Rayleigh Taylor Instability Challenge

SOLUTION-1:-

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 the taylor (1950) that two superimposed fluids with an interface the noramal to acceleration field could exit in a state of unstable equilibrium. However if small corrugations were to exit on the interface, a growing standing wave pattern could be expected. By considering an unbounded two-dimentional domain, incompressible potential flow and a sinusoidal interface, the velocity potentials `psi`in upper and lower layers are given by-

`psi_u=a*(e)^(-kz+nt)*cos(kx)`

`psi_l=-a*(e)^(kz+nt)*cos(kx)`

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 verticle cylinder with a initial flat free under surface. This is rayleigh taylor unstable with an Atwood number `A to 1`.The velocity potential must satisfy the Bernoulli\"s equation in cylindrical polar ordinates, where `alpha(t)` is arbitrary.

`(delphi)/(delt)-(1/2)*((delphi)/(delz))^2-(1/2)*((delphi)/(delr))^2-z=alpha(t)`

2. Buoyancy-drag balance:-

The complexity of directly evaluating a potential flow to calculate growth beheavior 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.

`(rho_l+C_a*rho_a*V)*(du)/(dt)=(rho_u-rho_l)*V*g-rho_u*S*u^2`

SOLUTION-2:-

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 0.2mm respectively.

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

Case1:Base mesh of 0.5 mm:-

Scaled residuals:-

Contours:-

Result animation:-

Case2:Mesh of 0.3 mm:-

Scaled residuals:-

Contours:-

Result animation:-

Case3:Mesh of 0.2 mm:-

Scaled residuals:-

Contours:-

Result animation:-

The steady state simulation is not usefull 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 with less dense fluid with time, therefore the transient simulation of such case is required.

SOLUTION-3:-

Atwood number:-

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

`A_t=(rho_w-rho_a)/(rho_w+rho_a)=(997-1.225)/(997+1.225)=0.9955`

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

1. Initially an exponential growth of infinitesimul perturbations that corresponds 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 saturations) 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 `g*t^2`