Numerical Simulation of Rayleigh Taylor Instability
Introduction:
The Rayleigh Taylor (RT) instability is a classical hydrodynamic instability and it plays an important role in many natural scenarios and industrial applications. It is one of the principal causes for fluid mixing induced by unstable stratification. The RT instability occurs when a light fluid accelerates through a heavy fluid or when the light fluid supports the heavy fluid against gravity. When the stable interface is disturbed, the light fluid moves upward due to buoyancy and at the same time, the heavy fluid moves down due to the hydrostatic forces due to gravity. The distortion of the interface depends on the difference in density between the two fluids. The numerical methods that try to capture the RT instability must focus on the evolution of the fluid interface. Some of the numerical methods are
Moving Particle Semi-implicit model:
It is based on a fully Lagrangian description and it is used to simulate the RT instability for viscous fluids. In the MPS method, the fluid is represented by particles which can move freely in the fluid field. Different phases are represented by different particles which help to capture the interface. In this method, the differential operators such as gradients, Laplacians are converted into equivalent particle interaction models using specific weight functions.
Potential flow model:
This model was developed by Layzer and it describes the movement of fluids in a circular tube. It assumes potential flow on the surface of the bubbles as it rises through the heavy fluid.
Vortex model:
It is efficient, flexible and easy to implement. This method represents the density interface by a vortex sheet.
The RT instability can be simulated in ANSYS Fluent using the Multiphase model
RT Instability simulation:
A 2D simulation of RT instability is performed in this report with two common fluids, air and water. For the RT instability to take place, the heavy fluid, in this case, water is placed on top of the air in a rectangular container as shown in figure 1.
Figure 1. Rectangular container for used for simulation
The model was meshed and the fluids were assigned to their corresponding patches. With a steady state simulation, the growth of RT instability cannot be visualized, hence a transient simulation is performed. To ensure the stability of the transient solution, the implicit method is used for solving.
Results and Discussion:
The simulation was performed for 3 different mesh sizes and the corresponding results attached.
Simulation 1:
Element size 0.5 mm
Number of Elements: 3200
Number of Nodes: 3321
Simulation 2:
Element size 0.3 mm
Number of Elements: 8978
Number of Nodes: 9180
Simulation 3:
Element size 0.2 mm
Number of Elements: 20000
Number of Nodes: 20301
It can be observed from the videos that the mesh size of 0.2 mm clearly defines the outlines of the bubbles more clearly. Also as the mesh size is decreased, the sense of continuity in the space is increased.
The RT instability depends on the difference in density between the two fluids considered which is represented by a dimensionless number, Atwood number that is defined as
At low values of Atwood number, close to 0, the up-welling (bubbles) and down-welling (spikes) disturbances form a symmetrical mixing region that expands away from the initial position of the density interface.
As values of Atwood number reach close to 1, the lighter fluid takes the form of large plumes.
The Atwood number in this particular study is calculated as follows:
Density of water = 997 kg/m3
Density of air = 1.1839 kg/m3
Atwood number =
= 0.9976
As the Atwood number is close to 1 in this case, the lighter fluid, air, takes the form of large bubbles as it can be seen from the videos attached in this report.