Point Vortex Dynamics in Two Dimensions
|
|
- Doreen Ward
- 6 years ago
- Views:
Transcription
1 Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 9 April to May, 9 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei Zhu Supervised by: Professor Keith Moffatt
2 Introduction Some theory on point vortices The vorticity of a fluid is defined as its circulation, thus it is given mathematically by the curl of the velocity vector field: ω = u In this project we consider only two dimensional flows, as such the vorticity vector is always directed perpendicularly to the plane of the flow. We denote the strength of the vortex as κ. Furthermore, we consider idealised point vortices such that the fluid is irrotational everywhere except at a point concentration. Stoke s theorem may be written: ω = u.dl. S Thus, for a circular loop around a single point vortex, the above integral may be evaluated as C κ = πrv θ where v θ is the tangential velocity arising from the point vortex along a circle of radius r. The vorticity equation is derived by taking the curl of the Navier-Stokes equation: Dω Dt = ω. u + ν ω However, in two dimensional inviscid flows both terms on the right vanish, yielding the important result that in such flows vorticity is carried with the fluid. Consequently, the motion of a point vortex is seen to depend only upon the velocity of the fluid in which it is embedded. If the fluid is stationary then the vortex will induce a circular velocity field, but will itself remain fixed in space. The interaction of point vortices, therefore, is governed by the fluid velocities that each induce at the position of the others. Given the radial velocity distribution induced by a point vortex earlier derived, the velocity of a vortex situated at (x α,y α ), may be found as the vector sum of the velocities induced by each of the vorticies at that point: dx α dt = β κ β (y α y β ) πr αβ, dy α dt = β κ β (x α x β ) πr αβ where the summation is performed for α β. A side-effect of the point vortex idealisation is that the velocity field at the location of the point vortex is singular. This has the unhappy consequence of causing the kinetic energy of the system to become infinite. Furthermore integrals of momentum and angular momentum become infinite in the limit as r. As such, new invariants must be devised in order to provide some reasonable physical constraints to the system. The first constraint is provided by the Hamiltonian for the system, which is given by: H = π N α,β= κ α κ β ln r αβ
3 This is a measure of the energy of interaction of the vortices. Momentum and angular momentum like quantities are given by: P = ρ(σκ αy α, Σκ α x α,) = cst M = ρ(,, Σκ α(x α + y α)) = cst In the case that Σκ α, a fixed centre of vorticity and dispersion, D, of the vortex distribution may be defined: x = Σκ αx α Σκ α, ȳ = Σκ αy α Σκ α D = Σκ α [(x α x) + (y α ȳ) ] = cst In general it is not possible to integrate a system of N point vortices where N >, however, the system is deterministic and solutions may be calculated numerically. In this project, code was written to numerically integrate the positions of an arbitrary number of vortices forward in time. The above invariants were used to confirm that the numerical procedure was working correctly. Results of numerical simulations Testing the code: Two point vortices In the case that each vortex is of similar strength, the motion of the vortex pair is determined by the relative signs of the vortices. Vortices of opposing sign are observed to propagate together along a line perpendicular to their bisector. Note that in this case Σκ α =, implying that the system is not constrained to have a fixed centre of vorticity or dispersion. However, similar signed vortices are constrained in this manner and they are observed to move in a circle around their fixed centre of vorticity. Figure shows the results of the numerical solutions for these cases (a) Two vortices, different signs (b) Two vortices, same signs Figure : Some numerical results
4 Three point vortices on three verticies of a square. κ =.8.. r r. κ = r κ = Figure : Initial configuration of vortices Three vortices were initially arranged as in Figure such that they formed a right angled triangle with two sides having equal length. The chosen values of vortex strength mean that the system has a fixed centre of vorticity, and this is clear from the resulting vortex tracks (Figure ) which show the vortices moving in a biperiodic manner around the point (, ). The periodic motion of the three vortices through phase space is illustrated in Figure b) where the values of the separations of the vortices have been plotted at each timestep r r r (a) Vortex tracks (b) Phase space Figure : Three point vortices around a square Instabilities in symmetrical three vortex configurations Three vortices arranged in an equilateral triangle will rotate around a fixed centre of vorticity provided that the sum of the vorticities is non-zero (i.e. the dispersion invariant is defined). However, the stability of their motion is determined by the value of the invariant J, a function of P and M given
5 by: J = ( ρ (κ + κ + κ ) M ρ P ) = R (κ κ + κ κ + κ κ ) Theory states that the configuration should be stable to perturbations when J >, unstable when J < and neutrally stable when J =. In the following simulations, the strengths of the point vortices were set such that J satisfied each of the above cases. In the unperturbed run, the vortices were arranged exactly at the vertices of an equilateral triangle. In the perturbed run, one of the vortices was shifted by. units in the x direction (a) Vortex tracks: note that two of the vortices follow identical trajectories.. r r..... r. (b) Separation space of vortices Figure : J > case, perturbed and unperturbed cases are identical In the case where J >, the system is theoretically stable. After one million iterations with a timestep of., the perturbed and unperturbed cases were indistinguishable by eye. In both cases the vortices exhibit periodic motion tracing circles with radii determined by their relative strengths. Note the gradual drift in the separations of the vortices arising from numerical error.... r r..... r.8 (a) Vortex tracks: unperturbed (b) Separation space: unperturbed Figure : Neutrally stable, unperturbed configuration: J =
6 (a) Vortex tracks: perturbed...8 r r r. (b) Separation space: perturbed Figure : Neutrally stable, perturbed configuration: J = J = is neutrally stable. The unperturbed vortex tracks are shown in Figure. Perturbations to this state caused the vortices to slowly spiral further apart, as shown in Figure, however the overall track pattern is largely unchanged. The configuration where (J < ) is unstable and a small perturbation to the initial position causes a dramatic change in the behaviour of the vortices over time. Note that in the case illustrated in Figure 8, κ =, κ = and κ =, therefore there is no fixed centre of vorticity and the dispersion of the system is not invariant r.. r r..... (a) Vortex tracks, unperturbed (b) Separation space, unperturbed r r.... r (c) Vortex tracks, perturbed (d) Separation space, perturbed Figure 7: Unstable, J <. Note dramatic change in vortex tracks and separations arising from small perturbation to original position of one vortex.
7 Instabilities in, 7 and 8 vortex configurations A further interesting case arises in the stability of N equal vortices which are regularly distributed around the circumference of a circle. If N equal vortices are placed at the vertices of a regular polygon, then a rotating steady state is obviously possible. The question is: what would happen if one of the vortices is perturbed? Detailed investigation shows that the configuration is stable if N and unstable if N 8. The case N = 7 is neutrally stable. The following figures illustrate the effects of perturbation on the stability of, 7 and 8 equal vortices located on the vertices of a regular polygon. It can be seen that a small perturbation has no significant effect on the trajectories of and 7 vortices, but it completely changes the trajectories of 8 vortices. (a) vortices, unperturbed (b) vortices, perturbed (c) 7 vortices, unperturbed (d) 7 vortices, perturbed (e) 8 vortices, unperturbed (f) 8 vortices, perturbed Figure 8: Vortex tracks for vortices arranged around circle. Note stability of and 7 vortex cases to small perturbations whilst the 8 vortex case demonstrates significant instability.
8 The three vortex collapsing configuration Figure 9: Initial configuration of vortices A special, neutrally stable (J = ) three vortex configuration exists wherein the vortices collapse to a point within a finite time. Let κ =, κ =, κ = be the strength of the vortices, and a, b and c be their initial separations r, r and r. It can be shown that the area of the triangle created by these vortices will go to zero in a finite time if the following constraints are satisfied: κ + κ + κ = a κ + b κ + c κ = These constraints correspond to setting J = and ensuring that the shape of the triangle is constant in time (though its area changes). In this configuration, each vortex follows a spiral path towards the centre of vorticity. Figure shows the vortex trajectories and the vortex separations as a function of time, and illustrates that the separations go to zero in finite time... separation (a) Vortex tracks t (b) Vortex separations Figure : Collapsing vortices, J =. Note that simulation stops prior to collapse due to numerical instability Coherent structures in a vortex system Let s investigate a system of four point vortices. The total vorticity of the system is not zero (Σκ α ). As you know, it means that we can define centre of vorticity by using earlier stated formulas. We can also define a momentum of the system and as you can see on Figure it does not coincide 7
9 . (,), κ =.. - (,), κ = (,), κ = (, ), κ = (a) Vortex configuration (b) Vortex tracks Figure : Initial vortex configuration and vortex tracks with direction of propagation of the vortices. This is the first but not last interesting feature of the system. Velocity of the first point vortex with κ = reduced to nearly zero, while another vortices continue moving with the same speed. It is well known that velocity of the point vortex depends only on interaction with the other ones within the system, in other words, it depends on structure of the system and vorticity. But we cannot say the same about acceleration of vortex. According to M. Rast and J-F Pinton work at smallest temporal increment acceleration can be modified by vorticity reconfiguration within the system. In modern theory of the systems of point vortices it has often been proposed that coherent structures play an important role in decreasing or increasing of velocities of the point vortices. On Figure there is exactly such structure created by vortices with total vorticity equal to zero. And we suppose to think that first vortices velocity reduction can be caused by the coherent structure created here. Much deeper analysis which includes solution of the system of four nonlinear equations is required in this problem. -vortex chaos and the Liapunov exponent The three vortex problem may be integrated using the system invariants to constrain the solution. However, there are insufficient invariants available to integrate the four vortex problem in general, and as such solutions must be computed numerically. Of particular interest is the emergence of chaotic behaviour in four vortex configurations. Though not all configurations are chaotic, many are, and in this section we identify a chaotic system and find the Liapunov exponent for that system. A chaotic system is one where tiny perturbations to the initial conditions of the system result in dramatically different solutions, where the difference between solutions grows quasi-exponentially with time. For point vortex systems as are considered here, we may define the distance between solutions as d(t) = r perturbed (t) r unperturbed (t), then, for a chaotic system this distance will grow exponentially at first until the physical constraints on the system (such as fixed dispersion) force the distance to 8
10 eventually decrease as /t. Initially at least, a chaotic system behaves as:: d(t) = d e µt. d is the distance between the solutions at t =, thus it is the initial perturbation. If µ then the distance between the solutions tends to zero (or a constant), and the system is not chaotic. However, for µ > the distance between the solutions grows exponentially and the system exhibits deterministic chaos. Rearranging the above, and taking the limit as t, we obtain the following expression for µ: µ = lim t t (ln d(t) ln(d )). Four vortices were arranged as in Figure and the solution was iterated for one million timesteps. The first vortex (having strength κ = ) was then perturbed in space by. and the solution was rerun. The resulting vortex tracks are shown in Figure and it is clear that the small perturbation has caused a significant change to the motion of the vortices. Figure : Initial configuration of vortices (a) Unperturbed vortex tracks (b) Perturbed vortex tracks Figure : Vortex tracks for perturbed and unperturbed vortex chaotic system The Liapunov exponent was evaluated at each timestep and is shown in Figure. Initially the exponent exhibits some high frequency variation, but over time it smooths out and reaches a positive non-zero limit, demonstrating that the system is chaotic. It should be noted that in this case the dispersion of the system is constrained, thus there is a maximum separation, d(t), that the vortices may attain. This means that in the limit that t, µ will be seen to decrease gradually towards zero in this case. 9
11 e-.e e- - 8 e+ (a) Separation between vortex tracks -e- 8 e+ (b) Liapunov exponent as a function of time Figure : Track separations exhibiting quasi-exponential behaviour in four vortex case References Professor Keith Moffatt: Notes and personal communication. Phys. Rev. E 79, (9): Point-vortex model for Lagrangian intermittency in turbulence (Mark Peter Rast, Jean-Francois Pinton)
On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data
On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data Monika Nitsche Department of Mathematics and Statistics University of New Mexico Collaborators: Darryl
More informationVortices in Superfluid MODD-Problems
Vortices in Superfluid MODD-Problems May 5, 2017 A. Steady filament (0.75) Consider a cylindrical beaker (radius R 0 a) of superfluid helium and a straight vertical vortex filament in its center Fig. 2.
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More information3 Generation and diffusion of vorticity
Version date: March 22, 21 1 3 Generation and diffusion of vorticity 3.1 The vorticity equation We start from Navier Stokes: u t + u u = 1 ρ p + ν 2 u 1) where we have not included a term describing a
More informationVorticity and Dynamics
Vorticity and Dynamics In Navier-Stokes equation Nonlinear term ω u the Lamb vector is related to the nonlinear term u 2 (u ) u = + ω u 2 Sort of Coriolis force in a rotation frame Viscous term ν u = ν
More informationLecture 3: The Navier-Stokes Equations: Topological aspects
Lecture 3: The Navier-Stokes Equations: Topological aspects September 9, 2015 1 Goal Topology is the branch of math wich studies shape-changing objects; objects which can transform one into another without
More informationHamiltonian aspects of fluid dynamics
Hamiltonian aspects of fluid dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS 01/29/08, 01/31/08 Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 1 / 34 Outline
More information(Jim You have a note for yourself here that reads Fill in full derivation, this is a sloppy treatment ).
Lecture. dministration Collect problem set. Distribute problem set due October 3, 004.. nd law of thermodynamics (Jim You have a note for yourself here that reads Fill in full derivation, this is a sloppy
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationInstabilities due a vortex at a density interface: gravitational and centrifugal effects
Instabilities due a vortex at a density interface: gravitational and centrifugal effects Harish N Dixit and Rama Govindarajan Abstract A vortex placed at an initially straight density interface winds it
More informationGFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability
GFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability Jeffrey B. Weiss; notes by Duncan Hewitt and Pedram Hassanzadeh June 18, 2012 1 Introduction 1.1 What
More informationTHE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS
ASIAN J. MATH. c 2009 International Press Vol. 13, No. 1, pp. 007 014, March 2009 002 THE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS Y. CHARLES LI Abstract. Nadirashvili presented a
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationFluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College
Fluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
More information3.5 Vorticity Equation
.0 - Marine Hydrodynamics, Spring 005 Lecture 9.0 - Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full Navier-Stokes equations (unsteady, viscous
More informationPHYS2330 Intermediate Mechanics Fall Final Exam Tuesday, 21 Dec 2010
Name: PHYS2330 Intermediate Mechanics Fall 2010 Final Exam Tuesday, 21 Dec 2010 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively
More informationSolution Set Two. 1 Problem #1: Projectile Motion Cartesian Coordinates Polar Coordinates... 3
: Solution Set Two Northwestern University, Classical Mechanics Classical Mechanics, Third Ed.- Goldstein October 7, 2015 Contents 1 Problem #1: Projectile Motion. 2 1.1 Cartesian Coordinates....................................
More informationVortex motion. Wasilij Barsukow, July 1, 2016
The concept of vorticity We call Vortex motion Wasilij Barsukow, mail@sturzhang.de July, 206 ω = v vorticity. It is a measure of the swirlyness of the flow, but is also present in shear flows where the
More informationmeters, we can re-arrange this expression to give
Turbulence When the Reynolds number becomes sufficiently large, the non-linear term (u ) u in the momentum equation inevitably becomes comparable to other important terms and the flow becomes more complicated.
More informationChapter 5. Sound Waves and Vortices. 5.1 Sound waves
Chapter 5 Sound Waves and Vortices In this chapter we explore a set of characteristic solutions to the uid equations with the goal of familiarizing the reader with typical behaviors in uid dynamics. Sound
More informationLifting Airfoils in Incompressible Irrotational Flow. AA210b Lecture 3 January 13, AA210b - Fundamentals of Compressible Flow II 1
Lifting Airfoils in Incompressible Irrotational Flow AA21b Lecture 3 January 13, 28 AA21b - Fundamentals of Compressible Flow II 1 Governing Equations For an incompressible fluid, the continuity equation
More informationLagrangian Coherent Structures (LCS)
Lagrangian Coherent Structures (LCS) CDS 140b - Spring 2012 May 15, 2012 ofarrell@cds.caltech.edu A time-dependent dynamical system ẋ (t; t 0, x 0 )=v(x(t;,t 0, x 0 ),t) x(t 0 ; t 0, x 0 )=x 0 t 2 I R
More informationClassical Mechanics III (8.09) Fall 2014 Assignment 7
Classical Mechanics III (8.09) Fall 2014 Assignment 7 Massachusetts Institute of Technology Physics Department Due Wed. November 12, 2014 Mon. November 3, 2014 6:00pm (This assignment is due on the Wednesday
More information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationChaos and Liapunov exponents
PHYS347 INTRODUCTION TO NONLINEAR PHYSICS - 2/22 Chaos and Liapunov exponents Definition of chaos In the lectures we followed Strogatz and defined chaos as aperiodic long-term behaviour in a deterministic
More informationRegularization by noise in infinite dimensions
Regularization by noise in infinite dimensions Franco Flandoli, University of Pisa King s College 2017 Franco Flandoli, University of Pisa () Regularization by noise King s College 2017 1 / 33 Plan of
More informationSession 6: Analytical Approximations for Low Thrust Maneuvers
Session 6: Analytical Approximations for Low Thrust Maneuvers As mentioned in the previous lecture, solving non-keplerian problems in general requires the use of perturbation methods and many are only
More informationProject Topic. Simulation of turbulent flow laden with finite-size particles using LBM. Leila Jahanshaloo
Project Topic Simulation of turbulent flow laden with finite-size particles using LBM Leila Jahanshaloo Project Details Turbulent flow modeling Lattice Boltzmann Method All I know about my project Solid-liquid
More informationAttractor of a Shallow Water Equations Model
Thai Journal of Mathematics Volume 5(2007) Number 2 : 299 307 www.math.science.cmu.ac.th/thaijournal Attractor of a Shallow Water Equations Model S. Sornsanam and D. Sukawat Abstract : In this research,
More informationLagrangian acceleration in confined 2d turbulent flow
Lagrangian acceleration in confined 2d turbulent flow Kai Schneider 1 1 Benjamin Kadoch, Wouter Bos & Marie Farge 3 1 CMI, Université Aix-Marseille, France 2 LMFA, Ecole Centrale, Lyon, France 3 LMD, Ecole
More informationOffshore Hydromechanics Module 1
Offshore Hydromechanics Module 1 Dr. ir. Pepijn de Jong 4. Potential Flows part 2 Introduction Topics of Module 1 Problems of interest Chapter 1 Hydrostatics Chapter 2 Floating stability Chapter 2 Constant
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationKirchhoff s Elliptical Vortex
1 Figure 1. An elliptical vortex oriented at an angle φ with respect to the positive x axis. Kirchhoff s Elliptical Vortex In the atmospheric and oceanic context, two-dimensional (height-independent) vortices
More information7 EQUATIONS OF MOTION FOR AN INVISCID FLUID
7 EQUATIONS OF MOTION FOR AN INISCID FLUID iscosity is a measure of the thickness of a fluid, and its resistance to shearing motions. Honey is difficult to stir because of its high viscosity, whereas water
More informationAGAT 2016, Cargèse a point-vortex toy model
AGAT 2016, Cargèse a point-vortex toy model Jean-François Pinton CNRS & ENS de Lyon M.P. Rast, JFP, PRE 79 (2009) M.P. Rast, JFP, PRL 107 (2011) M.P. Rast, JFP, P.D. Mininni, PRE 93 (2009) Motivations
More informationMAE 101A. Homework 7 - Solutions 3/12/2018
MAE 101A Homework 7 - Solutions 3/12/2018 Munson 6.31: The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression ψ = 2(x y) where the stream function has
More informationThe Liapunov Method for Determining Stability (DRAFT)
44 The Liapunov Method for Determining Stability (DRAFT) 44.1 The Liapunov Method, Naively Developed In the last chapter, we discussed describing trajectories of a 2 2 autonomous system x = F(x) as level
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationChaos in the Hénon-Heiles system
Chaos in the Hénon-Heiles system University of Karlstad Christian Emanuelsson Analytical Mechanics FYGC04 Abstract This paper briefly describes how the Hénon-Helies system exhibits chaos. First some subjects
More informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More informationSelf-Organization of Plasmas with Flows
Self-Organization of Plasmas with Flows ICNSP 2003/ 9/10 Graduate School of Frontier Sciences,, National Institute for Fusion Science R. NUMATA, Z. YOSHIDA, T. HAYASHI ICNSP 2003/ 9/10 p.1/14 Abstract
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
More information18.325: Vortex Dynamics
8.35: Vortex Dynamics Problem Sheet. Fluid is barotropic which means p = p(. The Euler equation, in presence of a conservative body force, is Du Dt = p χ. This can be written, on use of a vector identity,
More informationThe vorticity field. A dust devil
The vorticity field The vector ω = u curl u is twice the local angular velocity in the flow, and is called the vorticity of the flow (from Latin for a whirlpool). Vortex lines are everywhere in the direction
More informationPhysics 141 Rotational Motion 1 Page 1. Rotational Motion 1. We're going to turn this team around 360 degrees.! Jason Kidd
Physics 141 Rotational Motion 1 Page 1 Rotational Motion 1 We're going to turn this team around 360 degrees.! Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid
More informationQueen s University at Kingston. Faculty of Arts and Science. Department of Physics PHYSICS 106. Final Examination.
Page 1 of 5 Queen s University at Kingston Faculty of Arts and Science Department of Physics PHYSICS 106 Final Examination April 16th, 2009 Professor: A. B. McLean Time allowed: 3 HOURS Instructions This
More informationSHORT WAVE INSTABILITIES OF COUNTER-ROTATING BATCHELOR VORTEX PAIRS
Fifth International Conference on CFD in the Process Industries CSIRO, Melbourne, Australia 13-15 December 6 SHORT WAVE INSTABILITIES OF COUNTER-ROTATING BATCHELOR VORTEX PAIRS Kris RYAN, Gregory J. SHEARD
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationNonlinear Evolution of a Vortex Ring
Nonlinear Evolution of a Vortex Ring Yuji Hattori Kyushu Institute of Technology, JAPAN Yasuhide Fukumoto Kyushu University, JAPAN EUROMECH Colloquium 491 Vortex dynamics from quantum to geophysical scales
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation
More informationPoint vortices in a circular domain: stability, resonances, and instability of stationary rotation of a regular vortex polygon
Point vortices in a circular domain: stability, resonances, and instability of stationary rotation of a regular vortex polygon Leonid Kurakin South Federal University Department of Mathematics, Mechanics
More informationNONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis
LECTURE 3: FLOWS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharr Sstems Analsis, Modelling & Prediction Group www.eng.o.ac.uk/samp patrick@mcsharr.net Tel: +44 83 74 Numerical integration Stabilit analsis
More information= w. These evolve with time yielding the
1 Analytical prediction and representation of chaos. Michail Zak a Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109, USA Abstract. 1. Introduction The concept of randomness
More informationA RECURRENCE THEOREM ON THE SOLUTIONS TO THE 2D EULER EQUATION
ASIAN J. MATH. c 2009 International Press Vol. 13, No. 1, pp. 001 006, March 2009 001 A RECURRENCE THEOREM ON THE SOLUTIONS TO THE 2D EULER EQUATION Y. CHARLES LI Abstract. In this article, I will prove
More informationQUALIFYING EXAMINATION, Part 1. Solutions. Problem 1: Mathematical Methods. r r r 2 r r2 = 0 r 2. d 3 r. t 0 e t dt = e t
QUALIFYING EXAMINATION, Part 1 Solutions Problem 1: Mathematical Methods (a) For r > we find 2 ( 1 r ) = 1 ( ) 1 r 2 r r2 = 1 ( 1 ) r r r 2 r r2 = r 2 However for r = we get 1 because of the factor in
More informationDynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0
More informationPrimary, secondary instabilities and control of the rotating-disk boundary layer
Primary, secondary instabilities and control of the rotating-disk boundary layer Benoît PIER Laboratoire de mécanique des fluides et d acoustique CNRS Université de Lyon École centrale de Lyon, France
More informationChaotic Billiards. Part I Introduction to Dynamical Billiards. 1 Review of the Dierent Billiard Systems. Ben Parker and Alex Riina.
Chaotic Billiards Ben Parker and Alex Riina December 3, 2009 Part I Introduction to Dynamical Billiards 1 Review of the Dierent Billiard Systems In investigating dynamical billiard theory, we focus on
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationHorizontal buoyancy-driven flow along a differentially cooled underlying surface
Horizontal buoyancy-driven flow along a differentially cooled underlying surface By Alan Shapiro and Evgeni Fedorovich School of Meteorology, University of Oklahoma, Norman, OK, USA 6th Baltic Heat Transfer
More informationJ. Szantyr Lecture No. 4 Principles of the Turbulent Flow Theory The phenomenon of two markedly different types of flow, namely laminar and
J. Szantyr Lecture No. 4 Principles of the Turbulent Flow Theory The phenomenon of two markedly different types of flow, namely laminar and turbulent, was discovered by Osborne Reynolds (184 191) in 1883
More informationThe Euler Equation of Gas-Dynamics
The Euler Equation of Gas-Dynamics A. Mignone October 24, 217 In this lecture we study some properties of the Euler equations of gasdynamics, + (u) = ( ) u + u u + p = a p + u p + γp u = where, p and u
More informationProblems in Magnetostatics
Problems in Magnetostatics 8th February 27 Some of the later problems are quite challenging. This is characteristic of problems in magnetism. There are trivial problems and there are tough problems. Very
More information2.25 Advanced Fluid Mechanics Fall 2013
.5 Advanced Fluid Mechanics Fall 013 Solution to Problem 1-Final Exam- Fall 013 r j g u j ρ, µ,σ,u j u r 1 h(r) r = R Figure 1: Viscous Savart Sheet. Image courtesy: Villermaux et. al. [1]. This kind of
More informationQuantum vortex reconnections
Quantum vortex reconnections A.W. Baggaley 1,2, S. Zuccher 4, Carlo F Barenghi 2, 3, A.J. Youd 2 1 University of Glasgow 2 Joint Quantum Centre Durham-Newcastle 3 Newcastle University 4 University of Verona
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More informationAC & DC Magnetic Levitation and Semi-Levitation Modelling
International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 24-26, 2003 AC & DC Magnetic Levitation and Semi-Levitation Modelling V. Bojarevics, K. Pericleous Abstract
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationLecture 1. Hydrodynamic Stability F. H. Busse NotesbyA.Alexakis&E.Evstatiev
Lecture Hydrodynamic Stability F. H. Busse NotesbyA.Alexakis&E.Evstatiev Introduction In many cases in nature, like in the Earth s atmosphere, in the interior of stars and planets, one sees the appearance
More informationLinear stability of small-amplitude torus knot solutions of the Vortex Filament Equation
Linear stability of small-amplitude torus knot solutions of the Vortex Filament Equation A. Calini 1 T. Ivey 1 S. Keith 2 S. Lafortune 1 1 College of Charleston 2 University of North Carolina, Chapel Hill
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationReminders: Show your work! As appropriate, include references on your submitted version. Write legibly!
Phys 782 - Computer Simulation of Plasmas Homework # 4 (Project #1) Due Wednesday, October 22, 2014 Reminders: Show your work! As appropriate, include references on your submitted version. Write legibly!
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationFluid Animation. Christopher Batty November 17, 2011
Fluid Animation Christopher Batty November 17, 2011 What distinguishes fluids? What distinguishes fluids? No preferred shape Always flows when force is applied Deforms to fit its container Internal forces
More informationMax Planck Institut für Plasmaphysik
ASDEX Upgrade Max Planck Institut für Plasmaphysik 2D Fluid Turbulence Florian Merz Seminar on Turbulence, 08.09.05 2D turbulence? strictly speaking, there are no two-dimensional flows in nature approximately
More informationB r Solved Problems Magnetic Field of a Straight Wire
(4) Equate Iencwith d s to obtain I π r = NI NI = = ni = l π r 9. Solved Problems 9.. Magnetic Field of a Straight Wire Consider a straight wire of length L carrying a current I along the +x-direction,
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationSingle Particle Motion
Single Particle Motion C ontents Uniform E and B E = - guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad -B drift, B B invariance of µ. Magnetic
More informationTHREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
More informationLecture 10. Example: Friction and Motion
Lecture 10 Goals: Exploit Newton s 3 rd Law in problems with friction Employ Newton s Laws in 2D problems with circular motion Assignment: HW5, (Chapter 7, due 2/24, Wednesday) For Tuesday: Finish reading
More informationTHE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT.
ISTP-6, 5, PRAGUE 6 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT. Mei-Jiau Huang Mechanical Engineering Department, National Taiwan University
More informationPHYS 432 Physics of Fluids: Instabilities
PHYS 432 Physics of Fluids: Instabilities 1. Internal gravity waves Background state being perturbed: A stratified fluid in hydrostatic balance. It can be constant density like the ocean or compressible
More informationMaxwell s equations for electrostatics
Maxwell s equations for electrostatics October 6, 5 The differential form of Gauss s law Starting from the integral form of Gauss s law, we treat the charge as a continuous distribution, ρ x. Then, letting
More informationAn Inverse Mass Expansion for Entanglement Entropy. Free Massive Scalar Field Theory
in Free Massive Scalar Field Theory NCSR Demokritos National Technical University of Athens based on arxiv:1711.02618 [hep-th] in collaboration with Dimitris Katsinis March 28 2018 Entanglement and Entanglement
More informationAn Overview of Fluid Animation. Christopher Batty March 11, 2014
An Overview of Fluid Animation Christopher Batty March 11, 2014 What distinguishes fluids? What distinguishes fluids? No preferred shape. Always flows when force is applied. Deforms to fit its container.
More informationINTERNAL GRAVITY WAVES
INTERNAL GRAVITY WAVES B. R. Sutherland Departments of Physics and of Earth&Atmospheric Sciences University of Alberta Contents Preface List of Tables vii xi 1 Stratified Fluids and Waves 1 1.1 Introduction
More informationHomework Two. Abstract: Liu. Solutions for Homework Problems Two: (50 points total). Collected by Junyu
Homework Two Abstract: Liu. Solutions for Homework Problems Two: (50 points total). Collected by Junyu Contents 1 BT Problem 13.15 (8 points) (by Nick Hunter-Jones) 1 2 BT Problem 14.2 (12 points: 3+3+3+3)
More informationA Summary of Some Important Points about the Coriolis Force/Mass. D a U a Dt. 1 ρ
A Summary of Some Important Points about the Coriolis Force/Mass Introduction Newton s Second Law applied to fluids (also called the Navier-Stokes Equation) in an inertial, or absolute that is, unaccelerated,
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationA scaling limit from Euler to Navier-Stokes equations with random perturbation
A scaling limit from Euler to Navier-Stokes equations with random perturbation Franco Flandoli, Scuola Normale Superiore of Pisa Newton Institute, October 208 Newton Institute, October 208 / Subject of
More information1.12 Stability: Jeans mass and spiral structure
40 CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA 1.12 Stability: Jeans mass and spiral structure Until this point we have been concerned primarily with building equilibrium systems. We
More informationGyrokinetic simulations of magnetic fusion plasmas
Gyrokinetic simulations of magnetic fusion plasmas Tutorial 2 Virginie Grandgirard CEA/DSM/IRFM, Association Euratom-CEA, Cadarache, 13108 St Paul-lez-Durance, France. email: virginie.grandgirard@cea.fr
More informationPhysics 3323, Fall 2016 Problem Set 2 due Sep 9, 2016
Physics 3323, Fall 26 Problem Set 2 due Sep 9, 26. What s my charge? A spherical region of radius R is filled with a charge distribution that gives rise to an electric field inside of the form E E /R 2
More informationNumerical Simulations of N Point Vortices on Bounded Domains
Cornell University Mathematics Department Senior Thesis Numerical Simulations of N Point Vortices on Bounded Domains Author: Sy Yeu Chou Bachelor of Arts May 2014, Cornell University Thesis Advisor: Robert
More informationLooking Through the Vortex Glass
Looking Through the Vortex Glass Lorenz and the Complex Ginzburg-Landau Equation Igor Aronson It started in 1990 Project started in Lorenz Kramer s VW van on the way back from German Alps after unsuccessful
More informationThe motions of stars in the Galaxy
The motions of stars in the Galaxy The stars in the Galaxy define various components, that do not only differ in their spatial distribution but also in their kinematics. The dominant motion of stars (and
More informationNUMERICAL SIMULATION OF THE FLOW AROUND A SQUARE CYLINDER USING THE VORTEX METHOD
NUMERICAL SIMULATION OF THE FLOW AROUND A SQUARE CYLINDER USING THE VORTEX METHOD V. G. Guedes a, G. C. R. Bodstein b, and M. H. Hirata c a Centro de Pesquisas de Energia Elétrica Departamento de Tecnologias
More informationThe Hydrostatic Approximation. - Euler Equations in Spherical Coordinates. - The Approximation and the Equations
OUTLINE: The Hydrostatic Approximation - Euler Equations in Spherical Coordinates - The Approximation and the Equations - Critique of Hydrostatic Approximation Inertial Instability - The Phenomenon - The
More informationGeometry of particle paths in turbulent flows
Journal of Turbulence Volume 7, No. 6, 6 Geometry of particle paths in turbulent flows W. BRAUN,F.DELILLO and B. ECKHARDT Fachbereich Physik, Philipps-Universität Marburg, D-353 Marburg, Germany We use
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More information