A calculus for flows in periodic domains
- PDF / 4,675,510 Bytes
- 24 Pages / 595.276 x 790.866 pts Page_size
- 2 Downloads / 167 Views
O R I G I NA L A RT I C L E
Peter J. Baddoo
· Lorna J. Ayton
A calculus for flows in periodic domains
Received: 1 December 2019 / Accepted: 17 August 2020 © The Author(s) 2020
Abstract Purpose: We present a constructive procedure for the calculation of 2-D potential flows in periodic domains with multiple boundaries per period window. Methods: The solution requires two steps: (i) a conformal mapping from a canonical circular domain to the physical target domain, and (ii) the construction of the complex potential inside the circular domain. All singly periodic domains may be classified into three distinct types: unbounded in two directions, unbounded in one direction, and bounded. In each case, we use conformal mappings to relate the target periodic domain to a canonical circular domain with an appropriate branch structure. Results: We then present solutions for a range of potential flow phenomena including flow singularities, moving boundaries, uniform flows, straining flows and circulatory flows. Conclusion: By using the transcendental Schottky-Klein prime function, the ensuing solutions are valid for an arbitrary number of obstacles per period window. Moreover, our solutions are exact and do not require any asymptotic approximations. Keywords Potential flow · Conformal mapping · Periodic domains
1 Introduction Spatially periodic domains arise in almost every area of fluid dynamics. To name but a few applications, flows through periodic domains have been used to model the beating of motile cilia [70], vesicle suspensions in confined flows [55], and sedimentation of small particles [63]. Modern aerofoil designs exploit periodic features in the form of drag-reducing riblets [40] and noise-reducing serrations [4]. In other aerospace applications, individual turbomachinery stages may be modelled as periodic “cascades” of aerofoils [8,64], thereby permitting both aerodynamic [5,62] and aeroacoustic [42,58] analyses. Superhydrophobic surfaces are often manufactured with patterned longitudinal periodic arrays of ridges [23,35,51], and porous media can be represented as arrays of periodically spaced pores [11,65]. In summary, the accurate and versatile mathematical modelling of flows through periodic domains has applications in a wide range of fluid mechanical scenarios. Communicated by Jeff D. Eldredge. P. J. Baddoo (B) Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, UK E-mail: [email protected] L. J. Ayton Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
P. J. Baddoo , L. J. Ayton
In this article, we provide a constructive procedure for the calculation of such flows, i.e. a calculus for flows in periodic domains. Typically, the solution of a 2-D potential flow problem require two steps [1]: (i) a conformal mapping from a (multiply connected) canonical circular domain to the physical periodic target domain of interest, and (ii) the solution of the potential flow problem inside the circula
Data Loading...
