Fish Swimming in a Kármán Vortex Street: Kinematics, Sensory Biology and Energetics

Abstract

Fishes often live in environments characterized by complex flows. To study the mechanisms of how fishes interact with unsteady flows, the periodic shedding of vortices behind cylinders has been employed to great effect. In particular, fishes that hold station in a vortex street (i.e., Kármán gaiting) show swimming kinematics that are distinct from their patterns of motion during freestream swimming in uniform flows, although both behaviors can be modeled as an undulatory body wave. Kármán gait kinematics are largely preserved across flow velocities. Larger fish have a shorter body wavelength and slower body wave speed than smaller fish, in contrast to freestream swimming where body wavelength and wave speed increases with size. The opportunity for Kármán gaiting only occurs under specific conditions of flow velocity and depends on the length of the fish; this is reflected in the highest probability of Kármán gaiting at intermediate flow velocities. Fish typically Kármán gait in a region of the cylinder wake where the velocity deficit is about 40% of the nominal flow. The lateral line plays a role in tuning the kinematics of the Kármán gait, since blocking it leads to aberrant kinematics. Vision allows fish to maintain a consistent position relative to the cylinder. In the dark, fish do not show the same preference to hold station behind a cylinder though Kármán gait kinematics are the same. When oxygen consumption level is measured, it reveals that Kármán gaiting represents about half of the cost of swimming in the freestream.

Introduction

Understanding how fishes swim in unsteady flows has attracted attention from many disciplines, ranging from biologists interested in fish ecology to engineers working on the principles of efficient propulsion (Streitlien & Triantafyllou, 1996; Heggenes, 2002; McMahon & Hartman, 1989; Triantafyllou et al., 2004; Pavlov et al., 2000). Because the hydrodynamics of a cylinder wake is well characterized (Williamson, 1996), examining how fish interact with cylinder vortices has provided a tractable way to begin to understand fish-fluid interactions under complex, yet predictable flow conditions (Webb, 1998; Liao et al., 2003a; Sutterlin & Waddy, 1975). This is because the relationships between flow velocity, cylinder diameter, and vortex shedding frequency are already known. Flow moving past the cylinder creates vortices that shed alternately from each side of the cylinder, with the vortices staggered as two columnar arrays (Williamson, 1996; Zdravkovich, 1997). For Reynolds numbers of 40–100,000 in which many fishes swim, the Strouhal number (St) for cylinders is 0.2, where f is the vortex shedding frequency, d is the diameter of the cylinder, and U is the nominal flow velocity.

$$St=\frac{fd}{U}$$ (1)

The spacing of the vortices or wake wavelength (λ) can be calculated from the flow velocity divided by the vortex shedding frequency.

$$\lambda =U/f$$ (2)

This enables experimenters the ability to control the frequency and spacing of vortices by altering the flow speed and cylinder size and has provided a unique opportunity to study how fish behave in an unsteady, periodic environment.

Previous studies revealed that fishes adopt novel body kinematics behind a cylinder, termed the Kármán gait. Kármán gaiting can save energy for station holding fishes and occurs under certain conditions of flow velocity, body length, and cylinder size (Liao et al., 2003a, 2003b; Liao, 2004; Taguchi & Liao, 2011). To identify Kármán gaiting, five criteria are used: (1) the fish is holding station and not drifting upstream or downstream, (2) there is a traveling wave along the body, (3) the body displays a large lateral displacement (>½L), (4) the body posture adopts a long wavelength (>1L), and (5) there are no transient small-amplitude, high-frequency tail beats.

Body Wave Kinematics of Kármán Gaiting Versus Freestream Swimming

A traveling wave along the body can describe both freestream swimming and Kármán gaiting kinematics for subcarangiform swimming fishes. This equation, with an arbitrary initial phase (ϕ), takes the form:

$$h(x,t)=A(x)*\text{sin}\left(\frac{2\pi }{\lambda }x-2\pi ft+\varphi \right)$$ (3)

where t and x denote time and position along the body, respectively. The wave initiation point varies with the loco-motor mode (thunniform, carangiform, anguilliform), which is defined according to how much of the body participates in the undulatory wave (Breder, 1926). Tail beat frequency (f) and body wavelength (λ) define the temporal and spatial periodicity of the equation. The speed of the traveling wave (V) is defined by

$$V=\lambda f$$ (4)

The amplitude envelope (A(x)) also depends on the locomotor mode. For subcarangiform swimmers such as rainbow trout and mackerel, it is described by a second order polynomial A(x) = c₁x + c₂x² (Videler & Hess, 1984).

Whether the traveling wave is generated actively through muscular activity or passively due to flow-induced motions varies depending on the flow regime. In freestream swimming, an anteroposterior wave of red muscle activity drives the propagation of the wave (Jayne & Lauder, 1995). In contrast, during Kármán gaiting, undulatory waves are generated passively as a result of lateral acceleration while the fish is being buffeted from side to side by the fluid (Liao et al., 2003a). Several findings support this argument. First, muscle recordings indicate that Kármán gaiting fish activate only the anterior red axial muscles (Liao, 2004). Second, dead trout temporarily generate a mechanical wave similar to live fish (Beal et al., 2006). Third, there is a high correlation between the lateral acceleration and tail beat amplitude of the fish.

Freestream swimming and Kármán gaiting differ in that they are separated in the parameter space; the amplitude, wavelength, and frequency values of the traveling wave equation are substantially different for each behavior (Akanyeti & Liao, 2013b). During Kármán gaiting, the wave is initiated at the body center, which is 0.2L (where L = total body length) further down the body compared to the initiation point in freestream swimming. Fourier analysis on the motions of a dead trout towed behind a cylinder shows that, in a completely passive body, the wave starts at the base of the cranium similar to freestream swimming (Figure 1). This suggests that the location of the wave initiation point during Kármán gaiting is not due to the passive fish-fluid interactions. When live fish Kármán gait, they activate their anterior muscles in order to adopt a straight posture in the mid-body region; as a result, this arrangement changes the location of the initiation point. In this way, the interaction between fish and fluid in the mid-body region is more critical than the posterior region. Fish appear to keep the mid-body region from bending in order to provide a local axial control surface to harness the appropriate fluid forces. In addition to undulation, Kármán gaiting fish also exhibit substantial lateral translations and body rotations, which can constitute up to 75% of the behavior.

FIGURE 1.

Normalized frequency spectrum of a dead trout (A) towed in a vortex street compared to a live Kármán gaiting trout (A_i), where black = frequency with smallest lateral amplitude and white = largest amplitude. (B) Body amplitudes of dead trout were smaller than those of live trout (B_i). (C) The traveling wave along the body of dead trout was initiated more anteriorly than in live trout (C_i). (A_ii) Freestream swimming exhibited a different dominant frequency than for Kármán gaiting. (B_ii) During freestream swimming and Kármán gaiting, the amplitude of lateral oscillations was smallest at the mid-body region and increased gradually toward tail, with lower overall values for freestream swimming. (C_ii) Freestream swimming fish exhibit a traveling wave that is initiated about 0.2L anterior to the point of initiation in Kármán gaiting fish, with relatively constant wave speed along the posterior body. Solid line and gray shaded area represents the mean ± SEM.

Body Wave Speed Increases With Flow Speed

Kármán gaiting fish respond to increasing flow speed by increasing the speed of their traveling body wave. To do so, fish increase tail-beat frequency while keeping body wavelength and amplitude constant. Why does tail-beat frequency change and not body wavelength? Kármán gaiting is a flow-dominated behavior, where cylinder wake wavelength and vortex shedding frequency drives the body wavelength and tail-beat frequency. The cylinder vortex shedding frequency, but not wavelength, increases with flow speed, setting up a condition where the body wavelength is preserved across 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 toward forward thrust. This concept is less useful in a vortex street environment, where the contributions of passive versus powered thrust generation are harder to differentiate. The traveling wave is not entirely passive, given that across flow velocities body wave speed is consistently 25% greater than the speed of the vortices drifting at the nominal flow speed (Akanyeti & Liao, 2013a).

The Effect of Fish Length on Body Wavelength

Fish relate to the vortex street in different ways, depending on their body length. Kármán gaiting fish require cylinder-to-body length ratios that range from 1:2 to 1:4 (Liao et al., 2003b; Akanyeti & Liao, 2013a). The body wavelength of the fish affects its ability to properly interact with vortices to produce thrust. In order to Kármán gait, small fish interact differently with cylinder vortices than larger fish. When small fish Kármán gait, they have a longer body wavelength than larger fish, reflecting the relatively larger size and spacing of the cylinder vortices. Hypothetically, a larger fish may be able to adopt a shorter wavelength because the body spans to interact with two successive vortices, whereas a smaller fish must adopt a longer wavelength because its body can only interact with one vortex. This is the opposite of what is found in freestream swimming, where body wavelength increases with fish size (Webb et al., 1984). As mentioned above, during the Kármán gait less of the body participates in the traveling wave than observed for freestream swimming. Therefore, a longer body length does not correspond to a longer body wavelength, as it seems to do for freestream swimming (Figure 2).

FIGURE 2.

Body wavelength across speed and body size. (A) Body wavelength (solid line) starts lower than the cylinder wake wavelength (dashed line) and then rises above as flow speed increases (r² = 0.28, n = 9 fish). At the lowest swimming speeds, the absence of a strong vortex street likely requires use of a shorter body wave similar to freestream swimming fish. (B) At a flow speed of ~50 cm s⁻¹, smaller fish have a longer body wavelength than larger fish (p < 0.05, n = 15 fish). Values shown represent the mean ± SEM.

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

Probability of Kármán Gaiting Depends on Flow Speed

One of the most important factors determining how often fish prefer to Kármán gait is the nominal flow speed to which the cylinder is exposed. Figure 3 shows that the highest probability of Kármán gaiting occurs at intermediate flow speeds between two and five body lengths per second (Akanyeti & Liao, 2013a). The body center of the fish is typically located four to six cylinder diameters downstream from the cylinder regardless of flow velocity level (Figure 4A). This region corresponded to a velocity deficit of about 40% of the nominal velocity (Figure 4B). 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, 2003b; Liao, 2004; Taguchi & Liao, 2011). Kinematic results support this interpretation; at lower flow speeds, the tail-beat frequency was considerably higher than the vortex shedding frequency (Akanyeti & Liao, 2013a). The tail-beat frequency at the low flow speed was identical to that of a freestream swimming fish (Webb et al., 1984). Furthermore, body wavelength and tail-beat amplitude were very similar to those found in freestream swimming fish. At the highest speeds, trout do not hold station continuously and are either drawn upstream into the suction zone behind the cylinder or ejected laterally from the vortex street. Under these higher Reynolds number flow conditions, the wake can adopt complex, three-dimensional vortex dynamics such as braid vortices and other hydrodynamic instabilities (Wei & Smith, 1986). Merging and pairing between initial shear layer vortices can give rise to secondary structures that contribute to irregular vortex shedding frequencies and amplitudes (Blevins, 2001; Williamson, 1996). The resulting turbulent vortex street exceeds the stabilization abilities of fishes.

FIGURE 3.

The probability of Kármán gaiting changes with flow speed. Fish Kármán gait the most at intermediate flow speeds (~30–70 cm s⁻¹ for trout with total body length of 15.7 ± 0.8 cm) and the least at extreme speeds. Videos were binned into five flow speed categories, where each category consisted of a minimum of 50 videos from at least five different fish.

FIGURE 4.

(A) Location of the body center (black circles) of the fish relative to the D-cylinder in the flow tank. (B) Downstream position of the body center relative to the cylinder, superimposed on a heat map illustrating the magnitude of the velocity deficit behind the cylinder as a percentage of the freestream velocity (red represents the greatest relative flow reduction). Note that this plot does not distinguish the reversal in flow direction that is established in the suction region directly behind the cylinder.

Sensory Feedback During the Kármán Gait

What roles do vision and the lateral line play in the ability to exploit vortices in a cylinder wake? Theoretically, the hydrodynamic conditions of an oscillating wake make it possible for any foil-shaped object of the appropriate size to generate thrust passively (Bose & Lien, 1990; Wu & Chwang, 1975). Experimental evidence shows that a dead trout towed behind a cylinder can momentarily synchronize its body kinematics to the oscillating flow of a vortex street to generate thrust (Liao, 2004; Beal et al., 2006). However, for a fish to remain in the cylinder wake for sustained periods requires sensory feedback from the visual and lateral line systems.

In general, fishes rely heavily on both visual (Roeser & Baier, 2003; Ingle, 1971) and hydrodynamic (Dijkgraaf, 1963; Coombs et al., 2001; Engelmann et al., 2000) cues to adapt their swimming movements to their immediate environment. Kármán gait kinematics change when the lateral line is blocked, indicating that hydrodynamic feedback is used to alter motor output accordingly in turbulent flows. The greater variability in body wavelength for trout with a blocked versus intact lateral line underscores the importance of detecting local flow along the body in adjusting Kármán gait kinematics to maintain a favorable posture to facilitate vortex capture. In addition, trout with a blocked lateral line hold station further downstream from the cylinder than fish with an intact lateral line. These lines of evidence reiterate that a proportion of Kármán gait kinematics are under active control and are not the sole result of passive buffeting of the body by vortices. Longer body wavelength and faster wave speed suggest that Kármán gaiting is less efficient or more energetically costly without a functional lateral line. The fact that trout in the light with a blocked lateral line do not spend as much time Kármán gaiting as trout with a functional lateral line (Figure 5) provides behavioral evidence in support of this hypothesis. Whether altered Kármán gait kinematics reflect muscle activity and changes in energy expenditure for the individual is currently not known. When lateral line functionality is held constant (i.e., within fish with an intact or blocked lateral line), the presence or absence of light does not change Kármán gait kinematics. This provides further evidence that when trout hold station in a vortex street the lateral line, rather than vision, plays a larger role in body kinematics.

FIGURE 5.

Regions around a cylinder in flow that trout will either entrain or Kármán gait. In the light, fish prefer to Kármán gait in the vortex street downstream from the cylinder (black fill) for the majority of the time during a 60-min experiment, especially when the lateral line is intact (V+L+₁). Values for fish in the light with an intact lateral line exposed to the cylinder for 2 consecutive days (V+L+₂) are almost identical to those exposed for 1 day (V+L+₁), indicating that previous experience in the flow tank does not alter the preference to Kármán gait. In contrast to experiments performed in the light, fish in the dark do not spend much time in the vortex street regardless of lateral line functionality (V−L+₁ or V−L−₂), preferring to entrain (gray fill) just downstream and to the side of the cylinder. The time that fish spent exploring other regions of the flow tank (white) is similar across treatments.

One exception occurs where vision alone can alter Kármán gait kinematics. Fish with a blocked lateral line in the dark have a greater variability in body wavelength than fish in the light. These fish seem to have more difficulty exploiting vortices, often drifting position within the vortex street, displaying “corrective” motions, or switching to traditional undulatory swimming such as seen in uniform flow. Vision allows fish to maintain a consistent position relative to the cylinder. This may minimize the exposure to flow variation and thus variation in body wavelength, since the predictability and energy of the vortices decreases with downstream distance from the cylinder.

Energetics of Kármán Gaiting

The cost of Kármán gaiting can be measuring directly and noninvasively in live fishes by employing the technique of respirometry, which measures the oxygen consumed during a particular behavior. There are several regions around a cylinder that fish choose to hold station. Figure 6 illustrates that, for a given flow speed, oxygen consumption during Kármán gaiting is higher than entraining near the suction region, but lower than bow waking in the front of the cylinder (Taguchi & Liao, 2011). Kármán gaiting represents about half of the cost (47%) of swimming in the freestream away from the cylinder. When compared to free-stream swimming in flow equivalent to the reduced velocity behind the cylinder (which is about 40% of the nominal flow velocity), Kármán gaiting represents about 79% of the cost (Liao et al., 2003b). Therefore, by exploiting vortices Kármán gaiting fish used significantly less oxygen than predicted if there were only benefiting from swimming in the reduced velocity of the cylinder wake.

FIGURE 6.

Statistical comparison of the rate of oxygen consumption (MO₂) values between Kármán gaiting (KG) and bow waking (BW), entraining (EN) and freestreamswimming (FS) at 3.5L s⁻¹. Kármán gaiting requires significantly less oxygen than swimming in the freestream (47%) and bow waking (73%) but requires more oxygen than entraining (116%). Kármán gaiting fish use less oxygen (79%) compared to fish swimming in the freestream at 1.8L s⁻¹ (asterisk). Values are reported as the mean ± the standard error.

Future Directions

Our work on the midline kinematics of Kármán gaiting, sensory biology, and energetics can be used by roboticists to develop control algorithms that can move bioinspired robots and by computational fluid dynamics modelers to simulate fluid-structure interactions. Given that the majority of the body waves during Kármán gaiting are generated passively, it is more important for a flexible robot to control its head and the anterior body than control its posterior body. If hydrodynamic forces are harnessed appropriately at the anterior body, a traveling wave is generated passively at the posterior body starting from the body center. This represents a paradigm shift in the field of autonomous robotics locomotion which traditionally emphasize the control of the posterior body (Alvarado, 2007; Liu & Hue, 2006; Stefanini et al., 2012). What Kármán gaiting studies teach us is that head control is critical for steering and improving stability in unsteady flows by counterbalancing body rotations and lateral translation and that control functionality can be outsourced to the viscoelastic properties of the body itself.

The study of biological locomotion in unsteady flow regimes is a promising one that stands to shed light on new mechanisms of hydrodynamic propulsion. Though vortex streets generated by cylinders has proven to be a production experimental system, overall the responses of fishes to unsteady flows remains largely unexplored. Around a single cylinder, already three distinct energy-saving behaviors are observed. What if the wakes behind three-dimensional objects, arranged in aggregations or varying in flexibility, were investigated in more species? It is apparent that the diversity of maneuvering, wake exploitation and drag reduction behaviors would reflect the inexhaustible number of scenarios between over 33,000 species of fishes and unsteady flow conditions. Along this vein, experiments into more natural flow conditions stand to be well rewarded. For example, investigating the behavior of fish holding station behind two cylinders in tandem, only a slightly more involved experiment than a single cylinder but closer to mimicking flows from more natural object aggregations such as large woody debris in streams already reveals new principles of fluid-solid interactions and wake exploitation that could not be predicted (Stewart et al., 2016). Another promising topic is the investigation of how fishes navigate waves in the surf zone during foraging. With the application of Digital Particle Image Velocimetry, high-speed videography and physiological techniques such as respirometry and electro-myography, new insights into the mechanisms of fish locomotion are now more accessible to marine technology applications.