The seminar room at 15 South College Street has blackboards and a data projector.
Wireless access is available throughout the building.
Please note that the programme is subject to changes.
| Presentation Details |
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| Barkley, Dwight |
| Stability analysis for timesteppers |
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I will describe methods for branch following, eigenvalue stability, and transient growth analysis for flows with arbitrary geometric complexity, where in particular the flow is not required to vary slowly in the streamwise direction. The methods employ the `timestepper's approach' in which a Navier-Stokes code is modified to provide appropriate evolution operators. I will present the underlying mathematical treatment in primitive flow variables, then give some details of the inner and outer level code modifications, and finally I will show results from a variety of flow configurations.
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| Chini, Greg |
| Multiscale analysis of Langmuir circulation in the ocean surface boundary layer |
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Under wind-forced seas, vertical transport and mixing of fluid within the O(100) meter deep ocean surface boundary layer is dominated by a convective flow known as Langmuir circulation (LC). Ocean observations, numerical simulations and theoretical considerations all suggest that LC is characterized by counter-rotating vortical structures elongated in the wind direction [1,2,3]. These structures arise as an instability of a wind-driven shear flow on which surface waves propagate. The prevailing theoretical description of this phenomenon is attributable to Craik and Leibovich, who derived a surface-wave filtered version of the Navier--Stokes (NS) equations.
Formally, the Craik--Leibovich (CL) equations are identical to the instantaneous NS equations apart from the occurrence of a "vortex-force'' term, which is given by the cross-product of the surface-wave Stokes drift velocity (i.e. the Lagrangian mass drift associated with the filtered waves) and the vorticity vector. In this investigation, formal multiscale asymptotic analysis is used to capture the nonlinear structure and dynamics of strongly anisotropic LC. First, by exploring the CL equations in the physically relevant strong vortex-force limit, we show that vortices aligned with the wind direction are preferred. Using multiple scale asymptotics, we leverage this limit to derive a reduced set of PDEs governing anisotropic ``Langmuir turbulence'' [4]. Next, we employ the method of matched asymptotic expansions to construct semi-analytical solutions of the reduced PDEs that describe laminar (steady, 2D) but strongly nonlinear LC convective states arising in the weak-diffusion limit [5]. Using Floquet theory, we then carry out a secondary stability analysis to show that the reduced PDEs economically capture the dominant long-wavelength 3D instability of these 2D solutions, which leads to a bending of the wind-aligned roll vortices [4]. The secondary stability analysis also suggests that the mechanics of ``pure'' Langmuir turbulence differs from the ``self-sustaining process'' believed to govern wall-bounded shear flow (e.g. plane Couette flow) turbulence [6,7,8]. Finally, by introducing additional slow spatiotemporal scales into our asymptotic analysis of the CL equations, we are able to derive a multiscale system of PDEs that systematically couples the dynamics of ``fine-scale'' Langmuir turbulence with hydrostatic flows having much larger horizontal scales, including long-wavelength internal waves. This analysis follows the approach of Majda and Klein [9,10] for atmospheric flows.
REFERENCES
LC review articles:
1. Leibovich, S. The form and dynamics of Langmuir circulations.
Annual Reviews of Fluid Mechanics (1983). 15:391--427.
2. Thorpe, S. Langmuir circulation. Annual Reviews of Fluid
Mechanics (2004). 36:55--79.
Secondary stability analysis of the CL equations:
3. Tandon, A. and Leibovich, S. Secondary instabilities in
Langmuir circulations. Journal of Physical Oceanography (1995).
25:1206--1217.
Multiple-scale and matched-asymptotic analysis of LC:
4. Chini, G. P., Julien, K. and Knobloch, E. An asymptotically
reduced model of turbulent Langmuir circulation. Geophysical
and Astrophysical Fluid Dynamics (2009). 103:179--197.
5. Chini, G. P. Strongly nonlinear Langmuir circulation and
Rayleigh--Benard convection. Journal of Fluid Mechanics (2008).
614:39--65.
Mechanics of Langmuir and Couette-flow turbulence:
6. Tejada-Martinez and Grosch, C. Langmuir turbulence in shallow
water. Part 2. Large-eddy simulation. Journal of Fluid Mechanics
(2007). 576:63--108.
7. Teixeira, M. A. C. and Belcher, S. E. On the distortion of
turbulence by a progressive surface wave. Journal of Fluid
Mechanics (2002). 458:229--267.
8. Waleffe, F. On a self-sustaining process in shear flows.
Physics of Fluids (1997). 9:883--900.
Systematic multiscale PDE systems for geophysical flows:
9. Klein, R. An unified approach to meteorological modelling
based on multiple-scales asymptotics. Advances in Geosciences
(2008). 15:23--33.
10. Majda, A. and Klein, R. Systematic multiscale models for
the tropics. Journal of the Atmospheric Sciences (2003).
60:393--408.
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| Chomaz, Jean-Marc |
| Global instabilities in spatially developing flows: The convective Modoki |
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The objective of this lecture is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. In the global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This approach is more and more resorted to as a result of increased computational capability. We will demonstrate how global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. This convective nonnormality, the Modoki of the convective instability, will be discussed on different flow grometries.
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| Doering, Charles |
| Variational methods for turbulent transport and dissipation bounds |
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A central characteristic of turbulent flows is enhanced mixing and transport. Exact solutions are not available to study these properties, so analysis focuses on the derivation of rigorous estimates for physically relevant quantities. In this lecture series we discuss the formulation and solution of variational problems derived from the equations of motion that produce bounds on turbulent transport.
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| Friedlander, Susan |
| Lecture 1: Localized instabilities in fluids ( Linear instability). Lecture 2: Nonlinear instability for the fluid equations |
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Lecture 1.
Localized Instabilities in Fluids ( Linear Instability)
We discuss definitions of stability and instability for the fluid equations. We describe describe "the geometric optics" method which is specifically designed for studying highly localized short wave perturbations of an arbitrary, steady background flow. These perturbations are localized wave envelopes moving along the trajectories of the fluid elements. The evolution of a particular envelope is governed by a characteristic system of ordinary differential equations along the relevant trajectory. This system provides an effective tool for detecting fluid instabilities. It also allows us to understand the structure of the continuous spectrum of the linearized Euler equations. Further recent results concerning short wave instabilities will be given in the lectures of Roman Shvydkoy.
Some survey articles which contain detailed references are:
* Friedlander and Yudovich, Instabilities in fluid motion. Notices of the AMS, 46, pp 1358-1367, 1999.
* Friedlander and Shnirelman, Instability of steady flows of an ideal incompressible fluid. Advances in Mathematical Fluid Dynamics, pp 143-172, Birkhauser, 2001
* Friedlander and Lipton-Lifschitz, Localized instabilities in fluids. Handbook of Mathematical Fluid Dynmics, Vol 2, pp 289-354, Elsevier, 2003.
Lecture 2.
Nonlinear Instability for the Fluid Equations
We describe various partial results whereby it can be shown that certain classes of fluid flows are unstable in the sense of the full nonlinear equations. In particular, we stress the sensitivity of instability to the norm in which growth of a perturbation is measured. We discuss a boot-strap method which proves, under certain conditions, that linear instability implies nonlinear instability.
Sample references are:
* Friedlander, Strauss and Vishik, Nonlinear instability in an ideal
fluid. Ann Inst H. Poincare, Anal.Nonlineaire, 14, pp 187-209,1997.
* Vishik and Friedlander, Nonlinear instability in 2-dimensional
ideal fluids. Comm Math Phys, 243, pp 261-273,2003
* Friedlander, Pavlovic and Shvydkoy, Nonlinear instability for
the Navier-Stokes equations. Comm Math Phys, 264, pp 335-347,2006.
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| Galdi, Paolo |
| Steady-state solutions to the Navier-Stokes equations past an obstacle: Functional properties and related stability questions |
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The exterior Navier-Stokes problem is a most fundamental topic of classical fluid dynamics, and consists in finding the flow characteristics of a viscous liquid that occupies the infinite region outside a moving rigid body. After the pioneering and seminal works of Stokes, Oseen and Lichtenstein (to mention a few), a modern mathematical analysis of the problem was initiated in the early 1930's by the work of Leray, via one of the first applications of the Leray-Schauder degree, and it was continued and deepened several decades laters, by Hopf, Finn and Babenko. The investigation carried out by these authors focuses on basic results such as existence, uniqueness, regularity and qualitative properties of solutions to the (steady-state) boundary-value problem.
The objective of this short course is to give an updated description of the above results, as well as to present recent ones of more advanced nature. These latter focus on properties of solutions valid at arbitrary Reynolds numbers, and include geometric properties of the solution manifold, and control" of solutions by a finite number of parameters. The long time behavior of dynamical perturbations around these solutions will also be investigated.
The course will end with a number of open questions.
References.
1. G.P. Galdi, Determining Modes, Nodes and Volume Elements for Stationary Solutions of the Navier-Stokes Problem Past a Three-Dimensional Body, Archive Rat. Mech. Anal., 180, 97-126 (2006)
2. G.P. Galdi, Further properties of Steady-State Solutions to the Navier-Stokes Problem past a Three-Dimensional Obstacle, J. Math. Phys., 48 (2007)
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| Gibbon, John |
| Extreme events in solutions of the 3D Navier-Stokes equations and primitive climate models |
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The experiments of Batchelor and Townsend in 1949 were the first to suggest that the clustering of vorticity and strain gives rise to both spatial and temporal intermittency. Modern computations vividly demonstrate how the vorticity field accumulates on a time-evolving tangle of tubes and sheets. My lecture will discuss the possible analytical origins of this complicated space-time set in solutions of the Navier-Stokes equations. The methods employed will then be used to consider the hydrostatic & non-hydrostatic Primitive Equations that form the basis of most modern climate models. The occurrence of extreme events, such as fronts, will be discussed.
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| Healey, Jonathan |
| Destabilizing effects of confinement of shear layers |
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A review of recent results will be presented building on the ideas of absolute and convective instabilities that were introduced by Patrick Huerre in his lectures on Tuesday 30th June. It is well known that an unstable flow can remain laminar in a region of interest (like a boundary layer on an aerofoil) if growing disturbances exit downstream before reaching a large enough amplitude to produce a transition to turbulence. Such behaviour can occur if the flow is `convectively unstable', i.e. if growth of disturbances only occurs in frames of reference moving away from a disturbance source. On the other hand, some flows have the property that disturbances grow upstream, downstream and in the neighbourhood of a disturbance source; this is called `absolute instability', and can lead to self-sustained oscillations (like vortex shedding behind a cylinder). Briggs' method (developed in plasma physics in the 1960s) provides a means for determining whether a flow is absolutely or convectively unstable. The response of a flow to a localized impulsive disturbance is written down as a superposition of normal modes in the form of an integral over wavenumbers. This integral is dominated at large times by contributions from certain saddle points located in the complex wavenumber plane. Behaviour at the dominant saddle (often called a `pinch-point') determines the absolute/convective character of the flow. This type of investigation has become very important in hydrodynamic stability theory.
Many flows, e.g. jets, wakes, mixing layers, boundary layers, are modelled as being of infinite, or semi-infinite, extent. We call such flows `unconfined'. Practical flows are always confined in some way, but the effect of confinement is usually assumed to be weak, and slightly stabilizing, when the confining boundary is far from the shear layer. However, in this lecture we will show how the presence of confinement can create absolute instability, even when the confining boundary is arbitrarily far from the shear layer. Indeed the limit where the distance of the boundary tends to infinity is different from the unconfined case for a class of flows.
These findings arose through the resolution of a difficulty that had been noted by a number of authors studying different unconfined flows in which the pinch-point can cross into the left half of the complex wavenumber plane. Waves in the right half plane decay exponentially with distance from the shear layer and obviously satisfy homogeneous boundary conditions far from the shear layer. Waves in the left half plane grow exponentially with distance from the shear layer and so appear to be disallowed. Healey (2006, J. Fluid Mech. 560, 279-310) described what happens to disturbances when the pinch-point crosses, or approaches close enough to, the imaginary axis in the complex wavenumber plane. They propagate and grow exponentially with distance outside the shear layer. This propagation and growth means they eventually encounter any confining boundary, and are strongly affected by it, see Healey (2007, J. Fluid Mech. 579, 29-61), where it is shown that confinement creates a new family of saddle points close to the imaginary axis. This family of confinement saddles has been shown to create absolute instability in models of swirling jets used to study vortex breakdown (Healey, 2008, J. Fluid Mech. 613, 1-33), and in plane mixing layers (Healey, 2009, J. Fluid Mech. 623, 241-271).
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| Huerre, Patrick |
| Flow amplifiers and flow oscillators: fundamental concepts |
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Spatially developing shear flows such as wakes, jets, mixing layers, swirling flows are experimentally observed to behave as amplifiers of external perturbations or as intrinsic oscillators beating at a specific frequency. The objective of the lecture is to introduce the main theoretical concepts which are needed to account for these two distinct types of dynamics: absolute/convective instability, linear/nonlinear global mode and global frequency selection mechanisms. These notions will be illustrated on specific examples of shear flows.
Background references:
1. Huerre P. (2000) "Open Shear Flow Instabilities" in "Perspectives in Fluid Dynamics", G.K. Batchelor, H.K. Moffatt & M.G. Worster (eds.), Cambridge University Press, pp.159-229
This chapter will form the core of my lectures and it is therefore useful reading.
2. Ablowitz M.J. & Fokas A.S.(1997) "Complex Variables", Cambridge University Press.
This book provides a good background on complex functions of a complex variable. I will use such methods quite extensively in my lectures.
3. Bender C.M. & Orszag S.A. (1978) "Advanced Mathematical Methods for Scientists and Engineers", McGraw-Hill, chapter 6 on "Asymptotic Expansion of Integrals".
This chapter is also useful for those who need to become familiar with the method of steepest descent, which will be used in the lectures.
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| Joly, Laurent |
| Vortex and shear flow stability skewed by a large density contrast |
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This presentation is intended to span the purposes of a worshop lecture on fundamentals to a research talk on recent results on fluid stability when biased by variable-density effects.
We start with an introduction to variable-density low-speed flows with a focus on the baroclinic vorticity source in full NS equations as well as their linearized counterpart. We illustrate how adopting a vortex-dynamics point of view helps understanding inertia effects in vortical and shear flows. The contrast between stratified geophysical flows and infinite Froude number industrial flows is highlighted. Then we turn to the isolated vortex linear stability when the fluid is radially stratified within the vortex. The Rayleigh-Taylor instability of a massive vortex is analysed. The non-linear simulation of the instability is illustrated both for an even (m=2) and an odd mode (m=3). In the last part, we consider the linear stability analysis of a shear flow with a large density gradient. Both the primary and the 2D/3D secondary stability of the plane mixing-layer are eventually addressed in the frame of a modal stability analysis.
Joly, Laurent and Fontane, Jérôme and Chassaing, Patrick ( 2005) The
Rayleigh–Taylor instability of two-dimensional high-density vortices.
Journal of Fluid Mechanics, vol. 537 . pp. 415-431.
Sipp, D.; Fabre, D.; Michelin, S. & Jacquin, (2005) L. Stability of a
vortex with a heavy core. Submitted to Journal of Fluid Mechanics,
vol. 526, pp. 67-76
Fontane, Jérôme and Joly, Laurent ( 2008) The stability of the
variable-density Kelvin-Helmholtz billow. Journal of Fluid Mechanics,
vol. 612 . pp. 237-260.
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| Kerswell, Rich |
| Transition and coherent structures |
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The transition to turbulence in shear flows which are linearly stable must be triggered by finite-amplitude disturbances. As a result, it has proven hard to unpick the mechanisms for this. Recently, however, a constructive technique for identifying finite amplitude solutions to the Navier-Stokes equations has emerged which appears to help in our understanding.
Using the context of pipe flow which is always linearly stable yet typically exhibits transition at a Reynolds number of O(2000), I will describe this technique and its results. Evidence will then be presented for the physical relevance of these solutions to the transition process using carefully directed numerical computations and ideas from dynamical systems.
Reading list
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Waleffe Phys. Fluids 9, 883-900, (1997).
Wedin & Kerswell J. Fluid Mech. 508, 333-371 (2004).
Schneider, Eckhardt & Yorke Phys. Rev. Lett. 99 034502 (2007).
Tutty & Kerswell J. Fluid Mech. 584, 69-102 (2007).
Duguet, Willis & Kerswell J. Fluid Mech. 613, 255-274, (2008).
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| Kopiev, Victor |
| Mechanisms of vortex ring instability |
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Long-wave neutrally stable oscillations of a thin vortex ring with piecewise distribution of vorticity in incompressible ideal fluid are considered (Kopiev&Chernyshev 1997, JFM). The oscillations with positive and negative energy were recognized (Kopiev&Chernyshev 2000, Physics-Uspehi). The oscillation properties appear to be qualitatively changed with account for such factors as slight variation of vorticity profile (a smoothed profile instead of piecewise one, critical layer appearance) or weak compressibility (sound wave radiation). In compressible medium the vortex ring looses its energy due to sound wave radiation and therefore the oscillations with negative energy become unstable (Kopiev&Chernyshev 1987, Fluid Dynamics); increment is small (Broadbent&Moore 1979) due to ineffectiveness of sound radiation by vortex. In the case of a smoothed vorticity profile the energy of the mean flow comes through the critical layers to the disturbances (Kopiev&Chernyshev 2000, Fluid Dynamics) and the oscillations with positive energy become unstable; increment is proportional to the gradient of smooth vorticity field in the critical layer (Kopiev&Chernyshev 2000, Physics-Uspehi). It means that in addition to the short-wave instability (Kerswell 2002), vortex ring has multiple unstable oscillations of the other types including the long-wave spectrum part. Some experimental evidences are available to support this approach (Vladimirov&Tarasov 1979).
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| Le Dizes, Stephane |
| Vortex stability |
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In these two lectures, I shall consider the linear stability properties of a single vortex in an incompressible fluid. Classical results (centrifugal instability, shear instability) will be first reviewed [see Drazin & Reid (1981) on "Hydrodynamic stability" for background].
Then we shall focus on the properties of the neutral (Kelvin) waves living on stable vortices. These waves are the building blocks of other instabilities such as the elliptical instability when the vortex is subject to an external forcing. For the purpose of the study, I shall develop an asymptotical analysis using WKBJ approximations. Some background on WKBJ approximations can be found in Bender & Orzsag (1978) on "Advanced mathematical methods for scientists and engineers". This part will mainly follow Le Dizes & Lacaze, JFM 542, 69-96 (2005).
In the last part of the lecture, we shall assume that the fluid is stably stratified along the vortex axis. By using the same asymptotical technique, we shall analyse two recently discovered instabilities involving either wave resonance or internal wave emissions. The asymptotic results will be compared to recent numerical and experimental results obtained for a Lamb-Oseen vortex and a Taylor-Couette flow.
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| Lebovitz, Norman |
| Shear-flow transition: the structure of the basin boundary |
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The basin of attraction of the stable, laminar point is investigated for finite-dimensional models of shear flows, with an emphasis on understanding transition and relaminarization. The region complementary to the basin of attraction is shown to be extraordinarily thin in certain regions of phase space. The 'edge of chaos' is also discussed and mathematically consistent descriptions of it are proposed in finite-dimensional settings.
References:
N. Lebovitz Shear-flow transition: the basin boundary, arXiv:0904.2764.
J.D. Skufca, J.A. Yorke, and B. Eckhardt. Edge of chaos in a parallel
shear flow. Phys. Rev. Lett., 96:174101â1,174101â4, 2006.
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| Lin, Zhiwu |
| Two mathematical results related to transient turbulence of Couette flows |
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Couette flows are known to be linearly stable for any Reynolds number but become turbulent for large Reynolds numbers. In recent years, there have been lots of work using ideas from dynamical systems to explain the turbulent structures near Couette flows, mainly numerically. I will describe two related mathematical results. First, (with Charles Li) we proved that Couette flows are structurally unstable, more precisely, there exist unstable shear flows arbitrarily close to Couette flows, in both inviscid and slightly viscous cases. These unstable shear flows are also related to traveling wave solutions near Couette flows. Second, (with Chongchun Zeng) we showed the existence of unstable manifolds for linearly unstable shear flows in the inviscid case. This result might suggest the existence of unstable manifold of non-vanishing size for Navior-Stokes equations when the viscosity tends to zero.
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| Moffatt, Keith |
| Transient instability in shear flow: control by spanwise magnetic field. |
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It is known that if a random perturbation is superposed on a uniform shear flow, then, on linearised analysis, streamwise structures (described as a superposition of Kelvin modes) are subject to transient instability, the growth being ultimately controlled by viscosity. It is evident that, in a fluid of low conductivity, such instability should be damped by a spanwise magnetic field. A simple analysis reveals the nature of this damping process.
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| Renardy, Michael |
| Some new results on stability of Newtonian and viscoelastic shear flows |
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I shall present two lectures on the following topics:
1. Ill-posedness of the hydrostatic approximation and long wave instability of inviscid and viscous parallel flows.
2. Stability of viscoelastic shear flows in the high Weissenberg number limit.
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| Rousset, Frederic |
| Instability of line solitary water waves |
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We study rigorously the linear and nonlinear destabilization of line solitary capillary gravity water waves when submitted to transverse perturbations
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| Serre, Denis |
| Self-similar flows of an inviscid fluid with the equation of state of von Karman |
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The von Karman equation of state implies that the shock waves are reversible. Some calculus can be made explicitly, using geometrical tools. This applies to the supersonic domain of steady or self-similar flows. The supersonic part is studied with techniques from degenerate elliptic PDEs.
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| Shnirelman, Alexander |
| On the long time behavior of fluid flows |
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Euler equations define a dynamical system in the space of incompressible vector fields in a 2-dimensional domain. We show in this talk that there exists a nontrivial attractor for this dynamical system. It is an infinite-dimensional compact set containing stationary, time-periodic, quasiperiodic and possibly chaotic flows. This attractor was first discovered in the numerical simulations; their results will be shown.
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| Shvydkoy, Roman |
| On shortwave instabilities and the Fredholm spectrum of advective equations |
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Shortwave instabilities unlike the classical modal instabilities of ideal fluid are related to presence of the essential spectrum of the linearized Euler equations. The spectral radius of the essential spectrum can be described via the maximal Lyapunov exponent of a finite dimensional system of ODEs, so called bicharacteristic-amplitude system. This system governs evolution of the leading order amplitude and frequency of a shortwave perturbation and can be derived via a geometric optics type analysis (this will be described in detail in S. Friedlander's lectures). In recent years the structure of the Euler semigroup generated by the linearized Euler equations has been examined from the point of view of microlocal analysis. This analysis shows that the Euler semigroup can be understood as an element of a C^*-algebra of pseudodifferential operators with shifts on the cotangent bundle of the fluid domain. This algebra is isomorphic to the algebra generated by the corresponding linear skew-product flows.
Subsequently, the question of description of the essential spectrum of the Euler semigroup falls into the classical Sacker-Sell spectral theory of cocycles.
In these two lectures we will give a detailed introduction into these new tools of the spectral analysis of the Euler semigroup. The end result of the lectures will be a proof of the fact that in both two and three dimensions the essential spectrum fills a solid annulus where radii of the annulus are the corresponding minimal and maximal Lyapunov exponents of the bicharacteristic-amplitude system. We will also discuss recent attempts to give a geometric optics description to the nonlinear evolution of shortwave perturbations as a possible approach to the problem of nonlinear instability in 2D flows.
The lectures are mainly based on the following three papers:
Y. Latushkin, R. Shvydkoy: Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics, to appear in JFA.
R. Shvydkoy: Cocycles and Mañe sequences with an application to ideal fluids,/ Journal of Diffenretial Equations/, *229/1 *(2006) 49-62.
R. Shvydkoy: The essential spectrum of advective equations, /Communications in Mathematical Physics/, *265/2 *(2006) 507-545.
All are available at http://www.math.uic.edu/~shvydkoy/publications.html
Other technical material used in the lectures is contained in these books:
M. A. Shubin, Pseudodifferential operators and spectral theory, second ed., Springer- Verlag, Berlin, 2001.
Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, American Mathematical Society, Providence, RI, 1999.
Antonevich, A. Linear functional equations. Operator approach. Operator Theory: Adv. Appl. 83. Birkhauser Verlag, Basel, 1996.
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| Stuart, Charles |
| Stability of waves |
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Solitary gravity-capillary waves on the free surface of an ideal fluid constitute one example (and there are many others in hydrodynamics) of relative equilibria of a Hamiltonian system in which the Hamiltonian is invariant with respect to a group of symplectic isometries. In general, relative equilibria are neither asymptotically stable nor even stable and orbitial stability is the best that can be hoped for. General criteria which imply this orbitial stability are known, but checking that they are satisfied in particular cases involving partial differential equation remains a very challenging task.
From a pedagogical point of view, it seems preferable to present first the underlying structures and criteria in the particular case of a Hamiltonian system of ordinary differential equations where they are not obscured by numerous technical difficulties inherent in the treatment of partial differential equations. This is what I intend to do in the first part of my lectures which will only require a basic knowledge of ODE. Then I shall try to show how the same structures and criteria appear in the context of PDE by presenting Mielke's work on the stability of solitary water waves.
References:
[1] M. Grillakis, J. Shatah and W. Strauss: Stability theory for solitary waves in the presence of symmetry, I: J. Functional Analysis, 74 (1987), 160-197
This is the fundamental paper on the stability theory of relative equilibria. It deals with Hamiltonian systems on a phase space which may have infinite dimension so as to cover pde. I shall follow more closely the following notes which also cover an infinite dimensional phase space.
[2] C.A. Stuart: Lectures on the orbital stability of standing waves and application to the nonlinear Schr"odinger equation, Milan J. Math., 76 (2008), 329-399
To illustrate an interesting situation in hydrodynamics, I shall present a summary of the following paper.
[3] A. Mielke: An the energetic stability of solitary water waves, Phil. Trans. R. Soc. Lond. A, 360 (2002), 2337-2358
The starting point for this line of attack is a seminal paper by Zakharov.
[4] V. E. Zakharov: Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys, 9 (1968), 190-194
For some recent information about this problem, see
[5] W. Craig and C.E. Wayne: Mathematical aspects of surface water waves, Russian Math. Surveys, 62 (2007), 453-473
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| Vladimirov, Vladimir |
| Stability of inviscid flows with Yudovich's boundary conditions: Washing-off, trapping, and self-oscillations of vorticity |
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To the memory of Victor Yudovich
The paper addresses the nonlinear dynamics of the plane inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are: the normal component of velocity is given at both inlet and outlet cross-sections, while vorticity is prescribed at an inlet only. YBC are fully justified mathematically: the well-posedness of the problem is proven. In this paper we study general non-linear properties of channel flows with YBC. There are nine main results in the paper:
(i)~ the phenomenon of the trapping of a point vortex has been discovered, explained, and generalized to continuously distributed initial vorticity such as vortex patches, harmonic, and Gaussian-type perturbations;
(ii)~ the sufficient conditions for the decreasing of Arnold's and enstrophy functionals have been found; that leads us to the establishing of the washing-off property of channel flows;
(iii)~ it appears that YBC are exceptional ones, since only they provide the decreasing of Arnold's functional;
(iv)~ three new criteria of the nonlinear stability of steady channel flows have been formulated and proven;
(v)~ the counterbalance between the washing-off and the trapping has been recognized as the main factor responsible for the formation of recirculation zones;
(vi)~ a physical analogy between the properties of the channel flows with YBC and dissipative dynamical systems has been proposed; this analogy allows us to formulate three major Conjectures which are related to: (C1)-- asymptotic stability of flows, (C2)-- the relaxation to a steady and stable separated flow, and (C3)-- the self-oscillations of a flow;
(vii)~ a sufficient condition for the complete washing-off of fluid particles in a perturbed non-separated flow is established;
(viii)~ the nonlinear asymptotic stability of some selected steady channel flows is proven and the related perturbation thresholds are evaluated;
(ix)~ a number of computational solutions are obtained, that support the Conjectures and discover qualitatively different flow regimes.
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| Zumbrun, Kevin |
| Stability and bifurcation of shock and noncharacteristic boundary layers |
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Lecture one: stability and bifurcation of a shock in an infinite cylinder. we develop the mathematical theory of stability and bifurcation of a shock in an infinite rectangular duct with periodic boundary conditions, and use this to make an explicit connection between exchange of inviscid stability of planar shocks in the whole space and cellular instability of shocks in a finite cross sectional duct.
Lecture two: numerical verification of stability of shock and noncharacteristic boundary layers.
By a combination of numerical Evans function techniques and asymptotic ODE estimates, we establish stability of ideal gas and parallel MHD shocks across the full range of physical parameters. Notably, this involves rigorous demonstration of stability in the infinite shock strength limit.
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Analysis of Fluid Stability