Rheophilic fish commonly experience unsteady flows and hydrodynamic perturbations. Instead of avoiding turbulent zones though, rheophilic fish often seek out these zones for station holding. A behaviour associated with station holding in running water is called entraining. We investigated the entraining behaviour of rainbow trout swimming in the wake of a D-shaped cylinder or sideways of a semi-infinite flat plate displaying a rounded leading edge. Entraining trout moved into specific positions close to and sideways of the submerged objects, where they often maintained their position without corrective body and/or fin motions. To identify the hydrodynamic mechanism of entraining, the flow characteristics around an artificial trout placed at the position preferred by entraining trout were analysed. Numerical simulations of the 3-D unsteady flow field were performed to obtain the unsteady pressure forces. Our results suggest that entraining trout minimise their energy expenditure during station holding by tilting their body into the mean flow direction at an angle, where the resulting lift force and wake suction force cancel out the drag. Small motions of the caudal and/or pectoral fins provide an efficient way to correct the angle, such that an equilibrium is even reached in case of unsteadiness imposed by the wake of an object.
Rheophilic fish commonly experience unsteady flows and hydrodynamic perturbations associated with physical structures like rocks and branches. How rheophilic fish deal with unsteady flow conditions is of ecological and commercial importance and is the subject of many field and laboratory studies (Fausch, 1984; Fausch, 1993; Gerstner, 1998; Heggenes, 1988; Heggenes, 2002; McLaughlin and Noakes, 1998; Pavlov et al., 2000; Shuler et al., 1994; Smith and Brannon, 2005; Webb, 1998). It has recently been investigated how rainbow trout, Oncorhynchus mykiss, interact with unsteady flow regimes (Liao, 2004; Liao, 2006; Liao et al., 2003a; Liao et al., 2003b). At Reynolds numbers (Re) beyond 140 (see Eqn 1), flow behind a bluff 2-D body object, such as a cylinder, generates a staggered array of discrete, periodically shed, columnar vortices of alternating sign. This flow pattern is called a Kármán vortex street (Vogel, 1996). Trout exposed to a Kármán vortex street hold station by either swimming with undulating motions in the region of reduced flow behind the object (drafting) or by swimming directly in the vortex street displaying a swimming kinematic that synchronises with the shed vortices (Kármán gait) (Liao, 2004; Liao, 2006; Liao et al., 2003a; Liao et al., 2003b). For station holding, trout also use the high-pressure, reduced-flow bow wake zone in front of the object (Liao et al., 2003a). Compared with trout swimming in undisturbed, unidirectional flow, there is a reduction of locomotory costs during Kármán gaiting and swimming in the bow wake. This is evident from a reduced tail-beat frequency during these behaviours and from a decrease in red muscle activity during Kármán gaiting (Liao, 2004; Liao et al., 2003a; Liao et al., 2003b).
Another behaviour rheophilic fish may use to save locomotory costs is entraining. Entraining fish move into a stable position close to and sideways of an object, where they hold station by irregular axial swimming and/or fin motions (Liao, 2006; Liao, 2007; Sutterlin and Waddy, 1975; Webb, 1998). This suggests that fish try to optimise their position in response to unsteadiness in the wake of the object. Another observation is that entraining fish angle their bodies away from the object and the main flow direction, which exists upstream of the object. The hydrodynamic reason for this is still unknown (Liao, 2007). Therefore, we reinvestigated the entraining behaviour of trout. In our study, entraining was the most common behaviour of trout that were confronted with a D-shaped cylinder or with a semi-infinite flat plate displaying a rounded leading edge. Based on the results of computational fluid dynamics (CFD), we propose a hydrodynamic mechanism that can explain the fish's movements during entraining.
MATERIALS AND METHODS
Thirty-six rainbow trout, Oncorhynchus mykiss (Walbaum 1792, total body length L 14.1±2.1 cm, mean ± s.d.), were used for the experiments. Trout were purchased from a local dealer, held individually in 45 l aerated freshwater aquaria (water temperature 13±1°C, light:dark cycle 10 h:14 h) and fed daily with fish pellets.
Experiments were performed in a custom-made flow tank (1000 l, water temperature 13±1°C), consisting of an entrance cone (constriction ratio 3:1) and a working section (width 28 cm, height 40 cm, length 100 cm, water level 28 cm) made of transparent Perspex®, delineated within the flow tank by upstream and downstream nets (mesh width 1 cm, fibre diameter 0.02 cm). Flow was generated by two propellers (Kaplan propellers, 23 cm × 30 cm, Gröver Propeller GmbH, Köln, Germany), located downstream of the working section. These were coupled to a DC motor (SRF 10/1, Walter Flender Group, Düsseldorf, Germany). To avoid coarser-scale eddies in the working section, the water was directed through a collimator (inner tube diameter 0.4 cm, tube length 4 cm) and two turbulence grids (mesh widths 0.5 cm and 0.25 cm, wire diameters 0.1 cm and 0.05 cm). In previous experiments trout often swam close to the bottom of the working section. To avoid this, a net (mesh width 1 cm, fibre diameter 0.02 cm) was mounted 4 cm over the bottom of the working section. Two 150 W halogen lights were installed above the working section.
A solid black polyvinyl chloride cylinder (diameter 5 cm) was lengthwise cut in half (D-shaped cylinder, hereafter referred to as ‘cylinder’) and placed vertically in the working section of the flow tank, approximately 30 cm downstream of the upstream net, to generate hydrodynamic perturbations (cf. Fig. 1A, Fig. 2). Trout use the water motions caused by the cylinder not only for Kármán gaiting (Liao, 2004; Liao, 2006; Liao et al., 2003a; Liao et al., 2003b) but also for entraining (Liao, 2006; Montgomery et al., 2003; Sutterlin and Waddy, 1975; Webb, 1998) and swimming in the bow wake (Liao et al., 2003a).
A Kármán vortex street will only be generated over a certain range of Reynolds numbers, typically above 140. The Reynolds number Re is a dimensionless index that gives a measure for the ratio of inertial forces to viscous forces. The Reynolds number is calculated according to the formula: (1) where ρ is the fluid density (999.38 kg m–3 at 13°C), D the cylinder diameter (5 cm) or fish length (14.1±2.1 cm, mean ± s.d.), U the actual flow velocity in the region of the cylinder (see Eqn 3) and μ the fluid viscosity (1.2155×10–3 Pa s–3, 13°C) (Vogel, 1996).
The frequency of vortex detachment is called vortex-shedding frequency (VSF). The VSF is a function of the Strouhal number St (a dimensionless index), the diameter D of the cylinder and the actual flow velocity U (Vogel, 1996): (2)
The Strouhal number for the cylinder depends on the Reynolds number; however, it reaches an almost constant value of 0.2 for Re>2000. The projected area of the cylinder was 17.8% of the cross-sectional area of the working section. To account for flow constrictions near the cylinder due to blocking effects, the VSF was calculated using the actual flow velocity U in the region of the cylinder, according to an ansatz typically used for vortex flow meters (Igarashi, 1999; Liao et al., 2003a): (3) where U∞ is the nominal flow velocity (42 cm s–1 in our experiments) and W the width of the flow tank (28 cm). For the behavioural experiments a Reynolds number of 20,000, based on the diameter of the cylinder, was chosen.
Particle image velocimetry (PIV) was used to monitor the flow field in the working section of the flow tank and to verify the calculated VSF (cf. Eqns 2 and 3). Neutrally buoyant polyamide particles (diameter 50μm, Dantec Dynamics, Skovlunde, Denmark) were seeded in the flow tank and illuminated with a light sheet generated with a high-speed PIV laser (Newport, LaVision Inc., Göttingen, Germany). The laser sheet was oriented parallel to the water surface, approximately 10 cm above the bottom of the working section. Particles were recorded with a high-speed camera (250 frames s–1, FlowMaster HSS-4, LaVision Inc.; recording software: Photron Fastcam Viewer PFV, Photron USA Inc., San Diego, CA, USA) and their movements analysed with the software DaVis Imaging 7.1 (LaVision Inc.). Both the measured and the calculated (U∞=42 cm s–1, St 0.2) VSF were 2 Hz.
For a better understanding of the hydrodynamic mechanism of entraining, experiments with the D-shaped cylinder were compared with experiments with a semi-infinite flat plate displaying a rounded leading edge (width 5 cm, length 35 cm, height 40 cm), hereafter referred to as ‘plate’ (cf. Fig. 1B). The flow field around such a plate is more stable than the flow field around a cylinder (cf. Fig. 3) and it is of interest to compare the behaviour of trout during entraining. When the plate was positioned in the working section (Fig. 1B), trout preferred the bow wake zone (see below) or the region of reduced flow behind the plate. To prevent trout from entering these zones, the upstream and downstream nets were placed 10 cm upstream and 1 cm downstream of the plate.
Before each trial, trout were accustomed to the working section of the flow tank for at least 20 min. A mirror was mounted at 45 deg below the working section to film trout ventrally. Each trout was videotaped (25 frames s–1, WV-BP 100/6, Panasonic, Hamburg, Germany) for about 30 min to assess its spatial preference in the working section. For kinematic analysis, sequences of the ventral view of trout swimming in undisturbed, unidirectional flow (free stream, FS) and of trout entraining close to and sideways of the submerged objects were videotaped (62 frames s–1, RedLake MotionSCOPE M-1, Stuttgart, Germany).
The position of the trout was determined with the video analysis program VidAna (M. Hofmann, www.vidana.net), which recognises the fish's silhouette on the basis of intensity contrasts. According to Liao, four zones were specified in the working section of the flow tank (Liao, 2006). Entraining zones: the two zones close to and adjacent to the right and left side of the cylinder or the plate. Bow wake zone: the zone centred along the midline, adjacent to and upstream of the cylinder or the plate. Kármán gaiting and drafting zone: the zone centred along the midline downstream of the cylinder (see also Fig. 2).
To measure kinematic variables, a custom-made program (GraphicMeasurer, Ben Stöver) transferred points along the calculated body midline of the trout to a coordinate system using the centre of the cylinder's rear edge (or the corresponding point of the plate) as point of origin (cf. Fig. 1A,B). The variables tail-beat frequency, maximum lateral excursion at three body points (snout, 50% L, tail tip), mean distance from snout to the submerged object along the x-(lx) and y-axes (ly; for axes see Figs 1, 2), mean body angle β relative to the x-axis (= mean direction of the undisturbed, unidirectional flow), difference between the maximum and minimum body angle within each sequence Δ, maximum difference in chord length and the displacement of the fish's body along the x- and y-axes were determined.
Kinematics were only analysed for fish swimming within the entraining zone, using sequences (high-speed camera, 62 frames s–1) that lasted for at least 0.16 s (10 frames). To clearly identify no-motion sequences, i.e. sequences in which trout did not move their fins and/or body (see below), we used a time window of 0.16 s. During continuous swimming trout perform at least half a tail-beat cycle during this time. Tail-beat frequencies of entraining trout and of trout swimming in undisturbed, unidirectional flow were calculated by averaging at least five consecutive tail beats. Maximum lateral excursions of trout were calculated at three body points (snout, 50% L, tail tip) and defined as the distance between the extreme y-positions at each body point. To calculate the maximum difference in chord length, no distinction was made between U-shaped or S-shaped body profiles. Irrespective of the body profile, the distance between the snout and the tail tip was measured and expressed as a percentage of total body length L (= maximum chord length). The difference between L and the measured value was defined as the maximum difference in chord length. To calculate the displacement of the trout with respect to the x- and y-axes, the x- and y-coordinates at 21 body points, spaced equally along the midline of the trout, were measured (see above). Mean x- and y-positions were then calculated for each frame. Displacement was defined as the distance between the most extreme means calculated for x- and y-positions within each analysed sequence. Distances to the cylinder, displacement of the trout's body along the x- and y-axes and lateral excursion at three body points are given in cylinder diameter D. In addition, it was measured how long the pectoral fins were extended during swimming. Pectoral fin extension was defined as any visible abduction of the pectoral fins from the body. We did not distinguish whether only one or both pectoral fins were extended. Sequences, in which pectoral fin movements were counted, had a duration of 66 s (62 frames s–1). As the swimming behaviour of trout varied between individuals, not all trout contributed to the results with the same number of kinematic sequences.
For all kinematic variables mean values and standard deviations are determined. Selected maximum (max.) and minimum (min.) values are given in the text. Non-parametric tests were conducted because of violations of the assumption of normality and homogeneity of variance. Statistical tests were performed at an α-level of 0.05. The number of experimental animals (N) and the number of single observations (n) are given for all behavioural experiments.
The model applied to the numerical calculations is based on the fundamental equations of fluid mechanics, i.e. the Navier–Stokes equations, which describe the conservation of mass and momentum. Results show the distribution of all flow variables in each position for the calculated range over time. In consideration of the geometry, all calculations were made three-dimensionally. The fundamental equations are: (4) (5)
Turbulence, represented by the turbulent viscosity ηt, is described by the k–ϵ model (Launder and Sharma, 1974). It is a two-equation model, which includes two transport equations describing the turbulent properties of the flow. In this model, k is the turbulent kinetic energy whereas ϵ is the turbulent dissipation. ϵ determines the scale of the turbulence. The k–ϵ model gives good results for wall-bounded and internal flows. The turbulent viscosity ηt is computed from k and ϵ: (6)
The model constant Cη is computed by the model itself. The equations for k and ϵ are: (7) (8) Here, Pk is the generation of k due to the mean velocity gradients and Pb is the generation of turbulent kinetic energy due to buoyancy. C1ϵ, C2ϵ and C3ϵ are constants of the model. σk and σϵ are the turbulent Prandtl numbers for k and ϵ, respectively (OpenFOAM User Guide) (Bardina et al., 1997; Chen and Patel, 1988; Shih et al., 1995; Versteeg and Malalasekera, 1995).
The equations of the numerical model were solved with the code OpenFOAM 1.5.1 (OpenFOAM User Guide). The result is based on the Finite volume method. The code used the UDS (upwind differencing scheme) interpolation scheme. For the discretisation of derivatives the CDS (central differencing scheme) scheme was used. The procedure employs the PISO (pressure implicit with splitting of operators) algorithm for pressure correction (Issa, 1985; Patankar, 1980).
The solution domain and its dimensions were taken from the behavioural experiments. In the computational experiments the domain consisted of an inlet, an outlet, the fish and a cylinder. A constant velocity of 42 cm s–1 was set at the inlet. In the model, it is assumed that turbulent flow is fully developed when a turbulent intensity of 5% is appropriate for calculating values for k and ϵ. This is a typical value (3–10%) for fully developed turbulent channel flow. The pressure outlet was set to ambient pressure. All rigid walls fulfil the no-slip condition. The free surface was modelled as a flat wall with zero shear stress. The values for all boundaries are given in Table 2.
An unstructured tetrahedral mesh was generated with the commercial mesh generator ICEM CFD (ANSYS Inc., Canonsburg, PA, USA). In the mesh, most of the cells were concentrated in the area where the cylinder and the fish were present. Wall refinement was respected at the walls of the cylinder and the fish to resolve vortex shedding and flow structure between both objects. The number of cells was approximately 1.9 million.
The normalised pressure coefficient (Cp) is a dimensionless number, typically used for representation of pressure forces around bodies and is defined as: (9) where p∞ is the pressure in the far field of the flow and q the dynamic pressure of the inflow.
Results that varied from the data obtained in the model experiment as well as all relevant parameters are given in Table 1. The position of the model is given by coordinates, normalised with the diameter of the cylinder. The model was placed at an x-distance of 0.55 D and a y-distance of 1.38 D. The model was tilted to 11 deg. According to the behavioural experiments (cf. Table 3) it was assumed that these values coincide with the critical point at which the trout just manages to hold station. Here, the influence of the cylinder displacement flow is minimal and the drag force on the trout is assumed to be at its maximum for the observed positions (see Table 1).
To validate the numerical results, a model experiment was performed. Validation was necessary to account for the numerical error introduced by the turbulence model. An artificial fish was used to determine the velocity field around a trout while entraining. The model fish was enlarged by a ratio of 2:1, compared with the mean of the total body length L of the trout. The enlargement of the model was necessary to assure a sufficiently high optical resolution of the model and the flow field. The diameter of the cylinder was also enlarged by a factor of two, to assure the correct ratio between the length of the fish and the diameter of the cylinder. Otherwise, the experimental setup was identical to the setup used for the behavioural experiments (see Fig. 1). In the model experiments the fish was placed at the same position and body angle as the fish in the numerical calculations. A steel wire (diameter 0.3 cm) was used to hold the model at a constant position. The cylinder and the fish model were placed in a flow tank with a cross section of 40 cm× 40 cm. The cylinder was slightly shifted to one side of the flow tank to maintain the correct distance between the fish model and the wall of the flow tank. The velocity of the incoming flow was pre-set at 21 cm s–1, leading to a Reynolds number of 20,000, based on the diameter of the cylinder. The Strouhal number was calculated using the VSF and was in agreement in experiment and simulation, see Table 1. Standard PIV was used to determine the velocity field around the fish model. The light sheet was formed by a 120 mJ Nd:YAG laser (Solo-PIV Nd:YAG laser systems; Polytec GmbH, Waldbronn, Germany). The images were recorded via a surface mirror using a PCO 1600 camera (PCO AG, Kehlheim, Germany). The camera had a resolution of 1600 pixels×1200 pixels at a frequency of 8 Hz and a pulse separation of 0.002 s. To visualise the flow, VESTOSINT 2158 particles (diameter 20μm) were seeded into the flow. The software Dantec Dynamic Studio (Dantec Dynamics, Skovlunde, Denmark) was used to evaluate the velocity field. At first, an adaptive cross-correlation in 32 pixels×32 pixels sub windows and an overlap of 75% were used to evaluate the velocity vectors. In a second step, a peak validation algorithm was used to validate the correlation peaks. Thereafter, the velocity vectors were locally smoothed via a moving average filter over a 5×5 kernel. Finally, the results were averaged over the whole measurement period, i.e. 150 image pairs.
As a reference for the new experimental setup we first repeated some of the experiments performed by Liao (Liao et al., 2003a), using the cylinder. The trout (N=30, total observation time 890 min) swam in the entraining, bow wake and Kármán gait or drafting zone (cf. Fig. 2). Trout showed a significant difference in their preference for these three zones in the vicinity of the cylinder (Kruskal–Wallis test, P=0.018). The entraining zone was preferred (27.7±31.0% of the observation time) over the Kármán gait zone (7.9±17.2%; Mann–Whitney U-test, P=0.003). Trout spent 13.6±22.6% of the observation time entraining on the right and 14.2±25.0% entraining on the left side of the cylinder, neither preferring either side of the cylinder (Mann–Whitney U-test, P=0.739). Trout spent 28.3±36.3% of the time in the bow wake zone. Trout spent significantly more time in these three zones (cumulated size 465 cm2, total size of the working section 2800 cm2; cf. Fig. 2), than expected by random distribution (expected percentage 16.65; Mann–Whitney U-test, P<0.001). In undisturbed flow, trout (N=13, observation time 383 min) preferred to swim close to the walls of the working section and spent only 9.3±13.3% of the time in the fictitious entraining, bow wake and Kármán gait zone (Mann–Whitney U-test, P<0.001). This was less than expected by random distribution (Mann–Whitney U-test, P=0.001).
As expected (Liao, 2006), entraining trout positioned their head (body) downstream of and close to the sides of the cylinder without touching it (for mean entraining position see Fig. 1A). During entraining, sequences where trout had a stretched-straight body posture with the pectoral fins not extended and the body and tail fin showing no motions (no-motion sequences) interchanged with sequences of irregular body and/or pectoral and tail fin motions (motion sequences). No-motion sequences were used for kinematical analysis only, if they lasted for more than 0.16 s. The maximum duration of the no-motion sequences was 0.37 s. Motion sequences of equal length were analysed to allow for a direct comparison of the two types of entraining behaviour (no-motion vs motion behaviour). Trout (N=6, n=39) positioned themselves 0.96±0.4 D (min. 0.27 D, max. 2.15 D) downstream and 1.33±0.2 D (min. 0.93 D, max. 1.63 D) sideways of the cylinder (Mann–Whitney U-test, Pdistance along x-axis=0.955, Pdistance along y-axis=0.465; Fig. 1A and Table 3). During no-motion sequences (N=6, n=19), the maximum difference in chord length was 0.23±0.13% (min. 0.1%, max. 0.6%). During the motion sequences (N=6, n=20), the maximum difference in chord length reached significantly higher values of 2.95±1.0% (min. 1.5%, max. 4.9%; Mann–Whitney U-test, P<0.001; Table 3). No-motion sequences terminated after a displacement of 0.07±0.04 D (min. 0.03 D, max. 0.14 D; x-axis) and/or 0.06±0.04 D (min. 0.02 D, max. 0.17 D; y-axis) was reached (Table 3). Displacement along the x-axis did not differ between no-motion (N=6, n=19) and motion sequences (N=6, n=20) but differed along the y-axis (Mann–Whitney U-test, Px-axis=0.555, Py-axis=0.046; Table 3).
Entraining trout kept their body angled into the wake downstream of the cylinder with respect to the x-axis (Fig. 1A, Fig. 4 and Table 3). The mean body angle β did not differ between no-motion and motion sequences (Mann–Whitney U-test, P=0.177; Fig. 4, Table 3). However, the variation of body angle Δ was significantly larger in the motion sequences (the difference between maximum and minimum body angle within each sequence, Δ, varied between 14.16 deg and 3.36 deg) than in the no-motion sequences (5.0 deg and 1.15 deg, respectively; Mann–Whitney U-test, P<0.001; Fig. 4 and Table 3). In entraining trout, the distances between the most extreme y-positions at three body points (snout, 50% L, tail tip) increased from anterior to posterior. For 50% L and the tail tip, differences were significant between no-motion and motion sequences (Mann–Whitney U-test, P≤0.003; Table 3).
The tail-beat frequencies of entraining trout were measured only for motion sequences (N=6, n=20). In the cylinder experiment the average tail-beat frequency was 4.07±0.58 Hz (min. 2.81 Hz, max. 5.08 Hz). This was significantly less than the tail-beat frequency of trout swimming in undisturbed flow (4.9±0.46 Hz, min. 4.4 Hz, max. 5.89 Hz; N=6, n=19; Mann–Whitney U-test, P<0.001; Fig. 5).
Swimming trout sometimes extended their pectoral fins. During entraining (including no-motion and motion sequences), the pectoral fins were more often extended (72.9±30.9% of the observation time of 858 s; N=8, n=13; Fig. 6) than during swimming in the bow wake zone (25.8±25.2% of the observation time of 990 s; N=7, n=15; Mann–Whitney U-test, P<0.001) or in undisturbed flow (8.6±10.4% of the observation time of 462 s; N=5, n=7; Mann–Whitney U-test, P=0.001).
Trout (N=6) spent most (70.8±26.9%) of the observation time (177 min) entraining close to the sides of the plate. With respect to the flow direction, trout were released on the left side of the working section. Trout therefore preferred entraining on the left side of the plate (69.7±29.4%; right side: 1.1±2.6%; Mann–Whitney U-test, P=0.004).
Entraining trout positioned their head (body) sideways of the plate without touching it (for mean entraining position see Fig. 1B). Trout (N=5, n=47) positioned themselves 0.78±0.08 D (min. 0.59 D, max. 1.09 D) downstream and 0.88±0.09 D (min. 0.72 D, max. 1.04 D) sideways of the point of origin (Mann–Whitney U-test, Pdistance along x-axis=0.983, Pdistance along y-axis=0.478; Fig. 1B and Table 3). In both the no-motion and motion sequences, significant differences were observed in the trout's y-position when comparing experiments with the plate (N=5, n=47) and the cylinder (N=6, n=39; Mann–Whitney U-test, P<0.001). In the plate experiment, trout (N=5, n=47) swam almost parallel to the plate (Fig. 1B, Fig. 4 and Table 3). The difference between the mean body angle β in the cylinder and the plate experiment was significant (Mann–Whitney U-test, P<0.001; Fig. 4, Table 3). Additionally, in the plate experiment, mean body angles β were significantly smaller in the no-motion sequences (N=5, n=27) than in the motion sequences (N=5, n=20; Mann–Whitney U-test, P=0.039). As in the cylinder experiment (see above), the variation of body angle Δ was significantly larger in the motion (the difference between maximum and minimum body angle within each sequence, Δ, varied between 11.6 deg and 2.46 deg) than in the no-motion sequences (3.67 deg and 0.1 deg, respectively; Mann–Whitney U-test, P<0.001; Fig. 4, Table 3). The variation of body angle Δ was significantly larger during no-motion sequences in the cylinder experiment than in the plate experiment (Mann–Whitney U-test, P<0.001; Fig. 4, Table 3).
Maximum duration of the no-motion sequences was 0.81 s. No-motion sequences lasted significantly longer in the plate experiment (0.36±0.14 s; N=5, n=27) than in the cylinder experiment (0.22±0.07 s; N=6, n=19; Mann–Whitney U-test, P<0.001) but terminated after a displacement of 0.05±0.03 D (min. 0.01 D, max. 0.14 D; x-axis) and/or 0.03±0.01 D (min. 0.01 D, max. 0.06 D; y-axis) was reached (cf. Table 3). As in the cylinder experiment (see above), displacement along the y-axis was larger during motion sequences (N=5, n=20) than during no-motion sequences (N=5, n=27; Mann–Whitney U-test, Px-axis=0.966, Py-axis=0.001; Table 3). For both, no-motion and motion sequences, displacement was larger in the cylinder experiment (see above) than in the plate experiment (N=11, n=86; Mann–Whitney U-test, P≤0.016; Table 3).
The maximum difference in chord length during no-motion sequences (N=5, n=27) in the plate experiment was 0.19±0.09% (min. 0.01%, max. 0.4%). During the motion sequences, the maximum difference in chord length reached significantly higher values of 1.12±0.78% (min. 0.1%, max. 2.8%; N=5, n=20; Mann–Whitney U-test, P<0.001; Table 3). Maximum difference in chord length during the no-motion sequences in the experiment with the plate did not differ from the maximum difference in chord length during the no-motion sequences in the experiment with the cylinder (Mann–Whitney U-test, P=0.569; Table 3). But maximum difference in chord length during the motion sequences was significantly larger in the cylinder experiment than in the plate experiment (Mann–Whitney U-test, P<0.001; Table 3).
As in the cylinder experiment (see above), the differences between the most extreme y-positions at three body points (snout, 50% L, tail tip) increased from anterior to posterior. Differences in the lateral excursion at the snout, 50% L and the tail tip were significant between no-motion and motion sequences (Mann–Whitney U-test, P≤0.019; Table 3). In the plate experiment, the lateral excursion at three body points were smaller in the motion and the no-motion sequences (N=5; n=47), than in the cylinder experiment (N=6, n=39; Mann–Whitney U-test, P≤0.051).
In the plate experiment, the average tail-beat frequency of entraining trout was 3.87±0.59 Hz (min. 2.56 Hz, max. 4.66 Hz; N=5, n=18). This was significantly less than the tail-beat frequency of trout swimming in undisturbed flow (4.9±0.46 Hz, min. 4.4 Hz, max. 5.89 Hz; N=6, n=19; Mann–Whitney U-test, P<0.001; Fig. 5). Tail-beat frequencies of entraining trout did not differ between experimental setups (Mann–Whitney U-test, P=0.334; Fig. 5).
In the plate experiment, trout (N=5, n=15) extended their pectoral fins in 43.4±34.3% of the observation time (990 s). This was significantly less than in the cylinder experiments (Mann–Whitney U-test, P=0.009; Fig. 6). However, entraining trout extended their pectoral fins more often than trout swimming in undisturbed flow (Mann–Whitney U-test, P=0.031) or in the bow wake in front of the cylinder (Mann–Whitney U-test, P=0.081; Fig. 6).
Model experiments and numerical calculations
The behavioural experiments indicated that entraining trout can hold their position at one side of a flow disturbing stationary object for distinct periods of time, with little or no body or fin movement. At this position, the flow-induced forces on the body on average cancel to zero. The flow-induced forces are specified by the wall friction and pressure distribution around the trout. To confirm this, a model trout was placed at the position preferred by real trout during entraining (cf. Table 1). To determine the characteristics of the cylinder wake flow in the presence of an entraining trout, the flow field was calculated by CFD (see Materials and methods). In addition, the velocity and pressure distribution across the surface of the trout and hence the forces acting on the trout were calculated. Being in the order of micro-Newtons, these forces were difficult to measure. Fig. 7 illustrates the normalised magnitude of the velocity and the sectional streamline pattern for a momentaneous record of unsteady flow. Fig. 7A and B compare the numerical results with the model experiment for a similar flow state. The position of the cylinder and the trout model are indicated in black. The von Kármán type periodical vortex shedding can be identified in both images and is in agreement with the numerical calculations. The Strouhal number was approximated to be 0.21 in both the experiment and the numerical investigation, which is close to the Strouhal number for a cylinder of 0.2, given by Vogel (Vogel, 1996).
In the following, the focus is on the time-averaged flow field. Typically, the time-averaged velocity field contains most of the flow energy while the vortex shedding contributes to less than 20%. Fig. 8 illustrates sectional streamlines and the magnitude of the velocity of the time averaged flow field (simulation Fig. 8A and measurement Fig. 8B) in the sagittal plane of the model. One can identify the wake region downstream of the cylinder in the form of an attached separation bubble with recirculating flow. Again, simulation and experiment are in agreement. Due to the disturbance of the streamlines by the cylinder, the fish experienced a flow at its head that was already directed upwards. Therefore, the effective body angle (denoted as the angle of attack in the airfoil theory) is the sum of the geometric angle of the body and the angle of the mean flow direction near the front of the fish. The differences in wake flow between the experimental and the numerical models, especially regarding the length of the time averaged separation bubble, resulted from shortcomings of the realisable k–ϵ turbulence models, predicting the correct contours of the separation region. The turbulent kinetic energy k was overestimated by the computational model. This leads to an elongated separation bubble.
To evaluate the full, 3-D pressure distribution on the model, the numerical results were investigated in detail. Fig. 9 shows the time averaged Cp values as discrete points on the surface of the trout over the dimensionless length x/L for both the inside (side facing the cylinder) and the outside surface of the fish. The 3-D structure of the fish is evident in the scattered distribution of the Cp values. Fig. 9A shows the values for both sides of the fish, averaged for each surface and plotted over the dimensionless length x/L. The course of this line indicates a large region of relatively small pressure differences at the mid-body region of the entraining fish (0.4<x/L<1). The described configuration is compared with a fish swimming in undisturbed, unidirectional incoming flow. The body angle was assumed to be 11 deg, the typical body angle of entraining trout in our experiments (Fig. 9B)
The results for the undisturbed flow are plotted in Fig. 9 as a solid black line. The course of Cpo in Fig. 9B is comparable with the Cpo course in Fig. 9A. The obvious shift between both pressure profiles is caused by the effect of the cylinder on the flow around the fish. The major difference between the two configurations is evident in the course of Cpi. For undisturbed flow (Fig. 9B), the course of Cpi is shifted upwards to a higher pressure for most of the length of the trout (0.1<x/L<0.8). This results in a considerable difference between Cpi and Cpo. As a measure of total force we took the average of the difference Cp=Cpi–Cpo over the dimensionless length of the model x/L. The resulting value for the configuration involving the cylinder is 0.06, for the configuration without any object in the undisturbed flow it is 0.16. This indicates that the influence of the cylinder on the flow, around the entraining trout, reduces the total force. As a result of the unsteady cylinder wake flow, the resulting pressure forces on the trout oscillate between a minimum value of Cp,min.=–0.7 and a maximum value of Cp,max.=1.02. The frequency corresponds to the VSF in the wake of the cylinder of fCp=1.8 Hz. Further fine-tuning of the balance between the outer and the inner pressure field, either by moving closer to the object or by changing the body angle, may even enable the trout to cancel out the differences to zero, resulting in a perfect station holding position. Such a large number of numerical simulations, needed for the fine-tuning of the position and angle, could not be realised up to now due to the large computational effort.
Simulation of the plate experiment
Fig. 10 shows the streamlines around a plate, with the fish's body in the entraining position. The fish was angled away from the undisturbed inflow with 1 deg, which was the value obtained in the behavioural experiments. Interestingly, similar to the flow around the cylinder, the fish experienced a flow that was directed upwards, due to the flow deflection around the front part of the plate. Therefore, the effective angle of attack relative to the flow is the sum of the body angle (relative to the main flow upstream of the object) and the angle of the flow direction near the front of the fish, which was about 5 deg in the simulations. In contrast to the cylinder flow, there is no vortex shedding and thus the fish – at least in theory – has no need for corrective body and fin motions once it has reached a stable position for entrainment. This is in agreement with the observation that trout entraining in the vicinity of the plate show less fin and body movements than trout entraining in the vicinity of the cylinder (see above).
Entraining, Kármán gaiting, drafting and swimming in the bow wake represent energetically favourable strategies for station holding in running water (for a review see Liao, 2007). As expected, the trout used in our experiments also showed these behaviours during exposure to the flow perturbations caused by a stationary cylinder. The total body length of the trout was 14.1±2.1 cm, the cylinder that generated the Kármán vortex street had a diameter of 5 cm (cylinder diameter to fish length ratio was about 1:3) and the bulk flow velocity was 42 cm s–1. Accordingly, the hydrodynamic requirements for Kármán gaiting were fulfilled in our experiment. Nevertheless, trout swam only 7.9±17.2% of the observation time in the Kármán gait zone whereas the trout investigated by Liao (Liao, 2006) Kármán gaited in about 80% of the time. The reason for this discrepancy is unknown, but Liao et al. (Liao et al., 2003a) and Liao (Liao, 2006) already pointed out that the swimming behaviour of trout can be highly individual-specific (see also Swanson et al., 1998) and – for a given individual – may even change from day to day. Station holding strategies may also differ across species (Liao et al., 2003b). Other factors, like food availability, also influence the preferred mode for station holding (see also Fausch, 1984) (B. Baier, unpublished).
If exposed to flow perturbations caused by a cylinder, the trout preferred entraining. A cylinder exposed to bulk water flow sheds vortices in a regular pattern; therefore, the instantaneous forces that act on entraining trout fluctuate. Nevertheless, for up to 0.4 s (cylinder) and 0.8 s (plate) trout were able to hold station without any axial body and fin motions (no-motion sequences). During no-motion sequences the pectoral fins rested against the body that had a stretched-straight profile. Furthermore, the no-motion sequences were characterised by only small variations in body angle Δ (Fig. 4, Table 3). Maximum lateral body excursions were small, suggesting that for short periods of time there was no need for corrective body and/or fin motions (Table 3). The no-motion sequences alternated with sequences where trout showed irregular axial body and/or fin motions (motion sequences). During motion sequences trout showed higher lateral body excursions and a larger difference in chord length. All this suggests that entraining trout repeatedly had to balance thrust and drag forces to hold station. Obviously, without corrective body or fin motions, entraining trout were unable to maintain their position for long periods of time, most likely due to the unsteadiness of the flow behind the cylinder.
In the plate experiment trout also showed entraining behaviour. The flow field around a semi-infinite flat plate displaying a rounded leading edge is more stable than the flow field around a D-shaped cylinder (cf. Fig. 3). This is probably the reason why the no-motion sequences in the plate experiment were significantly longer than in the cylinder experiment. A significantly reduced maximum difference in chord length, pectoral fin activity as well as a reduced tail-beat frequency during the motion sequences also indicate that entraining sideways of the plate required less corrective motions than entraining downstream and sideways of the cylinder (Table 3). In general, in the plate experiment less corrective motions were seen, which can be attributed to more stable flow conditions.
The highly reduced body and fin motions during the no-motion behaviour suggest (see also Liao et al., 2003a) that station holding in the vicinity of a cylinder or a plate is energetically favourable. This agrees with a reduced muscle activity observed during entraining (A. Klein and H.B., unpublished).
Model experiments and simulations
Our model experiments suggest that entraining reduces locomotory costs. Fig. 11 illustrates the resulting forces affecting a model trout while entraining, assuming a stretched-straight body posture as observed in the behavioural experiments and a body angle similar to the body angle measured in real trout. The force components were separated in analogy to the flow around an airfoil (see Liao et al., 2003a) tilted to the flow (which is usually referred to as the angle of attack) in a drag and lift force. An additional suction force Finside acts on the model on the inner side which is due to the increased velocities in the gap between fish and cylinder wake. This simplified model was used here to demonstrate the principal effects of the body angle on the pressure distribution and the resulting forces. Because of the flow deflection near the front part of the body, entraining fish experience an upward directed flow at the nose (if the fish is entraining on the upper side of the body) and therefore, the direction of the average flow with velocity U (see Fig. 11) is already angled against the undisturbed flow U∞ direction far upstream of the object. According to the airfoil theory, the drag force Fdrag is pointing in the mean flow direction of the flow U upstream of the leading edge, while the lift force Flift is perpendicular to it. This results in a force Foutside, similar to drawing the forces for a symmetrical profile at an angle of attack. In phases of no-motion during entraining, the fish seem to keep a stable position downstream and sideways of the cylinder because the lift force Foutside is balanced by the counter-acting suction force Finside. Thus, at a specific position next to the cylinder, and for a specific body angle, it is possible that these forces sum up to zero and fish therefore can maintain their position without any active body or fin movements. A similar hydrodynamic argument based on the aerodynamic theory has been used to describe the force balance of dolphin while riding on free surface bow waves (Hayes, 1953).
The behavioural observations have shown that trout keep their station holding position by adjusting their body angle relative to the main flow. Corrective motions for fine-tuning of the body angle result either from movements of the pectoral fins (see also Liao, 2006; Webb, 1998) and/or the caudal fin. The pectoral fins seem to play a specifically important role, because they were unusually often spread without any obvious pattern of activity during entraining (Fig. 6). Although, adjusting their body angle away from the flow by turning their caudal fin (effectively adjusting the angle of attack) may be of similar importance, as it was observed in most corrective actions. This may reflect a highly optimised adaptation in fish, as proposed by Brücker (Brücker, 2006) based on an aerodynamic theory. As a highly simplified model of the oblate fish body, he compared body and fin with an airfoil (the body) with a hinged flap (the fin). In this model, the flap length represents the caudal fin, measured from the peduncle to the trailing edge of the fin, and the chord length represents the total fish length L. In the airfoil model the zero moment coefficient, which is the rate of change of pitching moment of the foil at constant lift, is given by: (10) according to the solution of Glauert (Glauert, 1927), with E being the flap-to-chord ratio and γ being the flap deflection angle (cf. Fig. 12A). The results of Glauert demonstrated that ∂CM0/∂γ is maximum at approximately E=0.2 (cf. Fig. 12B). In other words, the rate of change of pitching moment and therefore the change of body angle against the flow (swimming direction) are highest for E=0.2 for any slight change in γ (caudal fin angle). Surprisingly, trout actually have a ratio of fin-to-body length of roughly E=0.2. Consequently, the highly simplified airfoil model gives an idea of the mechanisms trout may employ to correct their station holding position in a fast and efficient manner. Accordingly, during entraining at an optimum position, it is hypothesised that trout are able to correct body angle changes by performing only small caudal fin movements. Such corrective motions of the caudal fin were indeed observed during the present study. Future work will therefore focus on providing evidence for the hypothesis provided by the numerical flow simulations assessing a fish model at different caudal fin angles relative to the body.
We thank R. Zelick and V. Schlüssel for critically reading an early draft of the manuscript. A. Klein deserves thanks for his help with the PIV studies.
This work was supported by the DFG (BL 242/12 and BR 1494/10).
LIST OF ABBREVIATIONS
- bow wake
- D-shaped cylinder
- normalised pressure coefficient
- pressure on the inside of the model
- pressure on the outside of the model
- maximum pressure coefficient
- minimum pressure coefficient
- model constant
- C1ϵ, C2ϵ, C3ϵ
- model constants
- central differencing scheme
- computational fluid dynamics
- cylinder diameter
- flap-to-chord ratio
- drag force
- force on the inside of the model
- lift force
- force on the outside of the model
- free stream
- turbulent kinetic energy
- total body length of fish
- distance from snout to submerged object along x-axis
- distance from snout to submerged object along y-axis
- number of observations
- number of experimental animals
- semi-infinite plate displaying a rounded leading edge
- generation of turbulent kinetic energy due to buoyancy
- pressure in the far field of the flow
- pressure implicit with splitting of operators
- particle image velocimetry
- generation of k due to the mean velocity gradients
- dynamic pressure of the inflow
- Reynolds number
- Strouhal number
- actual flow velocity
- nominal flow velocity
- upwind differencing scheme
- vortex-shedding frequency
- width of flow tank
- mean body angle
- difference between maximum and minimum body angle
- turbulent dissipation
- turbulent viscosity
- fluid viscosity
- fluid dynamics
- turbulent Prandtl numbers for k
- turbulent Prandtl numbers for ϵ
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