Stability of Two-dimensional Potential Flows Using Bicomplex Numbers

TL;DR

This paper introduces a bicomplex number framework to unify complex potential flow representations and stability analysis, providing a generalized approach that encompasses classical vortex street results.

Contribution

It develops a novel bicomplex number-based framework that reconciles complex potential and velocity representations in stability studies of flows.

Findings

Unified bicomplex framework for flow stability analysis

Generalized formula for von Kármán vortex street stability

Classical results are special cases of the new formulation

Abstract

The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von K\'arm\'an vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von K\'arm\'an vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.