Intermittent versus continuous swimming: An optimization tale

Intermittent swimming, also termed “burst-and-coast swimming,” has been reported as a strategy for fish to enhance their energetical efficiency. Intermittent swimming involves additional control parameters, which complexifies its understanding by means of quantitative and parametrical analysis, in comparison with continuous swimming. In this study, we used a hybrid computational fluid dynamic (CFD) model to assess the swimming performance in intermittent swimming parametrically and quantitatively. A Navier-Stokes solver is applied to construct a database in the multidimensional space of the control parameters to connect the undulation kinematics to swimming performance. Based on the database, an indirect numerical approach named “gait assembly” is used to generate arbitrary burst-and-coast gaits to explore the parameter space. Our simulations directly measured the hydrodynamics and energetics under the unsteady added-mass effect during burst-and-coast swimming. The results suggest that the instantaneous power of burst is basically determined by undulatory kinematics. The results show that the energetical performance of burst-and-coast swimming can be better than that of continuous swimming, but also that an unoptimized burst-and-coast gait may become very energetically expensive. These results shed light on the mechanisms at play in intermittent swimming, enabling us to better understand fish behavior and to propose design guidelines for fishlike robots.

Figure 1

Features of burst-and-coast swimming. (a) Example observation on burst-and-coast swimming. (b) Schematic description of burst-and-coast swimming with regard to time and velocity, revised from Videler and Weihs [10].

Figure 2

Numerical methods. (Upper) Direct numerical approach: (a) Fish surface model; (b) kinematic model; (c) computational mesh; (d) an example of CFD flow field. (Lower) (e) Indirect numerical approach: Burst-and-coast gait assembly.

Figure 3

Flow field features of a nonrepeated burst-and-coast bout. The burst of cyclic swimming lasts for 15 tail-beat cycles during which the fish has accelerated to a relatively stable velocity. It then stops its undulation and the fish body is kept straight during the coast phase that lasts for another 1.5 s (equivalent to a 15 tail-beat cycle time), reaching an almost static condition. Flow features are marked by numbers and explained in the text. Panels (a–d) correspond to 5, 10.5, 17, and 30 tail-beat cycles, respectively.

Figure 4

Simulation results of a single burst-and-coast bout. (a) Body axis time sequences; (b) $U^{*}$ versus tail-beat cycles; (c) ${\mathrm{\Omega }}_{\mathrm{accumulated}}^{*}$; (d) $P^{*}$ and $\eta$ versus tail-beat cycles; (e) $P^{*}$ and $\eta$ versus $U^{*}$; (f) $T^{*}$ versus tail-beat cycles; (g) $T^{*}$ and $D^{*}$ versus $U^{*}$; (h) $D^{*}$ versus tail-beat cycles; (i) drag coefficient $C_D$. In (i), as a reference, a speed-specific drag coefficient curve of the fish steadily gliding with a straight body is presented by the gray dashed line. Note that this reference curve does not represent a dynamic process. Each point on the reference curve denotes a drag coefficient obtained in a simulation when the fish constantly glides at a given speed. ($U^{*}:$ Dimensionless velocity, defined as $U^{*}=U/U_{\mathrm{ref}}$; $P^{*}:$ Dimensionless power, defined as $P^{*}=P/\left(\rho U_{\mathrm{ref}}^3{L}^2\right);$ $T^{*}$: Dimensionless thrust, defined as $T^{*}=T/\left(\rho U_{\mathrm{ref}}^2{L}^2\right)$; $D^{*}$: Dimensionless drag, defined as $D^{*}=D/\left(\rho U_{\mathrm{ref}}^2{L}^2\right)$; $C_D$: Drag coefficient, defined as $C_D=2D/\left(\rho U^{2}S\right)$; $P$: Instantaneous power; $\rho$: The water density; $U$: Instantaneous velocity; $U_{\mathrm{ref}}$: The reference velocity, defined as the velocity at the end of burst, $3.9L$/s; $L$: The body length; $S$: The wetted area; $T$: Instantaneous thrust; $D$: Instantaneous drag; $P$: Instantaneous power; $\eta$: Froude efficiency, where $\eta =TU/P$.

Figure 5

Simulation results in 40 full burst simulations with various undulatory kinematics: The combinations between tail-beat frequency and amplitude. (a) Relationship between instantaneous velocity and dimensionless time for all burst simulations, where the dimensionless instantaneous velocity in each simulation is normalized by the terminal velocity $U_{\mathrm{terminal}}$. (b) Relationship between instantaneous power and dimensionless time in all burst simulations, where the instantaneous dimensionless power in each simulation is normalized by the terminal power $P_{\mathrm{terminal}}$. The dimensionless time in (a,b) is given by $t^{*}=t/\left(A^{-1/2}{L}^{1/2}{f}^{-1}\right)$. (c) Relationship between $\mathrm{Sw},$ ${\mathrm{Re}}_{\mathrm{wave}}$, and ${\mathrm{Re}}_{\mathrm{terminal}}$ in burst simulations, where $\mathrm{Sw}=fAL/\nu$, ${\mathrm{Re}}_{\mathrm{wave}}=f{L}^2/\nu$, $A^{*}=A/L$, and ${\mathrm{Re}}_{\mathrm{terminal}}=UL/\nu$. (d) Relationship between dimensionless burst-undulation frequency ${\mathrm{Re}}_{\mathrm{wave}}$ and power coefficient [$C_P=P_{\mathrm{terminal}}/\left(\rho L^3{f}^3{A}^2\right)$]; the relationship between ${\mathrm{Re}}_{\mathrm{terminal}}$ and $C_P$ is also shown in the inset; (e) ${\mathrm{Re}}_{\mathrm{terminal}}$ map in the ${\mathrm{Re}}_{\mathrm{wave}}\text{−}{A}^{*}$ plane; (f) $T_{\mathrm{terminal}}^{*}$ map in the ${\mathrm{Re}}_{\mathrm{wave}}\text{−}{A}^{*}$ plane; (g) $P_{\mathrm{terminal}}^{*}$ map in the $R{e}_{\mathrm{wave}}\text{−}{A}^{*}$ plane. (e–g) are recalculated from Li et al. [14]. The dimensional data of all burst simulations are shown in the SM, Sec. D [28].

Figure 6

Comparison between optimal burst-and-coast and continuous gaits at various speed levels. (a) Optimal tail-beat frequency; (b) optimal tail-beat amplitude; (c) upper and lower speed boundaries of optimal burst-and-coast swimming, and the fluctuation range around the average speed. Note that upper and lower velocity boundaries are not applicable for the continuous gait. (d) Cost of transport of optimal burst-and-coast gait and continuous gait. $1L/s=0.02$ m/s.

Figure 7

Optimal gaits at various target velocity levels with burst-and-coast and continuous gaits, as well as information about the worst burst-and-coast gaits scanned. (a) Optimal cost of transport in burst-and-coast gait (○) and continuous gait ($⧫$), as well as the CoT range between optimal and worst burst-and-coast gaits scanned. (b) Examples of scanned optimal and worst burst-and-coast gaits at $U_{\mathrm{target}}=3L/\mathrm{s}$. The worst gait adopted an extremely powerful burst to reach a high velocity and then coasted for a long period to exhaust momentum.