Rayleigh-Taylor Instability Simulation

Objective

To perform Rayleigh Taylor instability simulation for 3 different mesh sizes and compare the animations.

Introduction

Rayleigh Taylor instability is the theory in which instability at an interface when two fluids of different densities are placed such that the heavier fluid is on top of the lighter fluid in the presence of Earth\'s gravitational force.

Transient Simulation

Transient simulations are performed to capture the mixing of fluids at various intervals, A steady-state analysis only gives the final result and the intermediate values are not time dependent. Hence, to get a clear view of the mixing with the change in time, transient simulation is performed.

Mathematical models based on Instabilities

• Richtmyer–Meshkov instability (RMI) occurs when the acceleration of two fluids takes place by the passage of a shock wave. Instability is generated begins with small amplitude changes and grows linearly with time. Later, the formation of a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid. The two fluids mix finally.
• Kelvin–Helmholtz instability occurs when there is a velocity difference across the interface between two fluids. The theory takes into account of the surface tension and predicts the wave formation when fluids of different densities move in the same direction with different velocities.

• Plateau-Rayleigh instability is about a falling stream of fluid, breaking up into smaller packets with the same volume but less surface area. It is concerned about fluid thread breakup. The liquids, by virtue of their surface tensions, tend to minimize their surface area.

Simulation

Here, 3 simulations are performed for different mesh sizes.

Case 1

Element size: 0.5mm

Number of nodes: 3321

Number of elements: 3200

Fig 1.1

Fig 1.1 is the close-up shot of the meshed model. The 2-D model with an element size of 0.5mm and 3200 elements, is the baseline mesh.

Fig 1.2

The mesh metrics of the element quality is shown in Fig 1.2. As the geometry is simple, the software was able to manage with Quad elements only. The average element quality was 99.872%.

Fig 1.3

The solution was set up with a timestep size of 0.005seconds. The simulation seemed converged and the result obtained is shown in Fig 1.3. The animation of the phenomenon can be viewed using the link below.

https://drive.google.com/open?id=1FSCpuxqtWDEQ9l0u2_a7JLY6q6guo5DF

Case 2

Element size: 0.2mm

Number of nodes: 20301

Number of elements: 20000

Fig 2.1

In this case, the mesh is refined with a smaller element size of 0.2mm. The picture of the same model is shown in Fig 2.1.

Fig 2.2

The mesh metrics of the element quality is shown in Fig 2.2. As the geometry is simple, the software was able to manage with Quad elements only. The average element quality was 99.896%.

Fig 2.3

The final snapshot of the converged solution is shown in Fig 2.3. The solution was set up with a timestep size of 0.005seconds. The animation of the solution can be viewed using the link below.

https://drive.google.com/open?id=134TGt5gYtnJ4Z75XIYaE20EeYd57Gwj5

Case 3

Element Size: 0.048

Number of nodes: 349030

Number of elements: 347778

Fig 3.1

The mesh is even refined with a smaller element size of 0.048mm. The picture of the same model is shown in Fig 3.1. The model is zoomed in to have a clear view of the elements.

Fig 3.2

The mesh metrics of the element quality is shown in Fig 3.2. As the geometry is simple, the software was able to manage with Quad elements only. The average element quality was 99.835%.

Fig 3.3

The final snapshot of the converged solution is shown in Fig 3.3. The solution was set up with varying timestep sizes of 0.0005seconds, followed by 0.001seconds and finally by 0.0025seconds. The animation of the solution can be viewed using the link below.

https://drive.google.com/open?id=1vTY50tirawCHLzb0-7vfzSF8gfppRKnq

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 = ρ 1 − ρ 2 ρ 1 + ρ 2 {\\displaystyle \\mathrm {A} ={\\frac {\\rho _{1}-\\rho _{2}}{\\rho _{1}+\\rho _{2}}}} where, ρ\">ρ1= density of heavier fluid and ρ2=density of lighter fluid

ρ\">In this problem, the fluids used are air(lighter fluid) and water(heavier fluid).

ρ\">A=`(998.2-1.225)/(998.2+1.225)`=0.99

• ρ\">A decrease in the Atwood number also lowers the mixing speed of the fluids as the difference in densities decrease and vice versa.
• ρ\">In this case, the Atwood number is very large and hence the mixing should take place very rapidly, which is evident in the case since the mixing takes place within 8 seconds.

ρ\">Conclusion

ρ\">The objective was to observe the Rayleigh-Taylor Instability of the mixing of two fluids of varying density, which is achieved.

ρ\">