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# A Numerical Method for Schwarz-Christoffel Conformal Transformation with Application to Potential Flow in Channels with Oblique Sub-channels

P.M.J. Trevelyan1, L. Elliott1, D.B. Ingham1
1 Department of Applied Mathematics, University of Leeds, LS2 9JT, UK.

Computer Modeling in Engineering & Sciences 2000, 1(3), 117-122. https://doi.org/10.3970/cmes.2000.001.419

### Abstract

The potential flow in a semi-infinite channel with multiple semi-infinite oblique sub-channels is determined using the Schwarz-Christoffel transformation and complex potential theory. The standard iterative technique, i.e. the Newton-Raphson method with the Jacobian matrix approximated by a finite-difference quotient matrix, was employed with an alternative integration region to that found elsewhere in the literature is employed after integrating across the boundaries to determine the Schwarz-Christoffel transformation parameters which solely depend on the dimensions of the region being considered. Each semi-infinte channel permits integration at infinity perpendicularly across the channel and sub-channels, yielding some analytical relationships between these parameters. The remainder of the parameters requires the use of numerical integration, which in the present situation was the Runge-Kutta-Merson method. Once these parameters were determined the Schwarz-Christoffel transformation was integrated numerically using a variable-step Adams method and this completes the mapping from the region being considered to the upper half of the complex plane. This technique employed is illustrated by considering a semi-infinite channel with an inlet and an outlet attached to the top and the side of the channel, respectively, with both these sub-channels at prescribed oblique angles to the main channel. The whole region is modelled by a semi-infinite channel with two sub-channels attached, all of which have uniform flows at infinity.

### Keywords

Schwarz-Christoffel, boundary integral method, oblique channels, potential flow.

Trevelyan, P., Elliott, L., Ingham, D. (2000). A Numerical Method for Schwarz-Christoffel Conformal Transformation with Application to Potential Flow in Channels with Oblique Sub-channels. CMES-Computer Modeling in Engineering & Sciences, 1(3), 117–122.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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