3.1. Time-dependent flow

Figure 3.

The dimensionless vorticity fields at $t/T=19.75$ . The leading edge of the plate is at its highest vertical displacement: $(a)$ unbounded domain $(D_L\rightarrow \infty ,H_L\rightarrow \infty )$ ; $(b)$ smooth wall $(D_L\rightarrow \infty ,H_L=1.5)$ ; $(c)$ compliant blade array with $D_L=1.00, H_L=1.5$ ; $(d)$ compliant blade array with $D_L=0.50, H_L=1.5$ ; $(e)$ compliant blade array with $D_L=0.25, H_L=1.5$ ; $(f)$ smooth wall with a reduced plate–wall distance, $D_L\rightarrow \infty ,H_L=0.5$ . Grey area denotes solid wall. Here, $f^*=0.64, A_L=0.5, K_p=3.5, K_b=3.0$ . See supplementary movie 1 for the vorticity evolution.

Figure 3. Long description

A heat map displays dimensionless vorticity fields at a specific time point. The leading edge of the plate is at its highest vertical displacement. The heat map compares six different scenarios: an unbounded domain, a smooth wall, and compliant blade arrays with varying dimensions. The grey area denotes a solid wall. The color scale ranges from blue to red, indicating different vorticity intensities. The heat map highlights the vorticity evolution in these different conditions, with specific attention to the vorticity at the blade tips and on the wall.

A series of simulations is performed at flapping frequency $f^*=0.64$ for the self-propelled plate, considering different blade spacing $D_L$ and vertical plate-wall distance $H_L$ . The motion of the plate reaches a periodic state for $t/T\geqslant 15$ . Figure 3 presents the dimensionless vorticity fields ( $\omega _z^*=\omega _z L/U_{\textit{ref}}$ ) at $t/T=19.75$ , corresponding to the instant at which the leading edge of the plate reaches its maximum vertical displacement (see supplementary movie 1 for the vorticity evolution, available at https://doi.org/10.1017/jfm.2026.11712.). For comparison, three reference cases are also presented. The first is an unbounded-domain case with $D_L\rightarrow \infty$ and $H_L\rightarrow \infty$ , shown in figure 3 $(a)$ . The other two are smooth-wall cases with $D_L\rightarrow \infty$ : smooth-wall case 2 (SW2), with $H_L=1.5$ , shown in figure 3 $(b)$ ; and smooth-wall case 1 (SW1), with $H_L=0.5$ , shown in figure 3 $(f)$ . Specifically, SW1 is equivalent to raising the smooth wall to the tips of the undeformed blades.

From figure 3( $a$ ) to figure 3( $f$ ), the distance travelled by the self-propelled plate increases monotonically, indicating that reducing either the vertical plate–wall distance $H_L$ or the blade spacing $D_L$ enhances the propulsive velocity of the plate. Notably, the discrepancy between the case in the unbounded domain and the case SW2 is negligible, and both cases exhibit the classical reverse Bénard–von Kármán vortex street in the near wake. By contrast, the other cases exhibit a vortex-pair wake, which initially deflects upwards before gradually transitioning to a horizontal orientation. This upward-deflected vortex wake, which is also observed in previous studies with a smooth wall (Quinn et al. Reference Quinn, Moored, Dewey and Smits2014; Dai et al. Reference Dai, He and Zhang2016; Tang et al. Reference Tang, Huang, Gao and Lu2016; Park et al. Reference Park, Kim and Sung2017; Zhang et al. Reference Zhang, Huang and Lu2017), arises because the positive vortex generated during the downward stroke of the plate is stronger. In particular, for the case where the blade spacing equals the plate length $(D_L=1.00)$ , the far wake exhibits an irregular spatial arrangement of vortices. Beneath the plate, a layer of negative vorticity forms at the blade tips or on the smooth wall due to the plate’s flapping motion, as indicated by the black arrow in figures 3( $e$ ) and 3( $f$ ); downstream of the vortex pair shed during the previous period, this wall-interaction vorticity becomes negligible.

Figure 4.

$(a)$ Time histories of the vertical displacements of the leading edge $y_l$ and trailing edge $y_t$ of the self-propelled plate over one period. Blue dots indicate the instants $t_1=19.14T$ and $t_2=19.64T$ at which the plate is oriented horizontally. $(b)$ Time histories of the horizontal force $F_x$ acting on the plate. Red dot denotes the instant $t_3=19.44T$ corresponding to the maximum thrust in the case with $D_L=0.25$ . Here, $f^*=0.64, A_L=0.5, H_L=1.5, K_p=3.5, K_b=3.0$ .

Figure 4. Long description

The image contains two line graphs. The first graph on the left shows the vertical displacements of the leading edge and trailing edge of a self-propelled plate over one period. The y-axis represents the normalized vertical displacement, and the x-axis represents the normalized time. Two lines are plotted: one for the leading edge and one for the trailing edge. Blue dots indicate specific instants where the plate is oriented horizontally. The second graph on the right shows the horizontal force acting on the plate over the same period. The y-axis represents the normalized horizontal force, and the x-axis represents the normalized time. Two lines are plotted: one for the case with infinite distance and one for the case with a specific distance. A red dot denotes the instant corresponding to the maximum thrust in the case with the specific distance. The graphs illustrate the dynamic behavior of the plate’s movement and the forces acting on it over time.

Figure 5.

The dimensionless pressure fields at (a,b) $t_1=19.14T$ and (c,d) $t_3=19.44T$ for two typical cases: (a,c) case with $D_L\rightarrow \infty , H_L=1.5$ (SW2), (b,d) case with $D_L=0.25$ , $H_L=1.5$ . Here, $f^*=0.64, A_L=0.5, K_p=3.5, K_b=3.0$ .

Figure 5. Long description

A heat map showing dimensionless pressure fields for two typical cases of fish swimming near the seabed. The heat map is divided into four subplots labeled (a), (b), (c), and (d). Subplots (a) and (c) represent case SW2, while subplots (b) and (d) represent another case. The color scale ranges from −4 to 4, with blue indicating lower values and red indicating higher values. The pressure fields show distinct patterns of high and low pressure regions, with notable clusters and gradients. The seabed is represented by a horizontal line at the bottom of each subplot, and the fish’s swimming motion is depicted by the pressure variations above this line. The heat map illustrates how the ground effect influences the pressure distribution around the fish, affecting their swimming performance.

The trailing-edge displacement of the flexible plate is a representative response of the fluid–structure system and is closely related to its propulsive performance (Smits Reference Smits2019). To clarify how the compliant blades modify the plate deformation, figure 4 $(a)$ presents the time-dependent vertical displacements of the leading edge $y_l$ and trailing edge $y_t$ of the self-propelled plate for the case with $D_L=0.25, H_L=1.5$ and the case with $D_L\rightarrow \infty , H_L=1.5$ (SW2). After the leading edge passes its minimum vertical displacement at $t/T=19.25$ and starts moving upwards, the trailing edge continues moving downwards due to inertia; similarly, the trailing edge keeps going upwards after the leading edge passes its maximum displacement at $t/T=19.75$ . Consequently, the trailing edge exhibits a larger oscillation amplitude than the leading edge, with phase lag approximately $0.1T$ relative to the latter. At the instants $t_1=19.14T$ and $t_2=19.64T$ (marked with blue dots), corresponding to $y_l/L=-0.19$ and $0.19$ , respectively, the trailing edge and leading edge are at the same vertical position, and the plate is oriented horizontally. For convenience, the lower half-period is defined as $t/T\in (19,19.5]$ , during which the leading edge is below its average height $H$ , while the upper half-period is defined as $t/T\in (19.5,20]$ , during which the leading edge is above its average height. Compared to case SW2, the trailing-edge displacement of the plate in the case with $D_L=0.25, H_L=1.5$ remains largely unchanged during the upper half-period, but is notably suppressed after $t_1$ in the lower half-period due to the enhanced pressure beneath the plate. This is illustrated in figures 5 $(a)$ and 5 $(b)$ , which present the dimensionless pressure fields of the two cases at $t_1$ . As a result, the symmetry of the plate deformation between the lower and upper half-periods is broken.

Figure 6.

$(a)$ Time histories of the strain energy of the self-propelled plate for different cases. $(b)$ Time histories of the propulsive velocity. Blue dots indicate the instants $t_1$ and $t_2$ at which the plate is oriented horizontally. Here, $f^*=0.64, A_L=0.5, K_p=3.5, K_b=3.0$ .

Figure 6. Long description

The image contains two line graphs side by side. The left graph shows the time histories of the strain energy of the self-propelled plate for different cases. The right graph shows the time histories of the propulsive velocity. The x-axis for both graphs represents time normalized by the period (t/T), ranging from 19.00 to 20.00. The y-axis of the left graph represents the normalized strain energy (E_s,p/E_ref), ranging from 0 to 0.9. The y-axis of the right graph represents the normalized propulsive velocity (U/U_ref), ranging from 1.5 to 2.1. Different lines represent various cases with different parameters (D_L and H_L). Blue dots indicate the instants at which the plate is oriented horizontally. The graphs show periodic behavior with distinct peaks and troughs, highlighting the dynamic changes in strain energy and propulsive velocity over time.

To explore the influence of compliant blades on the hydrodynamic force, the time-dependent horizontal force $F_x$ , rescaled by $F_{\textit{ref}} = \rho U_{\textit{ref}}^2 L$ , exerted on the self-propelled plate is shown in figure 4 $(b)$ . The maximum thrust is achieved when the leading edge is close to the average height ( $t/T=19$ and 19.5), while the maximum drag occurs near the minimum and maximum vertical displacements ( $t/T=19.25$ and 19.75). For the case with $D_L=0.25, H_L=1.5$ , the maximum thrust occurs slightly earlier than that in case SW2, and the maximum drag occurs later than that in case SW2. In the lower half-period, although the plate exhibits reduced deformation, its thrust ( $-F_x$ ) is dramatically enhanced by the ground effect, especially when the plate moves from the minimum vertical displacement to the average position, i.e. $t/T\in (19.25,19.5)$ . This enhancement in thrust is primarily attributed to the increased suction within the low-pressure region between the plate and the bottom wall, as illustrated in figures 5 $(c)$ and 5 $(d)$ , which present the dimensionless pressure fields of the two cases at $t_3=19.44T$ . By contrast, in the upper half-period, the drag ( $F_x$ ) increases when the plate moves from its maximum vertical displacement to the average position, i.e. $t/T\in (19.75,20)$ .

To further quantify the deformation of the self-propelled plate, the time-dependent strain energy of the plate, normalised by $E_{\textit{ref}} = \rho U_{\textit{ref}}^2 L^2$ , is presented in figure 6 $(a)$ . The strain energy is defined as

(3.1) \begin{equation} E_{s,p}=\int _0^L 0.5 (EI)_p\left |\frac {\partial ^2\boldsymbol{X}_p(s,t)}{\partial s^2}\right |^2\textrm {d} s. \end{equation}

Since $E_{s,p}$ is the integration of the square of the curvature over the entire plate, it represents the overall deformation of the plate. For all cases, $E_{s,p}$ reaches zero at almost the same instants $t_1$ and $t_2$ . During the time intervals $t\in (19T,t_1)$ and $t\in (t_2, 20T)$ , the bottom wall or blade array has a negligible effect on the deformation of the plate; in the remaining period, the ground effect strongly suppresses the strain energy of the plate for $t\in (t_1, 19.42T)$ , whereas it slightly enhances the strain energy for $t\in (19.42T, t_2)$ .

The time-dependent propulsive velocity of the self-propelled plate, defined by (2.14), is presented in figure 6 $(b)$ . Compared with the smooth-wall case SW2, the compliant blades increase the propulsive velocity of the plate over the entire period, and the velocity enhancement intensifies as the blade spacing decreases. Furthermore, the local maximum velocity during the upper half-period is larger than that during the lower half-period, and the local minimum velocity during the upper half-period is smaller than that during the lower half-period. These results indicate that the compliant blades break the symmetry of the plate’s motion. After the instant $t_2$ in the upper half-period, the instantaneous propulsive velocity of the plate is enhanced, whereas its deformation remains largely unchanged; as a result, this leads to an increase in the drag exerted on the plate, as evidenced by figure 4 $(b)$ .

3.2. Propulsive performance and scaling law

Figure 7.

$(a)$ The average propulsive velocity and $(b)$ the cost of transport of the self-propelled plate as functions of $D_L^{-1}$ across different $H_L$ . Here, $f^*=0.64, A_L=0.5, K_p=3.5, K_b=3.0$ .

Figure 7. Long description

The image contains two line graphs side by side. The left graph shows the average propulsive velocity, represented by the ratio of U bar to U subscript ref, plotted against the inverse of D subscript L. The right graph shows the cost of transport, represented in joules per kilogram per meter, also plotted against the inverse of D subscript L. Each graph includes four data series, distinguished by different symbols: triangles, squares, diamonds, and circles. These symbols correspond to different values of H subscript L: 1.500, 1.750, 1.625, and 2.000, respectively. The x-axis for both graphs ranges from 0 to 4, representing the inverse of D subscript L. The y-axis of the left graph ranges from 1.5 to 2.0, while the y-axis of the right graph ranges from 4.2 to 5.4. The data series show how the average propulsive velocity and cost of transport vary with different values of D subscript L to the power of negative one and H subscript L. All values are approximated.

For $t/T\geqslant 15$ , the motion of the self-propelled plate is already independent of its initial conditions, and the notation $\widetilde {({\cdot })}$ is employed to denote the time averaging over the interval $t\in [15T,25T]$ . The average propulsive velocity $\widetilde {U}$ and average power consumption $\widetilde {P}$ are defined as

(3.2) \begin{equation} \widetilde {U}=\frac {1}{10 T}\int _{15T}^{25 T} U\, \textrm {d} t, \quad \widetilde {P}=\frac {1}{10 T}\int _{15T}^{25 T} \int _0^L {-\boldsymbol{\tau }(s,t)\boldsymbol{\cdot }\frac {\partial \boldsymbol{X}(s,t)}{\partial t} }\,\textrm {d} s\,\textrm {d} t, \end{equation}

where $\boldsymbol{\tau }$ denotes the fluid stress exerted on the plate. It should be noted that $\widetilde {P}$ , representing the average power delivered by the plate to the fluid, is equal to the average power supplied by the external force acting at the leading edge of the plate, because the net change of the total mechanical energy of the plate is zero from $t=15T$ to $t=25T$ . Furthermore, the cost of transport ( ${\textit{COT}}$ ), a parameter quantifying the energy consumption per unit mass and transport distance, is defined as

(3.3) \begin{equation} \textit{COT}=\frac {\widetilde {P}}{mL\widetilde {U}}. \end{equation}

Here, $mL$ is the mass of the plate.

Figure 7 presents $\widetilde {U}$ and ${\textit{COT}}$ of the self-propelled plate as functions of the blade spacing. Here, the reciprocal of the blade spacing, $D_L^{-1}$ , is employed, and $D_L^{-1}=0$ corresponds to the smooth-wall cases. For the smooth-wall cases, $\widetilde {U}$ and ${\textit{COT}}$ are insensitive to $H_L$ over the range 1.5–2.0, indicating that the ground effect provided by the smooth wall alone is negligible. By contrast, in the presence of compliant blades, $\widetilde {U}$ increases and ${\textit{COT}}$ decreases with increasing $D_L^{-1}$ , indicating enhancements in both propulsion velocity and energy efficiency. For $D_L^{-1}\geqslant 2$ , $\widetilde {U}$ and ${\textit{COT}}$ tend to plateau. Notably, the enhancements in propulsive velocity and energy efficiency are more pronounced when $H_L$ is decreased from 1.625 to 1.5 than when it is decreased from 2.0 to 1.625, revealing a nonlinear dependence of the ground effect on $H_L$ . Further reducing $H_L$ may lead to direct contact between the plate and blades; this situation is not considered in the present study.

Figure 8.

$(a)$ The average propulsive velocity and $(b)$ the cost of transport of the self-propelled plate as functions of $D_L^{-1}$ across different $K_p$ . Here, $f^*=0.64, A_L=0.5, H_L=1.625, K_b=3.0$ .

Figure 8. Long description

The image contains two line graphs side by side. The left graph shows the average propulsive velocity of a self-propelled plate as a function of the inverse of distance. The x-axis is labeled with the inverse of distance, and the y-axis is labeled with the ratio of average propulsive velocity to reference velocity. Four different data sets are represented by different symbols: triangles, squares, diamonds, and circles, each corresponding to different stiffness values. The right graph shows the cost of transport as a function of the inverse of distance. The x-axis is labeled with the inverse of distance, and the y-axis is labeled with the cost of transport in joules per kilogram per meter. The same four data sets are represented by the same symbols. The graphs illustrate how the propulsive velocity and cost of transport vary with distance and stiffness.

Figure 9.

$(a)$ The average propulsive velocity and $(b)$ the cost of transport of the self-propelled plate as functions of $D_L^{-1}$ across different $f^*$ . Here, $H_L=1.5, A_L=0.5, K_p=3.5, K_b=3.0$ .

Figure 9. Long description

The image contains two line graphs side by side. The left graph shows the average propulsive velocity of a self-propelled plate as a function of the inverse of the dimensionless distance between the plate and the wall. The x-axis is labeled with the inverse of the dimensionless distance, and the y-axis is labeled with the normalized propulsive velocity. Three different data series are represented by different symbols: downward triangles, diamonds, and squares, corresponding to different dimensionless frequencies. The right graph shows the cost of transport of the self-propelled plate as a function of the same parameter. The x-axis is labeled with the inverse of the dimensionless distance, and the y-axis is labeled with the cost of transport in joules per kilogram per meter. The same three data series are represented by the same symbols as in the left graph. The graphs illustrate how the presence of compliant blades modifies the propulsive performance of the self-propelled plate near the wall and how the flapping motion of the plate affects the vibration response of the compliant blade array.

Figure 8 shows the effect of the plate bending stiffness $K_p$ on the propulsive performance. Here, $H_L$ is set to $1.625$ so that the ground effect remains significant. For all values of $K_p$ considered, increasing $D_L^{-1}$ leads to a higher $\widetilde {U}$ and a lower ${\textit{COT}}$ , indicating simultaneous improvements in swimming speed and transport efficiency. However, the extent of this improvement depends on $K_p$ . In particular, the case $K_p=3.5$ , which yields the highest $\widetilde {U}$ overall, also exhibits the most pronounced enhancement as $D_L^{-1}$ increases. As a result, although its ${\textit{COT}}$ is higher than that of the $K_p=4.25$ case at $D_L^{-1}=0$ , it becomes lower when $D_L^{-1}\geqslant 1$ . Since $K_p=3.5$ provides relatively good overall propulsive performance, it is adopted in the remainder of the study.

The effect of the flapping frequency $f^*$ of the self-propelled plate on its propulsive performance is further examined. As shown in figure 9, across all $f^*$ , $\widetilde {U}$ and ${\textit{COT}}$ exhibit the same trends of variation: $\widetilde {U}$ increases monotonically and ${\textit{COT}}$ decreases monotonically with increasing $D_L^{-1}$ . As $f^*$ increases, both $\widetilde {U}$ and ${\textit{COT}}$ increase. Specifically, with $f^*$ increasing from 0.51 to 0.64, $\widetilde {U}$ increases by approximately $0.34$ (from 0.51 to 0.57) and $0.24$ (from 0.57 to 0.64), while ${\textit{COT}}$ increases by $0.72$ and $0.77$ , respectively. These results indicate that a higher flapping frequency of the plate enhances its propulsive velocity at the cost of decreased energy efficiency.

Figure 10.

The relationship between the propulsive Reynolds number ${\textit{Re}}_c$ and the flapping Reynolds number ${\textit{Re}}_f$ in $(a)$ linear coordinates and $(b)$ logarithmic coordinates. Here, $f^*=0.64, K_p=3.5, K_b=3.0, 0.1\leqslant A_L\leqslant 0.5$ .

Figure 10. Long description

The image contains two graphs. The first graph, labeled (a), plots the propulsive Reynolds number (Rec) against the flapping Reynolds number (Ref) in linear coordinates. Different symbols and colors represent various conditions, such as different values of HL and DL, and two specific cases labeled SW1 and SW2. The second graph, labeled (b), shows the same relationship in logarithmic coordinates, with the x-axis as Log10(Ref) and the y-axis as Log10(Rec). An annotation indicates a proportional relationship, Rec is proportional to Ref to the power of 3 over 2. The graphs illustrate how the propulsive Reynolds number varies with the flapping Reynolds number under different conditions.

Universal scaling laws have long been a core research focus in the study of biological and bio-inspired propulsive systems (Gazzola, Argentina & Mahadevan Reference Gazzola, Argentina and Mahadevan2014; Floryan et al. Reference Floryan, Van Buren, Rowley and Smits2017; Ayancik et al. Reference Ayancik, Zhong, Quinn, Brandes, Bart-Smith and Moored2019; Lin et al. Reference Lin, Wu and Zhang2021; Liu et al. Reference Liu, Liu and Huang2022). To investigate the scaling law for the self-propelled plate near the blade-covered wall, figure 10(a) presents the propulsive Reynolds number ${\textit{Re}}_c=\widetilde {U} L/\nu$ as a function of the flapping Reynolds number ${\textit{Re}}_f$ for different blade spacing $D_L$ and vertical plate–wall distance $H_L$ . Here, the varying $\widetilde {U}$ is achieved by changing the heaving amplitude of the plate leading edge in the range $0.1\le A_L\le 0.5$ . Here, ${\textit{Re}}_f$ is defined using the flapping amplitude $a_{t}$ at the trailing edge of the plate as ${\textit{Re}}_f=2\pi T^{-1} a_t L/\nu$ (Gazzola et al. Reference Gazzola, Argentina and Mahadevan2014; Liu et al. Reference Liu, Liu and Huang2022), in which $a_{t}$ is defined as

(3.4) \begin{equation} a_t = 0.5[\max (y_t)-\min (y_t)]. \end{equation}

For fixed $H_L$ and $D_L$ , ${\textit{Re}}_c$ (an indicator of propulsive velocity) increases monotonically with ${\textit{Re}}_f$ at a rate steeper than linear growth. Furthermore, for a given ${\textit{Re}}_f$ value, ${\textit{Re}}_c$ increases with decreasing $H_L$ and $D_L$ . The logarithmic plot (figure 10 $b$ ) further reveals an approximate power-law relationship between ${\textit{Re}}_c$ and ${\textit{Re}}_f$ , where the corresponding power-law exponent increases as $H_L$ and $D_L$ decrease.

For the plate moving above a smooth wall at $H_L=1.5$ (SW2), where the ground effect is negligible, the data follow the scaling law ${\textit{Re}}_c\sim {\textit{Re}}_f^{3/2}$ derived by Liu et al. (Reference Liu, Liu and Huang2022). This scaling law can be interpreted as follows. The thrust scaling for a self-propelled slender body follows $F_{\textit{thrust}}\sim (a_t/T)^2$ , as established by Gazzola et al. (Reference Gazzola, Argentina and Mahadevan2014) and Quinn et al. (Reference Quinn, Moored, Dewey and Smits2014). The drag scaling for the self-propelled plate is $F_{\textit{drag}}\sim \widetilde {U}^{4/3}$ , as thoroughly validated by Liu et al. (Reference Liu, Liu and Huang2022). By balancing the thrust and drag, we have $\widetilde {U}^{4/3}\sim (a_t/T)^2$ , i.e. $\widetilde {U}\sim (a_t/T)^{3/2}$ . Based on the definitions of ${\textit{Re}}_c$ and ${\textit{Re}}_f$ , this relationship can be further written as ${\textit{Re}}_c\sim {\textit{Re}}_f^{3/2}$ .

When the ground effect becomes significant, the trailing-edge displacement of the plate becomes asymmetric between the upper and lower half-periods. To distinguish the two half-periods, the maximum trailing-edge displacements during the upper and lower half-periods are defined as

(3.5) \begin{equation} a_{\textit{upper}} = \max (y_t-H),\quad a_{lower} = \max (H-y_t). \end{equation}

In the upper half-period, because the plate deformation remains largely unchanged, the thrust and drag obey the same scaling laws as in the unbounded domain:

(3.6) \begin{equation} F_{\textit{thrust}, \textit{upper}} \sim (a_{\textit{upper}}/T)^2,\quad F_{drag, upper}\sim \widetilde {U}^{4/3}. \end{equation}

In the lower half-period, when the trailing edge moves downwards after $t_1$ , the decrease in $a_{lower}$ is attributed to the enhanced vertical pressure force, which is generated by the high pressure between the plate and wall, as shown in figure 5 $(b)$ . Given the approximate symmetry of the plate motion during its approach towards and recession from the wall, it is reasonable to assume that the ground effect exerts an equivalent amplifying influence on both the positive and negative pressures beneath the plate. Specifically, the negative pressure induced during upward motion and the positive pressure induced during downward motion are both amplified by a factor $\delta$ . Therefore, the maximum trailing-edge displacement for the lower half-period is reduced by a factor $\delta$ , i.e.

(3.7) \begin{equation} a_{lower}= a_{\textit{upper}}/\delta , \end{equation}

whereas the thrust generated by the plate is enhanced by a factor $\delta$ , i.e.

(3.8) \begin{equation} F_{thrust, lower}\sim \delta\, F_{\textit{thrust}, \textit{upper}}. \end{equation}

In addition, since the drag is proportional to the effective frontal area of the plate, which scales linearly with the trailing-edge displacement, we have

(3.9) \begin{equation} F_{drag, lower}\sim \frac {a_{lower}\,F_{drag, upper}}{a_{\textit{upper}}}\sim \frac {F_{drag, upper}}{\delta }. \end{equation}

By taking the average of the two half-periods, the average thrust of the self-propelled plate over a full period scales as

(3.10) \begin{equation} F_{\textit{thrust}}\sim 0.5(F_{\textit{thrust},\, \textit{upper}} + F_{\textit{thrust},\, \textit{lower}})\sim 0.5(1+\delta )\big(a_{\textit{upper}}/T\big)^2. \end{equation}

Similarly, the drag over a full period scales as

(3.11) \begin{equation} F_{\textit{drag}}\sim 0.5(F_{\textit{drag},\, \textit{upper}} + F_{\textit{drag},\, \textit{lower}})\sim 0.5(1+1/\delta )\widetilde {U}^{4/3}. \end{equation}

By balancing the average thrust and drag over a full period, $F_{\textit{thrust}}\sim F_{\textit{drag}}$ , we have

(3.12) \begin{equation} 0.5(1+1/\delta )\widetilde {U}^{4/3}\sim 0.5(1+\delta )\big(a_{\textit{upper}}/T\big)^2\quad\Rightarrow\quad \widetilde {U}\sim \big(\sqrt {\delta }\,a_{\textit{upper}}/T\big)^{3/2}, \end{equation}

which can be further written as

(3.13) \begin{equation} {\textit{Re}}_c\sim \big({\textit{Re}}_{f,\delta }\big)^{3/2}. \end{equation}

Here, the corrected flapping Reynolds number is defined as

(3.14) \begin{equation} {\textit{Re}}_{f,\delta } = 2\pi T^{-1}\left(\sqrt {\delta }\,a_{\textit{upper}}\right) L/\nu . \end{equation}

Figure 11.

$(a)$ Relation between the propulsive Reynolds number ${\textit{Re}}_c$ and the corrected flapping Reynolds number ${\textit{Re}}_{f,\delta }$ in logarithmic coordinates. $(b)$ Relation between the propulsive Reynolds number ${\textit{Re}}_c$ and the corrected Strouhal number $St_{\delta }$ in logarithmic coordinates. Here, $f^*=0.64, K_p=3.5, K_b=3.0, 0.1\le A_L\le 0.5$ .

Figure 11. Long description

The image contains two scatter plots. The first plot (a) shows the relationship between the propulsive Reynolds number (Re_c) and the corrected flapping Reynolds number (Re_f,δ) in logarithmic coordinates. Different symbols and colors represent various blade spacing (D_L) and vertical plate-wall distances (H_L). The plot indicates that for fixed H_L and D_L, the propulsive Reynolds number increases monotonically with the flapping Reynolds number at a rate steeper than linear growth. The second plot (b) shows the relationship between the propulsive Reynolds number (Re_c) and the corrected Strouhal number (St_δ) in logarithmic coordinates. The data points follow an approximate power-law relationship. The trends and relationships are visually represented through the distribution of data points and the fitted lines in both plots.

The Strouhal number $St=a_t/T\widetilde {U}$ is also widely adopted to characterise self-propulsion (Gazzola et al. Reference Gazzola, Argentina and Mahadevan2014). Following the same correction strategy as for ${\textit{Re}}_{f,\delta }$ , a corrected Strouhal number $St_{\delta }$ is defined as

(3.15) \begin{equation} St_{\delta } = \frac {\sqrt {\delta }\,a_{\textit{upper}}}{T\widetilde {U}} \sim \frac {{\textit{Re}}_{f,\delta }}{{\textit{Re}}_c}. \end{equation}

Using the scaling relation (3.13), the relationship between ${\textit{Re}}_c$ and $St_{\delta }$ can be derived as

(3.16) \begin{equation} {\textit{Re}}_c\sim ({\textit{Re}}_{f,\delta })^{3/2}\quad\Rightarrow\quad {\textit{Re}}_c\sim ({\textit{Re}}_c\,St_{\delta })^{3/2}\quad\Rightarrow\quad {\textit{Re}}_c\sim St_{\delta }^{-3}. \end{equation}

The relations of ${\textit{Re}}_c$ with ${\textit{Re}}_{f,\delta }$ and $St_{\delta }$ are displayed in figures 11 $(a)$ and 11 $(b)$ , respectively. Here, $a_{\textit{upper}},\,a_{lower}$ and $\delta$ are calculated post hoc using (3.5) and (3.7). All data points collapse onto the black solid lines given by (3.13) and (3.16), confirming the validity of the proposed scaling laws in the current Reynolds number range.

3.3. Vibration characteristics of the wall-mounted blades

Figure 12.

Spatiotemporal distribution of the horizontal displacement ${\rm d}x$ at the tips of all blades in the computational domain. The horizontal coordinate denotes the $x$ -position of the corresponding blade root, and the vertical coordinate denotes the dimensionless time. Here, $(a)\,f^*=0.51$ , $(b)\,f^*=0.57$ , $(c)\,f^*=0.64$ . The two black dashed lines denote the trajectory of the plate leading edge and that shifted upwards by $T$ . The red dashed line marks a wavefront of the FPTW. Here, $A_L=0.5, D_L=0.25, H_L=1.5, K_p=3.5, K_b=3.0$ .

Figure 12. Long description

Three graphs illustrate the spatiotemporal distribution of horizontal displacement at the tips of all blades in a computational domain. The horizontal axis represents the position of the corresponding blade root, while the vertical axis denotes dimensionless time. The two black dashed lines indicate the trajectory of the plate leading edge and that shifted upwards. The red dashed line marks a wavefront of the FPTW. The graphs are divided into three regimes: Regime A, Regime B, and Regime C. The inset in the second graph shows a detailed view of the displacement patterns.

The vibration of the wall-mounted compliant blades induced by the heaving motion of the self-propelled plate is examined under the parameter settings $A_L=0.5$ , $D_L=0.25$ , $K_p=3.5$ , $K_b=3.0$ , $H_L=1.5$ . Figure 12 presents the spatiotemporal distribution of the horizontal displacement at the blade tip, denoted by ${\rm d}x$ , with positive and negative values indicating rightward and leftward deflections, respectively. Here, the tip displacements ${\rm d}x$ of all blades in the computational domain are shown. In addition, the horizontal coordinate denotes the $x$ -position of the corresponding blade root, and the vertical coordinate denotes the dimensionless time. The time interval $tU_{\textit{ref}}/L \in [24, 40]$ , corresponding to $t/T\in [15, 25]$ for $f^*=0.64$ , is selected because the entire blade array has reached a statistically periodic state.

Three distinct regimes, labelled A, B and C, are identified in figure 12, which are separated by the trajectory of the plate’s leading edge and that shifted upwards by one flapping period $T$ . In regime A, since the self-propelled plate has not yet arrived, the blades exhibit negligible vibration, except for those in the immediate upstream of the leading-edge trajectory. In regime B, as the plate passes over the blades within one flapping period, the blades experience intense excitation, leading to a sudden increase in their vibration amplitude. In regime C, after the plate has moved away, the blades exhibit free vibrations, with their amplitude gradually decaying over time due to viscous dissipation. Notably, a travelling-wave pattern exists in regime C, which propagates along the direction of plate motion at a speed significantly faster than the plate’s propulsive velocity $\widetilde {U}$ . This pattern is herein referred to as the forward-propagating travelling wave (FPTW), with one of its wavefronts marked by the red dashed line in the figures. The phase velocity of the FPTW is denoted by $U_w$ . As the flapping frequency $f^*$ of the plate increases, both $\widetilde {U}$ and $U_w$ increase, corresponding to reduced slopes in the spatiotemporal diagrams. Meanwhile, a higher $\widetilde {U}$ enables the plate to travel a greater distance within one flapping period, thereby enlarging the spatial extent of regime B.

From a spatial perspective, figure 13 $(a)$ presents the distribution of horizontal displacement at the blade tips at $t/T = 19.00$ . In regime A, a small number of blades upstream of the leading edge of the plate are pre-excited by the plate, a phenomenon hereafter referred to as the upstream influence, leading to leftward displacements at their tips. Moreover, the tip displacements of these blades decrease monotonically as the distance from the leading edge increases. The region where the displacement exceeds $5\,\%$ of that at the plate leading edge is indicated by a red line, whose streamwise extent is approximately $3L$ . By contrast, the blades in regime B exhibit a sharp increase in tip displacement over a narrow streamwise interval. In regime C, the tip displacement decays progressively with increasing distance from the plate.

Figure 13.

$(a)$ Spatial distribution of horizontal displacement at the blade tips at $t/T = 19.00$ . $(b)$ Time history of the horizontal displacement at the tip of a representative blade located at $x/L=-17.25$ . The red lines highlight regions with significant displacement in regime A. The dimensionless parameters and the definition of the black dashed lines are the same as in figure 12(c), i.e. $f^*=0.64, A_L=0.5, D_L=0.25, H_L=1.5, K_p=3.5, K_b=3.0$ .

Figure 13. Long description

The image contains two line graphs side by side. The left graph shows the spatial distribution of horizontal displacement at the blade tips as a function of x over L, with the x-axis labeled x over L ranging from −70 to 0 and the y-axis labeled dx over L ranging from −0.10 to 0.10. The right graph shows the time history of the horizontal displacement at the tip of a representative blade as a function of t times U ref over L, with the x-axis labeled t times U ref over L ranging from 0 to 40 and the y-axis labeled dx over L ranging from −0.10 to 0.10. Both graphs feature a black line representing the displacement data and red lines highlighting regions with significant displacement in regime A. The black dashed lines in both graphs mark specific dimensionless parameters as defined in figure 12(c).

Figure 14.

Spatiotemporal spectral distributions of horizontal displacements at the blade tips, for $(a)\,f^*=0.51$ , $(b)\,f^*=0.64$ . The slope of the red dashed line represents the group velocity $(U_g=-\textrm {d} f/\textrm {d} k)$ . The red circle marks the most energetic spatiotemporal frequency.

Figure 14. Long description

A scatter plot showing spatiotemporal spectral distributions of horizontal displacements at blade tips. The x-axis represents the dimensionless wavenumber kL, ranging from −1.0 to 1.0. The y-axis represents the dimensionless frequency fL divided by a reference velocity Uref, ranging from −1.6 to 1.6. The plot contains several data points forming distinct patterns. A red dashed line indicates the group velocity, sloping downward from left to right. A red circle highlights the most energetic spatiotemporal frequency, located near the center of the plot. The label ‘FPTW’ is present near the center. The color gradient bar on the right indicates the magnitude of the spectral distributions, with values ranging from 0 to 3 times 10 to the power of −3.

From a temporal perspective, figure 13 $(b)$ presents the time history of the horizontal displacement at the tip of a representative blade located at $x/L=-17.25$ . The vibration process is also partitioned into three regimes. In regime A, before the arrival of the leading edge of the self-propelled plate, the tip displacement exhibits slight oscillation due to the upstream influence. The most significant wave is highlighted in a red line, whose temporal extent is approximately $0.8T$ . In regime B, the tip displacement increases rapidly over a short time interval, and exhibits irregular oscillations. In regime C, the tip displacement exhibits a typical underdamped free vibration, characterised by a decaying amplitude profile and dimensionless vibration frequency 0.79, which is close to its natural frequency $(f_n^{inv}L/U_{\textit{ref}}=0.73)$ in inviscid fluid. Here, the natural frequency $f_n^{inv}$ is computed by

(3.17) \begin{equation} f_n^{inv}=\frac {\varLambda ^2}{2\pi L^2}\sqrt {\frac {EI}{m+0.25C_m\pi \rho L}}, \end{equation}

where $\varLambda = 1.875$ is the first positive root of the equation

(3.18) \begin{equation} 1+\cos {\varLambda }\cosh {\varLambda }=0, \end{equation}

and $C_m \approx 1$ represents the added-mass effect (Van Eysden & Sader Reference Van Eysden and Sader2006; Brunetto et al. Reference Brunetto, Fortuna, Graziani and Strazzeri2008; Xiong et al. Reference Xiong, Gao, Lu and Chen2024).

These observations reveal that the compliant blades exhibit qualitatively distinct vibration characteristics in regimes B and C. To gain deeper insight into these mechanisms, a 2-D spatiotemporal fast Fourier transform (FFT) analysis on the spatiotemporal displacement fields is performed, as presented in figure 14. Due to the real-valued nature of the displacement signals, the spectra exhibit conjugate symmetry about the origin in the $(k, f)$ plane, where $k$ denotes the wavenumber, and $f$ denotes the frequency.

The harmonic components extracted from the FFT align along straight lines, with the slope of these lines (indicated by the red dashed line) defining the group velocity:

(3.19) \begin{equation} U_g = -\frac {\textrm {d} f}{\textrm {d} k}. \end{equation}

The negative sign is added because the self-propelled plate moves in the negative $x$ -direction. Physically, the group velocity characterises the propagation velocity of wave packets formed by the superposition of multiple harmonic components, as well as the energy transfer velocity within the vibrating blade array. Notably, the group velocity $U_g$ exhibits good agreement with the average propulsive velocity $\widetilde {U}$ of the self-propelled plate, as summarised in table 2. From the spatiotemporal diagram of blade displacements shown in figure 12, only regime B, where the blade vibrations are directly driven by the flapping motion of the plate, exhibits a wave packet structure that follows the plate’s position. The group velocity of the wave packet thus corresponds to the propagation velocity of regime B, and the associated vibration pattern is hereafter termed the flapping-driven mode.

Table 2.

The dimensionless average propulsive velocity $\widetilde {U}/U_{\textit{ref}}$ of the self-propelled plate, the group velocity $U_g/U_{\textit{ref}}$ of the spatiotemporal spectrum, and the wave speed $U_w/U_{\textit{ref}}$ , frequency $f_wL/U_{\textit{ref}}$ and wavelength $\lambda _w/L$ of the FPTW.

Figure 15.

Schematics of the plate–blades interaction process: $(a)$ the plate moves downwards; $(b)$ the plate moves upwards.

Figure 15. Long description

A diagram illustrating the interaction process between a plate and blades with fluid dynamics. The left panel shows the plate moving downwards, creating high pressure as it squeezes the fluid, with red arrows indicating the direction of motion. The right panel shows the plate moving upwards, creating low pressure as it draws the fluid, with blue arrows indicating the direction of motion. The interaction between the plate and blades is depicted through the movement of the fluid.

Figure 16.

The dimensionless pressure fields at $t/T=19.00$ . The leading edge of the plate is positioned at the average height and is moving downwards. Here, $(a)\,f^*=0.51$ , $(b)\,f^*=0.57$ , $(c)\,f^*=0.64$ . Horizontal arrowed lines mark the wavelength $\lambda _w$ of the FPTW. Here, $A_L=0.5, H_L=1.5, D_L=0.25, K_p=3.5, K_b=3.0$ . See supplementary movie 2 for the pressure evolution.

Figure 16. Long description

The heat map displays dimensionless pressure fields around a plate with three distinct wavelengths. The plate’s leading edge is positioned at an average height and moves downward. The wavelengths are marked by horizontal arrowed lines. The color scale ranges from blue to red, indicating varying pressure intensities. The first section shows a wavelength of approximately 4.88L, the second section shows a wavelength of approximately 6.15L, and the third section shows a wavelength of approximately 9.44L. The pressure fields exhibit distinct patterns and intensities across the different wavelengths.

Figure 15 presents two schematics of the interaction between the compliant blades and the nearby flapping plate, thereby offering a physical insight into the flapping-driven mode. As illustrated in figure 15 $(a)$ , when the plate descends towards the blade array, it squeezes fluid into the inter-blade gaps, and induces a local high-pressure region beneath the plate. Due to fluid incompressibility and confinement imposed by adjacent blades, the fluid is forced to escape laterally through the blade tips. This gives rise to tip vortices of opposite signs (anticlockwise at the left tip, clockwise at the right tip), and induces outward blade deformation. Conversely, when the plate ascends and moves away from the blades, as illustrated in figure 15 $(b)$ , fluid is drawn from the inter-blade gaps, inducing a local low-pressure region beneath the plate. The replenishing flow enters primarily from the outer sides of the blade tips, generating vortices with reversed rotational directions, and inducing inward blade deformation.

The most energetic spatiotemporal mode, indicated by the red circle in figure 14, exhibits a phase velocity in close agreement with the wave speed $U_w$ of the FPTW, and is thus identified as the FPTW in regime C. Its key parameters, the wave frequency $f_w$ , wavelength $\lambda _w=1/k_w$ , and wave speed $U_w=f_w/k_w$ are listed in table 2. Notably, $f_w$ is almost independent of the flapping frequency $f^*$ of the plate, and approximately equals the natural frequency $(f_n^{inv}L/U_{\textit{ref}}=0.73)$ of the blades. The $\lambda _w$ value is significantly larger than the distance travelled by the plate in each flapping period, and it increases dramatically from 4.88 to 9.44 as the flapping frequency $f^*$ of the plate grows from 0.51 to 0.64.

Figure 16 presents the dimensionless pressure fields for different flapping frequencies $f^*$ of the self-propelled plate at $t/T=19.00$ , when the plate’s leading edge is positioned at the average height and is moving downwards. A high-pressure region is observed beneath the plate, resulting from the fluid-squeezing effect generated by the downward motion of the plate. Notably, as $f^*$ increases, the spatial extent of the high-pressure region expands. Downstream of the plate, alternating high- and low-pressure regions propagating in the negative $x$ -direction are observed within the blade array (see supplementary movie 2 for the pressure evolution). Furthermore, the spacing between two adjacent low-pressure regions is nearly uniform and exhibits good agreement with the wavelength $\lambda _w$ of the FPTW, as marked by the arrowed lines in the figures. This observation confirms the strong correlation between the FPTW and the downstream pressure distribution within the blade array. As $f^*$ increases, corresponding to an increase in the propulsive velocity of the plate, the wave speed $U_w$ of the FPTW increases, indicating that $U_w$ is not an intrinsic property of the blade array, but depends on external excitations. Moreover, the motion of wake vortices has little correlation with the FPTW, indicating that the FPTW originates from the cooperative underdamped vibrations of downstream blades.

3.4. Resonance between the self-propelled plate and the compliant blade array

Figure 17.

The average strain energy, kinetic energy and mechanical energy of $(a)$ the self-propelled plate and $(b)$ the compliant blade array as functions of the blade bending stiffness $K_b$ . Here, $f^*=0.51, A_L=0.5, H_L=1.50, D_L=0.50, K_p=3.5$ .

Figure 17. Long description

The image contains two line graphs labeled (a) and (b), each showing the relationship between blade bending stiffness (Kb) and energy components. Graph (a) displays the average strain energy (Es,p), kinetic energy (Ek,p), and mechanical energy (Et,p) of a self-propelled plate as functions of blade bending stiffness. Graph (b) shows the average strain energy (Es,b), kinetic energy (Ek,b), and mechanical energy (Et,b) of a compliant blade array as functions of blade bending stiffness. Both graphs have the x-axis labeled as Kb and the y-axis labeled as E/Eref. The data points are represented by different symbols and colors: blue circles for strain energy, red triangles for kinetic energy, and black triangles for mechanical energy. A vertical dashed line labeled Kr is present in both graphs, indicating a critical point. The trends show that as Kb increases, the mechanical energy remains relatively constant, while the strain and kinetic energies exhibit different behaviors. All values are approximated.

Resonance is a common phenomenon in the forced vibration of elastic structures. To investigate the resonance characteristics in the present system, the strain energy $E_s$ , kinetic energy $E_k$ , and total mechanical energy $E_t$ are computed for both the self-propelled plate and the compliant blade array. For a single elastic beam, $E_s$ is defined similarly to (3.1), and $E_k$ is defined as

(3.20) \begin{equation} E_{k}=\int _0^{L}0.5m\, \left |\frac {\partial \boldsymbol{X}(s,t)}{\partial t}\right |^2\textrm {d} s. \end{equation}

The corresponding energies for the plate are denoted as $E_{s,p}$ , $E_{k,p}$ and $E_{t,p}=E_{s,p}+E_{k,p}$ . For the blade array, the energies are computed by summing the contributions of all individual blades:

(3.21) \begin{equation} E_{s,b}=\sum _{i=1}^N E_{s,bi}, \quad E_{k,b}=\sum _{i=1}^N E_{k,bi}, \quad E_{t,b}=E_{s,b}+E_{k,b}. \end{equation}

It is noted that $E_{s,b}$ , $E_{k,b}$ and $E_{t,b}$ also vary periodically for $t/T\ge 15$ , and the reference energy is defined as $E_{\textit{ref}}=\rho U_{\textit{ref}}^2L^2$ .

Figure 17 presents the average strain energy, kinetic energy and total mechanical energy of the plate and the blade array as functions of the blade bending stiffness $K_b$ . For the plate, its average strain energy $\widetilde {E_{s,p}}$ (blue line) is significantly smaller than its average kinetic energy $\widetilde {E_{k,p}}$ (red line), contributing negligibly to the average total mechanical energy $\widetilde {E_{t,p}}$ (black line). As $K_b$ increases, both $\widetilde {E_{k,p}}$ and $\widetilde {E_{t,p}}$ first increase rapidly, then decrease to a local minimum, and finally rise gradually. In contrast, for the blade array, its average strain energy $\widetilde {E_{s,b}}$ , average kinetic energy $\widetilde {E_{k,b}}$ , and average total mechanical energy $\widetilde {E_{t,b}}$ exhibit the opposing trend: they initially decrease rapidly, then rise to a local maximum, and finally decay gradually to zero as the blades become increasingly rigid. Resonance occurs when $\widetilde {E_{t,b}}$ reaches a local maximum, where the corresponding $K_b$ (marked by the black dashed line) is defined as the resonant bending stiffness $K_r$ . Furthermore, $\widetilde {E_{t,b}}$ is significantly smaller than $\widetilde {E_{t,p}}$ , indicating that the energy stored in the blade array accounts for only a small fraction of the total energy in the system.

The resonance phenomenon is also confirmed from the frequency perspective. Table 3 summarises the dimensionless flapping frequency $f^*$ of the self-propelled plate and the natural frequency $f_n^{inv} L/U_{\textit{ref}}$ of the blades. Here, $f_n^{inv}$ is calculated through (3.17) by setting the dimensionless bending stiffness to $K_r$ . In all cases, $f^*$ is close to but approximately 20 % lower than $f_n^{inv} L/U_{\textit{ref}}$ , a phenomenon also documented in the work of Xiong et al. (Reference Xiong, Gao, Lu and Chen2024). This is because the resonance frequency of a strongly damped system is lower than its natural frequency without damping, as illustrated in the vibration theory (Rao Reference Rao2010).

Table 3.

The dimensionless flapping frequency $f^*$ and the natural frequency $f_n^{inv} L/U_{\textit{ref}}$ of the compliant blades when resonance occurs.

Table 3. Long description

The table presents a comparison between the dimensionless flapping frequency and the natural frequency of compliant blades when resonance occurs. It consists of two rows and three columns. The first row contains the headers ‘f*’ and three numerical values: 0.38, 0.51, and 0.64. The second row is labeled ‘f_n^inv L/U_ref’ and lists the values 0.50, 0.66, and 0.78. The data indicates that the dimensionless flapping frequency is consistently lower than the natural frequency, approximately 20 percent lower, which aligns with findings from previous studies. This discrepancy is attributed to the damping effects in the system, as explained by vibration theory.

Figure 18.

The average propulsive velocity $\widetilde {U}$ of the self-propelled plate, and average mechanical energy $\widetilde {E_{t,b}}$ of the compliant blade array, as functions of the blade bending stiffness $K_b$ , for $(a)\,f^*=0.38$ , $(b)\,f^*=0.51$ , $(c)\,f^*=0.64$ . Here, $A_L=0.5, H_L=1.5, D_L=0.5, K_p=3.5$ . The red dashed lines mark the resonance stiffness $K_r$ . The black dashed lines represent $\widetilde {U}$ of the corresponding cases without blades.

Figure 18. Long description

The image contains three line graphs labeled (a), (b), and (c). Each graph shows two data series: the average propulsive velocity of the self-propelled plate (black line with square markers) and the average mechanical energy of the compliant blade array (red line with circle markers). The x-axis represents the blade bending stiffness, while the y-axes represent the normalized propulsive velocity and mechanical energy. The red dashed lines mark the resonance stiffness, and the black dashed lines represent the corresponding cases without blades. In all three graphs, the propulsive velocity remains relatively stable, while the mechanical energy decreases as the blade bending stiffness increases. The graphs illustrate how the presence of blades affects the propulsive velocity and mechanical energy of the system.

Figure 19.

The spatiotemporal distribution of the total mechanical energy $E_{t, bi}$ of the compliant blades, with bending stiffnesses $(a)$ $K_b=1.5$ , $(b)$ $K_b=2.5$ (resonance bending stiffness), $(c)$ $K_b=10$ . The energy proportions of the three regimes (A, B and C) are marked in the figures. Here, $f^*=0.51, A_L=0.5, D_L=0.5, H_L=1.5, K_p=3.5$ .

Figure 19. Long description

Three graphs illustrate the spatiotemporal distribution of the total mechanical energy of compliant blades with varying bending stiffnesses. Each graph is divided into three regimes labeled A, B, and C, with their respective energy proportions marked. The x-axis represents the normalized position, while the y-axis represents the normalized time. The color gradient indicates the energy levels, with darker shades representing higher energy. The dashed lines in each graph denote specific boundaries or transitions between the regimes. Regime A occupies 1.4 percent, 1.0 percent, and 2.6 percent of the total energy in the three graphs respectively. Regime B occupies 54.2 percent, 42.1 percent, and 77.9 percent. Regime C occupies 44.4 percent, 56.9 percent, and 19.5 percent. The graphs show how the energy distribution changes over time and space for different bending stiffnesses.

Figure 18 presents the average propulsive velocity $\widetilde {U}$ of the self-propelled plate, and average total mechanical energy $\widetilde {E_{t,b}}$ of the blade array, as functions of the blade bending stiffness $K_b$ . The black dashed lines denote $\widetilde {U}$ for the corresponding cases without blades, and each of them lies below that for the blade-covered wall. Since the magnitudes of $\widetilde {U}$ and $\widetilde {E_{t,b}}$ differ dramatically across the three frequencies, different $y$ -axis ranges are adopted in each subplot to better visualise the respective trends. The variation of $\widetilde {U}$ with $K_b$ exhibits a negative correlation with that of $\widetilde {E_{t,b}}$ , i.e. $\widetilde {U}$ first increases rapidly, then decreases to a local minimum, and finally rises gradually with increasing $K_b$ . As the flapping frequency $f^*$ of the plate increases, although both $\widetilde {U}$ and $\widetilde {E_{t,b}}$ increase, the local minimum of $\widetilde {U}$ and local maximum of $\widetilde {E_{t,b}}$ are flattened, and the resonant bending stiffness $K_r$ (marked by red dashed lines) increases. Furthermore, at $f^*=0.38$ , the local minimum of $\widetilde {U}$ nearly coincides with the resonant stiffness $K_r$ ; at higher flapping frequencies ( $f^*=0.51$ and 0.64), $\widetilde {U}$ reaches a local minimum at a higher $K_b$ than $K_r$ .

Figure 20.

The dimensionless (a,b,c) vorticity field and (d,e,f) $Q$ field ( $Q^*=QL^2/U_{\textit{ref}}^2$ ) at $t/T=19.25$ for different blade bending stiffness values: (a,d) $K_b=1.5$ , (b,e) $K_b=2.5$ (resonance bending stiffness), (c,f) $K_b=10$ . Here, $f^*=0.51, A_L=0.5, H_L=1.5, D_L=0.5, K_p=3.5$ .

Figure 20. Long description

A heat map displays the vorticity field and field values for different blade bending stiffness values near a flat wall. The map is divided into six subplots, with the left column showing vorticity fields and the right column showing field values. Each row corresponds to a different blade bending stiffness value: low, resonance, and high. The vorticity field subplots (a, b, c) use a color scale ranging from blue to red, indicating negative to positive vorticity values. The field value subplots (d, e, f) use a color scale ranging from blue to red, indicating lower to higher field values. The subplots reveal distinct patterns and vortex formations, with notable differences in vortex structures and intensities across the different stiffness values. The presence of downstream tip vortices is highlighted in subplot (e). The overall trend shows variations in vortex interactions and field distributions as the blade bending stiffness changes.

Figure 19 presents the spatiotemporal distribution of the total mechanical energy $E_{t, bi}$ of individual compliant blades at $f^*=0.51$ for three bending stiffnesses: $K_b=1.5$ , $K_b=K_r=2.5$ and $K_b=10$ . Being consistent with figure 12, the boundaries of the three regimes A, B and C are marked by black dashed lines. The results indicate that the total mechanical energy of blades in regimes B and C accounts for nearly the entire energy of the blade array. However, their respective proportions vary with $K_b$ . For the blade with $K_b=1.5$ (softer than the resonant bending stiffness $K_r$ ), 54.2 % of the total mechanical energy is distributed in regime B, exceeding that (44.4 %) distributed in regime C. At the resonant stiffness $K_b=K_r=2.5$ , a larger proportion of energy (56.9 %) is distributed in regime C, because the blades undergo more intense excitation and thus store more mechanical energy. For the stiffer blades with $K_b=10$ , the energy is primarily concentrated in regime B, which accounts for $77.9\,\%$ of the total energy. These results demonstrate that resonance facilitates energy storage in downstream regions.

Figure 20 presents distributions of the vorticity and the second invariant of the velocity gradient tensor $Q$ at $t/T=19.25$ for different blade bending stiffnesses $K_b$ . Here, the positive $Q$ is also used to indicate concentrated vortices, known as the $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). Upstream of the leading edge and beneath the plate, the blades are deflected leftwards due to the high-pressure region between the plate and the bottom wall, resulting in the shedding of concentrated positive vortices from the blade tips. As $K_b$ increases, the horizontal displacement of the blade tips decreases progressively; however, the strength of the tip vortices increases. Beneath the trailing-edge vortex and its feeding shear layer, negative vortices are formed at the blade tips. For the resonant case with $K_b=K_r=2.5$ , concentrated tip vortices are observed beyond $2L$ downstream of the plate, confirming that resonance facilitates the vibration of blades in the wake.