Week 5 - Rayleigh Taylor Instability

AIM -

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 BEHAVIOR OF THE INSTABILITY.

THEORY -

Q1 ) PRACTICAL CFD MODELS THAT HAVE BEEN BASED ON MATHEMATICAL ANALYSIS OF RT WAVES...

1. KELVIN-HELMHOLTZ INSTABILITY
2. RICHTMYER-MESHKOV INSTABILITY
3. PLATEAU-RAYLEIGH INSTABILITY

KELVIN-HELMHOLTZ INSTABILITY -

THIS INSTABILITY 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 WHEN WIND BLOWS ACROSS A WATER SURFACE; THE INSTABILITY CONSTANT IS ABLE TO MANIFEST ITSELF THROUGH WAVES ON WATER SURFACE. THE KELVIN-HELMHOLTZ INSTABILITY IS NOT ONLY RESTRICTED TO A WATER SURFACE AS CLOUDS, BUT IS EVIDENT THROUGH OTHER NATURAL PHENOMENA AS OCEAN AND THE SUN'S CORONA.

RICHTMYER-MESHKOV INSTABILITY -

THIS 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 RAYLEIGH TAYLOR INSTABILITY.

RAYLEIGH TAYLOR INSTABILITY -

THE RAYLEIGH–TAYLOR INSTABILITY, OR RT INSTABILITY (AFTER LORD RAYLEIGH AND G. I. TAYLOR), 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 IN THE GRAVITY OF EARTH,^[3] 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^[8] INERTIAL CONFINEMENT FUSION.

Q2 ) PERFORMING THE RTI SIMULATION WITH DIFFERENT MESH SIZE (0.5MM AND 0.3MM)

SOLVING -

GEOMETRY -

FOR THIS SIMULATION, WE CREATE 2 COMPONENTS OF AIR & WATER.

CREATE 2 RECTANGLES OF SIZE 20X20 MM, LOWER ONE OF AIR & UPPER ONE FOR WATER.

CREATE CONFORMAL MESH TO HAVE COMMON INTERFACE BETWEEN 2 INTERFACE & TO SHARE INFORMATION.

MESH -

MESH SIZE IF 0.5MM ( NODES 3321 AND ELEMENTS 3200)

MESH SIZE OF 0.3MM (NODES 9180 ELEMENTS 8978)

FLUENT SETUP -

SETTING UP PHYSICS: LAMINAR MODEL

SOLVER: PRESSURE-BASED, ASSIGN TIME AS TRANSIENT (NON-STEADY STATE)

MATERIAL SELECTION: 1) WATER & AIR

PROPERTIES OF MATERIAL:

FOR AIR: DENSITY= 1.22 KG/M^3

VISCOSITY= 0.0000181 KG/M.S

FOR WATER: DENSITY= 997 KG/M^3

VISCOSITY= 0.0000890 KG/M.S

ENABLE GRAVITY= -9.81 M/S^2

SELECT VOLUME OF FLUID METHOD & NUMBER OF PHASES ARE 2, ONE IS WATER & ONE IS AIR.

SELECT IMPLICIT METHOD.

RENAME PHASES AS AIR & WATER.

STANDARD INITIALIZATION.

DO PATCH GIVE VALUE TO WATER OF 1 AND AIR OF 0.

CREATE PHASE CONTOUR.

CREATE SOLUTION ANIMATION.

ASSIGN TIME STEP SIZE 0.005S & NUMBER OF TIME STEPS 1000.

ANIMATION -

FOR MESH SIZE 0.5MM - (AIR AND WATER)

LINK :https://www.youtube.com/watch?v=nTxx19RWeb4

FOR MESH SIZE 0.3MM - (AIR AND WATER)

LINK :https://www.youtube.com/watch?v=XGs1USnmYo8

COMPARASION:

ELEMENT SIZE (MM)	CELL COUNTS	NODES
0.5	3200	3321
0.3	8978	9180

OBSERVATION -

• IN THE ABOVE CASES, THE SIMULATION STARTS WITH THE STATE OF HYDROSTATIC EQUILIBRIUM AND THE RAYLEIGH TAYLOR INSTABILITY IS OBSERVED AT THE INTERFERENCE WHEN A LOWER DENSITY FLUID PUSHES A HIGHER DENSITY 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 UPWARD.
• IT IS OBSERVED THAT THE MORE WE REFINE THE MESH, THE MORE THE SIMULATION RESULTS GET SMOOTHER ABOUT THE IRREGULARITIES THAT TAKE PLACE AT THE INTERFACE OF THE TWO FLUIDS.
• THE COMPUTATION TIME ALSO INCREASES WITH INCREASES IN FINER OF MESH

WHY A STEADY-STATE APPROACH MIGHT NOT BE SUITABLE FOR THE ABOVE TYPES OF SIMULATION?

THE DIFFERENCE BETWEEN THE STEADY AND TRANSIENT IS THAT YOU CAN'T SEE THE SMALL-TIME VARIATION OF INSTABILITY. THE STEADY-STATE SIMULATION IS PERFORMED IF WE ARE CONCERNED MORE ABOUT THE FINAL STATE RESULTS OR THE EQUILIBRIUM STATE. IN RT-INSTABILITY CFD MODELS, WE ARE MORE CONCERNED TO LEARN ABOUT THE TRANSITION OF THE IRREGULARITIES THAT STARTS DEVELOPING WHEN WE POUR HIGH DENSE FLUID UPON LOW DENSE FLUID 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 STATE RESULTS FOR BOTH THE STEADY-STATE AND TRANSIENT STATE WILL BE THE SAME.

Q3 ) RUN ONE MORE SIMULATION WITH WATER AND USER-DEFINED MATERIAL(DENSITY = 400 KG/M³, VISCOSITY = 0.001 KG/M-S) FOR REFINED MESH.

MESH -

FOR MESH SIZE 0.2MM(NODES 20301 AND ELEMENTS 20000)

ANIMATION -

FOR MESH SIZE 0.2MM (WATER AND USER DEFINED MATERIAL)

LINK : https://www.youtube.com/watch?v=5kUccfN9qHk

Q4 ) DEFINE THE ATWOOD NUMBER ?

THE ATWOOD NUMBER (A) IS A DIMENSIONLESS NUMBER IN FLUID DYNAMICS USED IN THE STUDY OF HYDRODYNAMIC INSTABILITIES IN DENSITY STRATIFIED FLOWS. IT IS A DIMENSIONLESS DENSITY RATIO DEFINED AS

where

= density of heavier fluid

= density of lighter fluid

• FOR AIR & WATER, A= 0.9975
• FOR WATER & USER-DEIFNED MATERIAL, A= 0.4273

ATWOOD NUMBER IS AN IMPORTANT PARAMETER IN THE STUDY OF RAYLEIGH-TAYLOR INSTABILITY. FOR ATWOOD NUMBER CLOSE TO 0, RT INSTABILITY FLOWS TAKE THE FORM ASYMMETRIC “FINGER” OF FLUID; FOR ATWOOD NUMBER CLOSE TO 1, THE MUCH LIGHTER FLUID “BELOW” THE HEAVIER FLUID TAKES THE FORM OF LARGER BUBBLE-LIKE PLUMES.