1 OBJECTIVE

Perform RT INSTABILITY simulation for base mesh size as 0.5mm & after base mesh simulation refine mesh & re-do the simulation. Use air & water as a fluid for above cases.

Now create a user defined fluid instead of air whose density is 400Kg/m³ & viscosity as 0.001kg/m-s & use refined mesh to perform simulation. Compare results for all 3 cases.

2 INTRODUCTION & THEORY

From our daily life observation, we can conclude that atmospheric pressure can’t support even the smallest possible drop of water within a glass after inverting but TORRICELLI showed that atmospheric pressure can produce a force that can hold few centimetres of water column. This contradiction can be explained with help of instability of equilibrium configuration of lighter & heavier fluid. The configuration in which a lighter fluid is expected to support a heavier fluid in presence of gravitational field is known as Rayleigh Taylor instability or RT-Instability. This name was given in honour of LORD RAYLEIGH because he was the first person, we explained corresponding mathematical phenomenon in 1883. It can occur under gravity and, equivalently, under an acceleration of the fluid system in the direction toward the denser fluid. A beautiful finger like pattern can be modelled with the help of these equation.

The initial phase of exponential growth is followed, first, by a transition phase of bubble formation and then by an asymptotic stage of rising air columns. RTI mainly develops in mainly in 4 stages. These can be classified as perturbation, formation of bubbles, Bubble merging takes place, & turbulent mixing, which is the fourth and final stage in the evolution. The cause of exponential instability growth is the smaller amplitude as compared to wavelength of perturbation. At initial stage, the sinusoidal perturbation can retain its sinusoidal shape. At the end of this stage, non linear effect can be observed clearly. This stage can be identified by start of formulation of mushroom shapes spikes & bubbles. These non linear effects arise due to growing of heavy fluid in light fluid (spikes) & growth of lighter fluid into heavier fluid (bubbles). These effects can be modelled using buoyancy drag where growth rate is constant wrt time. At the end of this stage, spikes & bubbles start interacting with each other & this phenomenon is known as bubble merging. This results in merging of smaller spikes & bubbles to form a larger one. Also, bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated. This is the stage of turbulence mixing.

3 CFD MODEL BASED ON RTWAVES - INSTABILITIES

• RICHTMYER-MESHKOV INSTABILITY

Richtmyer–Meshkov (RM) instability (RMI) is viewed as the impulsive counterpart of Rayleigh–Taylor instability. RMI arises when a shock passes through an interface between two fluids, but with two important distinctions with RTI. First, the RMI can occur when the acceleration is toward either side of the interface, whereas for the Rayleigh–Taylor case one has instability only for acceleration in the direction of the lighter fluid. Second, the RMI evolution occurs largely during a time where the external forcing is zero since the driving is impulsive. RMI occurs in nature in the situation of shock–magnetosphere interactions.

• KELVIN-HELMHOLTZ INSTABILITY

Kelvin–Helmholtz (KH) instability has developed due to shear flow along the walls of the bubbles (also called a Taylor cavity). The shear layer at the interface of the fluids becomes KH unstable if there is a jump in the tangential component of the velocity across the interface. This leads to a self-induced production of the vorticity in the mixing region, The Kelvin–Helmholtz instability (KHI) plays a critical role for the time dependent transition process. KHI is induced from an unstable perturbed interface between fluids subject to a parallel shear flow and is one of the classical problems in fluid dynamics. The development of KHI often constitutes the critical step toward the onset of turbulent mixing observed both in the atmosphere and in oceans. Also, the KH instability is the reason for the evolution of the mushroom structures during the RT and RM nonlinear process. Hereafter, RTI, RMI, and KHI will be collectively referred to as hydrodynamic instabilities.

• PLATUE RAYLEIGH INSTABILITY

This type of instability explains the reason behind formation of smaller packets of fluids with same volume but lesser surface area. Liquid jet falling under the action of gravity has constant radius initially. Later on, the length of jet increases & after some time it reaches to its critical value. It results in loosing of cylindrical shape of jet & formation of droplets occurs. This effect arises due to presence of surface tension. PLATUE noted the instability arose when the liquid column length exceeded the column diameter by a factor of about 3.13

Above figure shows the breaking of fluid jet into droplets & clearly explains the PLATUE RAYLEIGH INSTABILITY phenomenon.

4 GEOMETRY & MESHING

4.1 GEOMETRY

Following figures shows the geometry of flow domain. The upper domain contain water which acts as a heavier fluid & lower domain acts as air & user defined material in their respective cases which act as a lighter fluid. The dimension of each domain is 20X20 mm.

4.2 MESHING

The multizone mode of meshing ensures uniform & regular meshing which can predict better result. Along with multizone mode, an inflation layer has been created at fluid-fluid interface which is well capable of capturing the interface instability at the start of simulation.

Base line meshing has been done by using 0.5mm element size. This element size was not capable of capturing very small spikes & bubbles. So, we further refined mesh & re simulated for the same boundary condition.

The total number of nodes & elements for 0.5mm element size is 7421 & 7200 & for refined mesh (size=0.2mm) is 30401 & 30000.

5 SETUP & PHYSICS

This is a transient simulation in presence of gravitational field. Laminar viscus model was chosen to perform this simulation.

STEADY STATE approach is not a suitable approach for RT Instability because steady state approach is used for fully developed flow where parameters don’t change wrt time. In this case we are interested in observing the mixing phenomenon of heavier & lighter fluid. We want to observe all the 4 phenomena from start of bubble formulation to the end of turbulent mixing. These effects can be captured in only transient simulation.

6 RESULT & DISSCUSION

We have seen that for the base line mesh the small bubble & spikes can’t be captured & even we can’t visualise the merging of small bubbles to form large one.

In refined mesh we were clearly able to observe the small size bubble & merging phenomenon.

Following figure shows residue for base line mesh

FOLLOWING GIF SHOWS CORROSPONDING ANIMATION OF RT STABILITY

Following figure shows residue forrefined mesh

FOLLOWING GIF SHOWS CORROSPONDING ANIMATION OF RT STABILITY

Following figure shows residue for user defined matterial & refined mesh

FOLLOWING GIF SHOWS CORROSPONDING ANIMATION OF RT STABILITY

Following figure shows development of flow

In above figure (a) shows deformation of fluid-fluid interface, (b) shows formulation of bubble & streak (c) shows emerging of small bubbles, (d) shows emerging of small streak to form large one, (e) shows turbulence mixing of both fluids.

The density dependence can be expressed by the so-called ATWOOD NUMBER leading to appreciably lower growth rates for fluids with nearly equal densities. Atwood number is represented by alpha (α).

Atwood number for our case is

• For water & air is α

α = 0.9975485

• For water & user defined material is α

α = 0.42783

7 CONCLUSION

• Transient simulation is preferred over steady state simulation in order to observe all effect properly.
• Mesh size plays an important role in capturing small bubbles & formulation of larger bubbles by mixing of small bubbles.
• Effect of density can be explained by defining Atwood number.