Advanced CFD using ANSYS FLUENT-Rayleigh Taylor Instability Challenge
Rayleigh Taylor Instability
Rayleigh taylor instability is the instability of the interface between two fluids of different densities in hydrostatic equilbrium under acceleration. Here the lighter fluid pushes the heavier fluid. This would result in the fromation of spikes.Thi eventually grows in a non-linear fashion and changes into plumes like structures. Spike like structures are also developed by the denser fluid as it falls downward which also grows non-linearly.
In the linear phase, equations can be linearized and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for the non-linear terms to be neglected
The following are a few mathematical models that have been adapted for cfd calculations
Kelvin–Helmholtz instability
The Kelvin–Helmholtz instability can occur when there is velocity shear in a single continous fluid or where there is a velocity difference across the interface between two fluids. The theory predicts the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds
Richtmyer–Meshkov instability
The Richtmyer–Meshkov instability occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by 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. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleigh–Taylor instability
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.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
Procedure
Create the geometries in space claim as two 20mm*20mm squares. Make them share topologies.
Create the mesh at required refinement. Here the baseline mesh size was 0.5mm
Setup the mesh for calculation.Add gravity.Use implicit method during phase method. Give properties of water and air. Initialize the setup. Patch the two geometries in such a way that the air has 0 percent water and water phase has 100 percent water in it. Make sure that the air is below and water is above. Run the simulation.
PS: When the mesh is further refined,make sure to reduce the time step to avoid floating point errors.
Record the simulation
Case 1:0.5mm Mesh
Mesh
Simulation
Case 2:0.35 mm Mesh
Mesh
Simulation
Case 3:0.2 mm Mesh
Mesh
Simulation
The transient simulation is ideal for our scenario because our aim is to observe the system as changes take place in real time. Steady state simulation only mostly allow us to observe the initial and final state with the intermediate state not being of not much importance.
This is why Transient Simulation is ideal for our simulation.
Atwood Number
Atwood Number is a dimensionless number that describes the density difference between two adjacent fluids with a common interface.
the equation is
A=(rho1-rho2)/(rho1+rho2)
where rho1=density of heavier fluid
rho2=density of lighter fluid
The density of our simulation would be A=(997-1.25)/(997+1.25)
=0.997
Here the density of water = 997 kg/m3
density of air = 1.225 kg/m3
In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale,A*g*t*t, Where A is Atwood number,g is gravitational acceleration and t is time.
This indicates that as the Atwood number increases the penetartion of denser fluid into the lighter fluid increases. This would mean that the higher the Atwood Number the higher the instability of the system.
We can reduce the difference in the densities of the fluid to 0 and observe that there is no instability in the system as atwood number would be 0. This would validate our results.