A MACHINE LEARNING APPROACH TO FOLLOWING TRACE PARTICLES IN TURBULENT FLOW
CHAPTER 1 INTRODUCTION 1 CHAPTER 2 METHOD 7 CHAPTER 3 RESULTS 45 CHAPTER 4 CONCLUSION 111 CONTENTS CHAPTER 1 INTRODUCTION THE MOTIVATION FOR THIS THESIS IS TO MAKE USE OF THE ESSENTIALLY LAGRANGIAN METHOD OF THREE DIMEN SIONAL PARTICLE TRACKING VELOCIMETRY 3D PTV TO ELUDICATE SOME MECHANISMS OF TURBULENCE THAT PREVIOUS EULERIAN APPROACHES COULD NOT ACCESS IN PARTICULAR THIS MEANS TO FURTHER DEVELOP THE NON INTRUSIVE OW MEASUREMENT TECHNIQUE OF 3D PTV SUCH THAT THE FULL TENSOR OF VELOCITY DERIVATIVES CAN BE MEASURED ALONG TRAJECTORIES OF UID PARTICLES THIS LAGRANGIAN APPROACH SHOULD GIVE NEW INSIGHT INTO SOME ASPECTS OF STRAIN AND ENSTROPHY DYNAMICS MOREOVER IT WILL ADD TO THE UNDERSTANDING OF THE BEHAVIOR OF MATERIAL VOLUMES AND SURFACES AND THE DI ERENCES BETWEEN STRETCHING OF VORTEX LINES AND MATERIAL LINES 1 1 VELOCITY DERIVATIVES VELOCITY DERIVATIVES UIJ UI XJ1 PLAY AN OUTSTANDING ROLE IN THE DYNAMICS OF TURBULENCE FOR A NUMBER OF REASONS THEIR IMPORTANCE BECAME CLEAR SINCE THE PAPERS BY TAYLOR 1937 1938 AND KOLMOGOROV 1941A B TAYLOR EMPHASIZED THE ROLE OF VORTICITY CURL U I E THE ANTISYMMETRIC PART OF THE VELOCITY GRADIENT TENSOR UIJ UI XJ WITH THE STRESS ON ONE OF THE MOST BASIC PHENOM ENA AND DISTINCTIVE FEATURES OF THREE DIMENSIONAL TURBULENCE THE PREDOMINANT VORTEX STRETCHING WHICH IS MANIFESTED IN POSITIVE NET ENSTROPHY PRODUCTION H I JSIJI 0 KOLMOGOROV STRESSED THE IMPORTANCE OF DISSIPATION 2 SIJSIJ OR 2 S2 AND THEREBY OF STRAIN SIJ 1 2 UI XJ UJ XI I E THE SYMMETRIC PART OF THE VELOCITY GRADIENT TENSOR IT IS KNOWN THAT THE PRODUCTION OF THE ELD OF VELOCITY DERIVATIVES IS A SPATIALLY LOCAL SELF AMPLI CATION PROCESS GALANTI AND TSINOBER 2000 WHERE THE TYPE OF FORCING DOES NOT PLAY ANY ROLE THE LOCAL PRODUCTION TERMS I JSIJ AND SIJSJKSKI FOR VORTICITY AND STRAIN ARE REPORTED BY TSINOBER 2001 TO BE ORDERS OF MAGNITUDE HIGHER THAN THEIR CORRESPONDING TERMS ASSOCIATED WITH FORCING THERE ARE INDICATIONS THAT THE PROCESS OF SELF AMPLI CATION HAS A UNIVERSAL CHARACTER FOR A WIDE RANGE OF DI ERENT OW TYPES DURING THE WHOLE HISTORY OF TURBULENCE RESEARCH THE MAIN BULK OF EXPERIMENTAL RESULTS WAS ASSOCIATED WITH THE VELOCITY ELD ITSELF AND SOME OF ITS SIMPLE CONSEQUENCES AS SPECTRA STRUCTURE FUNCTIONS AND PROBABILITY DENSITY FUNCTIONS PDFS MORE FREQUENTLY THAN OTHERWISE ONE VELOCITY COMPONENT ONLY WAS OBSERVED THE MAIN REASON FOR SUCH A STATE OF MATTERS WAS THE RELATIVELY 1THE INDICES I J K USED THROUGHOUT THIS TEXT ARE APPLIED ACCORDING TO THE FOLLOWING RULES I SUMMATION OVER INDICES THAT IN A TERM APPEAR MORE THAN ONCE E G 1 JSIJ 1SI1 2SI2 3SI3 E G 2 I JSIJ AFTER SUMMATION OVER J AND I BECOMES A SCALAR II IF THERE IS ONLY ONE INDEX OR AFTER SUMMATION THERE REMAINS ONLY ONE INDEX IT IS A VECTOR E G UI U1 U2 U3 OR JSIJ WI W1 W2 W3 III IF THERE ARE OR AFTER SUMMATION REMAIN TWO INDICES IT IS A TENSOR E G SIJ S11 S12 S13 S21 S22 S23 S31 S32 S33 FOR VECTORS AND TENSORS BOLD NOTATIONS ARE USED E G UI U1 U2 U3 U 1 SIMPLE ACCESSIBILITY OF THE STREAMWISE VELOCITY COMPONENT EVEN WITH THE ABILITY TO MEASURE MORE THAN ONE COMPONENT EITHER BY HOTWIRE ANEMOMETRY LASER DOPPLER VELOCIMETRY LDV OR OTHER TECHNIQUES THE ELD OF VELOCITY DERIVATIVES REMAINED INACCESSIBLE UNTIL RECENTLY THE FULL SET OF VELOCITY DERIVATIVES WAS MEASURED BY THE GROUP OF TSINOBER 2001 BY EMPLOYING A MULTI HOTWIRE PROBE BOTH IN LABORATORY AND IN ELD EXPERIMENTS KHOLMYANSKY ET AL 2001A 2001B THIS APPROACH IS ESSENTIALLY EULERIAN ONE IN WHICH THE MEASUREMENTS ARE PERFORMED IN A XED POINT IN SPACE TO OUR KNOWLEDGE NO LAGRANGIAN MEASUREMENTS OF UIJ WERE CONDUCTED WHEN THIS WORK STARTED IN 1999 UNTIL THE TWO DIMENSIONAL EXPERIMENT BY VOTH ET AL 2002 PUBLISHED RECENTLY WITH THE ADVENT OF POWERFUL COMPUTERS REGIONS OF STRONG VORTICITY WERE DISCOVERED IN DIRECT NUMERICAL SIMULATION DNS OF HOMOGENEOUS ISOTROPIC TURBULENCE I E SIGGIA 1981 KERR 1985 ASHURST ET AL 1987 AND HOSOKAWA AND YAMAMOTO 1989 THEY ARE USUALLY REFERRED TO AS WORMS AND BROUGHT EXCITEMENT TO THE PART OF THE TURBULENCE COMMUNITY CONCERNED WITH ITS FUNDAMENTAL ASPECTS SEE I E JIM NEZ ET AL 1993 OR VINCENT AND MENEGUZZI 1994 THE IDEA WAS STRAIGHT FORWARD NOW WITH THE PROBLEM BROKEN DOWN TO SMALL REGIONS OF CONCENTRATED VORTICITY AND LARGE UN IMPORTANT REGIONS WITH BACKGROUND TURBULENCE WHICH IS ALSO REFERRED TO AS RANDOM SEA IT WOULD BE ONLY A MATTER OF TIME FOR SUBSTANTIAL PROGRESS IN THE UNDERSTANDING OF THE BASIC TURBULENT MECHANISMS AND PHENOMENA SUCH AS INTERMITTENCY AND ANOMALOUS SCALING WHILE THIS PROGRESS DID NEVER REALLY HAPPEN THE OPPOSITION OF THE WORM OR CONCENTRATED VORTICITY THEORIES ARGUED THAT THE BACKGROUND OF TURBULENCE IS VERY STRUCTURED AND DYNAMICALLY ACTIVE AND THAT WORMS WERE MORE THE PASSIVE RESULT THAN THE ACTIVE CAUSE OF ANYTHING THIS ARGUMENT WAS PUT IN WORDS SOMEWHAT PRONOUNCED BY BRASSEUR 1999 DESCRIBING TURBULENCE EVOLUTION IN TERMS OF DYNAMICAL INTERACTIONS AMONG VORTEX TUBES AND SHEETS IS AKIN TO DESCRIBING A HAPPY CHUBBY DOG AS BEING WAGGED BY ITS SKINNY VIBRATING TALE THOUGH FORMALLY ALL OF THE OW ELD IS DETERMINED ENTIRELY BY THE ELD OF VORTICITY OR STRAIN TSINOBER 2001 THERE ARE SEVERAL REASONS WHY ONE SHOULD STUDY BOTH THE VORTICITY AND STRAIN TOGETHER FORMALLY THE RELATION BETWEEN STRAIN AND VORTICITY IS STRONGLY NONLOCAL BUT THE MAIN REASON IS THAT VORTEX STRETCHING IS ESSENTIALLY A PROCESS OF INTERACTION OF VORTICITY AND STRAIN STRAIN IN SOME RESPECT IS EVEN MORE IMPORTANT SINCE STRAIN DOMINATED REGIONS APPEAR TO BE THE MOST ACTIVE NONLINEAR IN A NUMBER OF ASPECTS FOR EXAMPLE THE ENERGY CASCADE AND ITS NAL RESULT DISSIPATION ARE ASSOCIATED WITH PREDOMINANT SELF AMPLI CATION OF THE RATE OF STRAIN PRODUCTION AND VORTEX COMPRESSION RATHER THAN WITH VORTEX STRETCHING APART FROM VORTICITY AND STRAIN DISSIPATION THERE ARE OTHER REASONS FOR A SPECIAL INTEREST IN THE CHARACTERISTICS OF THE ELD OF VELOCITY DERIVATIVES IN TURBULENT OWS FOR EXAMPLE THE ELD OF VELOCITY DERIVATIVES IS MUCH MORE SENSITIVE TO THE NON GAUSSIAN NATURE OF TURBULENCE OR MORE GENERALLY TO ITS STRUCTURE AND HENCE RE ECTS MORE OF ITS PHYSICS TSINOBER 2001 THE POSSIBILITY OF SINGULARITIES BEING GENERATED BY THE EULER AND THE NAVIER STOKES EQUATIONS NSE AND POSSIBLE BREAKDOWN OF NSE ARE INTIMATELY RELATED TO THE ELD OF VELOCITY DERIVATIVES DOERING AND GIBBON 1995 IN THE LAGRANGIAN DESCRIPTION OF UID OW IN A FRAME FOLLOWING A UID PARTICLE EACH POINT IS A CRITICAL ONE I E THE DIRECTION OF VELOCITY IS NOT DETERMINED PERRY AND FAIRLIE 1974 SO EVERYTHING HAPPENING IN THE PROXIMITY OF A UID PARTICLE IS CHARACTERIZED BY THE VELOCITY GRADIENT TENSOR UIJ UI XJ FOR INSTANCE LOCAL GEOMETRY TOPOLOGY IS NATURALLY DESCRIBED IN TERMS OF CRITICAL POINTS TERMINOLOGY SEE CANTWELL 1990 1992 CHACIN ET AL 1996 CHERTKOV ET AL 1999 MARTIN ET AL 1998 A MACHINE LEARNING APPROACH TO FOLLOWING TRACE PARTICLES IN TURBULENT FLOW 2 1 2 MATERIAL ELEMENTS AND LAGRANGIAN DESCRIPTION ONE OF THE COMMON VIEWS IS THAT THE PREVALENCE OF VORTEX STRETCHING IS DUE TO THE PREDOMINANCE OF STRETCHING OF MATERIAL LINES THIS VIEW ORIGINATES WITH TAYLOR 1938 TURBULENT MOTION IS FOUND TO BE DI USIVE SO THAT PARTICLES WHICH WERE ORIGINALLY NEIGHBORS MOVE APART AS MOTION PROCEEDS IN A DI USIVE MOTION THE AVERAGE VALUE OF D2 D2 0 CONTINUALLY INCREASES IT WILL BE SEEN THEREFORE FROM 1 I E FROM 0 D D0 WHERE D0 IS THE INITIAL DISTANCE BETWEEN TWO NEIGHBORING PARTICLES AND D IS THE DISTANCE AT A SUBSEQUENT TIME THAT THE AVERAGE VALUE OF 2 2 0 CONTINUALLY INCREASES THIS VIEW IS WIDELY ACCEPTED E G IN HUNT 1973 THE INTERESTING PHYSICAL ARGUMENT THAT H I JSIJI IS POSITIVE BECAUSE TWO PARTICLES ON AVERAGE MOVE APART FROM EACH OTHER AND THEREFORE VORTEX LINES ARE ON AVERAGE STRETCHED RATHER THAN COMPRESSED HOWEVER THIS VIEW IS AT BEST ONLY PARTIALLY TRUE SINCE THERE EXIST SEVERAL ESSENTIAL QUALITATIVE DI ERENCES BETWEEN THE TWO PROCESSES THESE WERE LISTED BY TSINOBER 2001 AS FOLLOWS THE EQUATION FOR A MATERIAL LINE ELEMENT L IS A LINEAR ONE AND THE VECTOR L IS PASSIVE I E THE UID OW DOES NOT KNOW ANYTHING WHATSOEVER ABOUT L THE VECTOR L AS ANY PASSIVE VECTOR DOES NOT EXERT ANY IN UENCE ON THE UID OW EXPRESSION 3 28 THE MATERIAL ELEMENT IS STRETCHED COMPRESSED LOCALLY AT AN EXPONENTIAL RATE PROPORTIONAL TO THE RATE OF STRAIN ALONG THE DIRECTION OF L SINCE THE STRAIN IS INDEPENDENT OF L ON THE CONTRARY THE EQUATION FOR VORTICITY IS A NONLINEAR PARTIAL DI ERENTIAL EQUATION AND THE VECTOR IS AN ACTIVE ONE IT REACTS BACK ON THE UID OW THE STRAIN DOES DEPEND IN A NONLOCAL MANNER ON AND VICE VERSA I E THE RATE OF VORTEX STRETCHING IS A NONLOCAL QUANTITY WHEREAS THE RATE OF STRETCHING OF MATERIAL LINES IS A LOCAL ONE THEREFORE THE RATE OF VORTEX STRETCHING COMPRESSING IS DI ERENT FROM THE EXPONENTIAL ONE AND IS UNKNOWN THERE ARE MANY FEWER VORTICITY LINES THAN MATERIAL LINES AT EACH POINT THERE IS TYPICALLY ONLY ONE VORTEX LINE BUT THERE ARE IN NITELY MANY MATERIAL LINES THIS LEADS TO DI ERENCES IN THE STATISTICAL PROPERTIES OF THE TWO ELDS IN THE ABSENCE OF VISCOSITY VORTEX LINES ARE MATERIAL LINES BUT THEY ARE SPECIAL IN THE SENSE THAT THEY ARE NOT PASSIVE AS ALL THE OTHER PASSIVE MATERIAL LINES CONSEQUENTLY WHILE A MATERIAL ELEMENT L TENDS TO BE ALIGNED WITH THE LARGEST POSITIVE EIGENVECTOR OF SIJ VORTICITY TENDS TO BE ALIGNED WITH THE INTERMEDIATE MOSTLY POSITIVE EIGENVECTOR OF SIJ THE EIGENFRAME OF SIJ ROTATES WITH AN ANGULAR VELOCITY OF THE ORDER OF VORTICITY FOR A GAUSSIAN ISOTROPIC VELOCITY ELD THE MEAN ENSTROPHY GENERATION VANISHES IDENTICALLY H I JSIJI 0 WHEREAS THE MEAN RATE OF STRETCHING OF MATERIAL LINES IS ESSENTIALLY POSITIVE THE SAME IS TRUE OF THE MEAN RATE OF VORTEX STRETCHING H I JSIJ 2I AND FOR PURELY TWO DIMENSIONAL OWS THIS MEANS THAT IN TURBULENT OWS THE MEAN GROWTH RATE OF MATERIAL LINES IS LARGER THAN THAT OF VORTICITY THE NATURE OF THE VORTEX STRETCHING PROCESS IS DYNAMICAL AND NOT A KINEMATIC ONE AS THE STRETCHING OF MATERIAL LINES IS THE CURVATURE OF VORTEX LINES INCREASES WITH STRAIN AND POSITIVE RATE OF VORTEX STRETCHING WHEREAS THE CURVATURE OF MATERIAL LINES DECREASES WITH STRAIN AND POSITIVE RATE OF MATERIAL LINE STRETCHING AN ADDITIONAL DI ERENCE DUE TO VISCOSITY BECOMES ESSENTIAL FOR REGIONS WITH CONCENTRATED VORTICITY IN WHICH THERE IS AN APPROXIMATE BALANCE BETWEEN ENSTROPHY GENERATION AND ITS REDUCTION VORTEX RECONNECTION IS ALLOWED BY NONZERO VISCOSITY NO SUCH PHENOMENA EXIST FOR MATERIAL LINES COMPARING VORTICITY WITH A PASSIVE VECTOR IN THE PRESENCE OF THE SAME DI USIVITY AS VISCOSITY THE ANALOGY IS PARTIAL NOT JUST BECAUSE THE EQUATION FOR VORTICITY IS NONLINEAR BUT ALSO BECAUSE IN THE CASE OF VORTICITY THE PROCESS IS DUE TO SELF AMPLI CATION OF THE ELD OF VELOCITY DERIVATIVES WHEREAS IN CASE OF A PASSIVE VECTOR IT IS NOT A MACHINE LEARNING APPROACH TO FOLLOWING TRACE PARTICLES IN TURBULENT FLOW 3