Rayleigh Taylor Instability Challenge
Rayleigh Taylor Instability
Rayleigh–Taylor instability (or RT instability) is named after Lord Rayleigh and G. I. Taylor. The RT instability is the instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. There are situations where a heavy liquid layer is placed over a lighter fluid layer. This situation has engineering implications in several industries. For example in die casting, liquid metal is injected in a cavity filled with air. In poor designs or other situations, some air is not evacuated and stays in a small cavity on the edges of the shape to be cast. Thus, it can create a situation where the liquid metal is above the air but cannot penetrate into the cavity because of instability.
Rayleigh-Taylor (RT) instability takes place when a lighter fluid supports a heavy fluid, then any perturbation of the interface grows and leads to spikes of the heavier fluid penetrating into the lighter one and the interface becomes unstable. The contact discontinuity between the two fluids is unstable to perturbations that grow by converting potential energy to kinetic energy, causing bubbles of the low-density fluid to rise, and spikes of the high-density fluid to sink. If the lighter fluid is above the heavy fluid, the interface is stable. Examples for such systems are dense water over oil (liquid-liquid), or water over air(gas-liquid).
For example, consider a fluid of high density above fluid with lower density. For a perfectly straight interface, the heavy fluid will stay above the lighter fluid. If the surface will be disturbed, some of the heavy liquid moves down. This disturbance can grow or return to its original situation. This condition is determined by competing forces i.e. surface forces and buoyancy forces. The force that is acting to get the above fluid down is the buoyancy force of the fluid in the depression. The depression is returned to its original position if the surface forces are large enough. In that case, this situation is considered to be stable. On the other hand, if the surface forces (surface tension) are not sufficient, the situation is unstable and the heavy liquid enters into the liquid fluid zone and vice versa.
CFD model based on mathematical analysis of Rayleigh Taylor waves
Plateau-Rayleigh instability
This instability takes place when there is an impulsive acceleration between two fluids like shocks at greater speeds. This instability starting with small disturbance linearly grows linearly with time then non-linear part comes bubbles or spikes appear depending on which fluid penetrating into other.
Kelvin-Helmholtz instability
This instability predicts the starting os instability and transition to turbulent flow of fluids of different densities moving at various speeds. This occurs when there is velocity difference between the interface of two fluids
Geometry used
Two squares of (20 x 20)mm are created to represent two fluids separately.
Mesh
Case 1 Element size = 0.5mm
number of elements = 3200
number of nodes= 3321
Case 2 Element size = 0.1mm
number of elements = 80000
number of nodes= 80601
Case 3 user-defined fluid and water
The steady-state approach will not be suitable for these types of simulations because we want to capture the mixing process, interface behavior changing with time, which will be captured only if we don't neglect the time derivative terms and if we use steady-state then it will lead towards the end result which is already known that heavier fluid will settle down.
Atwood number
Atwood number, abbreviated as A, is a dimensionless number that describes density difference between two adjacent fluids with a common interface.
| ρ1 | Density of heavier fluid (kg/m3) |
| ρ2 | Density of lighter fluid. (kg/m3) |
For a positive Atwood number, the interface is unstable, and for a negative value it is stable
For Atwood number close to 0, RT instability flows take the form of symmetric “fingers” of fluid;
for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.
Case 1 Atwood number for the air-water interface
`rho 1= 998.2`
`rho2=1.225`
A=(998.2-1.225)/(998.2+1.225)= 0.998
Here the value is near 1, also the above corresponding animation confirms the large bubbles like plumes formation of air. Hence in tune with the Atwood number.
Case 2 Atwood number for the userdefined material-water interface
`rho 1= 998.2`
`rho2 = 400`
A=(998.2-400)/(998.2+400)= 0.43
References