Rayleigh Taylor Instability

Imagine two fluids one on top of another. If upper fluid is less dense than lower fluid, then nothing happens but if they are switched such that lower fluid is less denser than the upper fluid. The lighter fluid will rise whereas heavy fluid will sink. The interference between two fluids is going to get distorted and this phenomenon is known as Rayleigh Taylor Instability. Based on this observation few more phenomenons came into existance which are as follows:

Kelvin-Helmholtz Instability- It occurs when there is velocity shear in a continuous fluid or velocity difference across the interference between two fluids. Basically, two streams of fluid travelling at different speed will be responsible for this instability. This is the phenomenon which can be observed very oftenly in clouds.

Richtmyer Meshkov Instability- It arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers.

Plateau-Rayleigh Instability- It explains the phenomenon of formation of droplets from continuous falling stream as it conserves energy because fluid with same volume posses less surface area due to formation of droplets.

Rayleigh–Bénard convection- It is a type of natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems.

Following are some CFD models based on mathematical analysis of Rayleigh Taylor waves:

1. Early and late time evolution- It was noted by Taylor in 1950 tha two superimposed liquids with an interface normal to the acceeration field could exist in a state of unstable equilibrium. However, if small series of ridges and grooves were to exist on the interface, a growing standing wave pattern could be expected.

2.Fluid thread breakup model- It is the process by which a single mass of fluid breaks into several smaller fluid masses. The process is characterized by the elongation of the fluid mass forming thin, thread-like regions between larger nodules of fluid. The thread-like regions continue to thin until they break, forming individual droplets of fluid. Thread breakup occurs where two fluids or a fluid in a vacuum form a free surface withsurface energy. If more surface area is present than the minimum required to contain the volume of fluid, the system has an excess of surface energy. A system not at the minimum energy state will attempt to rearrange so as to move toward the lower energy state, leading to the breakup of the fluid into smaller masses to minimize the system surface energy by reducing the surface area.

3. Potential Flow- Development of unstable interfaces beyond the linear phase was first studied by the Davis and Taylor (1950) and generalized by Lyzer (1955). Both studies begin by considering a 2D incompressible potential flow in a vertical cylinder with a initially flat free under surface. This is Rayleigh Taylor instability with Atwood number A=1. The velocity potential must satisfy Bernoulli\'s equation in cylindrical polar co-ordinates

4. Buoyancy-drag Balance- The complexity of directly evaluating a potential flow to calculate growth behaviour can be avoided. From the first principles, the following equation describes how buoyancy and drag compete to accelerate a body, which in this case is a bubble of lower layer fluid penetrating the upper layer.

5. Vorticity- The Rayleigh Taylor instability can be seen as the result of baroclinic torque created by the misalignment of the pressure and density gradieny at the perturbed interface, as described by the 2D inviscid viscosity equation, where, ω = vorticity ρ = density p = pressure

In this case, the dominant pressure gradient is hydrostatic, resulting from the acceleration due to gravity.

6. Turbulence Models- Turbulence mixing models of different degree of complexity are investigated for Rayleigh-Taylor mixing flows with reference to high-resolution implicit large eddy simulations. The models considered, in order of increasing complexity, comprise the (i) two-equation K-L, (ii) three-equation K-L-a, (iii) four-equation K-L-a-b, and (iv) Besnard-Harlow-Rauenzahn (BHR-2).

Geometry-

The prepared geometry represents two square surfaces with length of side equal to 20 mm. The top surface represents oil and bottom surface water.

Figure 1. Geometry for Rayleigh Taylor Instability

Meshing-

The meshing of the geometry is done in three different element sizes which can be observed from Figure 2. The details about the no. of nodes and elements can be observed in Table 1.

Figure 2. Meshing of geometry in different element sizes

Table 1. No. of nodes and elements for different mesh sizing

Mesh Size	0.5	0.3	0.1
No. of Nodes	3321	9180	80601
No. of Elements	3200	8978	80000

Simulation-

Case1- For case 1, three simulations were performed for the mixture of air and water with different mesh sizes and volume fraction of water was observed. Graph of residuals and contours of volume fraction can be observed from images and animations:

For Element Size 0.5mm

Time Step: 0.005

No. of Time Step: 1500

Figure 3. Graph of Residuals for mesh size of 0.5 mm

Animation 1. Volume Fraction of Water for element size of 0.5mm

For Element Size 0.3 mm

Time Step: 0.005

No. of Time Step: 1000

Figure 4. Graph of Residuals for mesh size of 0.3 mm

Animation 2. Volume Fraction of Water for element size of 0.3mm

For Element Size 0.1 mm

Time Step: 0.001

No. of Time Step: 1000

Figure 5. Graph of Residuals for mesh size of 0.1 mm

Animation 3. Volume Fraction of Water for element size of 0.1mm

For the above simulations, transient state approach was used as observation for the behaviour of phenomenon has to be made throughout the process. If steady state approach was used then this observation would not be possible for each step as steady state approach yields only end result and does not show behaviour throughout the process.

Case 2- For case 2, simulation was performed for the mixture of oil and water. Oil was defined in the materials with Dynamic Viscosity = 1.06 kg/m-s and Density = 400 kg/m³. Another phase used was water with Viscosity = 0.001003 kg/m-s and Density = 998.2 kg/m³. 0.001 is time step for 10000 no. of time step with 0.1 mm element size of mesh for the simulation. Graph of residuals and contours of volume fraction can be observed from images and animations:

Figure 6. Graph of Residuals for mesh size of 0.1 mm

Animation 4. Volume Fraction of Oil for element size of 0.1mm

Atwood No.- Atwood number is a dimensionless number used for studying hydrodynamic instabilities in density stratified flow field. It is a dimensionless density ratio defined by equation,

where, A = Atwood Number = density of heavier fluid = density of lighter fluid

Therefore, Atwood number for case 2 is

= 0.4278

Atwood no. for case 2 (0.4278) is relatively less than case 1 (0.9975). Hence, the formation of Rayleigh Taylor instability is slow for case 2 compared to case 1. Also, the formation of bubbles and spikes is also relatively lower in terms of surface area in case 2.

Viscosity does have some effects but what it does is decrease the growth rate of instability, in other words, it slows everything down. In case 2 water is the denser mixture relative to oil but oil is more viscous. Even with the high viscosity, things go quickly. Oil is pushed upwards and water sinks. The high viscosity keeps the reynolds number lower and prevents turbulence which is why the mixed region isen\'t very wide. If we observe case 1, the mixture of air and water with lower viscosity is allowing more mixing and mixing region cover almost whole region of both the fluids.