A Numerical Method for Conformal Mapping

Challis, Neil; Burley, D.M.

doi:10.1093/imanum/2.2.169

Cited by 33 publications

(17 citation statements)

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“…This transformation is accomplished by numerically finding the conformal mapping F : (x, y) → (α, β), which keeps the differential operators of (2.4) in the simplest possible form. The numerical conformal mapping is calculated by the solution of a double Laplace problem, following the guidelines given by Challis & Burley (1982) but employing a Fourier-Chebyshev spectral collocation method to solve the coupled Laplace equations. The use of the same spectral grid to solve the conformal mapping and the fluid flow equations (as will be explained) is advantageous in terms of coupling the two problems and precision, as recognized by Kopriva (2009).…”

Section: Problem Formulation and Numerical Solutionmentioning

confidence: 99%

Global stability of flow in symmetric wavy channels

Rivera-Alvarez

Ordóñez

2013

J. Fluid Mech.

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A two-dimensional global stability analysis is numerically conducted for the basic fully developed steady flow inside symmetric wavy channels. The relative amplitude of channel modulation, defined as ratio of wall modulation amplitude to mean hydraulic diameter, is fixed via the analysis at a large value of 0.15. The relative channel wavelength, defined as the ratio of wall modulation wavelength to mean hydraulic diameter, is varied between 1 and 5. An important feature of the present approach is the detailed consideration of the streamwise conditions imposed on the flow, which allows a tailored restriction of the possible disturbances by modifying the number of channel sections that set the periodicity. Stability of the base flow is determined on the basis of the spectral structure analysis conducted, possible after calculation of the complete eigenvalue spectrum and facilitated by the spectral method employed. Two different destabilization mechanisms have been identified for the geometry studied. For small relative channel wavelengths, with values below approximately 2.7, the flow is destabilized via a Hopf bifurcation produced by a Tollmien-Schlichting wave at Reynolds numbers in the approximated range from 265 to 324. For large relative channel wavelengths, with values above approximately 2.5, a pitchfork bifurcation (or a Hopf bifurcation with very small oscillation frequency), not previously reported in the literature and produced by a symmetric stationary disturbance, is found at Reynolds numbers in the approximated range from 146 to 230.

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Section: Problem Formulation and Numerical Solutionmentioning

confidence: 99%

Global stability of flow in symmetric wavy channels

Rivera-Alvarez

Ordóñez

2013

J. Fluid Mech.

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“…Mathematically, this result corresponds to the bounds of the conformal module in conformal mapping. 82 The reason for this lateral shift in Figure 5b is because the meshes generated from quasi-conformal mapping are not exactly square, and, consequently, the required transformation medium is not exactly isotropic. However, the resulting weak anisotropy is dropped as an approximation in an isotropic carpet cloak.…”

Section: Fabrication-metamaterials or Natural Materials?mentioning

confidence: 99%

Electrodynamics of transformation-based invisibility cloaking

Zhang

2012

Light Sci Appl

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“…A schematic of such a mapping is shown in modulus for various shapes can be found in the papers of Challis & Burley [12] and Seidl & Klose [27]. In the time-dependent case, the conformal modulus will depend on time!…”

Section: Time-dependent Conformal Mappingmentioning

confidence: 99%

Variational Principles for Water Waves From the Viewpoint of a Time‐dependent Moving Mesh

Bridges

Donaldson

2010

Mathematika

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The time-dependent motion of water waves with a parametrically-defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic PDEs. The aim is to study the effect of transformation on variational principles for water waves such as Luke's Lagrangian formulation, Zakharov's Hamiltonian formulation, and the Benjamin-Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian, and some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

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A Numerical Method for Conformal Mapping

Cited by 33 publications

References 0 publications

Global stability of flow in symmetric wavy channels

Global stability of flow in symmetric wavy channels

Electrodynamics of transformation-based invisibility cloaking

Variational Principles for Water Waves From the Viewpoint of a Time‐dependent Moving Mesh

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