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

Exact evolution of time-reversible symplectic integrators and their phase error for the harmonic oscillator

The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series expansions. They are also less distorted than modified Hamiltonian of non-reversible algorithms. The analytical form for the modified angular frequency can be used to assess the phase error of any time-reversible algorithm.Comment: Submitted to Phys. Lett. A, Six Pages two Column

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

The double-facing foreign relations function of the executive and its self-enforcing obligation to comply with international law

How does the international Rule of Law apply to constrain the conduct of the Executive within a constitutional State that adopts a dualist approach to the reception of international law? This paper argues that, so far from being inconsistent with the concept of the Rule of Law, the Executive within a dualist constitution has a self-enforcing obligation to abide by the obligations of the State under international law. This is not dependent on Parliament’s incorporation of treaty obligations into domestic law. It is the correlative consequence of the allocation to the Executive of the power to conduct foreign relations. The paper develops this argument in response to recent debate in the United Kingdom on whether Ministers have an obligation to comply with international law–a reference that the Government removed from the Ministerial Code. It shows that such an obligation is consistent with both four centuries of the practice of the British State and with principle

Weaving the realities and responsibilities: Crossing culture and concepts of healing within co-existing mental health and addictive disorders.

Within ‘Mainstream’ western and Kaupapa Māori services, competing priorities often mean that cultural and clinical services are not implemented in unison. Without this ‘unison’, treatment of Māori with co-existing mental health and addictive disorders may be ineffective. This paper reflects the journey of a small social service agency ‘Pai Ake Solutions Limited’ (PASL) in integrating western clinical approaches within tikanga Māori values. Pai Ake Solutions Limited (PASL) provides a range of group and individual services to whānau who are affected by mental illness and co-existing substance use problems. The initial development of the service was based on the whakaaro of providing ‘pai ake’ (better) services for whānau in the greater Waikato. These services were initially founded on the strong values of founders Mihaka Hohua and the late Ritchie Re Cribb, which were influenced by the practices and experiences of Ngati Kahangungu, Ngati Haua, and Waikato-Maniopoto. From an initial non-clinical service provision contract, PASL began receiving referrals from individuals and whānau whose needs were not being met by mainstream service providers. Encouraging outcomes and an internal agency review of clientele utilising the service, identified that those accessing PASL services were experiencing socially and clinically significant mental and physical health, addiction and psychosocial problems

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

Integrated and situated academic development for all categories of staff: lessons for constructive alignment from an HEA-accredited Continuing Professional Development scheme

Higher Education Academy (HEA) Fellowship and the UK Professional Standards Frameworks (UKPSF) are increasingly used within British higher education institutions to support the professional formation of university teachers. This paper evaluates the use of practitioner inquiry within an HEA-accredited scheme (OpenPAD) to support the professional development of part-time Associate Lecturers at a large distance-learning institution. OpenPAD was available between 2013 and 2016 to all academic and academic-related staff in the Open University, including the 5000+ part-time teaching-only staff who are the main focus of this evaluation. OpenPAD used situated learning through practitioner inquiry to generate evidence against the UKPSF. Participant experience is evaluated and lessons drawn, which may have implications for similar schemes in other institutions. Opportunities for the further integration of academic development opportunities for all categories of staff in a successor scheme are identified in a proposed alignment between academic development; career-related processes such as induction and appraisal; institutional teaching and learning policies; and scholarship agendas