Week 5 - Rayleigh Taylor Instability

Rayleigh-Taylor Instability

Task:

1. 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.
2. Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations (Attach them from your google drive). Observe the differences in the animation and explain the effect of mesh size on simulation. Also, explain why a steady-state approach might not be suitable for these types of simulation. [ Choose Air and Water for this simulation]
3. Run one more simulation with water and user-defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
4. Define the Atwood Number. Find out the Atwood number for both the cases and explain how the variation in Atwood number in the above two cases affects the behaviour of the instability.

Theory:

• Rayleigh-Taylor Instability: It is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.
• Types of CFD Models:
  • Kevin-Helmholtz instability: This model can occur when there is velocity shear in a single continuous fluid, or where there is a velocity difference between two fluids. An example is wind blowing over water. The instability manifests as waves on the water surface. More generally, clouds, oceans, Saturn’s bands, and the sun’s corona show this instability.
  • Ritchmyer-Meshkov instability: This arises when a shockwave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers.
  • Plateau-Rayleigh instability: This model explains why and how a falling stream of fluid breaks up into smaller packets with same volume but less surface area. The fluid instability is explained in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets.
• Practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:
  • Single-Fluid Model: A typical approach used for the analysis of two-phase flows is a mixture model, i.e., the individual fluid phases are assumed to behave as a flowing mixture described in terms of the mixture properties. The applied single-fluid model is a five-equation model consisting of the mass, momentum, and energy equations for a vapor/liquid mixture, and two equations describing the formation and growth of the liquid phase.
  • Two-Fluid Model: In this model, separate sets of the governing equation for the vapour and liquid phases have been used. The interaction between the droplets and the heat exchange between the liquid phase and the solid boundary are not modelled here as well. Additionally, the velocity slip between vapor and the liquid phase is in this model considered.
  • Smoother Particle Hydrodynamics (SPH Method): SPH Method is a computational method used for simulating the mechanics of continuum media, such as mechanics and fluid flows. Smoothed-particle hydrodynamics is being increasingly used to model fluid motion as well. This is due to several benefits over traditional grid-based techniques. First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass. Second, SPH computes pressure from weighted contributions of neighbouring articles rather than solving linear systems of equations. Finally, unlike grid-based techniques, which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly the particles represent the denser fluid and empty space represents the lighter fluid.
  • Turbulence Model: They are needed to predict the average mixing behaviour in flows that are on average one or dimensional. The approach to the construction of tile turbulence model is guided by the experimental behaviour. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows, CFD simulation use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

Rayleigh-Taylor Instability Simulation Procedure:

Vol. Of Fluid Multiphase method is used to calculate the instability between a denser & lighter fluid.

Steps:

• Create two 20mm x 20mm squares, on over the other on SpaceClaim.
• Create a 2D mesh based on the case.
  • Case 1: Baseline Mesh 0.5mm
  • Case 2: Fine Mesh 0.1mm
  • Case 3: Baseline Mesh 0.5mm
• Laminar Viscous Model.
• Time: Transient; Gravitational Acceleration: Y= -9.81 m/s²
• Materials:
  • Solid – Aluminium
  • Fluid – Air, Water, User fluid (density = 400 kg/m³, viscosity = 0.001 kg/m-s)
• Multi-phase model Phases: Homogenous models = Vol. of Fluid; Formulation = Implicit
  • Case 1: Primary Phase: Air; Secondary Phase: Water
  • Case 2: Primary Phase: Air; Secondary Phase: Water
  • Case 3: Primary Phase: Water; Secondary Phase: User Fluid
• Standard Method of Initialization.
• Patch --> Phase = Volume Fraction --> zones to patch = water surface --> value = 0
• Patch --> Phase = Volume Fraction --> zones to patch = air surface --> value = 1
• Run calculation --> Type = Adaptive --> Duration Specification Method = Total Number of Time Steps --> Initial Time step size = 0.01; Total number of timesteps = 1000

Results:

Case 1 Residual plot:

Case 2 Residual plot:

Case 3 Residual plot:

Conclusion:

We can notice that Rayleigh Taylor Instability is seen at the interface between the 2 fluids, where the high-density fluid gets pushed by lower density fluid and therefore, shockwaves are formed. Air bubbles are formed which compresses the heavy fluid around it, due to which shock waves of multidimensional fashion generate and they get stronger as it moves upward.

The more we refine the mesh, the smoother are the irregularities in the fluid interface.

In the third case, detailed results about the formation of shockwaves can be seen. The air bubbles get trapped at the lower region during the instil stage and then travel upwards. In the end, the two phases reach equilibrium and separate themselves from each other.

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

A. Steady-state simulation is performed when the final results are what matter the most. In RT-Instability simulations, we are more concerned about the transition of the irregularities that start to develop when 2 fluids of different densities mix. We are interested to know the behaviour of the fluids in the simulation, the motion of air bubbles, vortices, shockwaves, etc. Hence, transient-state simulation is more applicable.

Atwood Number:

The Atwood number (A) is a dimensionless number used in the study of hydrodynamic instabilities in density stratified flows. It is given by:

In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale.

For Case 1: Density 1 = 998.2 kg/m³ and Density 2 = 1.225 kg/m³

Hence, A = 0.996

For Case 2: Density 1 = 998.2 kg/m³ and Density 2 = 400 kg/m³

Hence, A = 0.427