Rayleigh Taylor Instability Challenge using ansys fluent

Aim:

To study and perform R-T instability simulation between two different fluid medium with different grid size and to validate the results by Atwood numer.

Rayleigh–Taylor (R–T) instability:

The Rayleigh–Taylor instability occurs when a high-density (ρ₁) fluid is placed over a lower-density (ρ₂) fluid in a gravitational field. When the density interface is disturbed away from horizontal, the hydrostatic pressure acts to drive the heavier fluid into the lighter one, with the disturbance amplitude initially growing exponentially, but shortly afterwards saturating and forming a characteristic mushroom shape, at least for some time. Eventually, either other wavelengths become important and accelerate the development of the instability, or mixing and dilution of the density contrast will lead to a deceleration.
Such naturally occurring R–T instabilities are quite common e.g. in oceans, supernovae and oil-trapping salt domes.

The RT instability is widely used in the CFD experiments in terms of several models and some of the widely used models are:

1.Saffman-Taylor Instability:

The Saffman-Taylor instability occurs when a morphologically unstable fluids interfere in a porous medium. This is also called as Viscous fingering. This type of instability can be seen in drainage when a low viscosity fluid is pushed inside a pipe where there is a fluid with higher viscosity. Most of the research using this model is done in the Hell-Shaw cells, Boundary integral methods and phase field methods.

2.Richtmyer-Tayolr Instability:

The Richtmyer instability occurs when two fluids with different viscosity is impulsively accelerated. These effects can be seen in shock waves where initially the shock wave is developed small and grows linearly with time.

3.Kelvin-Helmholtz Instability or Rayleigh Taylor Hybrid Model:

This type of instability occurs when there is velocity shear in a continuous medium or when the velocity layer between two fluids are different. A classic example of this instability can be seen when atmospheric air is blow over a water. Also, this instability can be seen in sun's corona, clouds and Saturn’s bands.

4.Plateau-Rayleigh Instability:

This instability model is used to explain how a falling water stream breaks into small droplets with same volume but with less surface area. The fluid thread breaks up in fluid dynamics branch involves most of its study using the plateau-Rayleigh instability. An example to demonstrate this effect is using the falling of water droplets from a faucet/tap. When the faucet is closed or when the flow is interrupted a long stretching of fluids occur at the neck of the faucet and if the faucet diameter is bigger the fluid is not sucked back and breaks down into smaller water droplets.

5.Potential flow model:

This model was developed by Layzer and it provide info on the movement of fluids in a circular tube. A constant supporting pressure (Atwood number =1) is present, competition for both the Rayleigh Taylor and the Richtmyer-Meshkov instabilities.

Atwood number:

In fluid dynamics Atwood number is used in the study of hydrodynamic instabilities in density stratified flows (i.e. to characterize the density difference between two fluids). The Atwood number (A) is a dimensionless number

`A = (rho1-rho2)/(rho1+rho2)`

`rho1`= density of hevier fluid

`rho2`= density of light fluid

For big density differences At tends to 1. For small differences at tends to zero.
Also with Atwood number the penetration distance can be defined by Agt²

^{Effect of Atwood number on instability}

^{If A is nearer to 1, R-T instability takes place in the form of fingers
If A is nearer to 0, R-T instability takes place in the form of bubble-like plumes.}

Why Transient state simulation?

A Steady state approach was not used for the Rayleigh Taylor instability simulation because, the steady state approach is independent on the time progressed and it would benefit us only if we are interested in the end results only. But here, in this simulation, we have to observe the pattern of the instability wave formed from time to time from the starting time of the simulation. This could only be achieved once we use a transient simulation.

Geometry:

The geometry is made up of two 20x20 square surfaces one over the other as shown below assuming each square as container (2D) filled with water and air respectively.

Note: Share topology to get conformal meshing

Meshing:

CASE A : baseline mesh

element size: 0.5mm

no. of elements: 3200

case B:

element size :0.1mm

no. of elements: 80000

case C:

element size : 0.1mm

no. of elements: 80000

Setup & Solution:

Solver type: pressure based
Time state: Transient
Enable gravity with -9.81m/s on y-axis
model: Multiphase model with Volume of fluid method
Formulation method: implicit
No boundary conditions
Materials used: water as heavy fluid
Air as light fluid for caseA

set the required fluid for primary and secondary phases.
Initialization: Standard
Patch: phase - water
variable- volume fraction
heavier fluid value as 1 and lighter fluid as 0
*patching should be done individually for different fluids and should be done every time after initialization
Use appropriate time step based the case

RESULTS :

time step size:0.005

no. of time steps 1000

iterations per time step 20

CASE B:

time step size: 0.004

no. of time steps: 50

iterations per time step 20

CASE C:

time step size : 0.05

no. of time steps: 1000

conclusions:

Atwood no. for CaseA- instability of water-air refined mesh

Atwood no. for CaseC- instability of water-oil:

ρ₁ – Density of heavier fluid (water) – 998.2 kg/m³
ρ₂ – Density of lighter fluid (oil) – 400 kg/m³
A= 0.428

In case B we could see there are spikes and bubbles formed whereas in case A only sharp spikes are formed when the instability mixing takes place.

From the results we can see that when the Atwood number increases, there is an increase in the spike formation between the two liquids as in case A(refined) and it decreases as in case C the instability have lesser peak spikes which is evident from the simulation from case C.
Therefore, the results obtained are validated by calculation the Atwood number.

We can also notice that Water and air combination diffused faster than the Water and oil combination. From this we can conclude that penetration power of spike is greater than penetration power of bubble.