Rayleigh Taylor Instability Challenge

Q1. 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.

Models used for Rayleigh-Taylor instability

1. Early-time growth
2. Potential flow
3. Buoyancy-drag balance
4. Energy budget

1. Early-time growth

It was noted in Taylor (1950) that two superposed fluids with an interface normal to the acceleration field 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. By considering an unbounded two-dimensional domain, incompressible potential flow, and a sinusoidal interface, the velocity potentials φ in upper and lower layers are given by

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 flow in a vertical cylinder with a initially flat free under-surface. This is Rayleigh-Taylor unstable with an Atwood number A → 1. The velocity potential must satisfy Bernoulli’s equation in cylindrical polar ordinates: where α (t) is arbitrary

3. Buoyancy-drag balance

The complexity of directly evaluating a potential flow to calculate growth behavior can be avoided. From 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:

Q2. Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations ( Attach them from your google drive). Also, explain why a steady state approach might not be suitable for these types of simulation. (150 points)

Mesh Independency Study

The simulation is carried out for three different grid size. The base mesh is of 0.5 mm, it is further refined to 0.3 mm and 0.2 mm respectively as shown in table 1.

Table 1. Simulation for different grid size

Size	Nodes	Elements
0.5 mm	3321	3200
0.3mm	9180	8978
0.2mm	20301	20000

Images for Base Mesh of 0.5mm

Figure 1. Volume fraction contours for mesh element 0.5mm

Images for Mesh of 0.3mm

Figure 2. Volume fraction contours for mesh element 0.3mm

Images for Mesh of 0.2mm

Figure 3. Volume fraction contours for mesh element 0.2mm

The animation of all the three simulations is attached in google drive. The steady-state simulation is not useful in this case because at steady state we already know the end result i.e the denser fluid gets settle down and lighter density fluid rises above. We are interested in how the mixing of heavy dense fluid gets mixed with less dense fluid with time, therefore the transient simulation of such case is required.

Q3. Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behaviour of the instability. How can we use the Atwood number to validate our simulation result? (25 points)

A dimensionless number that describes the density difference between two adjacent fluids with a common interface.

Youngs (1984) described the development of an R–T mixing region as a three-step process

(1) Initially an exponential growth of infinitesimal perturbations that correspond to linear stability analysis.

(2) At amplitudes of about one-half of the wavelength, the instability saturates and longer wavelengths (that have yet to reach saturation) take over. Emmons et al. (1960) coined the term ‘bubble competition’ to describe this regime.

(3) Eventually, through nonlinear mode interaction and successive wavelength saturation, a self-similar R–T mixing layer is formed with a scale similarity as gt²

References:

1. Davies, R. M. & Taylor, G. I. 1950 The mechanics of large bubbles rising through liquids in tubes. Proc. Roy. Soc. A 200, 375.
2. Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astroph. J. 122-1, 1.
3. Youngs, D. L. 1984 Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12, 32–44. (doi:10.1016/0167-2789(84)90512-8)