Fluid dynamics and efficiency of colonial swimming via multijet propulsion at intermediate Reynolds numbers

Colonial physonect siphonophores swim via laterally distributed multijet propulsion at intermediate Reynolds numbers (Re's) on the orders of 1–1000. Here, a computational fluid dynamics approach that assumes steady axisymmetric flow is employed to investigate the underlying fluid mechanics and adaptive values of colonial swimming via laterally distributed multijet propulsion, with comparison with rear-jetting single-jet propulsion. Results show that imposed flow fields, drag coefficients, powers, and efficiencies all vary significantly depending upon Re, jet angle, and way of jetting. For a given Re, two types of optimal jet angles are determined: one in the range of 61°–70° that maximizes the quasipropulsive efficiency (i.e., to minimize the jet power), and another in the range of 34°–45° that maximizes the Froude propulsion efficiency (i.e., to minimize the wake). Comparison with values for a documented siphonophore, Nanomia bijuga, indicates that siphonophores rely upon a spectrum of jet angles between these two theoretical optima. Multiple, laterally directed jets produced by colonial forms are less energetically efficient for propulsion than single, posteriorly directed jets produced by solitary individuals; however, colonial swimming achieves energetic benefits for jetting individuals within the colony because they require significantly lower per-module power than that required by a lone jet module swimming at the same speed. Hence, by sharing propulsive duties, colony formation helps alleviate inherent power constraints that characterize cnidarian muscles. Importantly, multiple jets that are directed obliquely away from the central body axis exert less impact on other colony members within the siphosome that is towed in the wake of the jetting aggregation.

Figure 1

Jet propulsion animals: (a) the salp Salpa thompsoni solitary, by Laurence Madin, ©Woods Hole Oceanographic Institution (with permission), (b) the hydromedusa Sarsia tubulosa, licensed under CC BY 2.0, (c) the queen scallop Chlamys opercularis, by Merlin Charon, licensed under CC0, (d) the Palau Nautilus belauensis, by Manuae, licensed under CC BY-SA 3.0, (e) the bigfin reef squid Sepioteuthis lessoniana, by George Berninger Jr., licensed under CC BY-SA 3.0, and (f) the physonect siphonophore Marrus orthocanna, Credit: NOAA. In each picture, the red arrow indicates the swimming direction, while the white arrow(s) the jet direction(s) [also the intake flow direction in (a)].

Figure 2

(a) General body structure of the physonect N. bijuga, modified from a video-recorded sequence depicted in Fig. 2 of Ref. [23], where the whole colony swims forward at a speed U (the blue arrow) while being propelled by the laterally distributed multijets (the red arrows). Schematics of three CFD-simulated propulsion strategies: a self-propelled axisymmetric body swimming steadily via the laterally distributed multijet propulsion, where 1–7 jet modules are considered [(b) L1–L7]; a self-propelled axisymmetric body swimming steadily via the rear-jetting single-jet propulsion, where seven different body lengths respectively equal to those of the L1–L7 bodies are considered [(c) R1–R7]; and a towed axisymmetric body, where seven different body lengths respectively equal to those of the L1–L7 bodies are considered [(d) T1–T7].

Figure 3

Grid and boundary conditions for the axisymmetric CFD model: (a) the whole computational domain; (b) the near-body region.

Figure 4

CFD-simulated drag coefficients $C_D$ (a), pressure drag coefficients $C_{D\text{−}\mathrm{pressure}}$ (b), viscous drag coefficients $C_{D\text{−}\mathrm{viscous}}$ (c), and $C_{D\text{−}\mathrm{pressure}}$/$C_{D\text{−}\mathrm{viscous}}$ (d) plotted as functions of $\mathrm{Re}\times e$, for a steadily towed sphere and for seven steadily towed axisymmetric bodies, T1–T7, of sequentially decreasing aspect ratios $e$. $\mathrm{Re}\times e$ is the Reynolds number that is defined based on the cross-sectional diameter. Given the same cross-sectional diameter, increasing $\mathrm{Re}\times e$ is equivalent to increasing the towing velocity.

Figure 5

Grid refinement simulations of a video-recorded swimming of an N. bijuga colony as depicted in Fig. 2 of Ref. [23]. The simulated axisymmetric colony consists of seven laterally distributed jets that are prescribed with the observed jet angles, i.e., 68.4°, 57.2°, 52.8°, 49.8°, 48.9°, 45.3°, and 44.7°, starting from the one closest to the anterior of the colony. Plotted here are the simulated jet velocity $U_{\mathrm{jet}}$ (a), drag coefficient $C_D$ (b), quasipropulsive efficiency ${\eta }_{\mathrm{QPE}}$ (c), and Froude propulsion efficiency ${\eta }_{\mathrm{FPE}}$ (d) as functions of Re. Simulated $C_D$’s for a steadily towed axisymmetric body of the same body length are also plotted in (b).

Figure 6

$\mathrm{Re}=18.9$. CFD simulated flow fields imposed by an L4 body that adopts the same jet angle of 70° for all four lateral jets and swims at 0.001 m/s [(a)–(d)], an R4 body that swims at 0.001 m/s using a rear jet [(e)–(h)], and a T4 body that is towed at 0.001 m/s [(i)–(l)]. (a), (e), (i) Streamline patterns in a stationary frame of reference. (b), (f), (j) Contours of azimuthal vorticity scaled by U/R; red contour levels are 0.300, 0.443, 0.654, 0.965, 1.420, 2.100, 3.110, 4.590, 6.770, and 10.000; blue contour levels are −0.300, −0.443, −0.654, −0.965, −1.420, −2.100, −3.110, −4.590, −6.770, and −10.000. (c), (g), (k) Contours of velocity magnitude in a stationary frame of reference and scaled by U; red contour levels start from 1.0 with increment 0.1; blue contour levels start from 0.1 to 0.9 with increment 0.1. (d), (h), (l) Contours of pressure scaled by $0.5\rho U^2$; red contour levels start from 0.1 with increment 0.1; blue contour levels start from −0.1 with increment −0.1; black contour lines are 0.

Figure 7

$\mathrm{Re}=1886.1$. CFD-simulated flow fields imposed by an L4 body that adopts the same jet angle of 64° for all four lateral jets and swims at 0.1 m/s [(a)–(d)], an R4 body that swims at 0.1 m/s using a rear jet [(e)–(h)], and a T4 body that is towed at 0.1 m/s [(i)–(l)]. (a), (e), (i) Streamline patterns in a stationary frame of reference. (b), (f), (j) Contours of azimuthal vorticity scaled by U/R; red contour levels are 0.300, 0.443, 0.654, 0.965, 1.420, 2.100, 3.110, 4.590, 6.770, and 10.000; blue contour levels are −0.300, −0.443, −0.654, −0.965, −1.420, −2.100, −3.110, −4.590, −6.770, and −10.000. (c), (g), (k) Contours of velocity magnitude in a stationary frame of reference and scaled by U; red contour levels start from 1.0 with increment 0.1; blue contour levels start from 0.1 to 0.9 with increment 0.1. (d), (h), (l) Contours of pressure scaled by $0.5\rho U^2$; red contour levels start from 0.1 with increment 0.1; blue contour levels start from −0.1 with increment −0.1; black contour lines are 0.

Figure 8

An L7 body swims by adopting the same jet angle for all its seven lateral jets. The jet angle varies between 5° and 85° for each of the considered seven Re's (color coded). Line plots of (a) the quasipropulsive efficiency ${\eta }_{\mathrm{QPE}}$, (b) the Froude propulsion efficiency ${\eta }_{\mathrm{FPE}}$, (c) the drag coefficient $C_D$, (e) the useful power $P_{\mathrm{useful}}$, and (f) the jet power $P_{\mathrm{jet}}$ against the jet angle θ. (d) Line plot of the tow power $P_{\mathrm{tow}}$ against Re. In (a), (c), (e), or (f), the solid black line shows the optimal jet angles that maximize ${\eta }_{\mathrm{QPE}}$. In (b), (c), (e), or (f), the dotted black line shows the optimal jet angles that maximize ${\eta }_{\mathrm{FPE}}$.

Figure 9

Line plots of (a) the optimal jet angle ${\theta }_{\mathrm{maxQPE}}$, (b) the achieved maximum quasipropulsive efficiency ${\eta }_{\mathrm{QPE},\max }$, (c) the tow power $P_{\mathrm{tow}}$, and (d) the ${\eta }_{\mathrm{QPE},\max }$-associated jet power $P_{\mathrm{jet},\mathrm{maxQPE}}$ against $\mathrm{Re}\times e$. Line plots of (e) the optimal jet angle ${\theta }_{\mathrm{maxFPE}}$, (f) the achieved maximum Froude propulsion efficiency ${\eta }_{\mathrm{FPE},\max }$, (g) the ${\eta }_{\mathrm{FPE},\max }$-associated useful power $P_{\mathrm{useful},\mathrm{maxFPE}}$, and the ${\eta }_{\mathrm{FPE},\max }$-associated jet power $P_{\mathrm{jet},\mathrm{maxFPE}}$ against $\mathrm{Re}\times e$. The lines are color coded by the L1–L7 bodies.

Figure 10

Line plots of (a) $P_{\mathrm{jet}\text{−}\mathrm{module}}$ and (b) $P_{\mathrm{jet}\text{−}\mathrm{module}}/P_{\mathrm{solitary}}\times 100$ against the number of jet modules in the swimming body that is propelled by laterally distributed multijets. The lines are color coded by the swimming speed U. See the main text for details of the variables.

Figure 11

Line plots of (a) the achieved maximum quasipropulsive efficiency ${\eta }_{\mathrm{QPE},\max }$ and (b) the ${\eta }_{\mathrm{QPE},\max }$-associated mechanical power per jet module $P_{\mathrm{mj},\mathrm{maxQPE}}/N$ against $\mathrm{Re}\times e$, for the L1–L7 bodies (color coded) that swim via the laterally distributed multijet propulsion. Line plots of (c) the quasipropulsive efficiency ${\eta }_{\mathrm{QPE}}$ and (d) the jet power $P_{\mathrm{sj}}$ against $\mathrm{Re}\times e$, for the R1–R7 bodies (color coded) that swim via the rear-jetting single-jet propulsion.

Figure 12

Line plots of (a) $C_D$, (b) $C_{D\text{−}\mathrm{pressure}}$, (c) $C_{D\text{−}\mathrm{viscous}}$, and (d) $C_{D\text{−}\mathrm{pressure}}$/$C_{D\text{−}\mathrm{viscous}}$ against $\mathrm{Re}\times e$, for the L1–L7 bodies (color coded) that swim via the laterally distributed multijet propulsion. Line plots of (e) $C_D$, (f) $C_{D\text{−}\mathrm{pressure}}$, (g) $C_{D\text{−}\mathrm{viscous}}$, and (h) $C_{D\text{−}\mathrm{pressure}}$/$C_{D\text{−}\mathrm{viscous}}$ against $\mathrm{Re}\times e$, for the R1–R7 bodies (color coded) that swim via the rear-jetting single-jet propulsion.

Figure 13

Lateral boundary-layer velocity profiles, $u$/U against $r$/R, plotted for (a) the L4 body that swims via the laterally distributed multijet propulsion at the maximum quasipropulsive efficiency, (b) the R4 body that swims via the rear-jetting single-jet propulsion, and (c) the T4 body that is towed through water, respectively, at two Re values (color coded).

Figure 14

$\mathrm{Re}=917.0$. Flow velocity vector fields (in a frame of reference fixed on the body) for (a) the L7 body that swims at 0.03 m/s via seven laterally distributed jets that are prescribed with the observed jet angles, i.e., 68.4°, 57.2°, 52.8°, 49.8°, 48.9°, 45.3°, and 44.7°, starting from the one closest to the anterior of the colony (as in a video-recorded swimming of an N. bijuga colony as depicted in Fig. 2 of Ref. [23]), and (b) the R7 body that swims at 0.03 m/s via the rear-jetting single-jet propulsion. For clarity, only 4.5% of total vectors are shown.

Figure 15

$\mathrm{Re}=917.0$. Filled color contours of the rate of deformation field overlapping with black streamlines (in a frame of reference fixed on the body) for the same two cases as in Fig. 14.

Figure 16

$\mathrm{Re}=917.0$. Flow velocity vector fields (in a stationary frame of reference) for the same two cases as in Fig. 14. For clarity, only 5.2% of total vectors are shown.