The effect of flow speed and body size on Kármán gait kinematics in rainbow trout

Fish that live in moving water must contend with complex flows arising from current moving past objects. Understanding how fish swim in unsteady flows has attracted attention from many disciplines, ranging from stream ecologists investigating how fish relate to habitat, to engineers seeking the underlying principles of efficient propulsion (Streitlien and Triantafyllou, 1996; Heggenes, 2002; McMahon and Gordon, 1989; Triantafyllou et al., 2004; Pavlov et al., 2000). The hydrodynamics of a cylinder wake is rich in flow phenomena, as many studies demonstrate (for review, see Williamson, 1996). Examining how fish interact with vortices shed from a cylinder has provided a focused way to begin to understand how fish swim in unsteady flows (Webb, 1998; Liao et al., 2003a; Liao et al., 2003b; Sutterlin and Waddy, 1975). In particular, the ability to control the frequency and spacing of vortices by altering the flow speed and cylinder size, respectively, has provided a unique opportunity to study how fish behave in a complex yet predictable environment. Previous studies revealed that rainbow trout adopt novel body kinematics behind a cylinder, termed the Kármán gait, and pointed to common relationships between body kinematics and wake variables that were associated with energy savings (Liao et al., 2003a; Liao et al., 2003b; Liao, 2004; Taguchi and Liao, 2011). These studies largely concentrated on only one flow speed; we still know very little about how flow velocity influences the preference and ability to Kármán gait. This represents a large gap in our understanding of how fish relate to unsteady flows because the vortex street is dynamically remodeled as flow speed increases (Zdravkovich, 1997). In addition, the effect of body size on Kármán gait kinematics remains unexplored.

In the absence of flow there is no vortex street associated with a cylinder. When water moves past a cylinder, shear layer instabilities around the cylinder give rise to regions of concentrated vorticity that grow and take turns inhibiting each other to cause the alternate shedding into the wake that forms a classic von Kármán vortex street (Williamson, 1996). Vortex shedding increases with flow speed, while the downstream–upstream spacing of the vortices remains the same, as it is dictated only by the cylinder diameter. A strong, stable low pressure region is also present on the lee side of the cylinder, where flow can move upstream, and which extends up to two cylinder diameter lengths downstream (Zdravkovich, 1997). As flow increases even more in this regime the vortex street gets stronger, resulting in lower pressures at each vortex center. The three-dimensional complexity of these vortices also increases with speed (Wei and Smith, 1986). The empirical interactions between these fluid phenomena and the flexible body of a fish are virtually unknown.

For undulatory locomotion in uniform flow (i.e. freestream swimming), the relationships between kinematic variables and force generation have been established (Lighthill, 1971; Webb, 1992). As speed and size increase, body kinematics change predictably (Webb et al., 1984). In contrast, there are still fundamental questions that remain unanswered for fish swimming in a vortex street. What are the forces that a Kármán gaiting fish experiences as flow speed and body size increase, and how do body kinematics change to reflect the interactions with these forces? In order to establish a comprehensive understanding of Kármán gaiting, both of these questions need to be addressed. This study takes a crucial first step by characterizing the effects of flow speed and body size on the kinematics of trout swimming behind a cylinder.

Rainbow trout, Oncorhynchus mykiss (Walbaum) were obtained from the Chattahoochee Forest National Fish Hatchery in Suches, GA, USA. Fish were held in a 473 l circular freshwater tank maintained at 15±1°C (DS-4-TXV Delta Star Chiller, Aqua Logic, San Diego, CA, USA) on a 12 h:12 h light:dark cycle and fed commercial trout pellets daily. Nine trout (L=15.7±0.8 cm, where L is total body length) were used for the experiments that investigated the effect of increasing flow velocity on Kármán gait kinematics. In another experiment, three size classes were selected. We chose the smallest size based on hatchery availability, and the largest size to avoid solid blocking effects given the dimensions of our flow tank. We categorized the sizes classes as small (L=10.4±0.1 cm, N=6 fish), medium (L=14.3±0.3 cm, N=6 fish) and large (L=18.6±0.6 cm, N=3 fish).

Trout were placed into a 175 l recirculating flow tank custom designed to hold a 5 cm diameter D-section cylinder in the working area (25×26×87 cm height×width×length), and allowed to acclimate to low flow for several minutes. A Phantom V12 video camera (Vision Research, Wayne, NJ, USA) was aimed at a 45 deg front-surface mirror placed below the flow tank and recorded the ventral view of the fish silhouetted against a light sheet for all flow speed experiments (Fig. 1). Video images (60 frames s⁻¹, 1280×800 pixels) were recorded for fish swimming behind the cylinder at flow speeds ranging from 10 to 90 cm s⁻¹, which corresponds to body-normalized swimming speeds of 1.0–6.0 L s⁻¹. Kármán gaiting kinematics across fish size were compared at a single, intermediate flow speed (~50 cm s⁻¹).

The term Kármán gaiting was originally used to describe the behavior of a fish holding station in a vortex street for an extended period of time (Liao et al., 2003a). For the flow speed and body size selected for that study, Kármán gaiting fish displayed a large amplitude lateral translation and a decreased tail-beat frequency that was very different compared with when it was swimming in uniform flow.

For this study, we first selected video sequences based on the criterion that fish were holding station in the vortex street without drifting upstream or downstream. Video frames were captured for three to four tail-beat cycles for each fish and trial with 10 trials conducted for each flow speed. To identify videos in which fish were Kármán gaiting, we established the following five criteria: (1) the fish was holding station and did not drift upstream or downstream, (2) there was a traveling wave along the body, (3) the body displayed a large lateral displacement (>½L), (4) the body posture adopted a long wavelength (>1L), and (5) there were no transient small-amplitude, high-frequency tail beats. For the videos that met these criteria, we performed two analyses. First, we estimated a probability function of Kármán gaiting by dividing the number of Kármán gait videos by the total number of videos. We use the term ‘probability’ to reflect the percentage of time that the fish was Kármán gaiting. Second, we analyzed how detailed Kármán gaiting kinematics changed across flow speed and body size.

A customized digitizing program (Matlab vR2009b) was used to mark points on each side of the body silhouette in the video images (Liao et al., 2003a). After a series of cubic spline functions were fitted to these points, a midline spline was constructed and then divided into 30 segments. The longitudinal positions of each segment were then recorded as relative body lengths.

We measured the following variables: tail-beat frequency, body wave speed, body wavelength, lateral amplitude of the tail tip and body center (BC), lateral acceleration of the BC and maximum head angle. Tail-beat frequency was determined by averaging the number of tail-beat oscillations over a known time. Body wave speed was calculated by tracking the body crests moving from head to tail during Kármán gaiting. Body wavelength was derived from the body wave speed and tail-beat frequency using the relationship given in Eqn 2. Lateral amplitudes were measured as the maximum lateral excursion from the midline. BC refers to the center of mass of a dead fish, which was experimentally determined post-mortem for straight-stretched fish by iteratively moving bilateral pins down the body until we found the anteroposterior pin position for which the body balanced. We measured the distance of this point to the tip of the head. This distance was then used to determine a fixed point along the midline of each fish frame of the video. BC does not strictly refer to the actual center of mass, which can lie outside the body at times in freely swimming fish. Note that BC has been used to refer to the center of mass in previous work (Liao et al., 2003a; Liao, 2004). BC acceleration was calculated from the second derivative of BC lateral amplitude. To minimize error propagation, we first approximated the BC motion with a sine function as it oscillates with the vortex shedding frequency. We calculated the second derivative of the function and divided the amplitude by the square of the sampling interval. Head angle was calculated as the standard deviation of the head angle relative to the axis of the freestream flow.

Kármán gaiting kinematic variables were compared with wake variables (i.e. vortex shedding frequency and wake wavelength) at P<0.05. The values for kinematic variables were also compared with the values found for freestream swimming fish (Webb et al., 1984).

To analyze size effects on Kármán gait kinematics, one-way ANOVA were run on the individual kinematic variables (Bonferroni-corrected at P<0.05). To determine whether size groups were distinct from one another, we performed a principal component analysis (PCA) on all variables to reduce the dimensions of the parameter space. We then ran a MANOVA on principal component one (PC1) and principal component two (PC2) to see whether body size would explain the variance (P<0.05). The MANOVA took the simultaneous effects of all the kinematic variables into account, and thereby protected against a Type 1 error. All values are shown as the means ± s.e.m.

Not all fish that held station in the downstream wake of a cylinder were Kármán gaiting (i.e. met all five criteria). For fish positioned in the vortex street, the highest probability of Kármán gaiting (above 50%) occurred at intermediate flow speeds between ~30 and 70 cm s⁻¹ (Fig. 2).

All fish positioned their bodies in a specific location relative to the cylinder. The BC of the fish was located about 4–6 cylinder diameters downstream from the cylinder for all velocity trials (Fig. 3A). The velocity deficit in this region was about 40% of the nominal velocity (Fig. 3B). The location of this 40% deficit region did not change relative to the cylinder across speeds. At the lowest speed, we observed almost no flow in a region 3–5 cylinder diameters behind the cylinder.

We next looked at the kinematic variables of Kármán gaiting fish and compared them with relevant hydrodynamic variables. The vortex shedding frequency of a cylinder was dependent on flow speed. Our measurements showed that it increased linearly with speed according to y=0.05x (Fig. 4A, r²=0.98, P<0.05), similar to what is predicted by Eqn 1. Tail-beat frequency closely matched the vortex shedding frequency and increased with flow velocity according to y=0.05x (Fig. 4B, Table 1, r²=0.93, P<0.05). There was no statistical difference between the slopes of the vortex shedding frequency and tail-beat frequency. For fish Kármán gaiting at a flow speed of ~50 cm s⁻¹, tail-beat frequency did not differ across the fish lengths used in this study (Fig. 4C).

Body wave speed increased with flow velocity as y=1.26x (Fig. 5A, r²=0.85, P<0.05). The slope of the best fit line was greater for small fish than for medium or large fish (Fig. 5B, P<0.05). There was no difference between medium and large sized fish. This suggests that small fish control their body wave speed differently from large fish.

Lateral amplitude of the tail tip increased with flow velocity according to y=8.59x/(4.63+x) (Fig. 7A, r²=0.14). At low speed, it was similar to the expected 5 cm lateral spacing of the vortices (as dictated by the cylinder diameter), but as flow speed increased it rose above this value and plateaued at ~7.5 cm. The lateral BC amplitude stayed relatively constant across speeds according to y=1.59x/(2.91+x) (r²=0.07) and never exceeded the cylinder diameter value. Medium and large fish had a greater tail tip amplitude compared with small fish (Fig. 7B, P<0.05). There was no difference between medium and large sized fish. There was no effect of body size on BC amplitude.

The lateral acceleration of the BC increased quadratically with flow velocity according to y=0.14x²+0.11x (Fig. 8A, r²=0.69). Body size had no effect on BC acceleration (Fig. 8B).

Variation in head angle decreased with flow velocity according to y=−0.03x+9.47 (Fig. 9A, r²=0.13, P<0.05). There is no difference between the average head angle and the freestream flow axis. Large fish had a larger variation in head angle than small fish (Fig. 9B, P<0.05).

A principal components analysis showed that PC1 and PC2 accounted for 85% and 11% of the total variance, respectively (Fig. 10). The variable that loaded the highest on PC1 was body wavelength, while the variables that loaded the highest on PC2 were tail tip amplitude and head angle. A MANOVA run on PC1 and PC2 revealed a difference in group means (P<0.05). Small fish were different from medium and large fish along PC1 (Table 2, F=23.2256, P<0.05), while along PC2, there was no difference between groups (F=3.4710, P>0.05).

Distortion of uniform flow by a cylinder creates local regions that fish can exploit to hold station. Trout will bow wake in the front of the cylinder, entrain to the side or Kármán gait in the vortex street downstream (Liao, 2007). Fish can also choose to avoid the cylinder and swim in uniform flow. Of these behaviors, Kármán gaiting is a particularly interesting way for fish to hold station because the ability to synchronize with drifting vortices translates into substantial energetic savings (Taguchi and Liao, 2011).

We found that the highest probability of Kármán gaiting occurred at intermediate speeds (Fig. 2). At low flow velocities, fish did not Kármán gait often and their motions resembled freestream swimming. This is because vortical flows must be sufficiently developed before fish can exploit them (Liao et al., 2003a; Liao et al., 2003b; Liao, 2004; Taguchi and Liao, 2011). Kinematic results support this interpretation; at a flow speed of 13 cm s⁻¹ tail-beat frequency (2.5±0.2 Hz) was considerably higher than the vortex shedding frequency (0.7±0.2 Hz). Furthermore, tail-beat frequency at the low flow speed was identical to that of a freestream swimming fish (2.5±0.1 Hz) (Webb et al., 1984). Additionally, body wavelength (1.1±0.2 cm) and tail-beat amplitude (0.12±0.03 cm) were very similar to those found in freestream swimming fish (0.9±0.4 cm and 0.18±0.05 cm, respectively) (Webb et al., 1984). At the highest speeds, trout did not hold station continuously and were either drawn upstream into the suction zone behind the cylinder or ejected laterally from the vortex street. Under these high flow conditions the wake can adopt complex, three-dimensional vortex dynamics such as braid vortices and other hydrodynamic instabilities (Wei and Smith, 1986).

The position of the body in the vortex street is a key component of Kármán gaiting. We discovered that fish position is not correlated to an absolute flow speed, but instead to a region where the velocity deficit is about 40% of the nominal flow. The location of this region did not change relative to the location of the cylinder across flow speeds. Unlike freestream swimming, Kármán gaiting is less about generating thrust and more about producing drag at the appropriate times to hold station (Liao et al., 2003a). Fish position themselves in a saddle point region. This region corresponds to a local minimum in the cross-stream direction, but seems to be unstable in the direction of the flow. Thus, if a fish approaches the cylinder too closely, it will be drawn into the suction zone. If a fish drifts too far downstream from where the vortex street is strongest, it will likely have to use active undulation. We found that this region starts about 20 cm downstream from the cylinder and encompasses approximately a 10 cm diameter area for this study.

In both freestream swimming and Kármán gaiting, fish responded to increasing flow speed by increasing body wave speed. To do so, fish increased tail-beat frequency while keeping body wavelength and amplitude constant. Why, in both behaviors, does tail-beat frequency change and not body wavelength? One hypothesis is that limitations in body flexibility or muscle physiology may prohibit fish from increasing relative body wavelength during a given behavior. Alternatively, under certain conditions, keeping body wavelength constant and increasing tail-beat frequency may be beneficial. For freestream swimming this seems like an appropriate strategy simply because of drag costs; passing small amplitude body waves faster is more efficient than passing large amplitude waves more slowly. However unlike in freestream swimming, a fish with a longer body wavelength in a vortex street may not necessarily experience more drag than a fish with a shorter body wavelength. We believe that Kármán gaiting is a flow-dominated behavior and that wake wavelength and vortex shedding frequency drive the body wavelength and tail-beat frequency. As the vortex shedding frequency, but not wavelength, increases with flow speed, it makes sense that body wavelength is preserved across flow speed. Most interestingly, these two behaviors occur in very different hydrodynamic environments yet still share the same control strategy; tail-beat frequency increases to compensate for increasing flow speed.

Experiments have shown that even a rigid foil positioned in a vortex street can generate thrust (Beal et al., 2006). What, then, is the role of the traveling wave in Kármán gait? In uniform flow, the ratio of the traveling wave to the forward body speed (i.e. slip) approaches unity when swimming is efficient because more momentum is directed towards forward thrust. This concept is less useful in a vortex street because we do not yet know how to evaluate efficiency for Kármán gaiting. In particular, what are the contributions of passive lift generation versus powered undulation in holding station? The traveling wave could simply be a physical consequence of a flexible body being buffeted by vortices. This seems unlikely given that across flow velocities body wave speed is consistently 25% greater in value than the speed of the drifting vortices (i.e. nominal flow speed). We anticipate additional advantages generated from the control of body posture and stiffness. This would require a simultaneous analysis of kinematics and visualization of vortex location with greater temporal resolution in future studies.

Our results suggest that the lateral acceleration of the body increased non-linearly because it was driven by steeper pressure gradients caused by vortex interactions as flow increased. We believe that the BC acceleration rises so sharply with speed because the pressure gradient across the flow follows a quadratic function, which in turn is due to the fact that the constant velocity deficit percentage across speed sets up a continuously increasing velocity gradient. Direct pressure measurements across the vortex street would confirm this hypothesis. This relationship is inherently unstable because acceleration cannot increase indefinitely with flow speed. We observed this at the highest flow speeds, where the probability of Kármán gaiting decreased substantially.

Large head angles increase drag in freestream swimming. For Kármán gaiting, we can envision two mechanisms for head control strategy, but ultimately more experiments are needed to understand which one is at play. The first mechanism assumes head angle is not under active control. In a high flow, high vortex shedding frequency situation, frequent interactions with vortices allow less time for the head to yaw at large angles. The second mechanism assumes that fish actively control their head angle. At lower flow speeds, fish have more time to respond to vortices, perhaps to increase stability. If this results in a higher head angle, the potentially greater drag may enable fish to stay in a favorable position in the vortex street. If at the highest flow speeds the drag caused by head angle were reduced, it may explain why we observed fish being drawn into the suction region more often.

Kármán gaiting is different at lower flow speeds compared with higher speeds. At lower speeds, both body wavelength and tail tip amplitude are smaller than wake wavelength and cylinder diameter, respectively. When body wavelength is not greater than wake wavelength, this coincides with the fish having a lower probability of Kármán gaiting. This suggests that there is some specific relationship between body posture and vortices that facilitates station-holding, which is consistent with original observations (Liao et al., 2003a).

In this study, we found a linear increase in the vortex shedding frequency and assumed that the wake wavelength and lateral vortex spacing were constant across speed. However, these variables could change substantially if measured in the presence of a fish because of interactions between the fluid and the body.

Fish relate to the vortex street in different ways depending on their body length. In the original description of the Kármán gait, two cylinders with diameters of 2.5 and 5.0 cm and one fish size (10 cm) were used to produce cylinder to body length ratios of 1:2 and 1:4 (Liao et al., 2003a). Here, we reveal that larger fish can also Kármán gait with cylinder to body length ratios of 1:2.8 and 1:3.7. Although these ratios are similar, they reflect two very different hydrodynamic environments. In the original study, the main objective was to see whether fish can control tail-beat frequency and body wavelength independently. This was done by using two different flow speeds and cylinder diameters in order to hold vortex shedding frequency constant while we changed wake wavelength, and vice versa. Compared with fish Kármán gaiting in the vortex street generated by a 2.5 cm diameter cylinder at 2.5 L s⁻¹ (vortex shedding frequency=2.24±0.01 Hz, cylinder wake wavelength=1.12±0.01 L), fish in the vortex street generated by a 5.0 cm diameter cylinder at 4.5 L s⁻¹ (vortex shedding frequency=2.22±0.01 Hz, cylinder wake wavelength=2.03±0.01 L) increased their body wavelength from 1.57±0.04 L to 4.05±0.22 L while maintaining a similar tail-beat frequency (~2.2 Hz). In addition, fish matched the two vortex shedding frequencies created by the 2.5 cm diameter cylinder at 2.5 and 4.5 L s⁻¹ (vortex shedding frequency=2.24±0.01 and 4.02±0.02 Hz, respectively) while maintaining a constant body wavelength.

In contrast, one of the main objectives of this study was to investigate the effect of body size on Kármán gait kinematic variables. To do this, we kept vortex shedding frequency and wake wavelength constant by selecting only one cylinder size.

When small fish Kármán gait, they have a longer body wavelength than large fish. What is interesting is that in rainbow trout that are freestream swimming, body wavelength increases with size (Webb et al., 1984), but during Kármán gaiting the opposite is true. Using Webb's freestream equation (λ=1.43L^0.83) the body wavelengths for small, medium and large size fish are 10.0, 13.0 and 16.2 cm, respectively. These values are still lower than those of Kármán gaiting, with the biggest difference seen for small fish. In freestream swimming, the increased flexural stiffness of larger fish due to the larger second moment of area would create a longer body wavelength. We believe that during the Kármán gait less of the body participates in the traveling wave. Therefore, a longer body length did not correspond to a longer body wavelength as it seems to do for freestream swimming.

In order to Kármán gait, small fish interacted with cylinder vortices differently than larger fish. The present study, along with the original study (Liao et al., 2003a), indicates that as the cylinder diameter to fish length ratio changes from 1:2 to 1:4 and the size of the vortices gets smaller relative to body length, the body wavelength decreases and approaches the cylinder wake wavelength. A longer body wavelength seems to be the result of increasing the relative size and spacing of the vortices. Hypothetically, a larger fish may be able to adopt a shorter wavelength because the body spans to interact with two successive vortices, while a smaller fish must adopt a longer, less sinuous wavelength because its body can only interact with one vortex. Put another way, a small vortex size relative to the body causes the fish to conform to the wake wavelength because local regions of the body are bent by the flow, whereas a larger vortex may barely bend the body.

Regardless of body size, Kármán gaiting fish had a body wavelength that was longer than the wake wavelength at intermediate flow speeds. This relationship seems crucial to hold station and likely orients the body to create more thrust-generating interactions with passing vortices.

The tail-beat amplitudes of freestream swimming fish increase with fish length according to α=0.36L^0.74, where small, medium and large fish have values of 2.1, 2.6 and 3.1 cm, respectively (Webb et al., 1984). Tail-beat amplitude of Kármán gaiting fish also increases with fish size but these values are almost three times larger than the values of freestream swimming.

Fish size had no effect on tail-beat frequency, BC amplitude or acceleration. As the tail moved according to the encounter rate with shed vortices, the length of a fish should not prohibit it from synchronizing to the vortex shedding frequency. No change in BC amplitude suggests that the head and tail do not interact with vortices to amplify or constrict the lateral excursion of the body. The BC acceleration also does not change with fish size, presumably because larger forces resulting from greater surface area in larger fish are counteracted by a larger mass. We do see that larger fish show greater variation in their BC lateral acceleration, suggesting that they may be modifying their lateral motions by active control of their body and fins.

The hydrodynamics around a cylinder in flow are very rich in flow phenomena and provide a controlled environment to investigate the interactions between fish and unsteady flows. This study reveals new speed and size-related kinematic patterns for Kármán gaiting fish and substantially expands our understanding of how fish relate to vortices. As the ability to hold station depends specifically on how the body interacts with individual vortices in real time, comparisons of average values for kinematic variables and cylinder parameters cannot provide a complete understanding of Kármán gaiting. More detailed insights into the mechanisms of Kármán gaiting await simultaneous visualization of the flow and body kinematics on a cycle-by-cycle basis.

We would like to thank Masashige Taguchi and Christina Walker for assistance in data collection, preliminary analyses and fish care. We would also like to thank Melanie Haehnel, Rafael Levi and Matt McHenry for reading an earlier version of this manuscript, and George V. Lauder for providing equipment to collect some of the data. Two anonymous referees provided constructive criticisms to improve the manuscript.