One of the simplifying approximations necessary for Muller et al. (1982)
to formulate their model was neglecting friction. It is well know that for regions of the flow with negligible net viscous forces, the Navier–Stokes equations (the most general description of the relationship between force and flow in a fluid) can be simplified by removing the viscous term from these equations (Cimbala & Cengel 2008
). The loss of this term strongly reduces the mathematical complexity of the model, but inevitably results in fluid flows that are not physically meaningful near solid walls, since flow is allowed to slip there.
Muller justified this approximation by including estimates of the thickness of the boundary layer at the internal side of the mouth cavity for a number of suction feeding fish, which differed in head size and the duration needed to expand their mouth cavities during feeding (Muller & Osse 1984
). This estimation was based on a dimensional analysis of the Navier–Stokes equation, which resulted in the following formula (Muller et al. 1982
is the thickness of the boundary layer; ν
is the kinematic viscosity of the fluid; L
is the length of the mouth cavity; and t
is the duration of mouth cavity expansion. This theoretical analysis showed that, even for large fish heads (e.g. 0.1
m) expanding relatively slowly during feeding (e.g. 95
ms), the thickness of the boundary layer is probably less than 1
mm (Muller & Osse 1984
). For our study animal L. gibbosus
, which expands its head within approximately 50
ms, this model predicts a boundary layer of 0.32
CFD is a mathematical modelling technique that allows a numerical solution of the Navier–Stokes equations, even for complex unsteady flows. The water surrounding our L. gibbosus model was split into a large number of small cells (finite volumes) for which the flow equations are solved in an iterative process by algorithms embedded in FLUENT software (Ansys, Lebanon, NH, USA). Recent versions of this software allow simulating flow caused by deforming solid bodies. The details of our approach in using this software in modelling the radially expanding and forward translating fish can be found in the electronic supplementary material.
We tested the difference between an expanding cone model with and without viscous forces included (). To do so, we compared the output of two CFD models: the first model solving the inviscid approximation of the Navier–Stokes equations (i.e. the Euler equations; b) and the second model solving the full Navier–Stokes equations (c). The inviscid CFD model can be regarded as an improved version of Muller's analytical model, since it does not neglect the radial acceleration of the flow inside the expanding hollow cone, and the velocity profile in the plane of the circular mouth is not precisely identical to the (quasi-steady) vortex flow model.
Figure 1 (a) Lepomis gibbosus feeding on a bloodworm, and flow velocities (see colour code below) and streamlines from the corresponding (b) inviscid and (c) viscous axisymmetric CFD models. (c) Note that, in the CFD model solving the full Navier–Stokes (more ...)
Our results showed considerable differences between both models in the intra-oral flow velocities ( and ). To quantify this in more detail, we conservatively define the boundary layer as the zone where, for a given cross section, less than 90 per cent of the peak axial flow velocities occur. At 5
mm posterior of the mouth opening, the boundary layer grows rapidly from 0.27
mm after 10
ms to 0.95
mm after 20
ms. Afterwards, the boundary layer remains approximately constant around 1
mm after 30
ms and 0.98
mm after 40
ms; ). Given that the radius of the mouth cavity at this cross section increases from 1.98 to 3.68
mm, apart from the first 10
ms of suction, the boundary layer size exceeds 30 per cent of the radius. We conclude from this that, not only for larval fish (Drost et al. 1988
; Osse & Drost 1989
), frictional forces play an important role during suction feeding.
Figure 2 Axially directed flow velocities with respect to the buccal surface at 5mm posterior of the jaw tips. Data from different times within the feeding sequence (see also figure 1) are given, as well as a comparison between the results from the viscous, (more ...)
Figure 3 Axial flow velocities at time=30ms as a function of position (a) inside and in front of the mouth cavity and (b) outside the mouth cavity are compared between the CFD model solving the full Navier–Stokes equations (solid line), the inviscid (more ...) Muller & Osse (1984)
argued that negligibly small boundary layers, and thus negligibly low friction, imply a hydrodynamic advantage for the predator because the occurrence of friction drag is avoided. On the one hand, it is probably true that friction drag hinders mouth cavity expansion. The area-weighted mean pressure on the internal surface of the mouth cavity near the moment of maximal expansion velocity (time =40
ms) was −434
Pa in our CFD model, compared with −213
Pa for the model that neglects friction at boundaries. Since pressure forces clearly dominate shear forces in the (radial) direction of expansion (radial pressure forces were approximately three orders of magnitude higher than the calculated radial shear forces), and expansion velocities are equal in both models, we estimate from the previous that more than twice the power input (power=force×velocity) was needed from the fish to expand the mouth cavity in the viscous compared with the inviscid flow conditions.
On the other hand, friction at the boundaries of the mouth cavity might as well be regarded as a factor that increases suction performance (yet at the expense of metabolic power, cf. above) because higher flow velocities will be reached at the centre for the same volume increase per unit of time (c versus b and a). Indeed, it appears from our full CFD simulations that the boundary layer causes the unrestricted inflow of water into the mouth cavity to be constrained to a narrower region at the centre of the expanding mouth cavity, which explains the higher velocities at the centre ( and a) due to the law of continuity. Consequently, any prey entering close to the centre of the mouth aperture will be transported faster to the back of the mouth cavity in the more accurate CFD simulation (c) compared with the inviscid models (Muller's model or b).
The flow in front of the mouth is nearly identical for the inviscid and viscous CFD models (). The decrease in flow velocity away from the mouth along the model's rotational symmetry axis corresponds quite nicely to the relationship inferred from the ‘circular vortex filament’ model described by Muller et al. (1982)
). This is in accordance with the results from recent in vivo
flow visualization analyses (Day et al. 2005
), where a highly accurate fit between the measured drop in flow velocity away from the mouth aperture and Muller's circular vortex filament model was observed along the centreline.
The study by Muller & van Leeuwen (1985)
showed that Muller's vortex flow model overestimates the flow velocity near the edges of the mouth opening compared with experimental data obtained from flow visualization of feeding trout (Salmo gairdneri
). Since in the plane of the circular mouth opening, the vortex model predicted the highest flow velocities to be located near the centre of the vortex ring (i.e. near the edges of the mouth aperture) while the highest flow velocities were observed near the centre of the mouth, they argued that the vortex approximation does not fully capture the hydrodynamic details at the mouth region (Muller & van Leeuwen 1985
). Our CFD results also showed the highest flow velocities near the centre of the mouth during most of the time (c
), which is in line with the observations of Muller & van Leeuwen (1985)
and more recent PIV studies (Day et al. 2005
; Nauwelaerts et al. 2007
). Additionally, these results may imply that suction feeders can improve suction performance by positioning the prey item so that it will travel along a path through the centre of the mouth during suction.