Week 5 - Rayleigh Taylor Instability

AIM:

• To mention some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves and explain how these mathematical models have been adapted for CFD calculations.
• To perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm and compare the results by showing the animations.
• To run one more simulation with water and user-defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh. 
• To define the Atwood Number and find out the Atwood number for both the cases and explain how the variation in Atwood number in the above two cases affects the behavior of the instability

What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations.

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. Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the more dense on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.

Practical CFD models that have been based on mathematical analysis of RT waves...

1. Kelvin-Helmholtz instability
2. Richtmyer-Meshkov instability
3. Plateau-Rayleigh instability

Kelvin-Helmholtz instability- This instability typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen when wind blows across a water surface; the instability constant is able to manifest itself through waves on water surface. The Kelvin-Helmholtz instability is not only restricted to a water surface as clouds, but is evident through other natural phenomena as ocean and the sun's corona.

Richtmyer-Meshkov instability- This 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.

Plateau-Rayleigh instability- This 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 RTI and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets. The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tension, tend to minimize their surface area.

Rayleigh Taylor instability simulation (mesh size = 0.5)

Geometry:

• Draw a square of 20mm and use pull tool to name it as 'air'. Now draw another square of same dimension and use pull tool to name 'water'.
• Now select both the squares and select share for conformal mesh for 'Share topology'. Check whether the surfaces are shared in the 'Share topology' option.
• Make this geometry in 2D in X-Y plane.

setup 

Case1:

mesh size : 0.5mm

Porperties

water & air

Properties of Material:

for air: density= 1.22 kg/m^3

          viscosity= 0.0000181 kg/m.s

for water: density= 997 kg/m^3

          viscosity= 0.0000890 kg/m.s

                                                   0.5mm mesh animation

Case 2:

mesh size : 0.3mm

Porperties

water & air

Properties of Material:

for air: density= 1.22 kg/m^3

          viscosity= 0.0000181 kg/m.s

for water: density= 997 kg/m^3

          viscosity= 0.0000890 kg/m.s

                                                  0.3mm mesh animation

Case 3:

mesh size : 0.2mm

Porperties

water & air

Properties of Material:

for air: density= 1.22 kg/m^3

          viscosity= 0.0000181 kg/m.s

for water: density= 997 kg/m^3

          viscosity= 0.0000890 kg/m.s

                                                            0.2mm mesh animation

Case 4:

mesh size : 0.2mm

Porperties

User define & air

Properties of Material:

for air: density= 1.22 kg/m^3

          viscosity= 0.0000181 kg/m.s

for User define: density= 400 kg/m^3

          viscosity= 0.001 kg/m.s                      

Observation:

Finer mesh captures the instability better than the coarse mesh , but it takes more time for the heavier fluid to settle down. So it is computationally expensive but produces results with better accuracy.

Since the simulation involves capturing instabilities which are changing in an unpredictable manner as a function of time , transient approach is preferred over steady state approach. 

Atwood Number

It is a dimensionless number that is used to study hydrodynamic instabilities in density stratified flows.

A=ρ1−ρ2/ρ1+ρ2

Where,

${\rho }_1$ is density of heavier fluid 

${\rho }_2$ is density of lighter fluid

For Air-water RT instability,

${\rho }_1$ = 998.2 kg/m³ [Water density]

${\rho }_2$ = 1.225 kg/m³ [Air density]

A = (998.2-1.225)/(998.2+1.225)

A = 0.998

For water- UD material RT instability,

${\rho }_1$ = 998.2 kg/m³ [Water density]

${\rho }_2$ = 400 kg/m³ [UD material density]

A = (998.2-400)/(998.2+400)

A = 0.428

Behaviour of RT Instability on variation of Atwood number:

RTI evolves in three distinct stages: Linear stability , mushroom head(falling) or bubble(rising) formation , long term evolution due to bubble merging and mixing. For low Atwood number, bubble and mushroom head shape is relatively symmetrical, the disturbances or perturbations are linear for a larger time. In contrast, for large Atwood number, bubble and mushroom head shape is less symmetrical, also it occurs very rapidly at an accelerating rate which leads to a non linear growth rate.

CONCLUSION:

• It is observed from the above experiment that when mesh gets finer, the mixing pattern and phase change are clearly observed at each iteration and time step. 
• As the mesh size gets finer, the simulation time gets increased since more number of cells are there to perform simulation. So it is necessary to choose optimum mesh size.
• It is observed that as the Atwood number gets decreased, the acceleration rate varies linearly and the phase mixing is gradual and vice versa.
• In the mixing pattern bubbles formations are observed initially during mixing and disappears on later stage which leads to swirl pattern.