scholarly journals Conjugate Functions for Engineers: a Simple Exposition of the Schwarz-Christoffel Transformation applied to the Solution of Problems involving Two-Dimensional Fields of Force and Flux

Nature ◽  
1934 ◽  
Vol 134 (3379) ◽  
pp. 164-164
Keyword(s):  
Conjugate Functions ◽  
Two Dimensional ◽  
Christoffel Transformation ◽  
Simple Exposition

The starting mechanism of wave-induced flow through a sharp-edged orifice

1977 ◽  
Vol 82 (1) ◽  
pp. 115-128 ◽  
Author(s):  
R. A. Evans ◽  
M. I. G. Bloor
Keyword(s):  
Pressure Distribution ◽  
Growth Rates ◽  
Vortex Sheet ◽  
Plane Shock ◽  
Initial Growth ◽  
Two Dimensional ◽  
Flow Through ◽  
Weak Plane ◽  
Christoffel Transformation ◽  
Wave Induced

Following weak plane shock diffraction at a knife-edge situated in a duct, a two-dimensional vortex sheet springs from the salient edge. The method of ‘vortex discretization’ is used, in conjunction with a Schwarz-Christoffel transformation, to develop a two-dimensional potential model for this constrained form of vortex generation. The analysis is independent of empirical parameters and describes, qualitatively, the pattern of streamlines through the orifice.Flow-visualization photographs are presented which illustrate the spiral shape of the starting vortex. Although of a limited nature, quantitative experimental vortex growth rates have been obtained and are compared with initial growth rates predicted theoretically. The results are discussed together with other aspects of the problem, including the limitations of the theory.An extension of vortex discretization is developed whereby the pressure distribution remote from the vortex sheet can be calculated. The combination of flow separation and the associated static wall pressure distribution gives theoretical insight into the mechanism of flow through an orifice.

A Schwarz-Christoffel Method for Generating Two-Dimensional Flow Grids

1985 ◽  
Vol 107 (3) ◽  
pp. 330-337 ◽  
Author(s):  
K. P. Sridhar ◽  
R. T. Davis
Keyword(s):  
Internal Flow ◽  
Mesh Size ◽  
Simple Expression ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
Mesh Spacing ◽  
Flow Configurations ◽  
External Flows ◽  
Christoffel Transformation ◽  
High Degree

A new coordinate generation technique, developed by Davis for external flows, is extended to allow for accurate grid generation for a variety of complex internal flow configurations. The approach is based on numerical integration of the Schwarz-Christoffel transformation for polygonal surfaces. It is shown to be second-order accurate with mesh size due to analytic treatment of boundary singularities. The method is flexible enough to allow for treatment of severe internal geometries, for a high degree of control of mesh spacing, and for generation of either orthogonal or nonorthogonal grids. In addition, this technique directly provides the two-dimensional incompressible potential flow solution for internal flows, as well as a simple expression for calculating the grid metric coefficients. Sample cases include symmetric and asymmetric channel, diffuser, and cascade flows.

Two Dimensional Lateral Flow Past a Barrier

1979 ◽  
Vol 101 (4) ◽  
pp. 449-452 ◽  
Author(s):  
A. S. Ramamurthy ◽  
B. L. Carballada
Keyword(s):  
Lateral Flow ◽  
Two Dimensional ◽  
Numerical Techniques ◽  
Contraction Coefficient ◽  
Flow Past ◽  
Christoffel Transformation

The well-known Schwarz-Christoffel transformation and the free streamline theory are used to solve the problem of flow past a lateral outlet housed in a two-dimensional conduit. The solution presented can be applied to lateral outlets which are fitted with a barrier that can be set at arbitrary inclinations. For the more general case, where the barrier inclination is arbitrary, numerical techniques were used to obtain the solution. The contraction coefficient and the inclination of the flow issuing out of the outlet are obtained as functions of the velocity ratios.

Numerical Calculation of the Potential Distribution in Ion Slit Lens Systems I

1959 ◽  
Vol 14 (9) ◽  
pp. 809-816 ◽  
Author(s):  
A. J. H. Boerboom
Keyword(s):  
Numerical Calculation ◽  
Iteration Method ◽  
Arbitrary Number ◽  
Conformal Transformation ◽  
Potential Distribution ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Lens System ◽  
Slit System ◽  
Christoffel Transformation

The potential distribution is calculated in an ion lens, consisting of three parallel collinear slits in three parallel electrodes. The slit system is supposed to be infinite in the direction of the slits, so the problem becomes two dimensional in a plane perpendicular on the direction of the slits. In this plane the potential distribution is calculated by the method of conformal transformation.The SCHWARZ—CHRISTOFFEL transformation is used to map conformally the region between the projections of the electrodes of the slit system. It proves to be very simple to perform this transformation. Formulae are given for the case of an ion lens consisting of slits in three parallel plates. A series expansion and an iteration method are developed to find the necessary parameters. Both methods prove to be satisfactory if the slit widths are smaller than the distance to the neighbouring electrodes. Symmetrical lenses, not satisfying this condition will be treated in a second paper. In a third paper slit system will be treated with an arbitrary number of electrodes.In the transformed region LAPLACE'S equation is solved, having as boundary conditions the potentials on the electrodes. In this way the exact potential distribution in the lens system is found. In a typical example the potential distributions are calculated along the axis for several potentials on the electrodes, together with the corresponding fields.

MAGNETIC UPWARD CONTINUATION FROM AN UNEVEN TRACK

Geophysics ◽  
1972 ◽  
Vol 37 (4) ◽  
pp. 662-668 ◽  
Author(s):  
Robert L. Parker ◽  
Kim D. Klitgord
Keyword(s):  
Practical Problem ◽  
New Method ◽  
Magnetic Data ◽  
Two Dimensional ◽  
Horizontal Straight Line ◽  
Upward Continuation ◽  
Straight Line ◽  
Christoffel Transformation

A new method for continuing two‐dimensional potential data upward from an uneven track is developed with special emphasis on solving a particular practical problem, that of magnetic data taken near the bottom of the ocean. The method is based on the use of the Schwarz‐Christoffel transformation, which maps the original, irregular track into a horizontal straight line. It has been found to be very fast computationally and to suffer none of the restrictions found in some earlier two‐dimensional algorithms.

A discussion on advanced methods of energy conversion- Magnetohydrodynamic power generation - Flow of current in a cross connected m.h.d. generator with four electrodes

1967 ◽  
Vol 261 (1123) ◽  
pp. 455-460 ◽  
Keyword(s):  
Current Flow ◽  
Linear Equations ◽  
Cathode Side ◽  
Two Dimensional ◽  
Anode Side ◽  
Flow Conditions ◽  
Fluid Properties ◽  
Simultaneous Linear Equations ◽  
Christoffel Transformation ◽  
Four Electrodes

The current flow patterns in an m.h.d. generator with four electrodes are derived. There are two electrodes each side of the duct; the upstream one on the cathode side is connected to the downstream one on the anode side and a load across the other pair of electrodes produces a rudimentary cross-connected generator. The fluid properties, flow conditions and magnetic field are supposed uniform across the duct and a two-dimensional analysis is made. A Schwarz-Christoffel transformation is employed to find potential and current distributions. Three simultaneous linear equations must be solved to give explicit equations for currents and potentials. The method is extendable to the case of more electrodes.

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