simulation of Rayleigh Taylor Instability using ansys

Introduction

      In this Rayleigh – Taylor instability simulation the lighter fluids are above the lighter fluid and regime is accelerated by gravity. Also, phase between two fluids is marked H(x)/patch by 1 for heavier fluid and zero for lighter fluid.

Rayleigh–Taylor instability [RTI]

   Two fluids of different densities superposed one over the other (or accelerated towards each other); the instability of the plane interface between the two fluids, when it occurs, is called the Rayleigh-Taylor instability”.

    The Rayleigh–Taylor instability (RTI) is a fundamental instability of an interface between two fluids of different densities, which occurs when a heavy fluid is superposed over a light fluid in a gravitational field or when a hydrodynamic instability occurs in any accelerating fluid system in which the density and pressure gradients have opposite signs.

Used to an analysis of–

• Include the behaviour of water suspended above oil 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, Capillarity modelling
• Instabilities in plasma fusion reactors and inertial confinement fusion.
• Used to research on fuel atomization, fuel spray from nozzle modelling in IC engine, marine simulation
• Used to forecast rain, to predict the effect of cirrus clouds
• Spray drying, industrial painting, environmentally friendly combustion, inkjet printing, materials processing, fire suppression, and pharmaceutical coating, Geo-fluid dynamics, oil extraction plat which on surface or in sea
• Used in simulate boiling and condensation (design of heat exchangers), cavitation in gas engine, granular flows and wherever bubble and wall interaction happening, Electro-hydrodynamics,cavity filling in casting, turbulent mixing of chemicals, Porous flow modelling like water purifying plants to simulate osmosis process, Bubble breakup and wall or heated wall interaction modelling, Modeling of airplane in raining or in thunderstorm,air bubble formation in the blood of deep-sea diver,
• Used also in ; Liquid atomization such as agriculture, coatings, gasification, water scrubber, pharmaceuticals, metal powder production, 3D printing, spray drying, fire suppression, and cooling.

Below image shows that transient simulation of heavier fluid is over lighter fluids and gravity is on –ve Y Direction.

Practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves

Liquid Film Atomization on Wall Edges—Separation Criterion and Droplets Formation Model [4] 

  Used to predict fuel mixture preparation inside port fuel injection engine. In model for the aerodynamic stripping of the fuel film deposited on the manifold walls is discussed, and a model for the fuel film separation and atomization near the sharp edges is developed.

Direct contact condensation modeling in pressure suppression pool system [5]

  In this simulation Chugging condensation mode in drywell–wet well suppression pool system is simulated they have used Eulerian two phase compression wave solver used along with Raleigh Taylor instability in consideration and that is why this model gives appropriate results; which they have validated, using capture bubble volume data from cameras.

Two-Phase Flow Simulation with Lattice Boltzmann Method: Application to Wave Breaking [6]

  To efficiently simulate multiphase flows with high-density ratios, in order to study complex air-sea interaction problems, such as wind wave breaking and related sea-spray generation.

K-L turbulence model for the self-similar growth of the Rayleigh-Taylor [RT] and Richtmyer-Meshkov [RM] instabilities [7]

  This type of models used in a multidimensional multiphase flow which are comes under RT and RM flows. Application are in Inertial confinement fusion ; (ICF) is a type of fusion energy research that attempts to initiate nuclear fusion reactions by heating and compressing a fuel target, typically in the form of a pellet that most often contains a mixture of deuterium and tritium [9].

Discrete particle modelling (DPM) of granular Rayleigh–Taylor instability [8]

  This method simulates the gravitational air–grain Rayleigh–Taylor (RT) flow instability, Used in to simulate how trapped air is removed through casting sand mould.

CHU model (primary drop breakup)

  This model is formulated by Chu which is based on RTI to predict droplet size of fuel based on critical weber number presented in this model for stable radius of a droplet.

Reitz-Diwakar model (primary drop breakup) [10]

        This model describes the spray modelling where droplet break ups in bag and stripping breakup [10] which depend upon weber number.

         Hear this model comes under; Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen with the wind blowing over water, the instability constant is able to manifest itself through waves on a water surface.[13] Also, this model predicts how fluid droplet changes its radius over time.

Kelvin–Helmholtz instability [KH] Rayleigh Tylor [RT] - KHRT model (secondary drop breakup)

Above image shows spray breakup

Below images show RT break up of fluid droplet [14, 15] where ‘a’ is aerodynamic drag and u is velocity vector.

In KHRT model aerodynamic force on fluid flatted that drop into the flat shape of liquid sheet and decelerate it’s into large scale fragments due to RTI. KH wave with much shorter wavelength originates at the edge of fragments and these wave breaks into micrometre size drops.

Below image show, KH wave and RT wave and fluid droplet break up [12,14]

Modelling spray atomization with Kelvin–Helmholtz instability Rayleigh Tylor KHRT hybrid model (secondary drop breakup) [16]

In secondary break up of drop; only the drops beyond the breakup length are affected by RT breakup. Furthermore, a Rosin-Rammler distribution was used to specify the sizes of children drops after the RT breakup of a parent drop. The modifications made to the KH-RT hybrid model were found to give satisfactory results and to improve the temperature dependence of the liquid fuel penetration of the diesel sprays significantly. [16]

Simulation of heavy liquid over lighter liquid that is Rayleigh-Taylor instability

1- Geometry

2- Meshing

3- Setup and solution

4- Post-processing

 
1- Geometry

2- Meshing

• M1- hex mesh size 0.5 mm
• M2 – hex mesh size 0.4 mm
• M3- hex mesh size 0.2 mm, Number of Elements: 20000

3- Setup

• Setting transient pressure-based solver multiphase implicit simulation is setup
• Fluid is water and air with their default properties from fluent.
• Patching water region as 1 and Air region with 0.
• Fixed time steps,0.005 step size running for 1500 steps

4- Case 1 Solution and postposing of results

• M1- hex mesh size 0.5 mm residual and water fraction contours

• M2 – hex mesh size 0.4 mm residual and water fraction contours

• M3- hex mesh size 0.2 mm residual and water fraction contours

Case 2 Setup o heavier fluid has a density as 400 kg/meter cube and viscosity as 0.01kg/m/s Hexa mesh size of 0.02 mm

• Setting transient pressure-based solver multiphase implicit simulation is setup
• The fluid is user-defined fluid with properties of density as 400 kg/meter cube and kinematic viscosity of 0.01kg/m/s and air with its default properties from fluent.
• Patching water region as 1 and Air region with 0.
• 005 step size running for 1500 steps

M3- density as 400 kg/meter cube and viscosity as 0.01kg/m/s hex mesh size 0.2 mm residual and water fraction contours

Atwood number And transient simulation

`Atwood_number (A) = ((rho(h)-rho(l)))/(((rho(h)+rho(l))) )`

h- Heavy fluid

l – Light fluid

Rho – density of a fluid

Case 1 -Atwood number for water 998.2 kg/meter cube and air 1.225 kg/meter cube simulation is  0.9975

Case 2 - Atwood number for user-modified properties density as 400 kg/meter cube and kinematic viscosity of 0.01kg/m/s and air has a density of 1.225 kg/meter cube is   0.9938

•     Increase in Atwood number increase in the permutation and generate large length of spikes of heavy fluid in the lighter fluid at the instant of time and also it's can be visualized by above contours case 2 is vortices is for a larger time in lower compartment than case 1 setup.
•    Atwood number significance:  In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the lighter fluid is a function of acceleration time scale where g is the gravitational acceleration and t is the time.

Transient simulation

• From the above discussion, RTI is time-dependent phenomena which occurred at the face of two different density fluid, where at the face of two fluids that is lighter fluid and heavy fluid ;

         the heavy fluid is accelerated in perceptive to gravity and thus velocity or any other property of fluid can’t become constant with respect to time in the regime and also because of momentum and energy transfer of fluid from one fluid to other; that is why also this type of simulation comes under transient case though maybe it reached equilibrium in above simulation but due to air there will be wave are generated over the face of water when water comes down. [18]

• when the heavy fluid sits on top, the growth of a small perturbation (a deviation of a system) at the interface is exponential and takes place at the rate of

H=exp(A*G*alpha*time^2)

A - Atwood number

G - Gravitational acceleration

Alpha - wavenumber is the number of waves per unit distance.

Time – time

H= amplitude of wave when liner stability theory is valid after some time this theory gets invalid because of the non-linear behaviour of two fluid

So that spikes generated on face two-fluid is a time-dependent phenomenon and that is why also this type of simulation is transient.

•   If the heavier fluid compartment is pressurized and lighter fluid is a vacuum then at face after permutation is generated there will be shock waves are generated and in a regime of the shock wave, there is a change in fluid properties and that is not dependent on the time that is why this type of simulation is transient.

And also give good result in supernova type of simulation and external aerodynamics aeroplanes simulation where rainy clouds are formed; though RTI based transient simulation simulates the system and give results nearly same but not exact due chaotic nature of the universe.

• If this type of simulation try to simulate with steady-state solver it will not give downward acceleration to the fluid it may happen water doesn’t get at lower bound because numerically this simulation is steady only

After adding or setting transient; Due to time step phenomena that lighter fluid accelerates above with respect to time and heavier fluid gets at lower compartment with respect to time.

Conclusion

RTI type of simulation are basics of multiphase simulation which is discussed above.

 References          

1- Rayleigh Taylor_instability https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability

2- Rayleigh-Taylor_instability_and_mixing http://www.scholarpedia.org/article/Rayleigh-Taylor_instability_and_mixing

3- https://www.sciencedirect.com/science/article/pii/S0898122113005166

4- https://asmedigitalcollection.asme.org/fluidsengineering/article-abstract/124/3/565/444324

5- https://www.sciencedirect.com/science/article/pii/S0029549316302977

6- https://asmedigitalcollection.asme.org/OMAE/proceedings-abstract/OMAE2013/55416/V007T08A002/270708

7- https://www.researchgate.net/publication/230885678_Numerical_Simulation_of_Rayleigh-Taylor_Instability_With_Two-Fluid_Model_and_Interface_Sharpening

8- https://www.sciencedirect.com/science/article/abs/pii/S0301932215001822

9- https://en.wikipedia.org/wiki/Inertial_confinement_fusion

10- https://www.researchgate.net/figure/Reitz-Diwakar-break-up-model_fig1_257775042

11- https://www.politesi.polimi.it/bitstream/10589/58942/3/M.Sc._Thesis.pdf

12- https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcSbYMULh-6ZdlaFRMMjRGdIEvdA-uC1q7xbdSAu58HJvDBEX3yP&usqp=CAU

13- https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_instability

14-  BAUMGARTEN, C. (1996). “Mixture Formation in Internal Combustion Engines”. ISBN -103-540-30835 -0 Springer-Verlag Berlin Heidelberg New York.

15- https://www.google.com/imgres?imgurl=x-raw-image%3A%2F%2F%2F241d7e23e1c77c7881ebbe92aba988437e65a0b67d0bcfa8d242b64816e3f91b&imgrefurl=http%3A%2F%2Fjafmonline.net%2FJournalArchive%2Fdownload%3Ffile_ID%3D41381%26issue_ID%3D237&tbnid=eHzH4SN-HSSAxM&vet=12ahUKEwjI9r7KitbpAhUUcX0KHQQgALQQMygYegQIARA0..i&docid=m-lwkC9vRRSAAM&w=502&h=234&q=KHRT%20model&hl=en&ved=2ahUKEwjI9r7KitbpAhUUcX0KHQQgALQQMygYegQIARA0

16- MODELING SPRAY ATOMIZATION WITH THE KELVIN-HELMHOLTZ/RAYLEIGH-TAYLOR HYBRID MODEL Jennifer C. Beale

17-  Atwood_number https://en.wikipedia.org/wiki/Atwood_number

18- Review of theoretical modelling approaches of Rayleigh–Taylor instabilities and turbulent mixing  https://royalsocietypublishing.org/doi/full/10.1098/rsta.2010.0020#d20889331e6090s