3.1 The effect of travelling waves on the flow separation of the mean flow

In this section, the mean flow is visualized to see the impact of both frequency and wavelength of travelling waves on flow separation. The impact of frequency and wavelength of the travelling wave on flow separation is reported quantitatively by defining a reattachment length for the separated flow. The reattachment length is defined here as the distance between the leading edge and the point where the time-averaged streamwise velocity becomes greater than zero ( $u_{z}>0$ ) on the midplane. For oscillating plates, the reattachment point is found over the buffer zone of motion, which contains all the points that the moving plate has passed through at least once in a cycle. This buffer zone is shown schematically by the grey region in figure 4(a). This buffer zone is defined because the time averaging of values, e.g. velocities, is valid only for the fluid nodes and not the solid nodes. Therefore, the time-averaged values in this study, including the pressure and velocity values, are for the nodes that are out of this buffer zone. The reattachment point can be visually detected by the streamline normal to the plate, as shown schematically in figure 4(a). The exact reattachment length, denoted by $R$ , is reported in table 2.

Figure 3.

Out-of-plane vorticity contours and streamlines of the mean flow on the midplane for nine cases. The left and right columns show the effect of frequency and wavelength on flow separation, respectively.

Table 2.

Values of the mean drag coefficient $(C_{D})$ , relative drag coefficient $(C_{D_{rel}})$ , viscous drag $(C_{D_{f}})$ , form drag $(C_{D_{p}})$ , relative drag $(C_{D_{rel}})$ , lift coefficient $(C_{L})$ , reattachment length ( $R$ ) and the r.m.s. values of drag and lift coefficients for different cases. The relative drag $(C_{D_{rel}})$ is defined as the drag reduction relative to case 1 (flat inclined) along the plate chord ( $C_{D_{rel}}=(C_{D}^{flat\,inclined}-C_{D}^{undulating})/\cos \unicode[STIX]{x1D703}$ ), which is a measure of the thrust force produced by the wave. All force coefficients are non-dimensionalized with $\unicode[STIX]{x1D70C}U^{2}L^{2}$ , as the area of the plate is $2L^{2}$ . Case 1 is the non-moving flat inclined plate and case 9 is the pitching inclined plate, for which the wave speed is not defined.

To visualize the flow separation, the time-averaged streamlines and out-of-plane vorticity on the midplane of the nine cases (table 2) are presented in figure 3. Figure 3(a–d) shows the cases with similar wavelength but different frequency, i.e. the effect of frequency, while figure 3(e–h) shows the cases with similar frequency but different wavelength, i.e. the effect of wavelength. Owing to the sharp leading edge of the plate, the flow separates from the leading edge for all cases and forms a clockwise recirculatory region near the leading edge (figure 3), which is depicted schematically in figure 4(a). This recirculatory (separation) region is represented by the negative (clockwise) vorticity (figure 3).

Over the flat inclined plate (figure 3 a), the separated shear layer, which is inviscidly unstable, transitions to turbulence and reattaches near the trailing edge ( $0.93L$ ) as the turbulent flow energizes the near-wall flow (Mueller & Batil Reference Mueller and Batil1982). The travelling wave at low frequency $f^{\ast }=6$ does not affect the flow separation (figure 3 b) and the reattachment length is similar to that for the flat inclined plate (table 2). By increasing $f^{\ast }$ from 6 to 20 and 30 ( $C/U$ from 1.2 to 4 and 6), however, the separated shear layer reattaches earlier at $0.78L$ and $0.70L$ from the leading edge (table 2), respectively, as observed in figure 3(c,d). In fact, comparing the streamlines in the left column of figure 3 shows that increasing the wave speed by increasing the frequency in this range reduces flow separation monotonically.

Figure 4.

(a) Schematic of the average flow field. The grey zone is the buffer zone of the motion, which includes all points the plate passes through at least once in a cycle. (b) The non-dimensional time-averaged streamwise velocity profile $\overline{u_{z}^{\ast }}$ at midplane.

The wavelength variation at constant frequency did not produce a monotonic pattern for decrease/increase of the flow separation (figure 3 e–h). Increasing $\unicode[STIX]{x1D706}^{\ast }$ from 0.15 (case 3) to 0.2 (case 4) and 0.5 (case 6) at constant $f^{\ast }=20$ , i.e. increasing $C/U$ from 3 to 4 and 10, increases the separation length from $0.73L$ to $0.78L$ and $0.89L$ , respectively (table 2 and compare figure 3 e, c and f). By increasing the wavelength further from 0.5 (case 6) to 1.0 (case 7) and 2.0 (case 8) ( $C/U$ from 10 to 20 and 40), however, the reattachment length reduces to $0.57L$ and then increases to $0.68L$ (table 2), respectively, as observed in figure 3(f–h). For the pitching plate (case 9), the flow separation increases compared to case 8 ( $\unicode[STIX]{x1D706}^{\ast }=2.0$ ), i.e. reattachment length increases to $0.82L$ (table 2 and figure 3 h,i). In summary, when the wave speed increases by increasing the wavelength (figure 3 e–h), in contrast to the trend observed for frequency (figure 3 a–d), the flow reattachment length neither decreases nor increases monotonically.

Figure 5.

Streamwise velocity field over the plane adjacent to the top surface of the buffer zone of the motion: (a) flat inclined plate, (b) undulating plate case 2 ( $f^{\ast }=6$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), (c) undulating plate case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ) and (d) undulating plate case 6 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.5$ ). The green rectangle is the plate. (e) The corresponding reattachment line plotted along the span of the plate by detecting the zero streamwise velocity for these cases. Here $x=0$ and $x=2L$ are tips, and $x=L$ is the midplane of the plate. (f) Time-averaged vortices on the suction side, including LEV and tip vortex (TV) identified by $Q$ -criterion with $Q=10$ and coloured with non-dimensional streamwise vorticity ( $\unicode[STIX]{x1D714}_{z}^{\ast }$ ).

To clearly show the reattaching and reverse flows, figure 4(b) presents the time-averaged streamwise velocity profile at six sections on the midplane. The time-averaged streamwise velocity profile of cases 1 and 2 are approximately identical in all sections of figure 4(b), which agrees with the results of table 2, indicating that low $f^{\ast }$ (case 2) does not change the reattachment length compared to the flat inclined one (case 1). As $f^{\ast }$ increases to 20, the reverse flow increases slightly near the leading edge, in sections L1 and L2 of figure 4(b), for cases 6 and 8, whereas it decreases in section L2 for case 7, which has the minimum reattachment length (table 2). Near the trailing edge, at section L4 of figure 4(b), the time-averaged streamwise velocity of cases 3–8 is higher than for cases 1 and 2 as the flow has reattached earlier for cases 3–8 (table 2).

To investigate the flow reattachment at different spanwise locations, the time-averaged streamwise velocity field on the plane that is adjacent to the top surface of the buffer zone of the motion is shown for cases 1, 2, 4 and 6 in figure 5(a–d). The velocity vectors in figure 5(a–d) are the projected velocity field to this plane. It can be observed in figure 5(a–d) that the reverse flow is higher near the midplane ( $x=L$ ) for all cases and decreases towards the tips of the plate as the spanwise velocity increases. Figure 5(e) shows the reattachment lines of cases 1, 2, 4 and 6, which are plotted by detecting the boundary of zero streamwise velocity of figure 5(a–d), and the time-averaged tip vortex (TV) and LEV of the flat inclined plate are visualized using the $Q$ -criterion in figure 5(f). The profile of the reattachment length (figure 5 e) shows that the flow reattaches earlier as we move towards the tips from the midplane due to the impact of the tip vortex. The tip vortex, which has a higher streamwise vorticity compared to the LEV, induces a spanwise velocity near the tips (figure 5 a–d), which contributes to the earlier flow reattachment near the tips of the plate (Kim & Gharib Reference Kim and Gharib2010).

Figure 6.

Distribution of the non-dimensional turbulent kinetic energy in (a) flat inclined plate, (b) undulating plate case 2 ( $f^{\ast }=6$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), (c) undulating plate case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ) and (d) undulating plate case 6 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.5$ ). The undulations increase the TKE over the plate, but decrease it in the wake downstream of the trailing edge.

3.2 The effect of travelling waves on turbulence intensity

Travelling waves are known to reduce the turbulence intensity of turbulent boundary layers, e.g. turbulent channel flows (Shen et al. Reference Shen, Zhang, Yue and Triantafyllou2003; Nakanishi, Mamori & Fukagata Reference Nakanishi, Mamori and Fukagata2012; Mamori, Iwamoto & Murata Reference Mamori, Iwamoto and Murata2014), by decreasing the random component of the flow (Mamori et al. Reference Mamori, Iwamoto and Murata2014). However, the effect of travelling waves on the turbulent intensity in external flows, e.g. flow over an inclined plate, where the boundary layer is not fully turbulent, is unknown. Here, we investigate the effect of travelling waves on the turbulence intensity of the flow.

To investigate the effects of travelling waves on turbulence intensity, the turbulent kinetic energy (TKE) is shown for cases 1, 2, 4 and 6 in figure 6(a–d), respectively. The TKE of the flat inclined plate is non-zero on the plate’s suction side where the separated shear layer mixes with reverse flow near the wall, but zero on the pressure side. Similarly, the TKE is non-zero only on the suction side of the low-frequency case (case 2) (figure 6 b). The TKE of the flow adjacent to both sides of the undulating plate increases as $f^{\ast }$ and $\unicode[STIX]{x1D706}^{\ast }$ increase (figure 6 c,d). However, in case 4, which has the lowest reattachment length among these cases, the TKE over the plate at the separation zone is higher than in other cases (figure 6 c). In addition, the TKE is lower in the wake downstream of the plate for cases 4 and 6 compared to the flat inclined plate.

Figure 7.

Three-dimensional vortical structures visualized by the instantaneous isosurfaces of the $Q$ -criterion ( $Q=20$ ) from midplane to the tip. (a) Flat inclined plate, (b) undulating plate case 2 ( $f^{\ast }=6$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), (c) undulating plate case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ) and (d) undulating plate case 6 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.5$ ).

The decrease of TKE in the downstream wake for high-frequency undulatory cases (figure 6 c,d) can also be observed by the instantaneous 3-D vortical structures. Figure 7 shows the 3-D vortical structures visualized by the isosurface of the $Q$ -criterion. Comparing the wake of cases 4 and 6 with the flat inclined one (figure 7 a,c,d) reveals that the number of vortical structures have decreased for cases 4 and 6 compared to case 1, which confirms the decrease of turbulence intensity in the wake (figure 6 a,c,d). Furthermore, it can be observed that the shear layer separates from the sharp leading edge and breaks down into small vortices as it becomes unstable. The vortical structures of the flat inclined plate resemble the vortices visualized by Visbal (Reference Visbal2011) whose plate had the same aspect ratio ( $=2$ ) but the angle of attack was $8^{\circ }$ . It can also be observed that the flow is more spread laterally at the midplane compared to the regions near the tips of the plate.

3.3 The effect of travelling waves on the aerodynamic performance

The flow visualizations in the previous section showed that travelling waves can reduce flow separation, which can impact both the drag and lift forces. A reduction in drag force is expected because backward travelling waves are known to generate thrust, which propels aquatic swimmers forwards (Lighthill Reference Lighthill1960; Fish & Lauder Reference Fish and Lauder2006; Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2010). However, the effect of travelling waves on the lift coefficient cannot be predicted purely based on the thrust generation because travelling waves completely change the pressure field by creating a low-pressure region on both sides of the plate at high frequencies, as will be discussed below. In addition, the effects of travelling waves on the mean and instantaneous aerodynamic forces are discussed by comparing the trend of computed forces and the analytically derived ones using EBT (Lighthill Reference Lighthill1960).

To quantify the aerodynamic performance, the drag ( $C_{D}$ ), relative drag ( $C_{D_{rel}}$ ) and lift ( $C_{L}$ ) coefficients, and the r.m.s. values of fluctuations of drag and lift coefficients, are reported in table 2. The drag coefficient ( $C_{D}=F_{drag}/(\unicode[STIX]{x1D70C}U^{2}L^{2})$ ) is the total non-dimensional drag force, which consists of the viscous ( $C_{D_{f}}$ ) and form ( $C_{D_{p}}$ ) drag acting on the plate along the incoming flow ( $z$ direction); and the lift coefficient ( $C_{L}=F_{lift}/(\unicode[STIX]{x1D70C}U^{2}L^{2})$ ) is the total non-dimensional force acting normal to the incoming flow in the $y$ direction. The relative drag coefficient for each case is defined as the drag reduction relative to the flat inclined case along the plate ( $C_{D_{rel}}=(C_{D}^{flat\,inclined}-C_{D}^{undulating})/\cos \unicode[STIX]{x1D703}$ ), which is almost equivalent to the thrust force produced by the oscillations. The time-averaged results for the oscillating plates are averaged over 60 cycles and the averaging is started after 50 cycles.

Table 2 shows that the viscous drag is approximately 10 % of the form (pressure) drag. When the plate undulates at a high reduced frequency $(f^{\ast }\geqslant 20)$ , e.g. cases 3–8, the ratio of the viscous to the form drag increases because the form drag is reduced by the undulations (undulations generate thrust) while viscous drag is negligibly changed. In fact, the change in the viscous drag with respect to the flat inclined plate is less than 0.005 for all cases ( $|\unicode[STIX]{x0394}C_{D_{f}}|<0.005$ , where $\unicode[STIX]{x0394}C_{D_{f}}=C_{D_{f}}^{flat\,inclined}-C_{D_{f}}^{undulating}$ is the difference between the viscous drag of the flat inclined plate and an undulating one). Consequently, the reduction in form drag accounts for the majority of the reduction in drag coefficient ( $C_{D}$ ).

Comparing the time-averaged values, table 2 shows that in case 2 ( $f^{\ast }=6$ , $C/U=1.2$ ) the travelling wave does not improve the aerodynamic performance compared to the flat inclined plate. However, increasing the wave speed by increasing the reduced frequency to $f^{\ast }=20$ ( $C/U=4$ ) and $f^{\ast }=30$ ( $C/U=6$ ) increases the lift force by 16.7 % and 31.4 % and decreases the drag force by 50 % and 119.2 % (generates thrust), respectively. It is interesting to note that when the travelling wave generates thrust (positive relative drag in table 2) at high $f^{\ast }$ , e.g. cases 3–8, a jet-like profile velocity is observed in the wake (sections L5 of figure 4 b). In addition, the thrust produced in cases 5, 7, 8 and 9 is high, such that the drag ( $C_{D}$ ) is negative (thrust type). In these cases a streamwise flow can be observed in the middle of the plate (section L3 of figure 4 b), i.e. jet-like profile is initiated earlier (closer to the leading edge).

When the frequency is kept the same ( $f^{\ast }=20$ ), increasing the wave speed by increasing the wavelength from $\unicode[STIX]{x1D706}^{\ast }=0.15$ ( $C/U=3$ ) in case 3 to $\unicode[STIX]{x1D706}^{\ast }=0.2$ ( $C/U=4$ ) in case 4 decreases the lift by 10 % (case 3 is 29 % higher than the flat inclined plate) but increases the relative drag by 33 %. Similarly, when the wave speed is further increased by increasing the wavelength to 0.5 in case 6 ( $f^{\ast }=20$ , $C/U=10$ ), lift decreases but thrust (relative drag) increases compared to case 4. Nevertheless, $C_{L}$ is still 9.2 % higher than the lift of the flat inclined plate. Similar results such as the increase of thrust (relative drag) and the decrease of lift are observed by increasing the wavelength to 1.0 and 2.0 in cases 7 and 8, respectively. It can also be observed that the pitching motion (case 9) produces the maximum relative drag but it does not change the lift compared to the flat inclined case.

Theoretically, the time-averaged thrust (relative drag) increases with both frequency and wavelength as wave speed $C$ increases according to EBT (Lighthill Reference Lighthill1960) – see appendix C for the derivation:

(3.1) $$\begin{eqnarray}C_{T}=0.11\unicode[STIX]{x03C0}(2\unicode[STIX]{x03C0}f^{\ast }a_{max}^{\ast })^{2}\left(1-\left(\frac{1}{\unicode[STIX]{x1D706}^{\ast }f^{\ast }}\right)^{2}\right).\end{eqnarray}$$

Note that $1/(\unicode[STIX]{x1D706}^{\ast }f^{\ast })=U/C$ . The calculated relative drag (table 2) follows the trend predicted by EBT, i.e. the thrust (relative drag) increases with increasing wave speed through increasing $f^{\ast }$ and/or $\unicode[STIX]{x1D706}^{\ast }$ . However, the lift force does not follow the same trend.

Figure 8.

(a) Spanwise-averaged distribution of the lift coefficient along the plate with three points along the span of the plate. The symbols ○, ▪ and $+$ are the lift coefficient at $0.3L$ , $L$ (midplane) and $0.1L$ from the tip of the plate, respectively, for cases 1, 4 and 6. (b–e) Time-averaged pressure field at midplane for cases 1, 2, 4 and 6, respectively. (f) Time history of cycle-averaged lift (upper curve) and drag coefficient (lower curve), for cases 2, 4 and 6. Here, $T$ is the period of the wave for undulating cases, which is $1/f$ , and $T$ is defined as $T=L/U$ for the flat inclined plate.

The lift force of cases 3–7 (undulating at high $f^{\ast }$ ) has increased compared to the flat inclined plate (case 1) as observed in table 2. A simple explanation is that the travelling waves on the plates generate thrust along the $Z$ direction (figure 1 b), which has a component in the lift ( $y$ ) direction, i.e. enhances lift. The contribution of thrust force, generated by travelling waves, to the lift coefficient (denoted by $\unicode[STIX]{x0394}C_{L}$ here) can be approximated by the component of the thrust force in the lift ( $y$ ) direction, i.e. $\unicode[STIX]{x0394}C_{L}=C_{T}\sin \unicode[STIX]{x1D703}$ , where $\unicode[STIX]{x1D703}$ is the angle of attack (the angle between $Z$ and $z$ directions in figure 1). Therefore, for case 3, $\unicode[STIX]{x0394}C_{L}=0.06\times \sin (10^{\circ })\approx 0.005$ . This is two orders of magnitude smaller than the lift enhancement of cases 3–6 ( ${\approx}0.1$ ) relative to the flat inclined plate (table 2). Therefore, the $z$ component of the generated thrust is not the main reason for such lift enhancement. The lift enhancement is due to the modification of the pressure distribution around the plate, which is further investigated below.

To see how the lift force along the plate is modified by the travelling waves, the mean spanwise-averaged lift distribution along the plate for the flat inclined plate and three undulating cases (2, 4 and 6) is presented in figure 8(a). To see the spatial dependence of the lift force, the spanwise averages are shown along with the lift coefficient at three sections across the span of the plate, $X=0.1L$ , $X=0.3L$ and $X=L$ , where $X$ is the spanwise distance between the section and the tip of the plate (see figure 5 e). It can be observed that the range of spanwise variation of the lift is higher in the recirculatory (separated) zone (see the error bars (vertical lines) in figure 8 a) because the lift is high on the midplane but decreases more near the tips of the plate ( $X=0.1L$ ) due to the tip vortex (figure 5 f) (Kim & Gharib Reference Kim and Gharib2010; Beem et al. Reference Beem, Rival and Triantafyllou2012). In addition, it can be observed that these variations are higher for the undulating cases.

To show that the lift force along the plate has reached quasi-steady state, the time histories of the cycle-averaged lift and drag coefficients for flat inclined (case 1) and three undulating cases (2, 4 and 6) are shown in (figure 8 f). Figure 8(f) shows that the cycle-averaged forces for undulating cases become steady after approximately 12 cycles. The lift forces of undulating cases 2, 4 and 6 plotted in figure 8(a) are averaged over 20 cycles, and averaging is started after 50 cycles, as the forces are steady as shown in figure 8(f). The time averaging for the flat inclined plate is performed for $10t/T$ , which is equivalent to 4000 iterations.

Comparing the mean spanwise-averaged lift coefficient of undulating plates with the flat inclined one (figure 8 a) shows that $C_{L}$ does not change significantly for the low-frequency undulating case (case 2), but it becomes higher near the leading edge and slightly decreases near the trailing edge for high-frequency undulatory cases. This indicates that the total lift of cases 4 and 6 has increased compared to the lift of the flat inclined plate (case 1) due to the increase of the generated lift near the leading edge. In fact, case 4, which has the largest $C_{L}$ near the leading edge (figure 8 a), has the highest total lift coefficient among these four cases (table 2).

The lift of high- $f^{\ast }$ cases (4 and 6) has increased near the leading edge because of the lower pressure in the recirculatory zone relative to the pressure side (figure 8 d,e). The reason for the slightly lower lift coefficient of the high- $f^{\ast }$ cases in the vicinity of the trailing edge (figure 8 a) is the symmetric time-averaged pressure distribution (figure 8 d,e). It can be observed in figure 8(d,e) that a low pressure exists on both suction and pressure sides of the undulating plate near the trailing-edge area, which can be explained by the scaling of pressure on an undulating surface at high frequency. The pressure on an undulating inclined plate is generated by two mechanisms: (i) the angle of the plate, which pushes the incoming flow downwards (downwash), generating a higher pressure on the lower surface and a lower pressure on the upper surface; and (ii) the pressure created by the undulations of the plate, which accelerates the fluid near the plate (appendix D). The pressure due to undulations (undulating pressure) increases as frequency increases. In fact, for the undulating pressure, the normal component of the time-averaged pressure gradient along the lateral direction pointing outwards from the plate $\unicode[STIX]{x2202}\bar{p}/\unicode[STIX]{x2202}n$ , near the trailing edge, scales with $(a_{max}^{\ast })^{2}(f^{\ast }-1/\unicode[STIX]{x1D706}^{\ast })^{2}$ , which is derived by simplifying the time-averaged Navier–Stokes equations (see appendix D). Note that the positive $\unicode[STIX]{x2202}\bar{p}/\unicode[STIX]{x2202}n$ indicates that the pressure increases as we move away from the plate in the normal direction, i.e. a low pressure near the undulating plate. At high frequencies (cases 3–8), the undulating component dominates the pressure created by downwash (flat inclined plate). In fact, the magnitude of pressure near the trailing edge of an undulating plate at higher frequencies is one order of magnitude larger than the pressure over the trailing edge of the flat inclined plate (case 1). Therefore, the pressure near the trailing edge on both sides of the plate becomes similar at high $f^{\ast }$ , i.e. results in a low symmetric time-averaged pressure distribution (figure 8 d,e), because (i) the pressure at the trailing edge is dominated by the undulating pressure at high $f^{\ast }$ and (ii) the undulating pressure is similar (low) on both sides of the plate.

Figure 9.

(a) The effect of wavelength on the instantaneous lift coefficient. Instantaneous lift coefficients for three cycles at quasi-steady state for four oscillating cases with the same $f^{\ast }=20$ : case 3 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.15$ ), case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), case 6 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.5$ ) and case 7 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=1.0$ ). (b) The effect of frequency on the instantaneous lift coefficient. Instantaneous lift coefficient for three cycles at quasi-steady state for three undulating cases with the same $\unicode[STIX]{x1D706}^{\ast }=0.2$ and different $f^{\ast }$ : case 2 ( $f^{\ast }=6$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ) and case 5 ( $f^{\ast }=30$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ). Here, $T=1/f$ is the period of the wave for undulating cases.

Although EBT does not predict any mean lift generation when the plate is undergoing an undulatory oscillation, it predicts a sinusoidal profile for instantaneous lift and thrust, similar to aquatic swimmers (Triantafyllou et al. Reference Triantafyllou, Triantafyllou and Yue2000; Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2009), which increases the r.m.s. value of the forces. The r.m.s. values of lift and drag coefficients are presented in table 2. It can be observed that the r.m.s. values for fluctuations of both lift and drag coefficients are small for the flat inclined plate ( $C_{L_{rms}}=0.004,~C_{D_{rms}}=0.001$ ). When the plate undulates with a low frequency (case 2, $f^{\ast }=6$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), the r.m.s. values increase compared to the flat inclined plate, but the r.m.s. values are still less than the mean values ( $C_{L_{rms}}/C_{L}\approx 0.2$ and $C_{D_{rms}}/C_{D}\approx 0.25$ ). By increasing the reduced frequency, however, in case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), $C_{L_{rms}}/C_{L}$ and $C_{D_{rms}}/C_{D}$ increase approximately to 3 and 6, respectively. It can be observed that, when the wavelength increases, the r.m.s. values increase more intensely, e.g. in case 6 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}=0.5$ ) $C_{L_{rms}}/C_{L}$ increases to 50. The high r.m.s. values of the forces in high-frequency oscillating cases 3–9 can be explained by the instantaneous forces. The effect of $\unicode[STIX]{x1D706}^{\ast }$ and $f^{\ast }$ on the instantaneous lift coefficient after reaching quasi-steady state is shown in figure 9(a) and 9(b), respectively. When the frequency increases, the intensity of the fluctuations increases (figure 9 b) and consequently the r.m.s. value of the forces increases as well (table 2). Increasing the wavelength, however, increases the amplitude of the forces more intensely (figure 9 a), e.g. the amplitude of the lateral force has increased from 14 to 37 by increasing the wavelength from 0.5 to 1.0. It can be noted that, although the net reduced drag generated by the wave is higher for case 5 than for cases 6 and 7 (table 2), the amplitude of the fluctuation is higher for cases 6 and 7. The observed trends are consistent with the theoretical trend for the amplitude of instantaneous lateral (reactive) force, according to EBT (Lighthill Reference Lighthill1960). In fact, the amplitude of the lateral force of an undulating plate scales with $O(\unicode[STIX]{x1D706}^{\ast }(f^{\ast }-1/\unicode[STIX]{x1D706}^{\ast })^{2})$ , according to (C 6) derived in appendix C.

Figure 10.

The effect of frequency on the phase-averaged flow field of undulating cases with similar wavelength ( $\unicode[STIX]{x1D706}^{\ast }=0.2$ ) on the midplane. The first row for each case visualizes the contours of out-of-plane vorticity and the footprint of the $Q$ -criterion ( $Q=40$ ) by black lines for phase angle $t/T=0$ . The second to fifth rows for each case visualize the vorticity and velocity vectors of the flow close to the plate for four phase angles. (a0–a4) The undulating plate case 2 ( $f^{\ast }=6$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ), (b0–b4) the undulating plate case 4 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.2$ ) and (c0–c4) the undulating plate case 5 ( $f^{\ast }=30$ , $\unicode[STIX]{x1D706}^{\ast }=0.5$ ). The inset shows the shear layer vorticity over the pressure side, and the plate vorticity over the pressure and suction sides of the plate. The visualizations of the second to fifth rows of each case are for phase angle of (1)  $t/T=0.0$ , (2)  $t/T=0.25$ , (3)  $t/T=0.5$ and (4)  $t/T=0.75$ , respectively; and $T=1/f$ is the period of each case.

Figure 11.

The effect of wavelength on the phase-averaged flow field of undulating cases with similar frequency ( $f^{\ast }=20$ ) on the midplane. The first row for each case visualizes the contours of out-of-plane vorticity and the footprint of the $Q$ -criterion ( $Q=40$ ) by black lines for phase angle $t/T=0$ . The second to fifth rows for each case visualize the vorticity and velocity vectors of flow close to the plate for four phase angles. (a0–a4) The undulating plate case 6 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=0.5$ ), (b0–b4) the undulating plate case 7 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=1.0$ ) and (c0–c4) the undulating plate case 8 ( $f^{\ast }=20$ , $\unicode[STIX]{x1D706}^{\ast }=2.0$ ). The inset shows the shear layer vorticity over the pressure side, and the plate vorticity over the pressure and suction sides of the plate. The visualizations of the second to fifth rows of each case are for phase angle of (1)  $t/T=0.0$ , (2)  $t/T=0.25$ , (3)  $t/T=0.5$ and (4)  $t/T=0.75$ , respectively; and $T=1/f$ is the period of each case.

3.4 Travelling waves and the mechanisms of flow reattachment

Travelling waves were shown to reduce turbulence intensity in the wake but increase it near the plate (§ 3.2). In addition, they were shown to reduce flow separation of the mean flow at high frequency while the wavelength effect was not monotonic (§ 3.1). To explain the effect of wave parameters on turbulence intensity, reattachment and detachment of the flow, the phase-averaged flow along with their corresponding scalings, e.g. lateral velocity ( $u_{Y}$ ), are investigated in this section, which will reveal the main mechanisms governing flow reattachment by travelling waves.

Figures 10 and 11 represent the effect of frequency and wavelength, respectively, on the phase-averaged flow field visualized by out-of-plane vorticity and velocity vectors for four phase angles. The first row of each case (figures 10 and 11) shows the contours of vorticity and the footprints of the 3-D vortices identified by the isolines of the $Q$ -criterion for the first phase angle ( $t/T=0$ ). The four rows below it show the velocity vectors and vorticity contours near the plate at four other time instants in the cycle. In all cases, an attached and a separated shear layer are formed on the pressure and suction sides of the plate, respectively. The shear layer on the suction side, which has been separated at the leading edge, forms an LEV that sheds into the wake.

In addition to the vortices shed to the wake by the LEV, there are vortices on the plate that travel by the travelling wave along the plate and into the wake (see figure 10 a0). The vorticity of these vortices (the non-dimensional out-of-plane vorticity of the fluid on the plate) is composed of the plate vorticity ( $\unicode[STIX]{x2202}u_{Y}^{\ast }/\unicode[STIX]{x2202}Z^{\ast }$ ) and the shear layer vorticity ( $\unicode[STIX]{x2202}u_{Z}^{\ast }/\unicode[STIX]{x2202}Y^{\ast }$ ), i.e. $\text{out-of-plane vorticity}=-\text{in-plane vorticity}=-\unicode[STIX]{x1D714}_{x}^{\ast }=-(\unicode[STIX]{x2202}u_{Y}^{\ast }/\unicode[STIX]{x2202}Z^{\ast }-\unicode[STIX]{x2202}u_{Z}^{\ast }/\unicode[STIX]{x2202}Y^{\ast })$ . The plate vorticity is obtained by taking the derivative of the lateral velocity with respect to the streamwise length ( $\unicode[STIX]{x2202}u_{Y}^{\ast }/\unicode[STIX]{x2202}Z^{\ast }$ ) because the plate has no streamwise motion ( $u_{Z}=0$ ). The plate vorticity scales with $O((a_{max}^{\ast }/\unicode[STIX]{x1D706}^{\ast })(f^{\ast }-1/\unicode[STIX]{x1D706}^{\ast }))$ while the shear layer vorticity scales with $O(\sqrt{Re})$ for an attached boundary layer of the plate based on Blasius skin-friction coefficient (Schlichting & Gersten Reference Schlichting and Gersten2016) – see appendix C.

If the flow is separated, e.g. on the suction side, the shear layer vorticity is low and the plate vorticity dominates. Consequently, the vorticity of the fluid near the plate follows the same trend as the plate vorticity on the suction side as observed in figures 10 and 11, e.g. the vorticity on the suction side is anticlockwise (positive) and clockwise (negative) on the peaks and valleys of the wave, respectively, similar to the plate vorticity for all cases (figures 10 and 11). On the pressure side, however, the shear layer is attached and creates an anticlockwise (positive) vorticity. The shear layer vorticity is approximately $0.3\sqrt{Re}\approx 45$ , which is much higher than the plate vorticity at low frequency (case 2), which is approximately $4\unicode[STIX]{x03C0}^{2}(a_{max}^{\ast }/\unicode[STIX]{x1D706}^{\ast })(f^{\ast }-1/\unicode[STIX]{x1D706}^{\ast })\approx 2$ . Similarly, for case 8, which has a high wavelength ( $\unicode[STIX]{x1D706}^{\ast }$ ), the approximation of plate vorticity $(\unicode[STIX]{x2202}u_{Y}^{\ast }/\unicode[STIX]{x2202}Z^{\ast }\approx 4\unicode[STIX]{x03C0}^{2}(a_{max}^{\ast }/\unicode[STIX]{x1D706}^{\ast })(f^{\ast }-1/\unicode[STIX]{x1D706}^{\ast })\approx 5)$ is much smaller than the shear layer vorticity $(\unicode[STIX]{x2202}u_{Z}^{\ast }/\unicode[STIX]{x2202}Y^{\ast }\approx 0.3\sqrt{Re}\approx 45)$ . Therefore, no clockwise (negative) vorticity is observed on the valleys on the pressure side for case 2 (figure 10 a0–a4) and case 8 (figure 11 c0–c4). For other cases (4–6), however, the clockwise (negative) vorticity is observed on the pressure side for cases 4–6 on the valleys of the wave close to the plate while the anticlockwise (positive) shear layer is still present and stretches from the leading to the trailing edge on the pressure side (figures 10 b,c and 11 a). This agrees with the trend predicted by the scaling because the order of plate vorticity increases with increase of $f^{\ast }$ and decrease of $\unicode[STIX]{x1D706}^{\ast }$ (cases 4–6) and becomes similar to the order of the shear layer vorticity on the pressure side.

The anticlockwise and clockwise vortices, which are on the suction side for case 2 and on both sides for cases 4–8, travel on the peaks and valleys of the travelling wave, respectively, to the wake (figure 10 a0). The anticlockwise (positive) vortices, identified by the isolines of the $Q$ -criterion, are visible in the wake of case 2 (figure 10 a0), but neither the clockwise nor the anticlockwise vortices are observed in the wake (after the trailing edge) of cases 4–8 (figures 10 b,c and 11 a–c). The reason for the annihilation of these vortices in the wake of cases 4–8 is the interaction of the clockwise and anticlockwise vortices due to the sudden reduction in their speed in the wake. In fact, these vortices are convected on the peaks and valleys of the wave towards the wake with the wave speed $C$ . However, their speed suddenly reduces when they reach the wake, as the speed in the wake is much smaller ( $U\ll C$ ). Therefore, shortly after a clockwise vortex enters the wake and slows down, the adjacent anticlockwise one (travelling with the wave speed $C$ ) reaches the wake as well. Owing to the short distance between these vortices, afterwards, they interact and cancel each other out. For case 2, the clockwise (negative) vortices are dissipated in the wake due to the interaction with the turbulent flow of the wake. However, the attached shear layer on the pressure side amplifies the anticlockwise (positive) ones, so they are still visible in the wake (figure 10 a0). Note that, for values of $Q>10$ at which the coherent structures can be distinguished, the annihilation of these vortices is found to be independent of the value of $Q$ used for the visualization ( $Q=40$ in figures 10 and 11). The cancellation of these vortices in the wake for $C\gg U$ is the main reason for the observed lower turbulence intensity in the wake of travelling waves (§ 3.2).

The flow reattachment can better be observed by the phase-averaged velocity field, which is shown for four phase angles in figures 10 and 11. The velocity and vorticity above the trailing edge and into the wake of case 2 are very small (figure 10 a), which indicates that the plate does not impart enough momentum into the flow because of its low frequency ( $f^{\ast }=6$ ). By increasing the frequency to $f^{\ast }=20$ in case 4, both the velocity in the wake above the trailing edge and the plate vorticity (figure 10 b1–b4) are much higher than in case 2 (figure 10 a1–a4), which indicates a higher momentum transfer into the wake because of higher $f^{\ast }$ of the plate. By increasing the frequency further to $f^{\ast }=30$ in case 5, the streamwise velocity increases near the trailing edge (figure 10 c1–c4) compared to case 4 (figure 10 b1–b4), but the streamwise flow does not change near the leading edge, i.e. the reverse flow can be observed in all phase angles near the leading edge (figure 10). However, increasing the wavelength from 0.5 to 1.0 or 2.0 (figure 11) totally removes the reverse flow near the leading edge in case 7 (figure 11 b3) and case 8 (figure 11 c2), which is probably due to the high instantaneous reactive force generated at high wavelengths (figure 9 and (C 6)), which changes the flow field from the leading to the trailing edge. In addition, by increasing the wavelength, both the lateral and streamwise velocities increase at the trailing edge (figure 11 b,c) where the increase of streamwise velocity was also observed as a jet-like profile of the time-averaged velocities in figure 4(b) at section L5.

When $f^{\ast }$ increases from 6 to 20 or 30 in cases 3–8, considerable thrust is generated by the wave (see $C_{D_{rel}}$ in table 2) and flow reattaches earlier compared to cases 1 and 2 (see $R$ in table 2). Note that the increase in thrust is equivalent to the increase in the axial momentum of the wake here (equation (C 10)) as shown by a simple control volume analysis in appendix C. Such a reduction in flow separation by increasing the frequency is, therefore, due to the increase in the transfer of axial momentum to the fluid by the undulations similar to the steady blowing technique in flow control (Donovan, Kral & Cary Reference Donovan, Kral and Cary1998). Nevertheless, higher axial momentum is not the only consideration for reducing flow separation based on our results. Case 8 compared to case 7, and also case 6 compared to case 4, have a higher $\unicode[STIX]{x1D706}^{\ast }$ (similar $f^{\ast }$ ) but a higher separation as well (table 2). Higher $\unicode[STIX]{x1D706}^{\ast }$ indicates a higher momentum transfer to the fluid according to (3.1) and (C 10) as well as the results of relative drag in table 2, i.e. the lower axial momentum is not the reason for larger reattachment length for case 8 compared to case 7 and also for case 6 compared to case 4 (table 2). It is probably due to the higher lateral velocity as it affects the spreading of the wake.

Based on EBT, the lateral velocity of a fluid particle over an undulating plate scales with $O(a_{max}^{\ast }(f^{\ast }-1/\unicode[STIX]{x1D706}^{\ast }))$ – see appendix C for details. This scaling shows that, at a given reduced frequency and amplitude, the magnitude of lateral velocity increases as wavelength increases. When the magnitude of the lateral velocity increases, the lateral spreading of the flow increases, which tends to increase the separation. Here, the spreading of the wake is approximated as the ratio of the height ( $H$ ) of the wake at the trailing edge to the chord length. The height ( $H$ ) is defined as the distance between the plate trailing edge and the point above the trailing edge ( $y$ direction) at which the out-of-plane vorticity is close to zero. The spreading of the wake is 0.44, 0.48 and 0.51 for cases 2, 4 and 6, respectively, which confirms that the spreading of the wake increases as the lateral velocity is increased by either wavelength or frequency. Herein, the vorticity threshold used for distinguishing the height of the wake is a small value ( $\text{out-of-plane vorticity}=1.0$ ) to filter out the vorticity produced by numerical perturbations. Our observation reveals that choosing a higher threshold, such as $\text{out-of-plane vorticity}=3.0$ , does not change the pattern observed above, i.e. the spreading of the wake increases by the increase of either wavelength or frequency.