Convergent Flow Pattern at an Abrupt Contraction

The Schwarz‐Christoffel transformation formula for flow past an eccentric, normal constriction of zero thickness in a straight channel can be proved to have a simple form of arctangent function from the properties of elliptic integrals. Use of this mapping formula with the aid of another mapping formula between the complex potential and the transformed variable enables one to plot the corresponding flow pattern. Using such transformation formulas, flow graphics for convergent flow are thus developed to plot the flow net and the velocity distributions along the streamlines and across the channel width. To minimize the number of flow patterns in the presentation, the transformation formulas are further normalized by using the channel width and the discharge per unit depth. The normalized flow pattern so plotted is shown to vary only with the two physical parameters, the opening ratio and the eccentricity. Typical convergent flow patterns are plotted to show the validity and applicability of the transformation formulas.