RAYLEIGH TAYLOR INSTABILITY BETWEEN TWO LIQUIDS OF DIFFERENT DENSITIES

Aim

The aim of the experiment is to study and perform a RT instability simulation bewteen two medium with different grid sizes and report the results.

Rayleigh Taylor instability

RT instability is a state when fluids of different densities interfer with the heavy fluid on the top and lighter fluid on the bottom with a constat gravitational force and the heavier fluid is pushed down due to gravitational force the lighter fluid moves up due to the release of potential energy resulting in irregular patterns which grows as a factor of time. Examples where the RT instability can be seen are, water suspended above oil in earths gravity, erruption of volcano and nulcaer explosion. The RT instabilty is widely used in the CFD experiments in terms of several models and some of the widely used models

• Saffman-Taylor Instability
• Richtmyer-Tayolr Instability
• Kelvin-Helmholtz Instability
• Plateau-Rayleigh Instability

Saffman-Taylor Instability

The saffman-taylor instability occurs whenn a morphologically unstable fluids interfer in a porous medium. This is also called as Viscous fingering. This type of instability can be seen in drainge 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 intergral methods and phase field methods.

Richtmyer-Tayolr Instability

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

Kelvin-Helmholtz Instability

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 atomospheric air is blow over a water. Also, this instability can be seen in sun\'s corona, clouds and saturns bands.

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 break up in fluid dynamics branch involves most of its study using the plateau-rayleigh instability. An example to demostrate this effect is using the the falling of water droplets from a faucet/tap. When the faucet is closed or when the flow is intruppted a long streching of fluids occur at the neck of the faucet and if the facuet diameter is bigger the fluid is not sucked back and breaks down into smaller water droplets.

Atwood Number on Instabilities

In fluid dynamics, the hydrodynamic instabilities based on density stratisfication is determined with the help of a a dimensionless number called as the atwood number. The atwood number is given by the Ratio of Density of heavier fluid and lighter fluid( the difference in their density divided by its sum).

The Atwood number is represented as

`A = (ρ1 - ρ2) / (ρ1+ρ2)`

ρ1 = Density of heavier liquid (kg/m3)

ρ2 = Density of lighter liquid(kg/m3)

A = Atwood number

Simulations to Study the Rt-Instability

Case 1

The simulation is carried out with the help of Ansys fluent. A geomentry is created with water(heavy fluid) at the top and Air(light fluid) at the bottom using Spaceclaim which is represented in Figure 1. The simulation is carried out with three mesh sizes starting with 0.5mm, 0.35mm and 0.20mm and the instability formation is studied with the three cases. An Transient case solver is used since the instabilities that occur between the fluid mediums are time dependent and it varies for every timestep which cannot be captured using a steady state solver, as later is more concerned with the initial and the final results.. Gravity is acted upon in the negative Y- direction (falling down) with laminar case study, multiphase volume of fluid method and Implicit discritization.

Figure 1: Geomentry showing two different fluid placement.

Results

Coarse Mesh (0.5mm)

Figure 2: Convergence residuals for Coarse mesh

Moderate mesh (0.35mm)

Figure 3: Convergence residuals for Medium mesh

Refined Mesh (0.2mm)

Figure 4: Convergence residuals for Refined mesh

Discussion on the case 1 results

• The Density of air taken is 1.225 (kg/m3) and water is 997 (kg/m3), which makes us to conclude that the atwood number for this case is 0.99.
• From the three meshes, we could see that the coarse mesh produce bubbles pushing the heaving fluid down. Whereas in the medium and fine mesh we could see formation of sharp spikes which makes us to conclude that the efficiency in the coarse mesh is poor. Also 500 iterations were carried out in the three cases with a timestep of 0.0025s. In the coarse mesh case we could see that the instability come to stop after the mixing whereas in the medium and fine mesh the instability keeps occuring which makes us to conclude that the coarse mesh does not provide appropriate results. Also, from the convergence residuals comparing the coarse with medium and refined the later residual convergence are stable whereas the coarse mesh residual is unstable.
• In the mixing pattern, the refined mesh produces more circular mixing comapred to the medium mesh leading us o conclude the higher efficiency in refined mesh.

Case 2

In this simulation the same setup is carried out with water and ammonia liquid with water being at the top and ammonia liquid at the bottom since the density of water is higher than the ammonia liquid. In this case the density of Ammonia liquid is 667(kg/m3)which makes the atwood number to be 0.1983. The experiment is crried ou with refined mesh with a element size same as case 1.

Figure 5: Convergence residuals for Refined mesh (Ammonia liquid at the bottom)

Atwood number effect on the instability

• From the simulation we derived at two atwood number. The atwood number in case1 is 0.99 and for case2 is 0.1983.
• From the results we can see that when the atwood number increases the spike formation between the two liquids aswell increase and when it decrease as in case 2 the instability have lesser peak spikes which is evident from the simulation from case2. Also in case 2 we could see initially there are bubbles formed leading to formation of spikes whereas in case 1 only sharp spikes are formed when the instability mixing takes place.
• The small spikes in case 2 is also due to higher density of the lighter fluid (ammonia liquid = 667 kg/m3) comapred to the density of the lighter fluid in case1 (air = 1.225 kg/m3).
• Also with atwood number the penetration distance can be defined by Agt^2 where A is the atwood number.
• Thus when the density increase the atwood number decreases leading to small spikes(case1) and when it is vice versa the formation of sharp spikes are steep(case2).