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The University of Chicago Data Driven Modeling Of THE UNIVERSITY OF CHICAGO DATA DRIVEN MODELING OF THE LOW-ATWOOD SINGLE-MODE RAYLEIGH-TAYLOR INSTABILITY A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY MAXWELL HUTCHINSON CHICAGO, ILLINOIS DECEMBER 2016 Copyright c 2016 by Maxwell Hutchinson All Rights Reserved Dedicated to Tracey Ziev TABLE OF CONTENTS LISTOFFIGURES .................................... vii LISTOFTABLES..................................... xi ACKNOWLEDGMENTS ................................. xii ABSTRACT ........................................ xiii 1 INTRODUCTION ................................... 1 1.1 Formaldefinition ................................. 1 1.2 Instancesandmotivation. ... 2 1.3 Terminology.................................... 3 1.4 Noteonthestructureofthedissertation . ...... 9 2 BACKGROUND .................................... 10 2.1 Linearandweaklynon-linearmodels . ..... 10 2.1.1 LordRayleigh’slinearmodel. .. 10 2.1.2 Viscousanddiffusivelinearmodels . .. 11 2.1.3 Weaklynonlinearexpansions. .. 12 2.2 Potentialflowmodels.............................. 13 2.2.1 Layzer’sunitAtwoodmodel . 13 2.2.2 Goncharov’shighAtwoodmodel . 14 2.3 Buoyancy-dragmodels . 15 2.3.1 BubblemodelofDaviesandTaylor . 16 2.3.2 TubemodelofDimonteandSchneider . 16 2.3.3 Self-similarmodelofOron . 17 2.4 Problems with single mode Rayleigh-Taylor modeling . .......... 17 2.4.1 Pressureinthesingle-modeRTI . .. 18 2.4.2 Departurefrompotentialflow . 18 2.4.3 Historical inconsistency in buoyancy-drag models . ........ 19 2.5 Related work aimed at modeling stagnation and re-acceleration ....... 19 2.5.1 VortexringcorrectionofRamaprabhu . ... 20 2.6 Vorticityandviscosityinpotentialflow . ........ 20 3 DATA-DRIVEN MODELING OF THE LOW-ATWOOD SINGLE-MODE RAYLEIGH- TAYLORINSTABILITY................................ 21 3.1 Abstract...................................... 21 3.2 Introduction.................................... 21 3.3 Simplemodel ................................... 24 3.3.1 Dynamics ................................. 24 3.3.2 Mixing................................... 26 3.3.3 Coefficientconstraints . 28 iv 3.3.4 Coefficientestimation. 30 3.4 Numericalexperiments . 33 3.4.1 Observables ................................ 35 3.5 Growthstagesthroughlatetimes . .... 37 3.5.1 Exponentialgrowth............................ 39 3.5.2 Saturationregime............................. 40 3.5.3 Viscousregime .............................. 41 3.5.4 Diffusiveregime.............................. 41 3.5.5 Stagnationandre-acceleration . .... 45 3.6 Modelfitcoefficients ............................... 46 3.6.1 Fitting................................... 46 3.6.2 Scopeofthemodels............................ 49 3.6.3 Accuracyofthemodels.......................... 50 3.6.4 Fitcoefficients............................... 52 3.7 Conclusions .................................... 60 3.8 Acknowledgements ................................ 63 4 DIRECT NUMERICAL SIMULATION OF SINGLE MODE THREE-DIMENSIONAL RAYLEIGH-TAYLOREXPERIMENTS . 64 4.1 Abstract...................................... 64 4.2 Introduction.................................... 65 4.3 Numericalmethods................................ 67 4.3.1 Spectralelementformulation. ... 67 4.3.2 Simulationsetup ............................. 69 4.3.3 Post-processing .............................. 71 4.4 Results....................................... 71 4.4.1 Validation ................................. 71 4.4.2 Extension ................................. 75 4.5 Conclusions .................................... 82 4.6 Acknowledgements ................................ 83 5 EFFICIENCY OF HIGH ORDER SPECTRAL ELEMENT METHODS ON PETAS- CALEARCHITECTURES .............................. 84 5.1 Abstract...................................... 84 5.2 Introduction.................................... 85 5.2.1 Outline .................................. 86 5.3 Nek’sComputationalCore . 87 5.3.1 Governing equations and time-splitting . ...... 87 5.3.2 Spectralelementmethod . 88 5.3.3 Computationalprofile .......................... 89 5.3.4 Order-dependentkernels . 90 5.4 KernelAnalysisandOptimization . .... 91 5.4.1 SmallMatrixMultiplications. ... 91 5.4.2 Enhancing Element Update Performance by Streaming Stores .... 94 v 5.4.3 Discussion of Performance Reproducers . ..... 94 5.5 ScenariosandPerformance. ... 97 5.5.1 Architectures ............................... 97 5.5.2 Single mode Rayleigh-Taylor instability . ....... 98 5.5.3 Timetoaccuracy ............................. 103 5.5.4 Wholeapplicationperformance . 105 5.6 Conclusion..................................... 106 6 CONCLUSIONS .................................... 109 6.1 Newmodelforlow-Atwoodsinglemode. 109 6.2 ValidationofDNS ................................ 110 6.3 Convergenceandperformance . 112 6.4 Flowregimes ................................... 113 6.5 Modelcoefficients................................. 113 6.6 Openproblems .................................. 114 REFERENCES....................................... 116 vi LIST OF FIGURES 3.1 Slices of the scalar and vertical component of the velocity at early times and high Grashof number. The arrows indicate the dependence of the model terms on different span-wise length scales, and are identical in both figures. C1 is related to the maximum cross sectional diameter of the bubble in the velocity field. C2 is related to the nominal side-wall diameter of the bubble in the velocity field. C5 is related to the nominal side-wall diameter of the bubble in the scalar field. δ is related to the interface thickness in the scalar field. The slice is in the plane x = y,whichpassesthroughonlybubblecenters. ... 31 3.2 Comparison of height metrics at Gr = 4.8 104 andSc=1. ........... 36 3.3 Bubble Froude number vs. nondimensional× bubble height for Ra = 105.75, Sc = 4, simulation vs. model with successive terms enabled: first C4,C6 and C8, then C3, then C1, then C2, and finally C5 and C7. The dashed vertical lines divide the trajectory into three regimes: linear growth for H/λ < 0.05, saturation until H/λ 8, and viscosity beyond that. The stagnation and re-acceleration tran- sients≈ are seen beginning at H/λ =0.5 and ending by H/λ =1.5. ........ 38 3.4 Bubble Froude number vs. nondimensional bubble height for Ra = 104.5, Sc = 8, simulation vs. model with successive terms enabled: first C4,C6 and C8, then C3, then C1, then C2, and finally C5 and C7. Dashed vertical lines divide the trajectory into four regimes: linear growth for H/λ < 0.05, saturation until H/λ 0.1, viscosity until H/λ 1.3, and diffusion beyond that. The stagnation and re-acceleration≈ transients, seen≈ in Figure 3.3, are suppressed by the viscous regime, which onsets before the transient begins at H/λ =0.5. ......... 38 3.5 Penetration depth, nondimensionalized, vs. the Rayleigh and Schmidt numbers. The dashed line separates completed from incomplete trajectories, which are clipped at h/λ 23. ................................. 42 3.6 Penetration depth,≈ nondimensionalized, vs. the Rayleigh. The dashed line is a 4 best fit line h/λ =2.41 10− Ra 0.36. ..................... 43 3.7 Example fit of the bubble× height,− top, and mixed volume, bottom, at Ra = and Sc = 8. The dashed line in the center plot is the stagnation velocity from potential flow models. The Froude number, which is a nondimensional velocity, is not directly fit but shows agreement between the model and simulation. 47 3.8 Relative model errors versus Rayleigh and Schmidt numbers. Experiments on the left side of the dashed line completed when the bubble stopped rising. Experi- ments on the right side of the dashed line are incomplete, having approached the vertical boundaries of the simulated domain. ....... 51 3.9 Best fit for C1 versus Rayleigh and Schmidt numbers. Experiments on the left side of the dashed line completed when the bubble stopped rising. Experiments on the right side of the dashed line are incomplete, having approached the vertical boundariesofthesimulateddomain. .... 52 vii 3.10 Best fit for C2 versus Rayleigh and Schmidt numbers. Experiments on the left side of the dashed line completed when the bubble stopped rising. Experiments on the right side of the dashed line are incomplete, having approached the vertical boundariesofthesimulateddomain. .... 54 3.11 Best fit for C3 versus Rayleigh and Schmidt numbers. Experiments on the left side of the dashed line completed when the bubble stopped rising. Experiments on the right side of the dashed line are incomplete, having approached the vertical boundariesofthesimulateddomain. .... 56 3.12 Best fit for C5 versus Rayleigh and Schmidt numbers. Experiments on the left side of the dashed line completed when the bubble stopped rising. Experiments on the right side of the dashed line are incomplete, having approached the vertical boundariesofthesimulateddomain. .... 57 3.13 Best fit for C7 versus Rayleigh and Schmidt numbers. Experiments on the left side of the dashed line completed when the bubble stopped rising. Experiments on the right side of the dashed line are incomplete, having approached the vertical boundariesofthesimulateddomain. .... 59 4.1 Initial condition, φ(x,y, 0, 0), for 4.5 mode simulation. The bubbles are logically separated by the solid grid and each unique bubble is labeled. .......... 68 4.2 Froude number vs. height, nondimensionalized by the wavelength. Lines are the derivative of cubic splines through simulation outputs. Points are from experi- ment via direct measurement of the bubble velocity [31]. The
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