The role of optimal vortex formation in biological fluid transport

Abstract

Animal phyla that require macro-scale fluid transport for functioning have repeatedly and often independently converged on the use of jet flows. During flow initiation these jets form fluid vortex rings, which facilitate mass transfer by stationary pumps (e.g. cardiac chambers) and momentum transfer by mobile systems (e.g. jet-propelled swimmers). Previous research has shown that vortex rings generated in the laboratory can be optimized for efficiency or thrust, based on the jet length-to-diameter ratio (L/D), with peak performance occurring at 3.5<L/D<4.5. Attempts to determine if biological jets achieve this optimization have been inconclusive, due to the inability to properly account for the diversity of jet kinematics found across animal phyla. We combine laboratory experiments, in situ observations and a framework that reduces the kinematics to a single parameter in order to quantitatively show that individual animal kinematics can be tuned in correlation with optimal vortex ring formation. This new approach identifies simple rules for effective fluid transport, facilitates comparative biological studies of jet flows across animal phyla irrespective of their specific functions and can be extended to unify theories of optimal jet-based and flapping-based vortex ring formation.

Figure 1

Schematic of jet flow apparatus with a time-varying nozzle exit diameter. All components are depicted as they appear in a meridian cross-section of the axisymmetric device. Nozzle actuators are not shown. Jet flow at the exit plane (i.e. through the origin and normal to the symmetry axis) is from left to right. Jet was initiated upstream using a piston-cylinder mechanism (cf. Gharib et al. 1998) that impulsively created a constant volume flux of 8.8 cm³ s⁻¹, with a rise time of 300 ms. Nozzle exit diameter was varied temporally between 45% and 67% of the inlet diameter, D_inlet=2.54 cm, in each experiment. Variables indicated in the schematic are referred to in the text.

Figure 2

Vorticity profiles for different rates of temporal increase in jet exit diameter. Radial variable (R) is measured along section A–A′ indicated in figure 1. Square symbols indicate vorticity profile for constant exit diameter D=0.45 D_inlet. Circle symbols indicate increase from D=0.45 D_inlet to 0.67 D_inlet in 2.2 s. Triangle symbols indicate increase from D=0.45 D_inlet to 0.67 D_inlet in 1.2 s.

Figure 3

Tail-first swimming kinematics of Lolliguncula brevis. Jet exit diameter and velocity data from successive swimming cycles were compiled from Bartol et al. 2001. Dashed vertical lines indicate range of vortex-limiting formation time, 3.5<(L/D)*<4.5. Cross symbols indicate trends computed using $L/\overline{D}$ as defined in the text. Circle symbols and solid line indicate trend computed using (L/D)* as defined in the text. Formation time calculations include the effect of background flow as in Krueger et al. (2003).

Figure 4

Transmitral flow during normal and pathological early diastolic filling (E-wave). Jet exit diameter and velocity data for each patient were compiled from Verdonck et al. (1996). Diameter data are normalized by the time-averaged mitral valve exit diameter in each patient to facilitate quantitative comparison. Dashed vertical line indicates lower bound of vortex-limiting formation time, (L/D)*=3.5. Open circle symbols indicate points of correlation with optimal vortex formation time.