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The ability to accurately estimate swimming performance is crucial to predict whether river infrastructure is likely to negatively impact fish movement (see e.g.25). This is particularly relevant for the conservation of endangered species, including the European eel that strongly depends on longitudinal movements between the ocean and rearing habitat within rivers and streams. As eel are catadromous, it is the juvenile life-stage that embarks on the upstream migration, and, due to their small size, the rate and extent of their movement is restricted by their swimming performance and ability to negotiate in-stream barriers such as weirs and barrages15.
Current approaches used to estimate the swimming performance of glass eel suffer from several major limitations, including the selection of representative flow velocities and accommodation of temperature effects on fish swimming time-to-fatigue. Specifically, the commonly-employed average cross-sectional velocity is not representative of the hydrodynamics experienced by glass eel, since this fish swims close to the channel bed, where the flow velocity is lower. Moreover, water temperature may have an important influence on swimming performance, which generally peaks at an optimum temperature, and ceases at some critical minimum and maximum threshold. To address these shortcomings, the present study integrated constant velocity swimming-performance tests with Computational Fluid Dynamics (CFD) to facilitate the reliable estimation of the velocities experienced by eel in the near-wall region. The developed CFD model was relatively simple to calibrate and allowed for the construction of the reported swimming curves. The k-ε closure method permitted a time efficient modelling process to be adopted that enabled systematic exploration of the velocity magnitude in the spatial domain with excellent resolution. To our knowledge, this is the first time CFD tools are used to provide velocity data that are integrated with the results of experiments on swimming performance of fish.
The dimensions of the channel used to test fish swimming performance can have an important influence on swimming performance curves. McCleave23 studied the swimming activity of juvenile European eel (mean fish length, 7.2 cm, ranging from 6.9 to 7.5 cm) in a darkened, rectangular swimming chamber. The average cross-sectional velocity U ranged from 0.25 to 0.5 ms-1, whereby the total swimming time-to-fatigue decreased from 146 to 16 s, respectively. The water temperatures in that study (ranged from 11.1–13.3 °C) were quite similar to our 12 °C treatment, but the overall performance was greater in our study, with total swimming time (tf) at U = 0.25 and 0.5 ms-1 being approximately 220 and 40 s, respectively (based on regression analysis), despite the almost-identical mean fish length. The key difference between the two studies was the length of swimming area, which was 1.16 m in our study ( > 15 times the mean length of the fish), and 0.5 m in McCleave23. Likewise, cross-sectional area also appears to be influential. Based on tests using a swimming length of 1.8 m, Clough and Turnpenny33 measured the swimming performance of juvenile eel through a narrow, circular Perspex pipe (0.04 m diameter). Using the developed swimming curve in Clough and Turnpenny33, the burst swimming velocity (total swimming time equal to 20 s) for a 7.0 cm long juvenile eel at a water temperature of 11.1 °C was 0.41 ms-1. Based on the results of our study conducted at 12 °C, we predict glass eel to be able to swim 70 s at an average flow velocity U = 0.41 ms-1. These comparisons may indicate the importance of both the length and the total volume of the working section when testing swimming performance in hydraulic facilities.
In our study, burst-and-coast swimming was increasingly observed for near-bottom velocities (Ub) exceeding 0.2 ms-1(i.e., average cross-sectional velocity  > 0.3 ms-1). This suggests that the burst-and-coast swimming mode is beneficial under higher velocities because intermittent swimming bestows energetic benefits. Indeed, it is well known that gait transitions, including burst-and-coast swimming, enables recovery and thus enhanced swimming performance19,37. Failure to provide sufficient test space can prevent the subject fish from displaying behaviors that can enhance performance, resulting in conservative estimates of swimming capability19.
The average cross-sectional velocity is clearly not representative of swimming conditions of bottom-dwelling fish, like Anguillidae, that gain energetic advantages by exploiting the low velocities characterizing near-wall flow regions. The present study demonstrates that, contextually to the flow conditions explored herein, differences in swimming-curve intercepts ranged between 18 and 32%. These differences are essentially equivalent to the average observed differences between U and Ub, which can be explained with the following scaling arguments. In open channel flows, near-bottom velocities Ub scale with the friction velocity, ({rm{u}}_{*})(a fundamental scaling velocity equal to the square root of the shear stress, τ0, divided by the water density, i.e. ({rm{u}}_{*}=sqrt{{uptau }_{0}/uprho })), which is related to the average velocity U via the Darcy-Weisbach friction factor (f) as ({rm{u}}_{*}=rm{U}sqrt{frac{rm{f}}{8}}). The near bed velocities ({rm{U}}_{rm{b}}) were averaged over a 3 mm thick volume of water, which embraces the so-called viscous sub-layer, the buffer sub-layer and, for some experimental conditions it may capture the logarithmic layer38. This is easy to demonstrate by scaling the height of the averaging volume (i.e. 3 mm) by means of the viscous length scale (upupsilon/ {rm{u}}_{*}). This non-dimensional height reaches, at most, the value of 73 which is indicative of a flow region within the logarithmic layer. Therefore, within the averaging volume it is fair to state that ({rm{U}}_{rm{b}}approx Oleft(10{rm{u}}_{*}right)) and hence37,

$${rm{U}}_{rm{b}}propto {f}^{1/2}rm{U}.$$
(2)

Note that the friction factor (f) may depend on: (i) the Reynolds number (i.e., (rm{Re}=rm{U}4rm{R}/upnu), where U is the average cross-sectional velocity, R is the hydraulic radius defined as the ratio between the wet area and the wet perimeter of the channel cross-section and ν is the water kinematic viscosity); (ii) the relative roughness of the flow; or (iii) both, if the flow is in the hydraulically-smooth, hydraulically-rough and -transition regime, respectively. In the present paper, experiments were carried out in the hydraulically-smooth regime, therefore we can assume (f=0.316{rm{Re}}^{-1/4})39 and hence, from Eq. (2) we obtain

$${rm{U}}_{rm{b}}propto {(rm{Re}}^{-1/8})rm{U},$$
(3)

where the proportionality coefficient is of order 1. From Fig. 2b, it is possible to infer that the average velocity U can be expressed as

$$rm{U}={rm{alpha }}_{1}{rm{t}}_{rm{ac}}^{{beta }_{1}},$$
(4)

where ({rm{alpha }}_{1}=1.099) and ({beta }_{1}= -0.266). Coupling Eqs. (3) and (4) leads to

$${rm{U}}_{rm{b}} propto {left(frac{rm{R U}}{upnu }right)}^{- frac{1}{8}}{{alpha }}_{1}{rm{t}}_{rm{ac}}^{{upbeta }_{1}}= {left(frac{rm{R}}{upnu }right)}^{- frac{1}{8}}{rm{U}}^{-frac{1}{8}}{{alpha }}_{1 }{rm{t}}_{rm{ac}}^{{upbeta }_{1}}={left(frac{rm{R}}{upnu }right)}^{- frac{1}{8}}{{{alpha }}_{1 }}^{left( 1- frac{1}{8}right)}{rm{ t}}_{rm{ac}}^{left(1 – frac{1}{8}right){upbeta }_{1}}.$$
(5)

As observed in Fig. 2b, Eq. (5) demonstrates that referring to the near-bottom velocity Ub rather than the average cross-sectional velocity U, leads to a reduction of the intercept coefficient of a factor scaling as ({left(frac{rm{R}}{upnu }right)}^{- frac{1}{8}}{{rm{alpha }}_{1 }}^{left(- frac{1}{8}right)}), corresponding to a 24–26% reduction, which is very similar to that observed experimentally (i.e., 18–32%, see “Results” section). In addition, the exponent ({upbeta }_{1}) undergoes a reduction of about 1/8, i.e. 12.5%, which compares very well with the 14% variation of the power-law exponents associated with the swimming curves plotted in Fig. 2b and reported in the results section.
The active swimming time of glass eel (tac) decreases with increasing near-bottom velocity and, for a given bottom velocity, tac increases with increasing water temperature (in the range 8–18 °C). This implies that, at higher temperatures, eel can sustain prescribed flow-velocities for longer times. Interestingly, for any temperature, Ub scales with tac as ({rm{U}}_{rm{b}}sim {rm{t}}_{rm{ac}}^{upbeta }) with (upbeta cong -) 1/3, on average. This may have some interesting implications which are now discussed.
It can be speculated that the flow resistance experienced by glass eel can be quantified as a drag force ({rm{F}}_{rm{D}}) that scales as (rho {rm{C}}_{rm{D}}a{rm{U}}_{rm{b}}^{2}), where (rho) is the water density, ({rm{C}}_{rm{D}}) is the fish drag coefficient and (a) is the fish frontal area. The power used by glass eel (i.e. the energy spent per unit time) to hold their position against a current, can be expressed as (rm{P}= rho {rm{C}}_{rm{D}}it{a}{rm{U}}_{rm{b}}^{3}) and the total energy spent by the fish is therefore (rm{E}={rm{Pt}}_{rm{ac}}=rho {rm{C}}_{rm{D}}it{a}{rm{U}}_{rm{b}}^{3}{rm{t}}_{rm{ac}}). In the experiments reported herein, eel had an almost uniform body-size (i.e. a similar frontal area a) and, therefore, for a specific water temperature, (rho) and (a) can be considered approximately as constant. For a prescribed water-temperature, the drag coefficient ({rm{C}}_{rm{D}}) of the eel might retain some Reynolds number dependence (in general, the ({rm{C}}_{rm{D}}) of a slender body immersed in a moving fluid reduces with increasing Re due to the weakening of viscous forces with respect to pressure forces in the total drag force experienced by the body), which translates, essentially, into a dependence on ({rm{U}}_{rm{b}}). However, the range of Reynolds numbers experienced by glass eel in the experiments presented herein is too small to induce significant variations in ({rm{C}}_{rm{D}}), which can therefore be considered, in good approximation, as constant. Therefore, it can be also reasonably assumed that the energy spent by the eel during a fatigue experiment, scales as (rm{E}sim {rm{U}}_{rm{b}}^{3}{rm{t}}_{rm{ac}}). However, since the near-bottom velocity scales approximately as ({rm{U}}_{rm{b}}sim {rm{t}}_{rm{ac}}^{-1/3}) (Fig. 3a), it follows that the energy spent by the eel is equal to a constant which is a function of temperature only. This suggests that, for a specific water temperature, the energy spent by a fish in a fatigue test is constant and independent on flow intensity levels (i.e. ({rm{U}}_{rm{b}})) or, in other words, this means that the swimming performance of glass eel might be energy-limited. Clearly, this hypothesis needs to be further substantiated by more experimental work allowing for direct measurement of oxygen (and hence energy) consumption during fatigue tests, possibly carried out using the framework of analysis presented herein. It could be also important to find an experimental, non-invasive technique able to track transparent glass eel while moving in flumes or swimming chambers (e.g.40,41). This will allow a better understanding of the link between swimming speed variability at constant flow and energy consumption16.
In the analyzed range of water temperatures (which represents common temperatures experienced by glass eel during upstream migration32) and flow velocities (Fig. 3b), Eq. (1) can be used to design a fish-pass or to evaluate its effectiveness for glass eel migration in a prescribed river reach. For instance, Vowles et al.42 proposed eel tiles as a cost-effective solution for mitigating the impacts of anthropogenic barriers to juvenile eel migration. Equation (1) can be used to verify whether velocities in the fish-pass are in an acceptable range, depending on the flow stage and the length of the eel tiles. Equation (1) provides also an estimate of the time needed by glass eel to circumvent dams and weirs or possible delays during migration. However, care would be needed in extrapolating the proposed formula to different water temperatures and larger velocities, compared to those analyzed in the present study. In the domain of application of Eq. (1), it can be speculated that ({rm{U}}_{rm{b}}) scales with temperature as (sim {rm{T}}^{updelta }), where δ is, on average, 0.77. Various effects are lumped into this exponent. Overall, for one velocity, the drag force ({rm{F}}_{rm{D}}) may increase by decreasing temperature because the density and the dynamic viscosity of water increase and this leads to increased values of pressure and viscous forces, respectively. This means that, for a given near-bottom velocity ({rm{U}}_{rm{b}}), water viscosity may cause the fish to get tired sooner (i.e. lower ({rm{t}}_{rm{ef}})) at low temperatures than at high temperatures. Furthermore, the temperature exponent is probably dictated by fish-metabolism. In terms of temperature range and related effects on glass eel swimming performance, similar results were found by other authors in the literature. For instance, Harrison et al.32 reports that low temperatures (below 10 °C) are known to reduce glass eel activity and that, in general, there is a positive correlation between temperature and upstream-migration speed. Furthermore, it was demonstrated that European eel muscle contractility and efficiency decrease rapidly with water temperature below 10 °C43,44. Therefore, low temperatures in rivers may affect eel ecology through both hydrodynamics and physiology, by exerting a direct limiting effect on the movement of the individual. The mechanism controlling the entire breadth of temperatures over which glass eel can have the highest or the lowest swimming performance is still not clear and further research is needed to extend the selected temperature range to achieve comprehensive results for this species. Since European eel is widely distributed across different European climates, we highlight that further investigation can be directed to better determine whether and how water temperature may affect eel swimming performance in different climatic environments and in the context of climate change. More

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