Rayleigh Taylor Instability

Aim:

• To simulate Rayleigh-Taylor instabilities, with different mesh sizes
• To study the significance of Atwood Number and calculate the Atwood number, when the lighter fluid is air and heavier one being water.

Rayleigh-Taylor Instability

The Rayleigh-Taylor instability , is the instability of an interface which is present between two fluids of different densities.This can be viewed when, the lighter fluid tries to push the heavier fluid.

Some Practical CFD models based on RT instability are

1. Saffman-Taylor Instability : This type of instability is aslo defined as viscous fingering and this formation occurs when there is there is instability in the interface between the two fluids in a Porous medium. This is commonly seen during drainage process.
2. Kelvin-Helmontz Instability : This instability occurs when there is velocity difference across the interface between two fluids. Helmholtz studied the dynamics of two fluids of different densities when a small disturbance, such as a wave, was introduced at the boundary connecting the fluids. Kelvin-Helmholtz instability can thus be characterized as unstable small scale motions occurring vertically and laterally.
3. 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 instabilitybegins with small amplitude perturbations which initially grow linearly with time. This is followed by a non-linear 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.
4. Plateau-Rayleigh Instability : This 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 RT instability and is part of a greater branch of fluid dynamics concerned with fluid thead breakup.

Pre-processing :

1.Space Claim

• The required profile for both the lighter and heavier fluid are made using the rectangle and pull tool. the geometry is constructed in such a way that the lighter fluid is at the bottom and the heavier dense fluid is at the top.
• After the geometry is created, share topology is created inorder to form a conformal mesh, near the interface of the two mediums

2. Meshing:

• The meshing is done in the mesh module.
• Various mesh sizes are used with the base mesh being 0.5mm

Solution :

• The mesh is then updated and transitioned to fluent.
• In fluent, transient method is used in order to visualize the instabilities vary with time.
• There is no turbulent model used, thereby treating it as a laminar flow
• In the physics ribbon, multiphase is selected and Volume of Fluid model is chosen.Since the number of faces used is 2. The implicit method is used, as the solving does not depend directly on the Courant number.
• After selecting VOF method, the initialization is done using standard method.
• The patch option is use, inorder to patch the surface where the water will be present with a volume fraction of 1 and other area where air is present as 0.

Case 1 :

Mesh size : 0.5mm

Lighter fluid: Air

Denser fluid : Water

Initial Volume Fraction Contour

Scaled Residuals for 0.5mm Grid Size

Case 2 :

Mesh size : 0.3 mm

Lighter fluid: Air

Denser fluid : Water

Scaled Residuals for Mesh size 0.3 mm

Case 3 :

Mesh size : 0.2mm

Lighter fluid: Air

Denser fluid : Water

Scaled Residual for 0.2 mm mesh size

Case 4 :

Mesh size : 0.2mm

Lighter fluid : Density : 400kg/m^3, Viscocity 0.001 `(kg)/(m.s)`

Denser fluid : Water

Atwood Number :

The Atwood number is a dimensionless number which is used in the field of fluid dynamics to study the hydrodynamic instabilities.

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

where,

`rho1` = Density of Heavier fluid`

`rho2` = Density of light fluid`

Atwood number for the instability involving water and air =

`rho1 = 1000`

`rho2 = 1.225`

`A= (1000-1.225)/(1000+1.225) = 0.997`

Atwood number for the instability involving water and air =

`rho1 = 1000`

`rho2 = 400`

`A= (1000-400)/(1000+400) = 0.428`

Conclusion :

Based on the Atwood number , for A is closer to 0 then the instability begins with a finger formation whereas if the atwood number is closer to 1 , it follows a bubble formation.

This can be verified based on the run simulations.