Intermittent unsteady propulsion with a combined heaving and pitching foil

Inviscid computations are presented of a self-propelled virtual body connected to a combined heaving and pitching foil that uses continuous and intermittent motions. It is determined that intermittent swimming can improve efficiency when the dimensionless heave ratio is $h^{*}<0.7$ while it degrades efficiency for $h^{*}\ge 0.7$. This is a consequence of the physical origins of the force production for pitch-dominated $(h^{*}<0.5)$ and heave-dominated $(h^{*}>0.5)$ motions. Based on insight derived from classic unsteady thin airfoil theory, it is discovered that pitch-dominated motions are driven by added mass-based thrust production where self-propelled efficiency is maximized for high reduced frequencies, while heave-dominated motions are driven by circulatory-based thrust production where self-propelled efficiency is maximized by low reduced frequencies. Regardless of the dimensionless heave ratio, the reduced frequency is high for small amplitude motions, high Lighthill numbers, and low duty cycles and vice versa. Moreover, during intermittent swimming, the stopping vortex that is shed at the junction of the bursting and coasting phases becomes negligibly weak for $h^{*}<0.5$ and small amplitude motions of $A^{*}=0.4$. This study provides insight into the mechanistic trade-offs that occur when biological or bioinspired swimmers continuously or intermittently use combined heaving and pitching hydrofoils.

Figure 1

(a) Schematic of the idealized self-propelled swimmer with a combined heaving and pitching hydrofoil. Dashed line represents a virtual body, not present in the computational domain, but its effect is a drag force, $D$, acting on the hydrofoil. The wake element endpoints are colored red and blue for counterclockwise and clockwise circulation, respectively. (b) Schematic representation of the wetted surface area, $S_w$ (purple shaded area), and the propulsor planform area, $S_p$ (teal shaded area), for a generic swimmer.

Figure 2

(a) A sample trailing edge amplitude of motion, $z\left(t\right)$, normalized by the chord length and decomposed into its heaving, $h\left(t\right)$, and pitching, $\theta \left(t\right)$, components. (b) Variable range for the study shown as a function of dimensionless amplitude and heave ratio. The Lighthill number of a simulation is denoted by different marker types with $\text{Li}=0.05$, 0.1, and 0.2 represented by square, circle, and triangle markers, respectively. (c) Schematic of the variables affecting the angle of attack. The chord length is, $c=0.1$ m, the maximum thickness of the airfoil is 10% of the chord length, $U\left(t\right)$ is the swimming speed, $\dot{h}\left(t\right)$ is the heave velocity, and ${\alpha }_h$ is the angle of attack from the heave component of the motion. (d) Maximum angle of attack as a function of duty cycle for $\text{Li}=0.05$. The dashed line represents the cut off maximum angle of attack at ${\alpha }_{\text{max}}={10}^{\circ }$.

Figure 3

Snapshots of (a) two oscillation cycles for a hydrofoil continuously swimming to the left and (b) one oscillation cycle of a hydrofoil intermittently swimming to the left with $\mathrm{\Delta }=0.5$.

Figure 4

Discretization independence graphs showing the percent change in (a) the thrust coefficient and (c) efficiency when the number of time steps are held constant ($N_S=168$) and the number of body elements are doubled. Similarly, discretization independence graphs showing the percent change in (b) the thrust coefficient and (d) efficiency when the number of body elements are held constant ($N_P=150$) and the number of time steps per bursting period are doubled.

Figure 5

Comparison of the BEM potential flow simulations to the experiments of Ref. [21]. The comparison is shown for the angle of attack of ${\alpha }_{\text{max}}={10}^{\circ }$ with $h_0/c=0.75$ and $\psi ={90}^{\circ }$, which is the phase difference between pitch and heave. Presented are (a) the thrust coefficient and (b) efficiency as functions of the Strouhal number. The lines and markers represent the BEM numerical solutions and the experimental data, respectively.

Figure 6

[(a)–(c)] Dimensionless speed, dimensionless range, and efficiency as a function of $h^{*}$ at $\text{Li}=0.1$ with varying $A^{*}$. [(d)–(f)] Dimensionless speed, dimensionless range, and efficiency as a function of $h^{*}$ at $A^{*}=1.0$ and varying Li.

Figure 7

(a) Efficiency of intermittent swimmers for varying duty cycles, amplitudes, and heave ratios with $\text{Li}=0.05$. The efficiency curves are grouped for high, medium, and low $h^{*}$. (b) Efficiency of intermittent swimmers for varying duty cycles, amplitudes, and Lighthill numbers with $0.21\le h^{*}\le 0.24$. (c) The ratio of the maximum efficiency of an intermittent swimmer to its continuous swimming efficiency as a function of the heave ratio.

Figure 8

(a) Efficiency as a function of reduced frequency for a variety of Li and $\mathrm{\Delta }$, and for $h^{*}=0.24$, 0.52, and 0.84 with a fixed maximum amplitude of motion of $A^{*}=1$. The solid black lines represent Garrick's analytic solutions for the respective $h^{*}$ values. Thrust (left axis) and power (right axis) coefficients as functions of reduced frequency for (b) $h^{*}=0.24$ and (c) $h^{*}=0.84$.

Figure 9

The thrust coefficient as a function of the reduced frequency for (a) $h^{*}=0.24$ and (b) $h^{*}=0.84$. The total thrust (black solid line) is decomposed into the pitching added mass term (blue solid line), pitching circulatory term (red dotted line), heaving-pitching circulatory cross term (red dash-dotted line), and heaving circulatory term (blue dashed line).

Figure 10

The wake flow produced over several oscillation cycles by an intermittent swimmer with $\text{Li}=0.05$, $A^{*}=0.4$, $\mathrm{\Delta }=0.5$, and (a) $h^{*}=0.02$, (b) $h^{*}=0.19$, (c) $h^{*}=0.52$, (d) $h^{*}=0.76$, and (e) $h^{*}=0.83$. The location of the trailing edge of the foil is located at $x/c=0$. The dimensionless circulation of the wake vortices is defined as ${\mathrm{\Gamma }}^{*}=\mathrm{\Gamma }/fAc$.

Figure 11

Performance map for $A^{*}=1$, $\text{Li}=0.05$. Efficiency of swimmer presented as a function of dimensionless velocity and dimensionless range, with the color code for variation of (a) duty cycle ($0.3\le \mathrm{\Delta }\le 1$) and (b) heave ratio ($0\le h^{*}\le 1$). The solid line represents $\eta =100\%$ and dashed lines represent $\eta =80\%,60\%,40\%,$ and $20\%$.

Figure 12

Convergence plots for $\mathrm{\Delta }=0.3$ and $A^{*}=1$.

Figure 13

Convergence plots for $\mathrm{\Delta }=0.5$ and $A^{*}=1$.

Figure 14

Convergence plots for $\mathrm{\Delta }=0.9$ and $A^{*}=1$.