Schedule for: 16w5102 - Recent Advances in Hydrodynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.

This is joint work with Romain Nguyen van yen and Marie Farge. We revisit the problem posed by Euler in 1748 that lead d'Alembert to formulate his paradox and address the following question: does energy dissipate when boundary layers detach from solid body in the vanishing viscosity limit, or equivalently in the limit of very large Reynolds number $Re$? To trigger detachment we consider a vortex dipole impinging onto a wall. We compare numerical solutions of two-dimensional Euler, Prandtl, and Navier-Stokes equations. We observe the formation of two opposite-sign boundary layers whose thickness scales in $Re^{-1/2}$, as predicted by Prandtl's theory in 1904. After a certain time when the boundary layers detach from the wall Prandtl's solution becomes singular, while the Navier-Stokes solution collapses down to a much finer thickness for the boundary layers in both directions (parallel but also perpendicular to the wall), that scales as $Re^{-1}$ in accordance with Kato's 1984 theorem [1]. The boundary layers then roll up and form vortices that dissipate a finite amount of energy, even in the vanishing viscosity limit [2]. These numerical results suggest that a new Reynolds independent description of the flow beyond the breakdown of Prandtl's solution might be possible. This lead to the following questions: does the solution converge to a weak dissipative solution of the Euler equation, analog to the dissipative shocks one get with the inviscid Burgers equation, and how would it be possible to approximate it numerically [3]? References: [1] T. Kato, 1984 Remarks on zero viscosity limit for non stationary Navier-Stokes flows with boundary. Seminar on nonlinear PDEs, MSRI, Berkeley, 85-98. [2] R. Nguyen van yen, M. Farge and K. Schneider, 2011 Energy dissipative structures in the vanishing viscosity limit of two-dimensional incompressible flow with boundaries. Phys. Rev.Lett., 106(8), 184502. [3] R. Pereira, R. Nguyen van yen, M. Farge and K. Schneider, 2013 Wavelet methods to eliminate resonances in the Galerkin-truncated Burgers and Euler equations. Phys. Rev. E, 87, 033017. [4] R. Nguyen van yen et al., 2016 Energy dissipation caused by boundary layer instability at vanishing viscosity. Preprint.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Shell models of turbulence are simpler to deal with analytically and numerically than the full Navier-Stokes equations. In this work, we look the continuum limit of the DN and GOY shell models and reproduce results from Kolmogorov theory for the stationary case. The continuum limit allows us to derive these results analytically, which we also confirm numerically.

Applications like flows in karst aquifers and hyporheic flow require the coupling of conduit flow with porous media flow. We present a few coupled flow models based on variational principle for both single phase and two-phase models. The two-phase models are of phase-field type. We also explore the relationship between various coupled models and investigate the singular structures associated with certain limits.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk presents a dynamically adaptive wavelet method for the shallow water equations on the staggered hexagonal C-grid on the sphere. The adaptive grid hierarchy is a dyadic subdivision of the icosahedron, which is optimized to ensure good geometric properties. Distinct biorthogonal second generation wavelet transforms are developed for the pressure and the velocity, together with compatible restriction operators to ensures discrete mass conservation and no numerical generation of vorticity. Coastlines are introduced by a new volume penalization method of the shallow water equations which ensure inertia-gravity waves are reflected physically, and that no-slip boundary conditions are imposed for the horizontal velocity. The code is fully parallelized using mpi, and we demonstrate good weak parallel scaling to at least 1000 processors. The efficiency and accuracy of the method are verified by applying it to a tsunami-type inertia-gravity wave with full topography, to wind-driven gyre flow and to homogeneous rotating turbulence. Even in the unfavourable case of homogeneous turbulence significant savings in the number of degrees of freedom are achieved by the adaptivity. This project is an initial step towards developing a full dynamically adaptive climate model. I will also discuss some outstanding issues in sub-grid parameterization of multiphysics processes (e.g. unresolved turbulence, cloud formation, precipitation, effect of topography).

We will discuss a computational study of a new blow-up criterion for the 3D incompressible Euler equations, based on the 3D Euler-Voigt equations. Traditional computational searches for blow-up have looked to the enstrophy coming from the 3D Euler equations themselves, which are not known to be globally well-posed, and moreover, are extremely difficult to simulate accurately. In contrast, the new blow-up criterion we investigate relies only on analyzing the enstrophy of the 3D Euler-Voigt equations, which are known to be globally well-posed and are less computationally intensive to simulate accurately over a certain range of the regularization parameter. We also study an analogue of this approach in the setting of the well-understood Burgers equation.

We will discuss several cases of Onsager critical or supercritical (turbulent) solutions to the Euler system which still conserve energy. These are the cases that exploit additional mechanisms such as symmetries coming from Hamiltonian structure, maximum principle in 2D, particular organization of singularity set, etc.

This talk is concerned with global regularity of solutions of the Navier--Stokes equations. Let $\Omega(t) \assign \{x:|u(x,t)| > c\norm{u}_{L^r}\}$, for some $r\ge3$ and constant $c$ independent of $t$, with measure $|\Omega|$. It is shown that if $\int_\Omega|p+\mathcal{P}|^{3/2}\mathd x$ becomes sufficiently small as $|\Omega|$ decreases, then $\norm{u}_{L^{(r+6)/3}}$ decays and regularity is secured. Here $p$ is the physical pressure and $\mathcal{P}$ is a pressure moderator of relatively broad forms. The implications of the results are discussed and regularity criteria in terms of bounds for $|p+\mathcal{P}|$ within $\Omega$ are deduced. This is joint work with Xinwei Yu.

We consider the three dimensional magnetohydrodynamics (MHD) equations in the presence of stochastic forcing as a model for magnetostrophic turbulence. For scales relevant to the Earth's fluid core this MHD system is very rich in small parameters. We discuss results concerning the asymptotics of the stochastically forced PDEs in the limit of vanishing parameters. In particular we establish that the system sustains ergodic statistically steady states thus providing a rigorous foundation for magnetostrophic turbulence. This is joint work with Juraj Foldes, Nathan Glatt-Holtz and Geordie Richards.

In 1980, van Dommelen and Shen provided a numerical simulation that predicted generation of a singularity in the Prandtl boundary layer equations from a smooth initial datum, for a nonzero Euler background flow. We provide a proof of the blowup by showing that a quantity related to the boundary layer thickness becomes infinite in a finite time. We will also briefly survey available local and global existence results. The blowup result is joint with Vlad Vicol and Fei Wang.

In this talk I will introduce a new method for 2D incompressible Euler. The new approach evolves the flow map using the gradient-augmented level set method (GALSM). Since the flow map can be decomposed into submaps (each over a finite time interval), the error can be controlled by choosing the remapping times appropriately. This leads to a numerical scheme that has exponential resolution in linear time. This approach is general and lends itself to many transport problems. I will review the basic GALSM and its application to 2D Euler. Throughout this talk I will provide examples from two-phase flow in addition to those from 2D incompressible Euler.

Meet in foyer of TCPL to participate in the BIRS group photo. Please don't be late, or you will not be in the official group photo! The photograph will be taken outdoors so a jacket might be required.

This is the first part of a joint talk with Yasunori Maekawa. In this part, I shall review stability issues for the Prandtl boundary layer expansion. The stability depends on the monotonicity properties of the base flow and the regularity properties of the perturbations. I will discuss regularity classes in which we may hope for local in time justification of the Prandtl asymptotics. Then I will briefly present a recent work with Yasunori Maekawa and Nader Masmoudi in which we justify the asymptotics in Gevrey in the monotonic case (to be explained in part 2). This talk is the second part of the joint talk with David Gerard-Varet. The verification of the Prandtl asymptotic expansion for the Navier-Stokes equations in a Gevrey class will be discussed in details around a monotone and convex boundary layer profile.

Accurate measurement of turbulence statistics in the inner layer of wall flows is of fundamental and applied importance for development of passive and active flow control systems. Planar and volumetric particle image velocimetry (PIV) and particle tracking velocimetry (PTV) will be discussed and evaluated in a turbulent channel flow at Reτ = 190. The results are compared with channel flow DNS of Moser et al. (1999). The application of the measurement systems in development of passive drag reduction techniques including superhydrophobic surfaces and polymeric drag reducers will be briefly discussed.

Hike to Tunnel Mountain: we will meet in front of Corbett Hall at 14:30. The hike is approx 2 hours. It is moderate with some steep parts, hiking boots recommended, elevation gain is approx 300 meters (1000 feet). There is a chance of rain, still warm, lease bring rain jacket, water, sun screen/sun glasses, hat/scarf. In alternative, interested parties can inquire about a shuttle to Lake Louise. This is a regular service provided by Discover Banff Tours, and they said that there is a departure from Banff at 13:00 and 14:00. One-way price is about $30. If interested, please confirm directly with them: www.banfftours.com

In this talk, I will give some indications on how to find a solution of the Navier-Stokes equations with time-dependent Navier-type boundary conditions on a sufficiently smooth 3D domain. Among the tools used are the new developments on non-autonomous maximal regularity for operators associated to sesquilinear forms. This is a joint work with El Maati Ouhabaz, IMB, Université de Bordeaux, France.

In this talk we discuss recent global, strong well-posedness results for the 3-D viscous primitive equations within the $L^p$-setting. Our approach is based on $H^2$ a priori estimates as well as on $L^p$-smoothing properties of the hydrostatic Stokes semigroup, which is given by the solution of the corresponding linearized problem. This is joint work with A. Hussein and T. Kashiwabara.

In this talk, I will take the mild solution approach to obtain regularity and decay estimates of the incompressible Navier-Stokes equation. I will first introduce the notion of mild solution, and will explain how to obtain analyticity of mild solutions and how to use this regularity to obtain decay rates of weak solutions. I will finally show Log-Lipschitz regularity of mild solutions and Holder regularity of the corresponding flow map.

In this work in collaboration with J.-M. Coron and F. Marbach, we consider the incompressible Navier-Stokes equations in a smooth bounded domain, either in 2D or in 3D, with a Navier slip-with-friction boundary condition except on a part of the boundary. This under-determination encodes that one has control over the remaining part of the boundary. We prove that for any initial data, for any positive time, there exists a weak Leray solution which vanishes at this given time.

We investigate computationally the effectiveness of a simple finite-dimensional feedback control scheme for globally stabilizing solutions of infinite-dimensional dissipative evolution equations introduced by Azouani and Titi (2013). This feedback control algorithm overcomes some of the major difficulty in control of multiscale processes: It does not require the presence of separation of scales nor does it assume the existence of an inertial manifold. In this work we present a theorical framework for the control algorithm which allows us to give a systematic stability analysis for the algorithm and present the parameter regime where stabilization or control objective is attained. In addition, the number of observables and controllers that we have derived analytically and implemented in our numerical studies is consistent with the finite number of determining modes that are relevant to the physical system. We verify our results computationally in the context of Chafee-Infante reaction-diffusion equation, the Kuramoto-Sivashinky equation, and other applied control problems and observe that control strategy is robust and independent of the model equation describing the dissipative system. This is joint work with Edriss S. Titi.

We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity. This is a joint work with David G. Ebin.

How regular does a solution to the (incompressible or compressible) Euler system need to be in order to conserve energy? In the incompressible context, this question is the subject of Onsager's famous conjecture from 1949. We will review the elegant proof of energy conservation for the incompressible system in Besov spaces with exponent greater than 1/3 by Constantin-E-Titi, and explain how their arguments can be refined to handle the isentropic compressible Euler equations.

We consider the $\alpha$-Euler equations in a bounded domain and discuss various results about the limit $\alpha \to 0$. This is a singular limit where, as in the vanishing viscosity limit, one has to deal with boundary layer issues. In the case of the dimension three and the Navier boundary conditions, we prove the expected convergence result by using conormal Sobolev spaces. This is joint work with V. Busuioc, M. Lopes Filho and H. Nussenzveig Lopes.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.