Abstract : The objective of the mini-symposium is to survey recent progress regarding a number of problems in theoretical fluid mechanics and to foster an exchange of new ideas in this field. It will cover a range of topics related to the existence of equilibrium solutions and their stability, extreme behaviors realizable in fluid flows, regularity of solutions versus singularity formation, transport, and turbulence. Both vicious and inviscid flows will be considered as well as some other simplified models of fluid flow. The mini-symposium will emphasize insights obtained by exploiting connections between rigorous mathematical analysis, physics, and numerical computations.
[00695] Enforcing conservation laws in truncated fluid models: the effect on heavy-tailed statistics
Format : Online Talk on Zoom
Author(s) :
Mohammad Farazmand (North Carolina State University)
Zack Hilliard (North Carolina State University)
Abstract : A significant class of partial differential equations (PDEs) have conserved quantities, arising from conservation of mass, energy, momentum, etc. These conserved quantities are not necessarily preserved when the PDE is discretized for numerical simulations. A recent method called reduced-order nonlinear solutions (RONS) allows us to ensure these conserved quantities are preserved after a Galerkin-type truncation of the PDE. We apply RONS to the Euler equation for ideal fluids, Navier--Stokes equations for incompressible flow, and shallow water equation modeling tsunamis. In each case, we discuss the effect of conserved quantities on the extreme events and energy fluxes and compare the results to conventional Galerkin truncations.
[00308] Verifying global stability of fluid flows despite transient growth of energy
Format : Online Talk on Zoom
Author(s) :
David Goluskin (University of Victoria)
Federico Fuentes (Pontificia Universidad Católica de Chile)
Sergei Chernyshenko (Imperial College London)
Abstract : To verify nonlinear stability of a laminar fluid flow against all perturbations, all past results rely on monotonic decrease of perturbation energy or a similar quadratic generalized energy. This "energy method" cannot show stability if perturbation energy can grow transiently, as in parallel shear flows at moderate Reynolds numbers. I will describe a more general approach that uses sum-of-squares polynomials to computationally construct non-quadratic Lyapunov functions that verify stability. Computational implementation for the example of 2D plane Couette flow verifies global stability at Reynolds numbers above the energy stability threshold found by Orr in 1907.
[01836] Invariant solutions representing extreme behaviour in turbulence
Format : Talk at Waseda University
Author(s) :
Genta Kawahara (Osaka University)
Abstract : Invariant solutions to the incompressible Navier-Stokes equations are reviewed to theoretically interpret extreme behaviour observed in turbulent flows. Turbulent bursting in near-wall turbulence is characterised in terms of homoclinic orbits to the periodic edge state at low Reynolds numbers, while it is discussed using another vigorous turbulent saddle at high Reynolds numbers. The ultimate state, i.e. anomaly of energy and scalar dissipation, of turbulent thermal convection is represented by steady solutions.
[03543] Numerical simulation of the convex integration for the dissipative Euler flow
Format : Talk at Waseda University
Author(s) :
Takeshi Matsumoto (Department of physics, Kyoto university)
Abstract : Weak solutions to the three-dimensional, incompressible Euler equations, which can dissipate the energy, were constructed with the convex integration by De Lellis, Szekelyhidi and collaborators. We develop a numerical simulation of the physically appealing construction by Buckmaster et al. Specifically, we study the solutions with the standard tools in physics of analyzing turbulent flows, such as the structure functions. We discuss insights obtained from them and also limitations of the simulation.
[00597] Systematic search for singularities in 3D Euler flows
Format : Talk at Waseda University
Author(s) :
Bartosz Protas (McMaster University)
Xinyu Zhao (McMaster University)
Abstract : We consider the question about formation of singularities in incompressible Euler flows. Based on the local well-posedness result guaranteeing existence of smooth solutions in the Sobolev space $H^m$, $m>5/2$, we search for potentially singular flows systematically by solving a PDE optimization problem where the $H^3$ norm is maximized at time $T$. Solutions of this problem obtained using an adjoint-based gradient descent method indicate the possibility of singularity formation if the time $T$ is sufficiently long.
[00371] Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching and related topics
Format : Talk at Waseda University
Author(s) :
Tsuyoshi Yoneda (Hitotsubashi University)
Susumu Goto (Osaka University)
Tomonori Tsuruhashi (University of Tokyo)
Abstract : With the aid of direct numerical simulations of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov-Richardson energy cascade in terms of vortex stretching. More precisely, under the assumptions of the scale-locally of the vortex stretching/compressing process and the statistical independence between vortices that are not directly stretched or compressed, we can derive the -5/3 power law of the energy spectrum of statistically stationary turbulence without directly using the Kolmogorov hypotheses.
[03746] Structure and scaling of extremely large velocity gradients in hydrdynamic turbulence.
Format : Talk at Waseda University
Author(s) :
Alain Pumir (CNRS and Ecole Normale Supérieure de Lyon)
Dhawal Buaria (New York University)
Abstract : I will discuss extreme events in the velocity gradient tensor of turbulent flows, using data from Direct Numerical Simulations (DNS) of turbulent flows up to a Taylor-scale Reynolds number of 1300. I will review some essential properties of the velocity gradient tensor, and in particular, the dependence of the strain, conditioned on vorticity, and the dependence on the Reynolds number of the probability density functions of the vorticity and strain. These properties lead to the proposition of a simple framework to quantify the extreme events and the smallest scales of turbulence. This work accentuates the importance of the relation between strain and vorticity in developing an accurate understanding of intermittency in turbulence.
In exploring further this relation, I will discuss the unexpected role of strain for very intense vortices, and discuss the self-attenuation of intense vortices in DNS of turbulent flows.
[00303] A model of turbulent flows based on a random Constantin-Lax-Majda-DeGregorio equation
Format : Talk at Waseda University
Author(s) :
Takashi Sakajo (Kyoto University)
Yuta Tsuji (Kyoto University)
Abstract : The generalized Constantin-Lax-Majda-DeGregorio (gCLMG) equation with the viscous dissipation under a large-scale forcing is utilized as a one-dimensional model of turbulent flows generating the cascade of the conserved quantity. In this talk, we show the global existence of a unique solution of the gCLMG equation subject to random forcing functions that are chosen from a given distribution. Moreover, we numerically investigate the solutions' statistical properties by Galerkin approximation of random variables with generalized Polynomial Chaos.
[00786] Extending the Gibbon-Fokas-Doering stagnation-point-type ansatz to finite-energy initial conditions: A solution to the Navier-Stokes Millennium Prize Problem?
Format : Talk at Waseda University
Author(s) :
Miguel David Bustamante (University College Dublin)
Abstract : The stagnation-point-type solution to the 3D incompressible Navier-Stokes equations found in {Gibbon, Fokas and Doering, (Physica \, D) $\bf 132$, 497 {1999}} produced an infinite family of solutions to the 3D incompressible Euler equations that blow up in a finite time. There is an exact formula for the singularity time as a functional of the initial conditions {Constantin, (Int. \, Math.\, Res.\, Not.) $\bf 2000$, 455 {2000}; Mulungye, Lucas and Bustamante, (J.\, Fluid\, Mech.) $\bf 771$, 468 {2015}; ___ , (J.\, Fluid\, Mech.) $\bf 788$, R3 {2016}}, and the solutions to this and related models are best understood in terms of infinitesimal Lie symmetries {Bustamante, (Phil.\, Trans.\, R.\, Soc.\, A) $\bf 380$, 20210050 {2022}}. The main drawback of these solutions, $\textit{from the viewpoint of the Clay Millennium Prize}$, is that the velocity field depends linearly on the out-of-plane spatial coordinate, and thus the initial condition has infinite energy. In this talk, I will present a way to extend these solutions in order to have an arbitrary dependence on the out-of-plane coordinate, allowing in principle for finite-energy solutions. This extension seems to break the infinitesimal Lie symmetry structure inherent to the previous infinite-energy solutions, so a statement regarding finite-time blowup is not yet available analytically in the finite-energy case. However, the extension allows for a novel numerical attempt at the finite-energy solution, via a hierarchy of systems of coupled 2D partial differential equations, which are much easier to handle than a full 3D problem. I will present results and prospects, and discuss potential applications to real-life experiments.
[04968] Singularity detection via regularization: Blow-up criteria for 3D Euler and related equations
Format : Talk at Waseda University
Author(s) :
Adam Larios (University of Nebraska-Lincoln)
Edriss Titi (Texas A&M University)
Isabel Safarik (University of Nebraska-Lincoln)
Abstract : The 3D Euler-Voigt equations can be thought of as a regularization of the 3D Euler equations in the sense that they are globally well-posed, and the solutions approximate the solutions to the 3D Euler equations. We describe a blow-up criterion for the 3D incompressible Euler equations based on inviscid Voigt regularization. Therefore, the blow-up criterion allows one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the “better-behaved” 3D Euler-Voigt equations. Analytical and computational results will be discussed. We will also discuss a applications to Navier-Stokes and a recent Voigt-type regularization and blow-up criterion based on the Velocity-Vorticity formulation of the 3D Navier-Stokes equations.
[02897] Thermalisation in finite-dimensional, inviscid equations of hydrodynamics
Format : Talk at Waseda University
Author(s) :
Samriddhi Sankar Ray (International Centre for Theoretical Sciences, Tata Institute of Fundamental Research (ICTS-TIFR))
Sugan D. Murugan (International Centre for Theoretical Sciences, Tata Institute of Fundamental Research (ICTS-TIFR))
Abstract : The question of thermalisation of classical systems with many degrees of
freedom is a fundamentally important one in statistical physics. There are
several examples of such systems, with explicitly broken integrability, which
thermalise. A slightly different class of such systems which are even less
understood are the finite-dimensional (Galerkin-truncated) equations of ideal
hydrodynamics. The long time solutions of these equations
thermalise---characterised by a Gibbs distribution of the velocity field and
kinetic energy equipartition amongst its (finite) Fourier modes---by virtue of
a phase-space and energy conservation and a simple application of Liouville's
theorem. While this property has been long known, the precise mechanisms which
trigger such states have only been discovered recently. In
this talk we discuss this mechanism and show how there could be ways to prevent
the onset of thermalisation and provide a way to tackle the important questions
of finite-time blow-up and weak solutions numerically.
[00758] How advection delays singularity formation in the Navier-Stokes equations
Format : Online Talk on Zoom
Author(s) :
Koji Ohkitani (RIMS, Kyoto University)
Abstract : We numerically study the Navier-Stokes equations modified by depleting advection.
In the inviscid case some solutions blow up in finite time when advection is discarded,
Constantin 1986. We use a pair of orthogonally offset vortex tubes as initial data.
We show that: 1) blowup persists even with viscosity when advection is discarded,
and 2) the time of breakdown increases logarithmically as we reinstate advection,
consistent with the regularity of the Navier-Stokes equations.
