Rayleigh-Taylor instability analysis

The Rayleigh–Taylor instability, or RT instability, is instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.

Some practical CFD models that are based on mathematical analysis of RT waves:

1. Early-time growth
2. Potential flow
3. Energy budget

1. Early-time growth:

It was noted by Taylor (1950) that two superposed liquids with an interface normal to
the acceleration eld could exist in a state of unstable equilibrium. However, if small
corrugations were to exist on the interface, a growing standing wave pattern could
be expected.

2. Potential flow:

Development of unstable interfaces beyond the linear phase was first studied by
Davies & Taylor (1950), and generalised somewhat in Layzer (1955). Both papers
begin by considering a two-dimensional incompressible potential ow in a vertical
cylinder with an initially flat free under-surface. This is Rayleigh-Taylor unstable with
an Atwood number A = 1.

3. Energy budget

By considering energy exchange between potential and kinetic as the instability
develops, we can derive a scaling law for its evolution. We assume here that the initial
the state is at rest, and all the potential energy released after time t0 is transformed into
kinetic energy, and is not dissipated on a timescale over which the model is valid.

Rayleigh Taylor Instability simulation

GEOMETRY :

The image of the geometry is shown in the above figure. It consists of two squares one over another each of dimensions 20mm by 20mm.

Now, we carry out simulation for three different cases as follows:

Case I :

Element size of Mesh = 0.50mm (Baseline mesh)

Number of Elements = 3200

Here, a transient simulation is selected because the flow becomes turbulent. Pressure based solver is chosen as the flow velocity is less than 0.3 times the speed of sound in that medium. Velocity formation is \'Absolute\'. Further, gravitational forces are considered to be acting in the along negative y-axis and magnitude of it is 9.81 m/s2. A Laminar model with Multiphase option is selected because of two different phases. Initially, water is assumed in the upper phase/rectangle and air in the lower phase/rectangle.

Results obtained by running the solution for 1500 time steps with time-step size = 0.005 sec

Google drive link of animation file:

https://drive.google.com/open?id=12XPdmYmJp1HJjhb3A17jUjxlrFgvvY3P

Case II :

Element size of Mesh = 0.25mm

Number of Elements = 12800

Setup is the same as in the case I. Results obtained by running the solution for 1500 time steps with time-step size = 0.005 sec are

Google drive link of animation file:

https://drive.google.com/open?id=1z7dm9FoHlRhkZVmY67Q_E22qdMh5HFY-

Case III :

Element size of Mesh = 0.15mm

Number of Elements = 35378

Setup is the same as in the case I. Results obtained by running the solution for 1500 time steps with time-step size = 0.005 sec are

Google drive link of animation file:

https://drive.google.com/open?id=1UYYfWg-s9YcSlux8SUlGV3s5S9gP3jBZ

Why a steady state approach might not be suitable for these types of simulation?

A steady-state simulation ignores many of the cross terms and higher order terms dealing with time. These terms all go to zero in steady state so they don\'t affect the steady state result. The transient simulation includes all these terms. Usually, this means the steady state model has an easier convergence as there are fewer terms to model and some transient non-linearities are removed, but in a few models these non-linearities help convergence. This is the case in Rayleigh-Taylor instability model. RT instability model involves non-linearity which gets ignored during the simulation and we do not get accurate results.

ATWOOD NUMBER

The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as

here,

= density of heavier fluid (water) = 998.2

= density of lighter fluid (air) = 1.225

so,

A = 0.9975

The rate at which mixing takes place depends upon the Atwood ratio. For low \'A\', the mixing occurs slowly and the penetration of heavier liquid in the lighter liquid occurs slowly. As the value of \'A\' increases, the mixing occurs at a much higher rate. The maximum value of the Atwood number is 1.

We can use past experiments to verify our results because many researchers performed experiments with different Atwood number and different combination of fluids. The graphical representation of Atwood with other parameters like dimensionless numbers, time, instabilities, etc. is available. We can compare our plots with them to verify our results.