Rayleigh Taylor instability

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 

Kelvin–Helmholtz instability:

The Kelvin–Helmholtz instability can occur when there is velocity shear in a single continous fluid, or where there is a velocity difference across the interface between two fluids.

    Example: Wind blowing over water, The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn's bands, Jupiter's Red Spot, and the sun's corona show this instability.

Richtmyer–Meshkov instability (RMI):

The Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear 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. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleight-Taylor instability. 

Plateau–Rayleigh instability:

The Plateau–Rayleigh instability, often just called the Rayleigh 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 Rayleigh-Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets.

The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.

Rayleigh-Taylor instability:

The Rayleigh–Taylor instability, or RT instability  is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil or water suspended above air in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions  in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion

CASE 1:

GEOMETRY:

                       

          Water surface is the top surface = 20 mm X 20 mm

           Air surface is the bottom surface = 20 mm X 20 mm

The above Geometry indicates that both the surfaces are in sharing topology. 

MESH:

Element size = 0.5 mm

Number of Nodes = 3321

Number of Elements = 3200

SIMULATION SETUP DETAILS:

I. SETUP PHYSICS

SOLVER

     Typer --> pressure based

     velocity Formulation --> Absolute

     Time --> Transient

     2D space --> planar

     Gravity - Tick the check box

     Gravitational Acceleration 

            X (m/s2) = 0

            Y (m/s2) = -9.81

MODELS

 Viscous --> Laminar

 Multiphase

      Model --> volume of Fluid

      Formulation --> Implicit

MATERIALS

Create/Edit Materials

      Air: 

     Density = 1.225 kg/m^3

     Viscosity = 1.7894e-05 kg/m-s

     Water:

     Density = 998.2 kg/m^3

     Viscosity = 0.001003 kg/m-s

PHASES

List/show all phases

     air - Primary Phase

     water - Secondary Phase

II SOLUTION:

INITIALIZATION

Method --> Hybrid

Initialize at t = 0 

Patch 

Phase --> water

variable --> volume Fraction

zones to patch --> water_water_surface

value -->1

Phase --> water

variable --> volume Fraction

zones to patch --> air_air_surface

value -->0

III RESULTS

GRAPHICS

 Contour --> New 

     options --Filled

     contours of -->Phases

     surfaces - select all surfaces

     Display

IV SOLUTION

ACTIVITIES 

Create --> solutions Animation

        Animation object --> contour

INITIALIZATION

Method --> Hybrid

Initialize at t = 0 

Patch 

Phase --> water

variable --> volume Fraction

zones to patch --> water_water_surface

value -->1

Phase --> water

variable --> volume Fraction

zones to patch --> air_air_surface

value -->0

RUN CALCULATON

Time step size = 0.005

No. of Time Steps = 1500

 scaled Residuals:

Phase Contours:

Before simulation

After simulation

 

 

 Animation:

 

CASE:2

GEOMETRY:

                        

           Water surface is the top surface = 20 mm X 20 mm

           Air surface is the bottom surface = 20 mm X 20 mm

The above Geometry indicates that both the surfaces are in sharing topology. 

MESH:

Element size = 0.375 mm

Number of Nodes = 5778 

Number of Elements = 5618

SIMULATION SETUP DETAILS:

I. SETUP PHYSICS

SOLVER

     Typer --> pressure based

     velocity Formulation --> Absolute

     Time --> Transient

     2D space --> planar

     Gravity - Tick the check box

     Gravitational Acceleration 

            X (m/s2) = 0

            Y (m/s2) = -9.81

MODELS

 Viscous --> Laminar

 Multiphase

      Model --> volume of Fluid

      Formulation --> Implicit

MATERIALS

Create/Edit Materials

      Air: 

     Density = 1.225 kg/m^3

     Viscosity = 1.7894e-05 kg/m-s

     Water:

     Density = 998.2 kg/m^3

     Viscosity = 0.001003 kg/m-s

PHASES

List/show all phases

     air - Primary Phase

     water - Secondary Phase

II SOLUTION:

INITIALIZATION

Method --> Hybrid

Initialize at t = 0 

Patch 

Phase --> water

variable --> volume Fraction

zones to patch --> water_water_surface

value -->1

Phase --> water

variable --> volume Fraction

zones to patch --> air_air_surface

value -->0

III RESULTS

GRAPHICS

 Contour --> New 

     options --Filled

     contours of -->Phases

     surfaces - select all surfaces

     Display

IV SOLUTION

ACTIVITIES 

Create --> solutions Animation

        Animation object --> contour

INITIALIZATION

Method --> Hybrid

Initialize at t = 0 

Patch 

Phase --> water

variable --> volume Fraction

zones to patch --> water_water_surface

value -->1

Phase --> water

variable --> volume Fraction

zones to patch --> air_air_surface

value -->0

RUN CALCULATON

Time step size = 0.005

No. of Time Steps = 1500

Scaled Residuals:

Phase Contours:

Before simulation

After simulation

 

Animation

CASE:3

GEOMETRY:

                       

 

          Water surface is the top surface = 20 mm X 20 mm

           Air surface is the bottom surface = 20 mm X 20 mm

The above Geometry indicates that both the surfaces are in sharing topology. 

MESH:

Element size = 0.25 mm

Number of Nodes = 13041

Number of Elements = 12800

SIMULATION SETUP DETAILS:

I. SETUP PHYSICS

SOLVER

     Typer --> pressure based

     velocity Formulation --> Absolute

     Time --> Transient

     2D space --> planar

     Gravity - Tick the check box

     Gravitational Acceleration 

            X (m/s2) = 0

            Y (m/s2) = -9.81

MODELS

 Viscous --> Laminar

 Multiphase

      Model --> volume of Fluid

      Formulation --> Implicit

MATERIALS

Create/Edit Materials

      Air: 

     Density = 1.225 kg/m^3

     Viscosity = 1.7894e-05 kg/m-s

     Water:

     Density = 998.2 kg/m^3

     Viscosity = 0.001003 kg/m-s

PHASES

List/show all phases

     air - Primary Phase

     water - Secondary Phase

II SOLUTION:

INITIALIZATION

Method --> Hybrid

Initialize at t = 0 

Patch 

Phase --> water

variable --> volume Fraction

zones to patch --> water_water_surface

value -->1

Phase --> water

variable --> volume Fraction

zones to patch --> air_air_surface

value -->0

III RESULTS

GRAPHICS

 Contour --> New 

     options --Filled

     contours of -->Phases

     surfaces - select all surfaces

     Display

IV SOLUTION

ACTIVITIES 

Create --> solutions Animation

        Animation object --> contour

INITIALIZATION

Method --> Hybrid

Initialize at t = 0 

Patch 

Phase --> water

variable --> volume Fraction

zones to patch --> water_water_surface

value -->1

Phase --> water

variable --> volume Fraction

zones to patch --> air_air_surface

value -->0

RUN CALCULATON

Time step size = 0.005

No. of Time Steps = 1500

Scaled Residuals:

Phase Contours:

Before simulation

After simulation

 

Animation:

Conclusion:

  • The Rayleigh–Taylor instability is observed at the interface of two fluids, of different densities which occurs when the lighter fluid is pushing the heavier fluid due to which formation of shock waves at the interface takes place.

  • Formation of air bubbles starts taking place, which compresses the heavy fluid around it, due to which shock waves of multidimensional fashion generates and it gets more stronger as it moves upwards.

  • If the Mesh is refined from the baseline mesh, the simulation results become smoother. In case 2 and 3 we can see detail and smoother results as compared with baseline mesh.

  • The formation of air bubbles that gets trapped at the lower region during the initial stage and then they travel towards the upper region generating shock waves. At the end of the simulation, it is to be observed that the two phases gets separated from eachother. some diffusivity between them is a volume fraction of air and water. 

Explain why a steady state approach might not be suitable for these types of simulation

The steady state simulation is performed if we are concerned more about the final state results or the equilibrium state. In Rayleigh Taylor instability CFD model, we are more concerned to learn about the transition of the irregularities that starts developing when high dense fluid, water is suspended above low dense fluid, air under gravity effect. so, by using transient solver along with refined mesh of the model, we can compute the smooth transition of irregularities that takes place at the interface of the fluids. The final results for both the steady and transient state will be the same. 

3. 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?

Atwood number, abbreviated as A, is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows, it is defined as the ratio of the density difference between two adjacent fluids with a common interface to the sum of their densities. Atwood number is an important parameter in the study of Rayleigh–Taylor instability and Richtmyer–Meshkov instability. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale,

                        `A = {ρ_1−ρ_2}/{ρ_1+ρ_2}`

                 where        

                         `ρ_1` = density of heavier fluid

                         `ρ_2`  = density of lighter fluid

For Atwood number close to 0, RT instability flows take the form of symmetric “fingers” of fluid; for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale,

Atwood number for this case:

Atwood number  A = `{998.2-1.225}/{998.2+1.225}`

                             = 0.997549

The calculated Atwood number is approximately equal to 1, and from the simulation results it is found that when high dense fluid water poured is upon low dense fluid air then formation of large bubble like plumes takes place which travels towards upward region in the form of waves and some gets trapped at the lower regions during the initial stage, which afterward try to move towards upper region and at the end, both phases gets separated with some diffusivity left at the middle portion between them. Thus the calculated Atwood number is validated for our simulation results.                    


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