A steady separated viscous corner flow

An example is presented of a separated flow in an unbounded domain in which, as the Reynolds number becomes large, the separated region remains of size 0(1) and tends to a non-trivial Prandtl-Batchelor flow. The multigrid method is used to obtain rapid convergence to the solution of the discretized Navier-Stokes equations at Reynolds numbers of up to 5000. Extremely fine grids and tests of an integral property of the flow ensure accuracy. The flow exhibits the separation of a boundary layer with ensuing formation of a downstream eddy and reattachment of a free shear layer. The asymptotic (’triple deck’) theory of laminar separation from a leading edge, due to Sychev (1979), is clarified and compared to the numerical solutions. Much better qualitative agreement is obtained than has been reported previously. Together with a plausible choice of two free parameters, the data can be extrapolated to infinite Reynolds number, giving quantitative agreement with triple-deck theory with errors of 20% or less. The development of a region of constant vorticity is observed in the downstream eddy, and the global infinite-Reynolds-number limit is a Prandtl-Batchelor flow; however, when the plate is stationary, the occurrence of secondary separation suggests that the limiting flow contains an infinite sequence of eddies behind the separation point. Secondary separation can be averted by driving the plate, and in this case the limit is a single-vortex Prandtl-Batchelor flow of the type found by Moore, Saffman & Tanveer (1988); detailed, encouraging comparisons are made to the vortex-sheet strength and position. Altering the boundary condition on the plate gives viscous eddies that approximate different members of the family of inviscid solutions

Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite difference approximations on a fixed, arbitrary grid, which have analogues of any number of integrals of the partial differential equation and of some of its symmetries. A basis for the space of of such difference operators is constructed; most cases of interest involve a single such basis element. (The ``Arakawa'' Jacobian is such an element.) We show how the topology of the grid affects the complexity of the operators.Comment: 24 pages, LaTeX sourc

M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to {prove the existence of obstructions arising from} conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries

A note on the motion of surfaces

We study the motion of surfaces in an intrinsic formulation in which the surface is described by its metric and curvature tensors. The evolution equations for the six quantities contained in these tensors are reduced in number in two cases: (i) for arbitrary surfaces, we use principal coordinates to obtain two equations for the two principal curvatures, highlighting the similarity with the equations of motion of a plane curve; and (ii) for surfaces with spatially constant negative curvature, we use parameterization by Tchebyshev nets to reduce to a single evolution equation. We also obtain necessary and sufficient conditions for a surface to maintain spatially constant negative curvature as it moves. One choice for the surface's normal motion leads to the modified-Korteweg de Vries equation,the appearance of which is explained by connections to the AKNS hierarchy and the motion of space curves.Comment: 10 pages, compile with AMSTEX. Two figures available from the author