irregularities that takes place at the interface of the fluids. The final state results for both the steady-state and transient state will be the same.

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 = (1000 - 1.25) / (1000 + 1.25) = 0.9975

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 asymmetric “finger” of fluid; for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

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.

References:

• http://www.scholarpedia.org/article/Rayleigh-Taylor_instability_and_mixing
• https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability
• https://www.researchgate.net/publication/330266850_Modeling_of_Rayleigh-Taylor_mixing_using_single-fluid_models
• https://www.sciencedirect.com/science/article/pii/S0016236108004304#bbib20