A coupled far-field formulation for time-periodic numerical problems in fluid dynamics
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A coupled far-field formulation for time-periodic numerical problems in fluid dynamics EDMUND CHADWICK and RABEA EL-MAZUZI School of Computing, Science and Engineering, University of Salford, Salford M5 4WT, UK E-mail: [email protected] MS received 20 July 2011 Abstract. Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described. Keywords.
Oscillatory flow; Oseen flow; infinite elements; boundary elements.
1. Introduction The problem of uniform flow past an oscillating body is a general one, examples being the flapping flight of birds and insects, and the swimming of mammals, fish, micro-organisms, and micro-devices. It is also envisioned that future low energy submersible propulsion will be enabled by a swimming motion, for underwater unmanned vehicles engaged in exploration, and search and rescue. Similarly, it is expected that the manoeuvrability of flying unmanned autonomous vehicles will be enhanced by a flapping flight. However, one of the problems faced for any numerical method is unwanted errors due to the truncation of the numerical domain, which is necessary because the whole domain is infinite in extent for an exterior problem. This is particularly evident for time periodic problems where unwanted reflections occur at the truncation boundary. In order to alleviate this, in the present paper it is proposed to couple the numerical fluid dynamics formulation in the finite truncated domain in the vicinity of the body to a new far-field formulation, thus modelling the whole space. This is achieved through the use of the Oseen equations, which are obtained by a linearization of the Navier–Stokes equations to a uniform stream, which holds in the far-field. Two examples of formulations are then given, one for boundary elements and one for infinite elements. The literature on boundary element methods in fluid dynamics focus mainly on Stokes flow (low Reynolds number) and Oseen flow, because both are linearizations of the Navier–Stokes equations which mean that a Green’s integral representation is possible by the use of Green’s functions, from which a boundary integral formulation is obtained which can then be discretized to give a boundary element method. Time-dependent Oseen and associated Stokes flows subdivide into transient analysis and oscillatory analysis, 661
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Edmund Chadwick and Rabea El-Mazuzi
with the majority of work on transient rather than oscillatory analysis. Price and Tan [26] used transient oseenlets in order to model ship motions. Also, Chan and Chwang [19] and Lu and Chwang [23] described the unsteady (transient) stokeslet and oseenlet and give applications related to acceleration and free surface waves. Childress
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