Using computational fluid dynamics to calculate the stimulus to the lateral line of a fish in still water

Abstract

SUMMARY This paper presents the first computational fluid dynamics (CFD)simulations of viscous flow due to a small sphere vibrating near a fish, a configuration that is frequently used for experiments on dipole source localization by the lateral line. Both two-dimensional (2-D) and three-dimensional (3-D) meshes were constructed, reproducing a previously published account of a mottled sculpin approaching an artificial prey. Both the fish-body geometry and the sphere vibration were explicitly included in the simulations. For comparison purposes, calculations using potential flow theory (PFT) of a 3-D dipole without a fish body being present were also performed. Comparisons between the 2-D and 3-D CFD simulations showed that the 2-D calculations did not accurately represent the 3-D flow and therefore did not produce realistic results. The 3-D CFD simulations showed that the presence of the fish body perturbed the dipole source pressure field near the fish body, an effect that was obviously absent in the PFT calculations of the dipole alone. In spite of this discrepancy, the pressure-gradient patterns to the lateral line system calculated from 3-D CFD simulations and PFT were similar. Conversely, the velocity field, which acted on the superficial neuromasts (SNs), was altered by the oscillatory boundary layer that formed at the fish's skin due to the flow produced by the vibrating sphere (accounted for in CFD but not PFT). An analytical solution of an oscillatory boundary layer above a flat plate, which was validated with CFD, was used to represent the flow near the fish's skin and to calculate the detection thresholds of the SNs in terms of flow velocity and strain rate. These calculations show that the boundary layer effects can be important, especially when the height of the cupula is less than the oscillatory boundary layer's Stokes viscous length scale.

Hydrodynamic object recognition using pressure sensing

“…introduces two essential geometric quantities: R 0 and a 0 , respectively, the uniquely defined (outer) conformal radius (Pólya & Szegö 1951) and the conformal centre (Pommerenke 1975;Duren 1983). The (external) conformal mapping provides a unique but not explicit definition of R 0 and a 0 , which can be interpreted as the characteristic size and location of the obstacle.…”

“…The conformal centre, a 0 = −b 0 , may be calculated from the inverse mapping g by (Pommerenke 1975) …”

“…The analytic function h is, by definition, univalent (Pommerenke 1975;Duren 1983) and consequently satisfies theorems such as the so-called one-fourth theorem and the area theorem (Krantz 1999), hence imposing constraints on the coefficients {m k } k≥1 of the shape fingerprint, for instance…”

“…To express the complex potential (4.1) in the physical plane in terms of z and the obstacle geometric characteristics N = {a 0 , R 0 , q 0 } and S = {m k } k≥1 , a series inversion up to a given order (Morse & Feshbach 1953) of the Laurent expansion (3.2) is required. This inversion process is mathematically guaranteed by the univalent fingerprint h and the form of its Laurent series (Pommerenke 1975;Duren 1983). It yields the expression of the coefficients {b k } k≥1 appearing in equation (3.3) in terms of {a k } k≥1 which embody both S and {R 0 , q 0 }.…”

“…The vast majority of studies reported in the literature on these problems of detection and source encoding used a simple translating/vibrating sphere as a stimulus (e.g. the recent articles by Rapo et al (2009) and Goulet et al (2008) and the multiple references therein).…”