A Conformal Transformation Technique for Mapping an Open Channel Fluid Flow into a Two-Dimensional Plane

Résumé

Due to the emergence of many applications in areas of open-channel fluid flow, interest in this branch of fluid mechanics has increased considerably. These areas are wide-ranging, including electricity generation using tidal waves, control of floods and improvement of irrigation systems. Finance, thermal imaging, harnessing of melting of glaciers, flow of fluids over ramps and sluice gates, and quite notably, petroleum exploration are other areas where open channel flow mathematics find great relevance. Whether the free surface of the open channel is being predicted using known bottom characteristics (which is called the ‘direct approach’), or the bottom is being inferred from an observable surface (referred to as the ‘inverse approach’), researchers often find the need to make computations easier by transforming complex topologies into simpler, more relatable physical equivalents. In this paper, the channel flow of a gravity-influenced Newtonian fluid is treated as the physical plane. The fluid is flowing in the positive  direction. The upper half-plane is conformally equivalent to the interior domain determined by any polygon, and the interior points of the physical plane are transformed to the corresponding points above the real axis of the upper half-plane which is then mapped onto the auxiliary half-plane, (the  plane) by being treated as an infinite strip by use of the Schwarz-Christoffel theorem. Assumptions are made that the fluid has dimensional quantities such as uniform spee  far upstream, and velocity potential . Far upstream before the arbitrary obstacle is encountered, the fluid has a uniform height,  Then  and   are used to nondimensionalize the variables to enable computation in a completely non-dimensional environment.  With the fluid assumed as steady, inviscid, irrotational and incompressible, w representing the physical w-plane,  (and t) being points on the auxiliary the - plane,  the angle made by a tangent to this plane at designated points, the required mapping is found to be    .