Rayleigh Taylor Instability Challenge

AIM: Study of Rayleigh Taylor instability in 2 phase flow.

Methodology:

We can divide this project into given parts

• Two-phase flow and multiphase flow
• Hydrodynamic instability
• Models based on hydrodynamic instabilities
• Fluent setup
• Why the steady-state approach is not recommended
• Atwood number
• Atwood number calculation
• Learning outcome
• conclusion

• Two-phase flows & Multiphase flow

2 phase flow is an interactive of 2 phase of flows like gas-solid, gas-liquid, liquid-liquid, and liquid gas.

Some practical mathematical models based on hydrodynamic stability

Hydrodynamic instability:

• Hydrodynamic instability is a field where the stability of the flow is analyzed if the flow is converting to turbulent from laminar than it is unstable.If flow is unstable , than the reason behind the instabilities is studied.The most known scientists behind the studies are kelvin, Reynolds, rayliegh and Helmholtz.

KELVIN-HALMHOLTZ instability (effect of shear)

• The classical problem serving as an excellent introduction to hydrodynamic instability is that of Kelvin–Helmholtz. It is usually invisible and so can be dangerous as clear-air turbulence. But it can be detected by radar or seen by chance as billow clouds when the humidity is such that the rising air in a vortex leads to condensation as cloud and the falling air leads to evaporation.
• It was described by Kelvin and Helmholtz which is also called wind over water and clear air turbulence instability. It occurs when two different materials have different velocity. Imagine at micro view, two parallel incompressible streamflow has different velocities, due to different velocities and pressure difference (due to Bernoulli theory) , the flow will form a pattern, here lesser dense fluid will be at the top and higher dense fluid will be at the bottom, this phenomenon can be seen on Jupiter as well.

`

RAYLEIGH PLATEAU instability(effect of surface tension)

• Joseph Plateau first characterized this instability in 1873 through experimental observation, building on the work of Savart. He noted the instability arose when the liquid column length exceeded the column diameter by some factor.
• instability is depended on the viscosity of fluid width a liquid column length.

RAYLEIGH BERNARD Flow(Effect of Buoyancy)

• It is based on thermal convection, heating a plate with fluid, fluid will develop e bernard cell or natural convection pattern. due to high and low temperature difference , this instability will rise.

TAYLOR COUETTE flow(Effect of rotation)

• Taylor-Couette flow is the name of fluid flow and the related instability that occurs in the annulus between differentially rotating concentric cylinders, most often with the inner cylinder rotating and the outer cylinder fixed, when the rotation rate exceeds a critical value.

Rayleigh Taylor instability

• RT instability occurs when two different fluids with different density penetrate each other, It occurs when two light fluid pushes heavy fluid due to gravity and density difference. This was the set-up as studied by Lord Rayleigh.
• The contact discontinuity between the two fluids is unstable to perturbations that grow by converting potential energy to kinetic energy, causing bubbles of the low-density fluid to rise, and spikes of the high-density fluid to sink.

why the steady-state approach is not recommended for hydrodynamic instabilities

• The steady-state approach is independent of time, which is only depended on end-stage for i.e. for rt instability at the end steady-state approach will give us a result of mixing, but intermediate data will not be provided, for example at which time the pattern is changing from laminar to transient and transient to turbulent, at which time spikes and bubbles are generated for RT instability, at which time cloud pattern is generated in kelvin instability.steady-state is long term transient approach, But desire paths are different.
• For incompressible flow match number, more than 0.1 densities are going to change with time period,at that time transient approach is more suitable , because steady state approach is not able to track the data for intermediate time steps.

Fluent setup

Case 1: air-water (2 phase ) analysis with baseline mesh.

• Using space claim making a square channel of 20X20mm of water and air, and sharing the topology for interpolation errors.

Meshing

Element size :0.5mm

type: all quad

Fluent setup

Model: laminar

Multiphase model: volume of fluid

Material addition from library: Air & water

Patching criteria for initialization: water 1, air 0

Method: water ----->Volume of a fraction

Case 2: refined mesh: air-water(2 phase flow) simulation:

Fluent setup will be same for all cases.

Mesh size :0.1 mm

type: quad

Case 3: refined mesh: custom material -water(2 phase flow) simulation:

Mesh size: 0.1mm

type: quad

density = 400 kg/m³, viscosity = 0.001 kg/m-s

For patching: water 1, custom 0

Result of case 1: air-water (2 phase ) analysis with baseline mesh.

Transient approach

Time step size :0.001s

Mesh size: 0.5mm

• Residuals

Result of case 2: air-water (2 phase ) analysis with finer mesh.

Transient approach

Time step size :0.001s

Mesh size: 0.1mm

+

Mesh refinement result

element size :0.5mm

element size :0.1 mm

As we can see that for finer mesh there will be less diffusion. there will be more fraction of water at bottom level. Hence mesh refinement will give you accurate results( the half of the channel should be filled with water , which we can see in finer mesh.)

Result of case 3: refined mesh: custom material -water(2 phase flow) simulation:

Transient approach

Time step size :0.05s

Mesh size: 0.1mm

Residuals

For the case of air and water perturbations happening at the initial stage and spike and bubbles can be seen at the initial stage due to higher Atwood number, for custom material, due to lower Atwood umber height of spike is low and the bubble is high(height increases in Quadro function).more is explained below after Atwood number

Atwood number:

Atwood number is a dimensionless number, which describes the density difference between two adjacent fluids at a common interface.Atwood number is an important parameter in the study of Rayleigh–Taylor instability and Richtmyer–Meshkov instability. If density difference is very low than Atwood number is tends to zero

The formula for the Atwood number is given below.

`At=(rho1-rho2)/(rho1+rho2)`

`rho1=density of heavy fluid`

`rho2=density of light fluid`

• Atwood number for air water case :

Air density : 1.225 kg/m3

water density:997 kg/m3

Atwood number=0.998

• Atwood number for custom material-water case

custom material density : 400 kg/m3

water density:997 kg/m3

Atwood number=0.4273

• for the case of small Atwood number, the density difference between the two sides of the interface is small, and the density distribution of the upper and
  lower layers is nearly symmetrical. for small Atwood number initial perturbation starts late due to higher density difference and higher viscosity of the material.
• Once the Atwood number increases for incompressible flow the spike height increases faster & the development of the bubble height slows down. but for higher match number and impressionable flow, it is vice versa.
• For incompressible Rayleigh-Taylor instability at high Reynolds number, the development of two-dimensional (2D) single-mode RTI can be divided into five stages diffusive growth (DG), exponential growth (EG), potential flow growth (PFG), acceleration (RA), and chaotic development (CD)

Learning outcome

• Learning of hydrodynamic instability and relevant models
• Fluent analysis
• Atwood number and its significance
• how spikes and bubbles are related to Atwood number
• Why steady-state approach is more suitable

Conclusion

Rt instability is noted down with the effect of Atwood number for incompressible 2 phase flow.