Calculation of potential flows

Abstract

Methods for calculating potential flows based on the theory of functions of a complex variable have been widely used as fundamental investigation methods in many fields of engineering, including seepage theory, elasticity theory, continuum mechanics, heat dynamics, aero- and hydromechanics, electromagnetism, electroand radio engineering, etc. [1‐3]. In most cases, the application of these methods involves conformal mapping of the rectangle 1 — 2 — 3 — 4 of the complex domain W = ϕ + i ψ (Fig. 1a) onto a complex half-plane ζ = ξ + i η (Fig. 1b). It is well known [1‐5] that this mapping is performed by means of Jacobi elliptic functions, making use of the complete elliptic integrals of the first kind K and K ’ (Fig. 1) with the modulus λ and the complementary modulus λ ’ = , respectively. This generates considerable difficulties due to the necessity of series expansion of elliptic functions, interpolation of special nonograms and tables, solution of inverse table problems, etc., particularly when it comes to determining the current values of Jacobi functions for the rectangle interior [1‐4, 6, 7]. Moreover, the difficulty of expressing the elliptic functions in terms of elementary functions restricts the possibility of analytically representing the relationship between the physical parameters of the problem under consideration and the given boundary conditions, as well as the use of complicated calculation techniques. The above circumstances considerably constrain the further development of analytical methods for investigating engineering problems in the above-listed lines of inquiry. In this study, we present a new method for solving this problem based on the conformal mapping of a rectangle, one side of which has a vanishingly small convexity, onto a half-plane by means of elementary functions. For this purpose, in the complex half-band W = ϕ + i ψ of width H (Fig. 2), we introduce the function