DOCSLIB.ORG

Turbulent Complex Flows Reconstruction Via Data Assimilation in Large Eddy Models Pranav Chandramouli

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Turbulent Complex Flows Reconstruction Via Data Assimilation in Large Eddy Models Pranav Chandramouli Turbulent complex flows reconstruction via data assimilation in large eddy models Pranav Chandramouli To cite this version: Pranav Chandramouli. Turbulent complex flows reconstruction via data assimilation in large eddy models. Fluids mechanics [physics.class-ph]. Université Rennes 1, 2018. English. NNT : 2018REN1S035. tel-01951540 HAL Id: tel-01951540 https://tel.archives-ouvertes.fr/tel-01951540 Submitted on 11 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. i For my parents, who got me here and to Palak, who got me through ii iii Acknowledgements Etienne and Dominique, your guidance has ensured I traverse successfully the narrow, treacherous straits that is a PhD program. Your rigorous standards, academic familiarity and attitude towards research are traits I hope to inculcate and emulate. I would like to express my profound thanks to you for taking me in this program, and for guiding me, constantly, through it. To Sylvain, without whom I would not be here, I am thankful for your guidance in everything numerical. To my friends, past and present, in the corridor - ‘Vincent, Benoit, Sandeep, Anca, Antoine, Manu, Valentin, and countless others’, I thank you for all the special memories, fun and funny, we had over the past three years. Special mention to Long, for our mutual fifa addiction, Carlo, for our coffee, and Yacine, for our multitude of breaks. A big thank you to Huguette for taking care of all the administrative works that I would have, otherwise, been swamped with. A final thanks to my family whose constant support and never-ending belief in me always motivated me to work harder and do something more. iv Contents Introduction en français ix Introduction xv 1 Fluid flow fundamentals 1 1.1 Computational fluid dynamics . 2 1.1.1 A brief history of CFD . 2 1.1.2 Governing equations . 2 1.1.3 Discretisation techniques . 3 1.1.4 Boundary conditions . 5 1.1.5 Turbulence modelling . 6 1.2 Experimental fluid dynamics . 6 1.2.1 A brief history of EFD . 6 1.2.2 Hot wire anemometry (HWA) . 7 1.2.3 Laser doppler anemometry (LDA) . 7 1.2.4 Particle image velocimetry (PIV) . 8 1.3 Limitations . 10 1.4 Coupling EFD and CFD . 11 2Turbulencemodelling 15 2.1 Reynolds averaged Navier-Stokes . 17 2.1.1 RANS equations . 17 2.1.2 Turbulent-viscosity models . 18 2.1.3 Reynolds stress models (RSM) . 21 2.2 Large eddy simulation . 21 2.2.1 Filtered Navier-Stokes equations . 22 2.2.2 Smagorinsky models . 23 2.2.3 WALE model . 27 2.2.4 Models under location uncertainty . 28 2.2.5 Other Approaches . 32 2.3 Hybrid approaches . 34 3Largeeddysimulation-applications 37 3.1 Flow solver - Incompact3d . 37 3.2 Channel flow at Re⌧ =395 ........................... 45 3.2.1 Channel flow parameters . 45 3.2.2 Channel flow results . 46 3.3 Wake flow around a circular cylinder at Re = 3900 . 49 3.3.1 Introduction to wake flow . 49 v vi CONTENTS 3.3.2 Flow configuration and numerical methods . 53 3.3.3 Wake flow results . 56 3.4 Concluding remarks . 76 4StochasticSimulations-vortexforcesandLangmuircirculation 79 4.1 Introduction . 79 4.2 Mathematical formulations . 80 4.2.1 CL model . 80 4.2.2 Large-scale drift MULU . 80 4.3 Vortex force theory for streak formation . 83 4.4 Large-scale application . 88 5Dataassimilationtechniques 93 5.1 Mathematical Formulation . 94 5.2 Data assimilation - a review . 94 5.3 4D-Var approach . 98 5.3.1 4D-Var Algorithm Variants . 100 6 Flow reconstruction - snapshot optimisation method 105 6.1 Introduction . 105 6.2 Snapshot optimisation method . 107 6.2.1 SO model concept . 107 6.2.2 Mathematical formulation . 108 6.2.3 Reduced order formulation . 110 6.2.4 Averaging SO method . 111 6.3 Application and results . 111 6.3.1 Synthetic case-studies . 112 6.3.2 Algorithm Enhancement Techniques . 117 6.3.3 Experimental case-studies . 120 6.4 Conclusion . 127 7Dataassimilation-wakeflowatRe=3900 129 7.1 Code Formulation . 129 7.1.1 4D-Var . 129 7.1.2 4D-Varles .................................136 7.2 3D synthetic assimilation . 142 7.2.1 Flow configuration . 142 7.2.2 Observations construction . 144 7.2.3 Initial background construction . 146 7.2.4 Results and Discussion . 146 7.3 2D synthetic assimilation . 152 7.3.1 Flow configuration . 152 7.3.2 Observations construction . 152 7.3.3 Initial background construction . 156 7.3.4 Results and Discussion . 156 7.4 Background covariance . 162 7.5 Coefficient estimation . 165 7.6 Combined case study . 168 7.7 Conclusions . 172 CONTENTS vii Conclusions and perspectives 175 viii CONTENTS Introduction en français Contexte L’étude des fluides remonte à l’antiquité aux travaux d’Archimède tels que le fameux ‘principe d’Archimède’ sur la flottabilité. Au fil du temps, le domaine s’est diversifié dans l’étude des fluides stationnaires (ou de la statique des fluides) et des écoulement fluides (ou de la dynamique des fluides). La dynamique des fluides peut être subdivisée en l’étude de fluides newtoniens et non-newtoniens en fonction de la forme de la contrainte de frottement moléculaire. Les fluides newtoniens, qui sont les fluides étudiés dans cette thèse, ont un taux de déformation proportionnel à la contrainte appliquée et leur dynamique s’exprime mathématiquement au moyen des équations de conservation de la masse, de quantité de mouvement et d’énergie. L’étude de la mécanique des fluides a été la pierre angulaire de nombreuses avancées dans le domaine de l’aérospatial, de la prévision météorologique, des applications géophysiques, de l’automobile, du génie civil, etc. Des approches plus récentes et sophistiquées visent également à améliorer les conditions environnementales à l’intérieur de bâtiments (conception de salle blanche pour l’industrie pharmaceutique ou agroali- mentaire (Arroyo et al., 2016)). Ces nombreuses applications soulignent l’importance des écoulements fluides dans de larges pans de notre vie quotidienne. Cela justifie pleinement l’étude poussée de ces écoulements. L’analyse des écoulements fluides peut se faire de l’une des deux manières suivantes: Expérimentalement: l’étude expérimentale des fluides (EFD) s’appuie sur des méthodes de mesures expérimentales telles que la vélocimétrie par image de particule, la vélocimétrie par suivi de particules, la vélocimétrie doppler laser, etc. Ces méthodes peuvent être intrusives comme c’est le cas pour anémométrie filaire (HWA) ou non intrusive (PIV, LDA, PTV). Des progrès récents ont conduit à l’élaboration de techniques de mesures 3D telles que la PIV tomographique (tomo-PIV) qui permettent d’appréhender le champ des vitesses 3D3C (trois dimensions, trois composantes) dans un volume donné. Ces techniques restent cependant encore assez limitées dans leur étendue spatiale. Numériquement: Dynamique des fluides computationnelle (CFD) repose sur la simu- lation des équations mathématiques de la dynamique des fluides. Les équations de con- servation décrivant la dynamique des fluides sont résolues en utilisant une grande variété de modèles numériques. La précision de ces modèles reste limitée par le coût de calcul requis pour une simulation. Avec l’émergence et le progrès constant des superordinateurs et des processeurs, des écoulements de plus en plus complexes sont simulés. Néanmoins, une grande variété d’écoulement à haute vitesse restent encore pour longtemps hors de portée de ces simulations. Une combinaison de ces deux techniques a été utilisée efficacement, en particulier dans le domaine de la prévision météorologique où les données EFD sont utilisées afin de guider les simulations CFD. Ce couplage permet de s’affranchir ainsi partiellement des incon- vénients associées à ces deux techniques. Une telle combinaison, appelée assimilation de ix x INTRODUCTION EN FRANÇAIS données (AD), est maintenant utilisée de plus en plus fréquemment dans la communauté de la mécanique des fluides avec une augmentation significative de publications au cours de la dernière décennie qui décrivent différentes techniques de couplage (Papadakis and Mémin, 2008; Gu and Oliver, 2007; Artana et al., 2012; Yang, 2014). Cette thèse étudie l’extension d’technique d’assimilation proposée par Gronskis et al. (2013) pour des écoule- ments 2D et par Robinson (2015) pour des écoulements 3D. Le but de ce travail vise à étendre ces derniers travaux de Robinson (2015) à des écoulements plus complexes associés à des vitesses plus élevées en utilisant le concept de simulations aux grandes échelles de la turbulence (LES). Les techniques de type LES ont d’abord été initiées par Smagorinsky (1963) avec la motivation de réduire le coût de calcul d’une simulation atmosphérique, en ne simulant que les grandes échelles des vitesses (les grands tourbillons) tandis que les effets des plus petites échelles sont modélisés. Smagorinsky a introduit un modèle simple qui reste très populaire dans le domaine LES. Au cours des décennies suivantes, des modèles plus complexes ont été introduits par un certain nombre d’auteurs tels que Germano et al. (1991); Nicoud and Ducros (1999), plusieurs simulation LES, ce qui rend sou coût d’autant plus crucial. Dans le cadre des techniques d’assimilation de données s’avère nécessaire d’effectuer. L’objectif de cette thèse est de formuler et de valider des techniques LES pour l’assimilation de données permettant de réduire autant que possible les coûts de calcul associés.
Load more

Recommended publications
  • Numerical Analysis and Fluid Flow Modeling of Incompressible Navier-Stokes Equations
    UNLV Theses, Dissertations, Professional Papers, and Capstones 5-1-2019 Numerical Analysis and Fluid Flow Modeling of Incompressible Navier-Stokes Equations Tahj Hill Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Aerodynamics and Fluid Mechanics Commons, Applied Mathematics Commons, and the Mathematics Commons Repository Citation Hill, Tahj, "Numerical Analysis and Fluid Flow Modeling of Incompressible Navier-Stokes Equations" (2019). UNLV Theses, Dissertations, Professional Papers, and Capstones. 3611. http://dx.doi.org/10.34917/15778447 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. NUMERICAL ANALYSIS AND FLUID FLOW MODELING OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS By Tahj Hill Bachelor of Science { Mathematical Sciences University of Nevada, Las Vegas 2013 A thesis submitted in partial fulfillment of the requirements
    [Show full text]
  • Selective Energy and Enstrophy Modification of Two-Dimensional
    Selective energy and enstrophy modification of two-dimensional decaying turbulence Aditya G. Nair∗, James Hanna & Matteo Aureli Department of Mechanical Engineering, University of Nevada, Reno Abstract In two-dimensional decaying homogeneous isotropic turbulence, kinetic energy and enstrophy are respectively transferred to larger and smaller scales. In such spatiotemporally complex dynamics, it is challenging to identify the important flow structures that govern this behavior. We propose and numerically employ two flow modification strategies that leverage the inviscid global conservation of energy and enstrophy to design external forcing inputs which change these quantities selectively and simultaneously, and drive the system towards steady-state or other late-stage behavior. One strategy employs only local flow-field information, while the other is global. We observe various flow structures excited by these inputs and compare with recent literature. Energy modification is characterized by excitation of smaller wavenumber structures in the flow than enstrophy modification. 1 Introduction Turbulent flows exhibit nonlinear interactions over a wide range of spatiotemporal scales. In two-dimensional (2D) decaying turbulence, the rate of energy dissipation is considerably slowed by kinetic energy transfer to large-scale coherent vortex cores through the inverse energy flux mechanism [2,3,5, 13, 14], while the enstrophy dissipation rate is enhanced by enstrophy transfer to small scale eddies [21]. The identification and modification of collective structures in the flow that accelerate or decelerate the inverse energy flux mechanism or alter the enstrophy cascade is a fundamental question [9, 10]. This is unlikely to be addressed by flow modification strategies based on linearization of the Navier–Stokes equations and reduced-order/surrogate representations that lack strict adherence to conservation laws.
    [Show full text]
  • Enstrophy Transfers in Helical Turbulence
    Enstrophy transfers in helical turbulence Shubhadeep Sadhukhan,1, ∗ Roshan Samuel,2, y Franck Plunian,3, z Rodion Stepanov,4, x Ravi Samtaney,5, { and Mahendra Kumar Verma6, ∗∗ 1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India 2Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India 3Universit´eGrenoble Alpes, Universit´eSavoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France 4Institute of Continuous Media Mechanics UB RAS, Perm, Russian Federation 5Mechanical Engineering, Division of Physical Science and Engineering, King Abdullah University of Science and Technology - Thuwal 23955-6900, Kingdom of Saudi Arabia 6Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016 (Dated: April 23, 2020) In this paper we study the enstrophy transers in helical turbulence using direct numerical simu- lation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra (∼ k−5=3) and fluxes (constants). We also calculate the separate con- tributions to enstrophy transfers via velocity to vorticity and vorticity to vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the fluxes due to vorticity stretching are larger than that due to advection. These transfers too are insensitive to helicity injection. I. INTRODUCTION tensively studied [7{26]. Andr´eand Lesieur [9] studied the effect of helicity on the evolution of isotropic turbu- Turbulence is a classic and still open problem in fluid lence at high Reynolds numbers using a variant of Marko- dynamics.
    [Show full text]
  • DNS, Enstrophy Balance, and the Dissipation Equation in a Separated Turbulent Channel Flow P
    43rd AIAA Fluid Dynamics Conference, June 24-27, 2013, San Diego, California DNS, Enstrophy Balance, and the Dissipation Equation in a Separated Turbulent Channel Flow P. Balakumar Flow Physics and Control Branch R. Rubinstein and C. L. Rumsey Computational AeroSciences Branch NASA Langley Research Center, Hampton, VA 23681 The turbulent flows through a plane channel and a channel with a constriction (2-D hill) are numerically simulated using DNS and RANS calculations. The Navier-Stokes equations in the DNS are solved using a higher order kinetic energy preserving central schemes and a fifth order accurate upwind biased WENO scheme for the space discretization. RANS calculations are performed using the NASA code CFL3D with the k- omega SST two-equation model and a full Reynolds stress model. Using DNS, the magnitudes of different terms that appear in the enstrophy equation are evaluated. The results show that the dissipation and the diffusion terms reach large values at the wall. All the vortex stretching terms have similar magnitudes within the buffer region. Beyond that the triple correlation among the vorticity and strain rate fluctuations becomes the important kinematic term in the enstrophy equation. This term is balanced by the viscous dissipation. In the separated flow, the triple correlation term and the viscous dissipation term peak locally and balance each other near the separated shear layer region. These findings concur with the analysis of Tennekes and Lumley, confirming that the energy transfer terms associated with the small-scale dissipation and the fluctuations of the vortex stretching essentially cancel each other, leaving an equation for the dissipation that is governed by the large-scale motion.
    [Show full text]
  • Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows
    Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows Di Kang, Dongfang Yun and Bartosz Protas∗ Department of Mathematics and Statistics, McMaster University Hamilton, Ontario, L8S 4K1, Canada March 11, 2020 Abstract This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system — as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy 0 are sought such that the enstrophy in the resulting Navier-Stokes flow is maximizedE at some time T . Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of 0 and T , we demonstrate that the maximum growth of enstrophy is in E 3=2 fact finite and scales in proportion to 0 as 0 becomes large. Thus, in such worst-case scenario the enstrophy still remains boundedE E for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.
    [Show full text]
  • Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace
    Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace Adapted from Publisher: John S. Wiley & Sons 2002 Center for Scientific Computation and Applied Mathematical Modeling Institute for Physical Science Johns Hopkins University National Science Foundation and Technology Describe how properties of a fluid element change, as given by the substantial derivative, D( )/Dt, as the element is transported through the flow field. mean and fluctuating velocity components are divergence free (N-S) Decomposing into mean and fluctuating parts, invoking continuity and averaging using Reynolds rules yields Reynolds stress tensor This equation and together are called the Reynolds Averaged Navier-Stokes Equations (RANS). This equation describes the transport of the Mean momentum in the xi direction There are more unknowns than the four available equations. To “close” the set of equations the Reynolds stress term must be related to the velocity gradient in some way through a model equation. Reynolds Shear stress or velocity fluctuation covariance For i = j, and summing over i, half this covariance becomes the turbulent kinetic energy (TKE) when multiplied by the density K T The components of the TKE are related to the variances of the velocity fluctuations The square roots of these variance are the root-mean-square values of the velocity fluctuation components. They are a measure of the amplitudes of the fluctuations. By decomposing the N-S equation for momentum and subtracting the RANS equation from it, a transport equation for the momentum of the fluctuations is obtained Multiply the transport equation for the fluctuating momentum, term by term, by The fluctuating velocity components, uj.
    [Show full text]
  • Potential Enstrophy in Stratified Turbulence
    Potential enstrophy in stratified turbulence Michael L. Waite† Department of Applied Mathematics, University of Waterloo, 200 University Avenue W., Waterloo, ON N2L 3G1, Canada (Received 14 December 2012; revised 13 February 2013; accepted 9 March 2013) Direct numerical simulations are used to investigate potential enstrophy in stratified turbulence with small Froude numbers, large Reynolds numbers, and buoyancy Reynolds numbers (Reb) both smaller and larger than unity. We investigate the conditions under which the potential enstrophy, which is a quartic quantity in the flow variables, can be approximated by its quadratic terms, as is often done in geophysical fluid dynamics. We show that at large scales, the quadratic fraction of the potential enstrophy is determined by Reb. The quadratic part dominates for small Reb, i.e. in the viscously coupled regime of stratified turbulence, but not when Reb & 1. The breakdown of the quadratic approximation is consistent with the development of Kelvin–Helmholtz instabilities, which are frequently observed to grow on the layerwise structure of stratified turbulence when Reb is not too small. Key words: geophysical and geological flows, stratified turbulence, turbulence simulation 1. Introduction The Ertel potential vorticity (PV) is an important quantity in geophysical fluid dynamics. PV is conserved following the flow in adiabatic rotating stratified fluids, and PV conservation is fundamental to the theory of large-scale quasi-geostrophic (QG) motion in the atmosphere and ocean (e.g. Pedlosky 1987). In this paper, we consider the integrated squared PV, or potential enstrophy, in strongly stratified turbulence without rotation. This parameter regime is relevant at intermediate geophysical scales where QG breaks down.
    [Show full text]
  • Arxiv:2007.08269V1 [Physics.Flu-Dyn] 16 Jul 2020
    AIP/123-QED Anomalous Features in Internal Cylinder Flow Instabilities subject to Uncertain Rotational Effects Ali Akhavan-Safaei,1, 2 S. Hadi Seyedi,1 and Mohsen Zayernouri1, 3, a) 1)Department of Mechanical Engineering, Michigan State University, 428 S Shaw Ln, East Lansing, MI 48824, USA 2)Department of Computational Mathematics, Science, and Engineering, Michigan State University, 428 S Shaw Ln, East Lansing, MI 48824, USA 3)Department of Statistics and Probability, Michigan State University, 619 Red Cedar Road, Wells Hall, East Lansing, MI 48824, USA (Dated: 17 July 2020) arXiv:2007.08269v1 [physics.flu-dyn] 16 Jul 2020 1 We study the flow dynamics inside a high-speed rotating cylinder after introducing strong symmetry-breaking disturbance factors at cylinder wall motion. We propose and formulate a mathematically robust stochastic model for the rotational motion of cylinder wall alongside the stochastic representation of incompressible Navier- Stokes equations. We employ a comprehensive stochastic computational fluid dy- namics framework combining spectral/hp element method and probabilistic colloca- tion method to obtain high-fidelity realizations of our mathematical model in order to quantify the propagation of parametric uncertainty for dynamics-representative quantities of interests. We observe that the modeled symmetry-breaking disturbances cause a flow instability arising from the wall. Utilizing global sensitivity analysis ap- proaches, we identify the dominant source of uncertainty in our proposed model. We next perform a qualitative and quantitative statistical analysis on the fluctuating fields characterizing the fingerprints and measures of intense and rapidly evolving non-Gaussian behavior through space and time. We claim that such non-Gaussian statistics essentially emerge and evolve due to an intensified presence of coherent vortical motions initially triggered by the flow instability due to symmetry-breaking rotation of the cylinder.
    [Show full text]
  • Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation
    Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation Dongfang Yun and Bartosz Protas∗ Department of Mathematics and Statistics, McMaster University Hamilton, Ontario, L8S 4K1, Canada Abstract This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blow-up in these two regimes, this makes it a useful model to study the maxi- mum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy arXiv:1610.09578v2 [physics.flu-dyn] 15 Jan 2018 rate of growth on the fractional dissipation exponent has the same global form in the sub- critical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the frac- tional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small ∗Corresponding author.
    [Show full text]
  • 3.2 Energy and Enstrophy Spectra
    Copyright by Nikhil Padhye 1994 Statistical Mechanics of 2-D Fluids by Nikhil Padhye, B.Tech. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Arts The University of Texas at Austin May, 1994 Statistical Mechanics of 2-D Fluids APPROVED BY SUPERVISING COMMITTEE: Supervisor: µ!:10 1. ~ Philip J. Morrison , /7 ~uwL.fJ. J(~d6, Richard D. Hazeltine/ .• ...- Acknowledgements I am grateful to my advisor, Phil Morrison, for his enthusiastic guidance and support. I also wish to thank him, Neil Balmforth and Richard Hazeltine for critically reading preliminary forms of this thesis and making helpful suggestions. I also enjoyed the opportunity to interact with Brad, Diego, Raul, John and Bill during the Friday afternoon meetings. Nikhil Padhye April, 18th, 1994. IV ABSTRACT Statistical Mechanics of 2-D Fluids by Nikhil Padhye, M.A. The University of Texas at Austin, 1994 SUPERVISOR: Philip J. Morrison Various approaches to study the turbulent relaxation of 2-D fluids are discussed. The results are compared to an experiment on electrons in a magnetized column. v Contents 1 Introduction 1 1.1 The 2-D Euler Equation for the Ideal Fluid . 2 1.2 Hamiltonian Description of the 2-D Ideal Fluid 4 1.3 Constants of Motion 5 2 Statistical Methods 7 2.1 Point Vortex Approximation 7 2.2 Maximum Entropy Flows . 11 3 The Selective Decay Hypothesis 16 3.1 Effects of Viscosity and Truncation 16 3.2 Energy and Enstrophy Spectra .
    [Show full text]
  • Vorticity and Potential Vorticity
    Chapter 1 Vorticity and Potential Vorticity In this chapter we use potential vorticity to unify a number of the concepts we have previously encountered. We derive the general form of potential vorticity conservation, and then appropriate approximate forms of the potential vorticity conservation law, corresponding to the familiar quasi- geostrophic and planetary geostrophic equations. In order to keep the chapter essentially self- contained, there is some repetition of material elsewhere. 1.1 Vorticity 1.1.1 Preliminaries Vorticity is defined to be the curl of velocity, and normally denoted by the symbol ω. Thus ω =∇×v (1.1) Circulation is defined to be the integral of velocity around a closed loop. That is I C = v · dl. (1.2) Using Stokes’ theorem circulation is also given by Z C = ω · dS (1.3) The circulation thus may also be defined as the integral of vorticity over a material surface. The circulation is not a field like vorticity and velocity. Rather, we think of the circulation around a particular material line of finite length, and so its value generally depends on the path chosen. If δS is an infinitesimal surface element whose normal points in the direction of the unit vector nˆ, then I n ·∇×v = 1 v · l δS d (1.4) 1 2 CHAPTER 1. VORTICITY AND POTENTIAL VORTICITY where the line integral is around the infinitesimal area. Thus the vorticity at a point is proportional to the circulation around the surrounding infinitesimal fluid element, divided by the elemental area. (Sometimes the curl is defined by an equation similar to (1.4).) A heuristic test for the presence of vorticity is then to imagine a small paddle-wheel in the flow: the paddle wheel acts as a ‘circulation- meter’ and so rotates if vorticity is non-zero.
    [Show full text]
  • Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
    Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows Diego Ayala1;2 and Bartosz Protas2;∗ 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA 2 Department of Mathematics and Statistics, McMaster University Hamilton, Ontario, L8S 4K1, Canada March 2, 2017 Abstract In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy E0 which maximize the instantaneous rate of growth of enstrophy dE=dt. We provide an analytic characterization of these extreme vortex states in the limit of vanishing enstrophy E0 and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of dE=dt in this limit. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu & Doering (2008) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound dE=dt < C E3, for some constant C > 0. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy arXiv:1605.05742v2 [physics.flu-dyn] 28 Feb 2017 E0. However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.
    [Show full text]