Rayleigh Taylor Instability Challenge
OBJECTIVES
1) To understand Rayleigh Taylor Instability and some CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves.
2) To perform the Rayleigh Taylor Instability simulation for different mesh sizes and compare them.
3) To Find out the effect of Atwood number variation on Rayleigh Taylor Instability.
INTRODUCTION
Rayleigh-Taylor instability is the instability of the plane interface between the two fluids of different densities. Whenever light fluid pushes on a heavy fluid (or accelerated towards each other), it occurs. It is a dynamic process whereby the two fluids seek to reduce their combined potential energy.
RTI has a profound effect on Earth's climate. Winds off the coasts of Greenland and Iceland drive evaporation of the ocean surface making the upper layers saltier and heavier. This generates descending plumes that drive the abyssal ocean circulation, which in turn acts as a heat pump transporting warm equatorial waters up north. Without ocean overturning from RTI, northern Europe would likely face drastic drops in temperature.
RTI at the interface launches the shock wave into the upper fluid when the state of hydrostatic equilibrium beginning in which heavier fluid set on the top and lighter fluid at the bottom in a constant gravitational field. Rising bubbles of light fluid act like a piston, compressing the heavy fluid ahead of the fronts and generating shocks. These shocks coalesce in multidimensional fashion into a strong normal shock, which increases in strength as it propagates upwards.
Some examples,
I) Natural phenomena
a) Overturn of the outer portion of the collapsed core of a massive star.
b) The formation of high luminosity twin exhaust jets in rotating gas clouds in an external gravitational potential.
II) Technological applications
a) Laser implosion of deuterium-tritium fusion targets
b) electromagnetic implosion of a metal liner.
Some CFD models based on the mathematical analysis of Rayleigh Taylor waves.
1) Richtmyer-Meshkov Instability(RMI)
The Richtmyer-Meshkov instability (RMI) refers to the instability of an interface between two fluids that is impulsively accelerated, usually by a shock wave. It occurs over wide ranges of length and time scales in technological applications and natural phenomena, such as a supernova, combustion, hypersonic air-breathing engines and internal confinement fusion (ICF), a promising approach for fusion energy generation.
2) Potential flow models of Rayleigh Taylor and Richtmyer Meshkov for bubble fronts
A potential flow model of Rayleigh–Taylor, and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. They used ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two‐dimensional single‐mode evolution as well as two‐bubble competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities, the latter treated in an impulse approximation.
3) Kelvin-Helmholtz instability model
The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids.
4) Discrete Boltzmann model (DBM)
A discrete Boltzmann model (DBM) is developed to investigate the hydrodynamic and thermodynamic non-equilibrium (TNE) effects in phase separation processes. ... Based on the TNE features, we define TNE strength which roughly estimates the deviation amplitude from the thermodynamic equilibrium.
5) Buoyancy Drag model for the evolution of the Rayleigh–Taylor, and Richtmyer–Meshkov instabilities
The growth of a single-mode perturbation is described by a buoyancy–drag equation, which describes all instability stages (linear, nonlinear, and asymptotic) at time-dependent Atwood number and acceleration profile. The evolution of a multimode spectrum of perturbations from a short wavelength random noise is described using a single characteristic wavelength. The temporal evolution of this wavelength allows the description of both the linear stage and the late time self-similar behavior. Model results are compared to full two-dimensional numerical simulations and shock-tube experiments of random perturbations, studying the various stages of the evolution. Extensions to the model for more complicated flows are suggested.
6) K-L turbulence model (Two equation)
The k-kL model is a two-equation turbulence model based on Abdol-Hamid's closure and Menter's modification to Rotta's two-equation model. Rotta shows that a reliable transport equation can be formed from the turbulent length scale L and the turbulent kinetic energy k.
| Mesh | Element size | No of elements |
| Baseline mesh | 0.5 mm | 3200 |
| Refine 1 | 0.275 mm | 10658 |
| Refine 2 | 0.12 mm | 55778 |
3. Setting up the Physics (Pre-processing)
The transient type simulation selected for this simulation because we want to see how the interface between two fluids of different densities collapses as a function of time.
Solver: Pressure Based solver
Velocity formulation: absolute velocity
Gravity enabled and provided the value i.e. 9.81 in a downward direction (we construct the volume in the x-y plane so we provide y=-9.81 m/sec2)
Viscous model: laminar
Multiphase Volume of Fluid method is used to capture the mixing of two different density fluids.
Volume Fraction parameter: Implicit
Materials: select air and water two fluids for this simulation.
User-Defined Material: Density= 400kg/m3, viscosity = 0.001 kg/M-S.
After selecting the materials defined the primary phase as air and water as a secondary phase.
Method: Coupled Implicit solver (First-order Upwind schemes)
Initialization: Standard
After the initialization, it is necessary to patch the set up in Volume of fluid method
Here we patch air for 0 and water for 1.
and see the initialization in the contours plot.
4) Solving
Case 1: Primary phase: air,
Secondary phase: water
a) Mesh size 0.5 mm, time step size: 0.001, no of iteration per time step: 5, No of time step: 350
https://drive.google.com/file/d/1YlFgIfRkg7C0moVrJxPxfUV5BADEqumw/view?usp=sharing
b) Mesh size 0.275 mm, time step size: 0.005, no of iteration per time step: 5, No of time step: 600
https://drive.google.com/file/d/1fprjljNBt5OUux6zAiyJByrO6jctfceW/view?usp=sharing
c) Mesh size 0.12 mm, time step size: 0.005, no of iteration per time step: 5, No of time step: 800
https://drive.google.com/file/d/1IvQyBpyHOLgk9oSqgZM83VrWv3Y9u_X9/view?usp=sharing
Case 2: Primary phase: user-defined Material,
Secondary phase: water
a) Mesh size 0.5 mm, time step size: 0.001, no of iteration per time step: 5, No of time step: 350
https://drive.google.com/file/d/1rZhoUjTIPBMzV3Pl2Qy_Lm4qM2NhWRYg/view?usp=sharing
b) Mesh size 0.275 mm, time step size: 0.005, no of iteration per time step: 5, No of time step: 600
https://drive.google.com/file/d/1sPC5hBf65ayQZVe9AKWtQk6ZQiwHcsHa/view?usp=sharing
c) Mesh size 0.12 mm, time step size: 0.005, no of iteration per time step: 5, No of time step: 800
https://drive.google.com/file/d/1CyoWi1orySNjPhRXX60aACB1dMjtwxu9/view?usp=sharing
Result: As we reduced the mesh size we get the good distribution of instability in both the cases also vortices are captured very well but due to finer mesh it increases the computational time. Also, the time step size affects the simulation result and the floating-point exception occurs. So we should also take care of the time step. Also do not take too small a time step, it will increase the computation time.
a steady-state approach might not be suitable for these types of simulation because in Rayleigh Taylor Instability we want to see how the interface between two fluids of different densities collapses as a function of time and this time step was too small and need capture perfectly to understand this effect while in the steady-state simulation we care about only end result. So transient simulation is perfect for these types of simulation.
Atwood Number:
Atwood number is a dimensionless number in fluid dynamics which used in the study of hydrodynamics instabilities. It is an important parameter in the study of Rayleigh -Taylor instability. Atwood number is defined as the density ratio defined by the given below formula :
A = {Density of heavier fluid - Density of lighter fluid} - {Density of heavier fluid + density of lighter fluid}
`A = frac{rho_1 -rho_2}{rho_1+rho_2}`
Case 1: Atwood number= 0.99775, as we get the animation result for this case, we can say that when A is near 1 then it will produce fewer vortices and reach steady-state in less time.
Case 2: Atwood number= 0.422, get a large number of vortices from the simulation which causes turbulence.
Inference:
Transient simulation is used here because we want to see how the interface between two fluids of different densities collapses as a function of time.
Simulation is directly affected by the mesh size when we used the finer mesh, it shows the good distribution in the instability and capturing the vortices but increase the computational time.
Atwood number depends on the density of the fluid and must be in the range of 0 to 1.
Air and water fluid mixing has the Atwood number near 1 which shows the rayleigh Taylor instability and reaches the steady-state in less time while user material and water required more time to dissolve in each other.