# A Method for Estimating the Velocity at Which Anaerobic Metabolism Begins in Swimming Fish

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{s}

^{1.27}(R

^{2}= 0.948, p < 0.001) and the power value (1.27) of U

_{s}indicated high swimming efficiency. (3) Increased swimming speed led to increases in the tail beat frequency. (4) Swimming costs were calculated via rate of oxygen consumption and hydrodynamic modeling. Then, the drag coefficient of the crucian carp during swimming was calibrated (0.126–0.140), and the velocity at which anaerobic metabolism was initiated was estimated (0.52 m/s), via the new method described herein. This study adds to our understanding of the metabolic patterns of fish at different swimming speeds.

## 1. Introduction

^{−1}m

^{−1}, respectively). Since energy expenditure metrics did not appear to be predictive of successful fishway passage, these researchers were led to conclude that other endogenous or exogenous factors were influencing passage success. In the drag force hydrodynamic model, it is generally assumed that the drag coefficient of a fish is identical to that of an equivalent rigid body [19,26], an assumption that may be invalid while fish are swimming. Moreover, the estimation of energy consumption via oxygen consumption does not include the energy provided by anaerobic metabolism during exercise, and many studies show that fish enter anaerobic metabolism before reaching critical swimming speed [27,28,29]. The energy contribution of anaerobic metabolism is difficult to estimate from the swimming speed of live fish.

## 2. Materials and Methods

#### 2.1. Respirometer

#### 2.2. Experimental Design

**Fish.**Crucian carp (n = 31, body mass = 260.10 ± 7.93 g, total length = 24.73 ± 0.25 cm, spawning period) were obtained from a fish farm in Wuhan, Hubei, China. The fish were immediately transferred to laboratory aquaria (diameter 2.0 m, height 0.8 m), where they acclimated for at least two days. They were fed daily and any remaining food was removed after 1 hr. Testing took place from 1 July to 10 August 2019 and was carried out in the Key Laboratory of Ecological Impacts of Hydraulic-Projects and Restoration of Aquatic Ecosystem of the Ministry of Water Resource, Wuhan, China, with room temperature controlled by air conditioning. The test fish were kept temporarily in the pool on site for 2 d and not fed for the 24 h period prior to being transferred, in water, to the respirometer [15]. The duration of each test was around 6 h. Dissolved oxygen (DO) within the respirometer was maintained above 6.0 mg/L [29], and the water temperature maintained at 20(±1) °C. When the DO fell below 6.0 mg/L, testing was interrupted and the water was aerated [17]. Sample numbers and corresponding water velocities are shown in Table 1.

**Cameras**. The PIV laser was placed horizontally emitting a light surface parallel to the bottom surface of the device, in order to ensure that the two horizontally-oriented CCD cameras could photograph the test plane illuminated by the laser; the details were described in Tarrade’s paper [31]. The high-speed video camera was set above the chamber to record the swimming behavior of the fish (video was recorded during the test and the tail beat frequency could be obtained through frame-by-frame analysis by ProAnaly software). It was ensured that the three cameras were capturing the same plane. The requisite supplementary light sources were also set up alongside the test device (Figure 2).

**Calibration.**The relationship between the frequency setting of the variable-speed motor, x, and water velocity, y, was calibrated using the LaVision Particle Image Velocimetry system (PIV). The equation y = 0.03788x (R

^{2}= 0.9978) fitted this data. (See Appendix A.) During testing, the rotation speed of the motor was adjusted using its frequency setting, in order to control the water velocity in the swim chamber.

**Stepped Velocity Testing**. Stepped velocity tests were conducted to estimate the critical swimming speed (U

_{crit}), oxygen consumption rate (AMR), cost of transport (COT), and tail beat frequency (TBF) of the crucian carp; these tests were repeated n times (Table 1). During each test, water velocity was increased by 0.15 m/s, starting at 0.15 m/s, at 25-min intervals until the test fish was fatigued. At the same time, the DO (mg/L) and temperature in the respirometer were recorded every 5 min for 25 min. A fish was considered as fatigued when it stopped at the end of the test area, lightly patted the downstream wall for 20 s and still could not swim [32]. After testing was complete, the body length (cm), total length (cm), and body mass (g) of the fish were measured.

#### 2.3. Morphometric and Statistical Analysis

_{crit}was calculated using the flow velocities and step intervals recorded during the test using Equation (1) [14]:

_{2}·h

^{−1}·kg

^{−1}); St is the value of the slope of dissolved oxygen with respect to time with the test fish swimming (apparent oxygen consumption mg O

_{2}·min

^{−1}·L

^{−1}); Sb is the absolute value of the slope of dissolved oxygen per unit time in the absence of a test fish (bacterial oxygen consumption mg O

_{2}·min

^{−1}·L

^{−1}); 60 is the time constant (s/min); Vol is the total volume (L) of the experimental water system (90 L in this study), and m is the body mass of the fish (kg).

^{−1}·m

^{−1}) is the energy expended in swimming, which was obtained using Equation (3) [25]:

_{s}is swimming speed (m·s

^{−1}) and 13.54 is the oxycaloric value (J·mg O

_{2}

^{−1}) [35].

_{d}is the drag coefficient; ρ is the density of water (1000 kg·m

^{−3}); A

_{s}is the wetted surface area of the fish, which can be calculated using Equation (5); and U

_{s}is the swimming speed. C

_{d}is composed of two factors, the frictional drag coefficient C

_{f}and pressure drag coefficient C

_{p}. C

_{d}can be obtained using Equations (6)–(8) [19,20]:

_{s}/ѵ [33] (ѵ is the kinematic viscosity of water); α and β are empirical constants, α = 0.465, β = 2.11 [20]; and L is the body length of the fish.

_{0}(mg O

_{2}·h

^{−1}·kg

^{−1}) is the oxygen consumed per hour per kilogram of body mass in still water.

_{s}. The energy efficiency of fish swimming under aerobic metabolism can be obtained using Equation (10) [36]:

_{d}(W) is the power of the drag, P

_{d}= FU

_{s}.

#### 2.4. Data Analysis

## 3. Results

#### 3.1. Swimming Kinematics

_{crit}was 0.85 ± 0.032 m/s (4.40 ± 0.16 body lengths, BL/s). Correlation between swimming speed U

_{s}and tail beat frequency TBF are shown in Figure 3. The equation TBF = 0.82 + 5.2 U

_{s}(R

^{2}= 0.977, p < 0.001) was fitted to this data. For the highest water velocities, in excess of U

_{crit}, the fit was poorer than for lower velocities.

#### 3.2. Relationship Between Oxygen Consumption and Water Velocity

_{s}

^{1.27}(R

^{2}= 0.948, p < 0.001). For the highest water velocities, in excess of U

_{crit}, the fit was poorer than for lower velocities. The rate of oxygen consumption in still water was 131.24 ± 13.44 mg O

_{2}·h

^{−1}·kg

^{−1}. As water velocity increased, the rate of oxygen consumption also increased, varying over velocity intervals. However, the rate of increase varied over velocity intervals. With water velocity at 1.2 m/s, the rate of oxygen consumption reached its maximum, at 621.92 ± 111.74 mg O

_{2}·h

^{−1}·kg

^{−1}.

_{s}, is shown in Figure 5. Based on the relationship between COT and AMR, the relationship between COT and U

_{s}was calculated using Equation (3), as COT = 0.49 U

_{s}

^{−1}+ 1.73 U

_{s}

^{0.27}(R

^{2}= 0.594, p < 0.001).

#### 3.3. Drag Calculation Based on Hydrodynamic and Energy Efficiency

^{2}= 0.998, p < 0.001).

## 4. Discussion

#### 4.1. Swimming Kinematics

_{crit}is an important parameter used in fishway design [42]. The U

_{crit}is related to the duration and speed increments used during testing [34]. In this study, with water temperature at 21 ± 1 °C, 25 min duration stages, and 0.15 m/s speed increments, the critical speed for crucian carp of body length 19.32 ± 0.24 cm was 0.85 ± 0.032 m/s (4.40 ± 0.16 BL/s). The critical swimming speed of crucian carp determined by Li et al. [43] was 1.19 ± 0.024 m/s (5.91 ± 0.09 BL/s, with body length of 20.0 ± 0.80 cm). The fish used in both this and Li’s study were crucian carp of similar body length, but their source, habitat/environment and living habits were quite different. The crucian carp used in this study came from a fishing ground in the Wuhan area; those used in Li′s study came from a fishing ground in the Harbin area, and had higher critical swimming speeds.

#### 4.2. The Relation Between Metabolic Rate and Speed

_{s}

^{c}, the velocity index, c, is typically in the range 1.1–3.3 [49]. In this study, the velocity index for the crucian carp was 1.27, reflecting their energy efficiency during aerobic exercise. (The higher the index, the lower the swimming energy efficiency, and vice versa.) In Beamish’s [50] research on the velocity indexes of several bony fish species, their mean value was 2.3. The smaller speed index determined in this study indicates a higher energy efficiency.

#### 4.3. Drag Coefficient and Drag Force

_{d}ρ A

_{s}U

_{s}. At low values of Re, A

_{s}is usually based on frontal area whereas at higher values of Re, total wetted area is commonly used [51]. In this test, with 3476 < R

_{e}< 59583, the decision was taken to use total wetted area, calculated using Equation (5), which was formulated to describe the wetted area of a salmonid [33]. The drag coefficient C

_{d}is a complicated function of the Re number and its value depends on the flow conditions surrounding a body [33]. Under turbulent conditions, C

_{d}gradually decreases as Re increases, approximately in proportion with Re

^{−0.2}.

_{s}, or the drag coefficient, C

_{d}, used in the calculation was too small, resulting in excessively low energy efficiency η. However, the incorrectness of A

_{s}was excluded by using the measured surface area of the fish.

_{d}was then possible. The calibration principle follows the boundary layer separation theory of hydraulics under high Reynolds number values, i.e., for 103 < Re < 2 × 105, C

_{d}can be regarded as a constant, the position of the separation point remains basically constant, and F∝ U

_{∞}

^{2}[54]. With the swimming attitude at different water velocities remaining basically unchanged, along with the angle of attack, the drag coefficient C

_{d}can be treated as a constant. For the range of Re numbers mentioned above, drag F will be proportional to U

_{s}

^{2}, and the power needed to overcome this drag will therefore be proportional to U

_{s}

^{3}. During exercise entailing purely aerobic respiration, it can be assumed that η = 90%−100% (P

_{s}= 0.9−1.0 P

_{d}), and fitting of the relationship between P

_{s}and U

_{s}should assume proportionality to the third power of the latter; that is, P

_{d}= k U

_{s}

^{3}= 90%P

_{s}and P

_{d}= k U

_{s}

^{3}= 100%P

_{s}(k is a constant). According to Equations (4) and (10), a formula for C

_{d}can then be derived via inversion: C

_{d}= 2k/ρA

_{s}.

_{crit}) during testing, fish were relying purely upon aerobic respiration, and the oxygen consumption rate could be used to estimate the total energy consumption of fish at these swimming speeds. Power for low-flow aerobic swimming was then fitted (Figure 7) resulting in the equations P

_{s}= 1.718 U

_{s}

^{3}(P

_{s}= 0.9 P

_{d}, R

^{2}= 0.953, p < 0.001), and P

_{s}= 1.910 U

_{s}

^{3}(P

_{s}= 1.0 P

_{d}, R

^{2}= 0.954, p < 0.001); that is, k = 1.718–1.910. From this, the drag coefficient, C

_{d}= 0.126–0.140 can be derived, with a value larger than the estimate obtained using Equation (8) (Table 2), but smaller than the drag coefficient (0.45) [54] for flow past a round sphere at the corresponding Reynolds number. Considering that the bodies of fish are streamlined, it is reasonable that their drag coefficients are smaller than that of a sphere.

#### 4.4. Estimating the Velocity at which Anaerobic Metabolism Begins

_{d}, of the crucian carp began to exceed their aerobic power, P

_{s}, at a water velocity of 0.52 m/s. Since drag power exceeding aerobic power likely indicates the initiation of anaerobic metabolism, the swimming speed corresponding to the point of divergence between P

_{d}and P

_{s}can be viewed as the velocity at which anaerobic metabolism is initiated, and the excess of P

_{d}over P

_{s}as an estimate of non-aerobic cost. As water velocity increased to 0.90 m/s, the power provided by aerobic metabolism reached a maximum from which it scarcely increased further. However, as water velocity increased, more power was needed, and the non-aerobic cost continued to increase. This non-aerobic cost is unfavorable to the upstream migration of fish, so, during the design of fishways, it is very important to obtain well-founded estimates of the water velocity at which anaerobic metabolism begins [53], despite the difficulty of this task [29].

_{crit}[28]. This value may be different for different temperatures, species and environments. In this study, for crucian carp swimming at 60% U

_{crit}(0.51 m/s), drag power, P

_{d}, was almost equal to the capacity of aerobic respiration to generate energy for swimming (P

_{d}= 90%P

_{s}); for crucian carp swimming at 80% U

_{crit}, P

_{d}was almost two times P

_{s}; for carp swimming at U

_{crit}, P

_{d}was almost 2.5 times P

_{s}. However, as swimming speed surpassed U

_{crit}, sample sizes were significantly reduced, and the quality of these estimates inevitably deteriorated. Anaerobic costs become necessary when fish swim at high speeds, and when these exceed a certain limit, may result in irreversible damage to their health [29]. Further research is necessary to determine the maximum speeds at which fish can swim without incurring irreversible health damage.

## 5. Conclusions

_{d}, since the excess post-exercise oxygen consumption at high speed is not measured. A pressing need for accurate measurements of the energy efficiency and excess post-exercise oxygen consumption of a wide range of fish species remains.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Velocity-modulator frequency relationship. The flow velocity in the test area had a linear relationship with the frequency displayed by the motor, fitting the equation y = 0.03788x (where x is the modulator frequency and y is the water velocity).

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**Figure 3.**Tail beat frequency (TBF, mean ± SEM) as a function of swimming speed (linear function: TBF = 0.82 + 5.2 U

_{s}, R

^{2}= 0.977, p < 0.001).

**Figure 4.**Oxygen consumption rate (AMR, mean ± SEM) as a function of swimming speed (power function: AMR = 131.24 + 461.26U

_{s}

^{1.27}, R

^{2}= 0.948, p < 0.001).

**Figure 5.**Cost of transport (COT, mean ± SEM) as a function of swimming speed (function: COT = 0.49U

_{s}

^{−1}+ 1.73U

_{s}

^{0.27}, R

^{2}= 0.594, p < 0.001).

**Figure 6.**Drag, F, and efficiency, η, of crucian carp at different swimming speeds. Drag of fish (F, mean) as function of swimming speed (Function: F = 0.0055 + 0.13Us, R

^{2}= 0.998, p < 0.001, black solid line). Trend of F relative to Us (black dotted line). Trend of η relative to U

_{s}(blue solid line).

**Figure 7.**Power of crucian carp at different swimming speeds. The value of P

_{s}(black solid line) was obtained using Equation (9). P

_{d}(blue dotted line) were fitted using Equation P

_{d}= k U

_{s}

^{3}using oxygen consumption rate data for swimming speeds not greater than 0.45 m/s (P

_{s}= 0.9 P

_{d}, P

_{s}= 1.718 U

_{s}

^{3}, R

^{2}= 0.953; P

_{s}= 1.0 P

_{d}, P

_{s}= 1.910 U

_{s}

^{3}, R

^{2}= 0.954).

**Table 1.**Number of samples at each stage. When the velocity (m/s) reaches the critical swimming speed, the number of samples naturally decreases.

Number of Samples | 0.15 (m/s) | 0.30 (m/s) | 0.45 (m/s) | 0.60 (m/s) | 0.75 (m/s) | 0.90 (m/s) | 1.05 (m/s) | 1.20 (m/s) |
---|---|---|---|---|---|---|---|---|

n | 31 | 31 | 31 | 31 | 25 | 14 | 7 | 2 |

Drag Coefficients | 0.15 m/s | 0.30 m/s | 0.45 m/s | 0.60 m/s | 0.75 m/s | 0.90 m/s | 1.05 m/s | 1.20 m/s |
---|---|---|---|---|---|---|---|---|

C_{d} | 0.0149 | 0.0130 | 0.0120 | 0.0113 | 0.0108 | 0.0104 | 0.0101 | 0.0098 |

Calibration | C_{d} = 0.126–0.140 |

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**MDPI and ACS Style**

He, F.; Wang, X.; Li, Y.; Hou, Y.; Zou, Q.; Shen, D. A Method for Estimating the Velocity at Which Anaerobic Metabolism Begins in Swimming Fish. *Water* **2021**, *13*, 1430.
https://doi.org/10.3390/w13101430

**AMA Style**

He F, Wang X, Li Y, Hou Y, Zou Q, Shen D. A Method for Estimating the Velocity at Which Anaerobic Metabolism Begins in Swimming Fish. *Water*. 2021; 13(10):1430.
https://doi.org/10.3390/w13101430

**Chicago/Turabian Style**

He, Feifei, Xiaogang Wang, Yun Li, Yiqun Hou, Qiubao Zou, and Dengle Shen. 2021. "A Method for Estimating the Velocity at Which Anaerobic Metabolism Begins in Swimming Fish" *Water* 13, no. 10: 1430.
https://doi.org/10.3390/w13101430