Overview

Of the different problem categories in the remainder of this text, this is the simplest and, appropriately, a good starting point. A potential flow field is one where a single field variable suffices and a single flow-governing equation applies. This variable has typically been chosen as either the stream function ψ or the velocity potential ϕ. This apparent simplicity, nevertheless, may (in the larger picture) underestimate the critical role a potential-flow code often plays in a typical cascade-design setting, as well as the inherent analytical difficulty in securing a single-valued flow solution in a multiply-connected domain, with the latter being the focus of this chapter.

Beginning as early as the 1930s, several methods were devised for the problem of potential flow past a cascade of lifting bodies. Some of these methods were based on the use of conformal transformation [1–5], where one or more transformation step(s) are used in mapping the computational domain into a set of ovals or a flatplate cascade [4, 5]. A separate category of analytical solutions [6] is based on the so-called singularity method, whereby sequences of sources and sinks and/or vortices are used to replace the airfoil itself. Next, the streamline-curvature method was established [7] as a viable approach to the airfoil-cascade-flow problem. With advent of the computer revolution came several numerical models of the problem based on the finite-difference method [9], finite element [9–12], and finite-volume [13] computational techniques.

From an analytical viewpoint a flow passage will conceptually suffer one level of multiconnectivity at any point where two streams with two different histories are allowed to mix together.