Jonathan Halcrow

Bio: Jonathan Halcrow is an academic researcher from Google. The author has contributed to research in topics: Couette flow & Reynolds number. The author has an hindex of 8, co-authored 9 publications receiving 1056 citations. Previous affiliations of Jonathan Halcrow include Georgia Institute of Technology.

Visualizing the geometry of state space in plane Couette flow

Abstract: Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical 10 5 -dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

Visualizing the geometry of state space in plane Couette flow

Abstract: Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.

Equilibrium and travelling-wave solutions of plane Couette flow

Abstract: We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

Equilibrium and traveling-wave solutions of plane Couette flow

Abstract: We present ten new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number (Re) and two new traveling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low Re turbulence. Projections of these solutions and their unstable manifolds from their infinite-dimensional state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

Heteroclinic connections in plane Couette flow

Abstract: Plane Couette flow transitions to turbulence for Re~325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are quite distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. Computing such connections is essential for understanding the global dynamics of spatially localized structures that occur in transitional plane Couette flow. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.

Advanced Calculus I

Abstract: WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

The Significance of Simple Invariant Solutions in Turbulent Flows

Abstract: Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. A significant advance in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier-Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e., the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.

Visualizing the geometry of state space in plane Couette flow

Abstract: Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.

Coherent Structures in Wall-Bounded Turbulence

Abstract: The current knowledge about some particular kinds of coherent structures in the logarithmic and outer layers of wall-bounded turbulent flows is briefly reviewed. It is shown that a lot has been learned about their geometry, flow properties and temporal behaviour. It is also shown that, although the wall-attached structures carry the largest fraction of most flow properties, they are only extreme cases of smaller wall-detached eddies, and that the latter connect with the more classical behaviour of homogeneous turbulence away from walls. Nevertheless, it is argued that little is known about the dynamical origin of these structures, and that a concerned effort is required to quantitatively identify which one (or ones) of the plausible available dynamical models is a better representation of the observed behaviour.

The Significance of Simple Invariant Solutions in Turbulent Flows

Abstract: Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. One of the significant advances in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier--Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e. the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.