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First published online May 1, 2009
Journal of Experimental Biology 212, 1494-1505 (2009)
Published by The Company of Biologists 2009
doi: 10.1242/jeb.026732

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

Mark A. Rapo¹, Houshuo Jiang^1,*, Mark A. Grosenbaugh¹ and Sheryl Coombs²

¹ Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA
² Department of Biological Sciences and J. P. Scott Center for Neuroscience, Mind and Behavior, Bowling Green State University, Bowling Green, OH 43402, USA

View larger version (82K): [in this window] [in a new window]   Fig. 1. (A) 2-D computational domain for a circular cylinder (with cross-section identical to that of the sphere) vibrating nearby a sculpin cross-section. (B) A zoomed-in view of the meshes surrounding the cylinder and the sculpin. The vibrating motion of the cylinder was represented by a small rectangular deforming mesh zone that surrounds the immediate area of the cylinder. (C) Node locations around the sculpin cross-section where pressures at pore openings were evaluated. The convention for calculating the pressure difference is identified by the direction of the arrows. The pressure at the arrow tip (p1) is subtracted from the pressure at the arrow tail (p2). The distance between two consecutive pore openings is assumed to be 2 mm. dp/ds denotes the pressure gradient along the profile.

View larger version (50K): [in this window] [in a new window]   Fig. 2. (A) 3-D computational domain for a sphere vibrating nearby a sculpin body. (B) A zoomed-in view of the surface meshes of the sphere and of the sculpin, with nodes spaced approximately 1 mm apart on the sphere and 2 mm apart on the sculpin. The vibrating motion of the sphere was directly represented by a deforming mesh that surrounds the immediate vicinity of the sphere and is contained inside a small rectangular prism (not shown). (C) Node locations around the sculpin body where pressures were evaluated, including those points which fall directly on the sculpin's lateral line system (marked by various colors). (D) Points around the sculpin mid-plane cross-section, where pressures were interpolated from the pressure data computed on the 3-D sculpin surface. In both C and D, the convention for calculating the pressure difference is identified by the arrow direction. The pressure at the arrow tip (p1) is subtracted from the pressure at the arrow tail (p2). The distance between two consecutive pore openings is assumed to be 2 mm. dp/ds denotes the pressure gradient along the profile.

View larger version (33K): [in this window] [in a new window]   Fig. 3. Comparison between the 3-D computational fluid dynamics (CFD) simulations (symbols in C–F) and the potential flow theory (PFT) (solid lines in C–F) for a sphere vibrating above an infinite flat plane wall. The PFT solution consists of a dipole source above the wall and an image dipole source below the wall to satisfy the zero-normal-flow wall boundary condition. The axis of sphere vibration is either parallel (left column) or perpendicular (right column) to the wall. Iso-pressure contours of the instantaneous pressure field are depicted in (A) and (B). Line plots of the instantaneous pressure (C,D) and pressure gradient (E,F) along the wall are shown for three distances (1.8, 3.5 and 8.5 sphere diameters) between the sphere and the wall. U0, velocity amplitude of the sphere vibration; , fluid density; =2f; p1, p2, pressure values along the wall, which are used to calculate the pressure gradients, dp/ds.

View larger version (33K): [in this window] [in a new window]   Fig. 4. Comparison between the 3-D computational fluid dynamics (CFD) simulations (symbols) and the analytical solution (solid lines from Eqns 6, 7, 8) for the oscillatory boundary layer at the surface of an infinite flat plane wall, created by a sphere vibrating above and parallel to the wall. Vertical (y-) profiles are compared at the wall location directly below the equilibrium position of the sphere. The left column corresponds to the along-wall flow velocity, u, and the right column corresponds to the strain rate, S, and the shear rate, du/dy. A variety of sphere-to-wall distances (r), source vibration magnitudes (U0), frequencies (f), and sphere radii (a) are considered. Four examples are shown here: (A) f=30 Hz, r=10a U0=0.04 m s–1, phase t=0.4, where =2f and t the time. (B)f=45 Hz, r=20a, U0=0.1 m s–1, phase t=1.6. (C)f=50 Hz, r=3.67a, U0=0.007 m s–1, phase t=2. (D)f=75 Hz, r=40a, U0=0.03 m s–1, phase t=2. , Stokes viscous length scale; U, fluid velocity just outside the oscillatory boundary layer.

View larger version (182K): [in this window] [in a new window]   Fig. 5. Comparison between 2-D calculations (A) and 3-D calculations (B), and comparison between 3-D computational fluid dynamics (CFD) (columns 2 and 3 in B–D) and 3-D potential flow theory (PFT, column 4 in B–D) results when the pectoral fins are either included (column 2) or excluded (columns 3 and 4). The calculations correspond to a video-recorded sequence of a mottled sculpin's step-by-step approach towards an artificial prey – in this case a sinusoidally vibrating sphere [from Coombs and Conley (Coombs and Conley, 1997a)]. The first three steps of the prey-tracking sequence are illustrated, including the initial orienting response at signal onset (A and B are for the same step) followed by two subsequent approach steps (C,D). Each block in columns 2–4 contains a plot of the iso-pressure contours of the normalized instantaneous pressure field that surrounds the sculpin and the vibrating sphere. Plotted in the lower panel are distributions of normalized pressure gradient along the three major portions of the lateral line canal system: the trunk canal on the ipsilateral (I) side of the body with respect to the dipole source (blue curve), the trunk canal on the contralateral (C) side of the body (red curve) and the frontal canals (F) on the head (green curve). T stands for the tail position. The pressure contours in column 4 correspond to the solution of a 3-D (2-D in A) dipole source in an unbounded fluid, unperturbed by the presence of the fish. The pressure gradient values plotted in column 4 are calculated from the unperturbed pressure field but the magnitudes are doubled to account for the potential flow wall-boundary condition of the fish and to facilitate comparison with the CFD solutions. The red arrow in column 2 of A indicates the pressure gradient results for the pectoral fin insertion points. U0, velocity amplitude; , fluid density; =2f; a, sphere radius; dp/ds, pressure gradient; p, pressure.

View larger version (11K): [in this window] [in a new window]   Fig. 6. The complete dipole source pressure-gradient signal to the whole canal lateral line system, calculated from 3-D computational fluid dynamics (CFD) and evaluated for strike position No. 3 as shown in Fig. 5D. U0, velocity amplitude; , fluid density; =2f; dp/ds, pressure gradient.

View larger version (15K): [in this window] [in a new window]   Fig. 7. The maximum tip velocity experienced by the superficial neuromasts (SNs) as a function of the neuromast height. The plot was generated analytically based on the experimental conditions used by Coombs and Janssen (Coombs and Janssen, 1990) – sphere radius, r, of 3 mm vibrating in a direction perpendicular to the frontal plane of the fish at frequencies, f, of 10, 50 and 100 Hz and with velocity amplitude, U0, of 3.5 mm s–1, 5.3 mm s–1 and 6.3 mm s–1. The distance from the sphere center to the fish surface was 15 mm. The solid circle is the potential flow result for an SN with cupula height of 100 µm. u, x velocity component.

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