4.2.1. In-plane oncoming flow: $\gamma =0^\circ$

We start with in-plane plate oscillation ( $\alpha = 0^\circ$ ) in an in-plane oncoming flow ( $\gamma =0^\circ$ ). These are the only cases without significant vortex shedding and thus have the smallest change in heat transfer due to plate motion.

The top panels of figures 8(a) and 8(b) show the vorticity and temperatures fields for $A/|\boldsymbol{U}_{\infty }| =0.2$ and $A/|\boldsymbol{U}_{\infty }| =0.3$ , respectively, at the instants of maximum rightward and leftward velocity, shown by the red arrows. The attached shear layers and vortex wakes are slightly different from those without plate motion (figure 4 a), but still up–down symmetric. When the plate moves to the right (top row), the thermal boundary layer is more uniform than for the steady plate, and heat flux is reduced near the leading edge. When the plate moves to the left (second row), the larger velocity difference with the oncoming flow reduces the thermal boundary thickness near the leading edge, increasing the heat flux there.

Figure 9. Flows and heat transfer for oblique plate oscillation ( $\alpha = 30^\circ , 45^\circ$ and $60^\circ$ ) in an in-plane oncoming flow ( $\gamma =0^\circ$ ). Panels (a) and (b) show snapshots of the vorticity and temperature field at $\alpha = 45^\circ$ and $\gamma =0^\circ$ . The motion in panel (b) is shown in the supplementary movie 2, accessible here. The red arrows in the vorticity snapshots show the instantaneous plate velocity direction. Panels (c) and (d) show the time-averaged local $Nu$ and plate-temperature distributions versus $x$ (the horizontal axis) when the plate temperature or heat flux is fixed, respectively. The graphs are arranged in three columns, one for each $\alpha$ , and five rows, one for each $Re_f$ from 50 to 250, increasing downward. Pairs of subpanels are shown in each column, corresponding to $A/|\boldsymbol{U}_{\infty }| =0.2$ and 0.3 on the left and right, respectively. Panel (e) shows the global heat transfer quantities; here $\langle \overline {Nu} \rangle$ and $\langle T_{{avg}} \rangle$ are the spatial averages of the graphs in panels (c) and (d) (and summing the top and bottom values of $\langle Nu \rangle$ for $\langle \overline {Nu} \rangle$ ). In all cases the black dashed lines show the results for the non-oscillating plate.

Figure 10. The non-periodic case with $\gamma =0^\circ$ , $\alpha = 60^\circ$ , $Re_f=50$ and $A/{|\boldsymbol{U}|_{\infty }}=0.3$ . The three snapshots of the vorticity and temperature field correspond to times marked with red dots on the graphs of $ \overline {Nu}$ and $\langle T_{{max}} \rangle$ . The red arrows in the vorticity snapshots show the instantaneous direction of plate motion. The right three panels show the cycle-averaged $\langle Nu \rangle$ of the top and bottom walls for the isothermal plate, and the cycle-averaged $\langle T_{{plate}} \rangle$ with fixed heat flux. The first 5 cycles are omitted to avoid transient effects. The black dashed line shows the results for the non-oscillating plate. This motion is shown in the supplementary movie 3, accessible here.

The third and fourth rows of panels (a) and (b) show that the time-averaged quantities $\langle Nu \rangle$ and $\langle T_{{plate}} \rangle$ (red lines) are slightly worse near the leading edge and slightly better near the trailing edge than the steady case (black dashed lines), and the difference diminishes as $Re_f$ increases. Here, $Nu$ denotes $Nu_{{top}}$ and $Nu_{{bot}}$ , equal by symmetry. As $Re_f$ increases, the vorticity fields and $\langle \overline {Nu} \rangle$ (panel (c)) are closer to the steady values. The lower two subpanels of (c) show that, interestingly, $\langle T_{{avg}} \rangle$ and $\max (T_{max })$ have opposite trends with $Re_f$ , so smaller, higher-frequency oscillations mitigate the maximum temperature better, although the overall differences are small for both quantities.

Next, we discuss oblique plate oscillation ( $\alpha = 30^\circ , 45^\circ$ and $60^\circ$ ) in an in-plane oncoming flow ( $\gamma =0^\circ$ ), with the results shown in figure 9. As shown in figure 6, unlike $\gamma =45^\circ$ and $90^\circ$ , these flows are time periodic at both $A/|\boldsymbol{U}_{\infty }|$ values and all $Re_f$ values, except for a single case ( $\alpha =60^\circ$ , $Re_f=50$ and $A/|\boldsymbol{U}_{\infty }| = 0.3$ ) that is deferred to figure 10. The vortex dynamics and temperature fields of the cases in figure 9 have a common form. Panels (a) and (b) show two examples with $\alpha =45^\circ$ , at a lower $Re_f$ (100) and a higher $Re_f$ (200), respectively. In both cases, negative (blue) vortices shed from the left edge merge into a group that stays attached to the top of the left edge. We also observe that positive vortices (red) form at the bottom of the left edge when the plate moves upward (the first snapshot of the vorticity field). Meanwhile a series of dipoles sheds from the right edge and advects downstream. At the lower $Re_f$ in panel (a), the vortical structures are larger than in panel (b) because they have more time to grow.

These common patterns of the vortex dynamics lead to similar local heat transfer distributions. Local heat transfer is typically enhanced near the right edge but reduced near the left edge. Specifically, panel (c) shows the local $Nu$ distributions, which vary along $x$ , the horizontal axis, and are time averaged. All the subpanels share the same $y$ axis, and the local $Nu$ of the non-oscillating plate (black dashed lines) is provided as a reference. The blue curves showing $\langle Nu_{{top}} \rangle$ are lower than the non-oscillating-plate curves (black dashed lines) near the left edge but higher on the rest of the wall. Meanwhile the red curves showing $\langle Nu_{{bot}} \rangle$ are increased from the non-oscillating case on the right sides and less changed on the left sides. For the fixed-heat-flux plate, panel (d) shows that $\langle T_{{plate}} \rangle$ is higher than the steady values on the left sides and lower and more uniform on the right sides. These changes in local heat transfer are caused by the combined effects of the vortices (or vortex groups) formed at the left edge and the vortex shedding at the right edge. On one hand, the rotation of the negative vortex group advects hot fluid leftward along the top wall from the middle of the plate (or sometimes from the right edge if the vortex (group) is large enough), making the thermal boundary layer thicker than that of the non-oscillating plate near the left edge but thinner along the rest of the top wall. The positive vortex at the bottom of the left edge has a similar effect. However, since this vortex is smaller and weaker, the reduction of $\langle Nu_{{bot}} \rangle$ near the left edge is less significant than $\langle Nu_{{top}} \rangle$ . On the other hand, the dipoles shed from the right edge also enhance heat transfer near the right edge because they disrupt the layer of hot fluid that forms there in the non-oscillating-plate case.

The effect of $Re_f$ is more subtle but still noticeable. The dips at the left side of the blue $\langle Nu_{{top}} \rangle$ curves in panel (c) become more concentrated as $Re_f$ increases (moving downward), because the shed vortices are smaller in this high-frequency–low-amplitude regime. Meanwhile, the improvements in $\langle Nu_{{top}} \rangle$ and $\langle Nu_{{bot}} \rangle$ on the right sides become smaller. These effects are strongest at $\alpha =30^\circ$ (left column of panel (c)), still significant at $\alpha =45^\circ$ (middle column) and less noticeable at $\alpha =60^\circ$ (right column), particularly at $A/|\boldsymbol{U}_{\infty }|=0.3$ (right side of right column). In this case, higher $A/|\boldsymbol{U}_{\infty }|$ and more transverse oscillations make the vortices strong enough even at large $Re_f$ (small amplitude) to give strong heat transfer enhancement on the right side. Panel (d) shows the effects of $Re_f$ on $\langle T_{{plate}} \rangle$ in the fixed-heat-flux case. The results are similar to those in panel (c), with increases and decreases in $\langle T_{{plate}} \rangle$ relative to the steady case in panel (d) occurring where there are relative decreases and increases in $\langle Nu_{{top}} \rangle$ in panel (c). With stronger vortices, the location of the largest $\langle T_{{plate}} \rangle$ shifts farther from the non-oscillating-plate location at the right side towards the left side, due to advection of cold fluid by the vortex group near the left edge as well as the dipoles shed at the right edge. Despite these common characteristics of the variation of local heat transfer with $Re_f$ , the global heat transfer quantities (panel (e)) do not display a uniform trend as $Re_f$ increases (monotonic variation is only seen in some cases), indicating a complicated relationship between oscillation frequency and heat transfer.

The net effect, shown in $\langle \overline {Nu} \rangle$ in panel (e), is a maximum increase of $1$ %– $11\,\%$ at $A/|\boldsymbol{U}_{\infty }|=0.2$ and $23$ %– $35\%$ at $A/|\boldsymbol{U}_{\infty }|=0.3$ as $\alpha$ ranges from 30 to 60 $^\circ$ . For the fixed-heat-flux plate, the net result, $\langle T_{{avg}} \rangle$ in panel (e), ranges from worse to slightly better than the steady plate as $\alpha$ ranges from 30 to 60 $^\circ$ at $A/|\boldsymbol{U}_{\infty }|=0.2$ . There is always improvement at $A/|\boldsymbol{U}_{\infty }|=0.3$ , however, up to 11 %–19 % decreases in average temperature in the best cases over this $\alpha$ range. Panel (e) shows generally larger improvements in $\max (T_{max })$ . The improvements are larger at larger $\alpha$ because stronger vorticity is shed when the oscillation is more transverse (closer to 90 $^\circ$ ) than in-plane (0 $^\circ$ , where essentially no shedding occurs). Improvements are also larger at the larger $A/|\boldsymbol{U}_{\infty }|$ , where stronger vorticity is shed.

A single case was omitted in figures 9(c) and 9(d), in the upper right corners where blank spaces appear. This is the only non-periodic case in that parameter range, as can also be seen in figure 6(b) where, at $\gamma =0^\circ$ , $\alpha = 60^\circ$ and $Re_f = 50$ , it is marked with a filled circle rather than the open square of the other cases from figure 9, which have period 1. Figure 6 shows that non-periodicity is most common with $\gamma = 45^\circ$ and $90^\circ$ , but among those cases, the dynamics is farther from periodic with $\alpha$ close to $90^\circ$ and $A/|\boldsymbol{U}_{\infty }| = 0.3$ . For the case omitted from figure 9(c) and (d), $\gamma =0^\circ$ like the periodic cases discussed so far, but $\alpha = 60^\circ$ and $A/|\boldsymbol{U}_{\infty }| =0.3$ , values associated with stronger shed vorticity and non-periodicity. This case is shown in figure 10, in a format we use for other non-periodic cases shown subsequently.

The vortex dynamics has similarities to the period-1 cases shown in figure 9. However, the formation and shedding of vortices vary from one cycle to the next in an irregular fashion. The first and third vorticity snapshots of figure 10, one oscillation period apart, have some similarities but differences strong enough to give large differences in $\overline {Nu}$ , $T_{avg}$ , and $ T_{{max}} $ , corresponding to the first and third red dots on the graphs. Instead of long-time averages of their spatial distributions, as shown previously, in the three rightmost panels we show ensembles of distributions averaged over single oscillation cycles, from the 6th to the 30th cycles. The variations among the cycles are noticeable but not large compared with the mean (not shown). The distributions are close to those next to the omitted panels in figure 9. Like those cases, large negative vortex groups at the left edge and vortex shedding at the right edge increase $\langle Nu_{{top}} \rangle$ and $\langle Nu_{{bot}} \rangle$ near the right edge, and $\langle T_{{plate}} \rangle$ is maximum near the left edge.

For $\gamma = 0^\circ$ we have discussed $\alpha = 0^\circ , 30^\circ , 45^\circ $ and $60^\circ$ . The last $\alpha$ , $90^\circ$ , was previously studied for heat transfer (Rahman & Tafti Reference Rahman and Tafti2020a ) and is the most commonly studied case for propulsion (Alben Reference Alben2021a ). At $A/|\boldsymbol{U}_{\infty }| = 0.2$ , the dynamics transitions from up–down symmetric with period 1/2 to asymmetric with period 1 as $Re_f$ increases above 100. Figures 11(a) and 11(b) show the vorticity and temperature fields before and after the transition, respectively. Panel (c) shows that the local heat transfer quantities improve after the transition, moving from the red to the blue lines. The improvement is mainly on the left side. This corresponds to enlarged positive vorticity (through merging with the positive vortex of the previous cycle) and shrunken negative vorticity regions below and above the left edge, respectively, in panel (b). Panel (c) shows that the heat transfer is enhanced on both sides of the plate, with a smaller but broader region of enhancement on the bottom. Near the right edge, similar reverse von Kármán vortex streets form and give similar distributions of heat transfer quantities. Overall, $\langle \overline {Nu} \rangle$ increases by 11 % and (in the fixed-flux case) $\langle T_{{avg}} \rangle$ and $\max (T_{max })$ decrease by 11 % and 15 %, respectively, moving from $Re_f=100$ to 200.

Figure 11. Flows and heat transfer for transverse plate oscillation ( $\alpha = 90^\circ$ ) in an in-plane oncoming flow ( $\gamma =0^\circ$ ). (a) The case with period $ T_c=1/2$ at $Re_f=100$ and $A=0.2$ , also shown in the supplementary movie 4, accessible here. (b) The case with period $T_c=1$ at $Re_f=200$ and $A=0.1$ , also shown in the supplementary movie 5, accessible here. (c) Comparison of time-averaged local $Nu$ and plate temperature for (a) and (b). (d) The non-periodic case at $Re_f=200$ and $A=0.15$ . The three snapshots at the bottom left occur during the 26th, 56th and 84th cycles of oscillation, respectively, and the corresponding cycle-averaged local heat transfer quantities are shown with dash-dotted curves blue, green and orange curves, respectively, in the three rightmost subpanels. This motion is shown in the supplementary movie 6, accessible here. The red arrows in the vorticity snapshots show the instantaneous direction of plate motion. In all cases the black dashed lines show the results for the non-oscillating plate.

At $A/|\boldsymbol{U}_{\infty }| = 0.3$ , the dynamics instead transitions from periodic to non-periodic as $Re_f$ increases, again with significant improvement of the global heat transfer after the transition. Panel (d) shows examples of the vorticity and temperature fields at $Re_f=200$ , after the transition. Here, vortex grouping occurs in an irregular fashion, and the graphs of $\overline {Nu}$ , $T_{{avg}}$ and $T_{{max}}$ are not close to periodic. We observe three different stages in the vortex dynamics and show one snapshot from each stage. The first snapshot somewhat resembles those in panel (b), with a grouping of positive vortices near the left edge of the bottom wall and a reverse von Kármán vortex street emanating from the right edge. At this instant, blue dashed-dotted lines show the cycle-averaged local heat transfer quantities in the three subpanels on the right side of panel (d). We see some similarities with the blue lines in panel (c). The second snapshot shows the second stage, when the large positive vortex group detaches from the left edge. It continues absorbing positive vortices from the left edge, but is more diffuse than in the previous stage. The local heat transfer quantities for this snapshot are shown by the green dashed-dotted lines in the rightmost subpanels, and are somewhat worse at the middle and left side of the plate than the blue dashed-dotted lines of the first stage. At the right edge the vortex street is more strongly paired as dipoles, and $\langle Nu_{{bot}} \rangle$ is improved. The third snapshot approximates a mirror image of the first snapshot, with a region of negative vorticity above the left edge and an array of dipoles shed from the right edge, directed downward. The local heat transfer quantities are shown by the orange dashed-dotted line in the rightmost subpanels. The values of $\langle Nu_{{top}}\rangle$ and $\langle Nu_{{bot}} \rangle$ for this snapshot are approximately $\langle Nu_{{bot}} \rangle$ and $\langle Nu_{{top}}\rangle$ for the first snapshot (blue dashed-dotted lines), while the $\langle T_{{plate}} \rangle$ distributions are approximately equal, consistent with a mirror-image symmetry of the flows and temperature fields. In these cases and in figures 9 and 10, $\langle Nu \rangle$ has a local minimum near the leading edge on the side facing the large region of merged vortices, and is larger (relative to the steady case, the black dashed line) near the trailing edge. On the other side of the plate, $\langle Nu \rangle$ is instead larger near the leading edge and smaller near the trailing edge. In figure 11(d) and in other non-periodic cases discussed next, vortices merge near one edge into a large cloud over many cycles, then the cloud detaches, and then vortex merging occurs at the other edge. The switching between the edges occurs at somewhat irregular intervals. During these changes in the local heat transfer distribution, the global (spatially averaged) heat transfer does not change much; increases in heat transfer in one region are balanced by decreases in other regions. At lower $Re_f$ , 50 and 100, the dynamics is periodic and similar to panel (b). In these two cases $\langle \overline {Nu} \rangle$ values are 28 % and 20 % lower than the non-periodic $Re_f=200$ case, while $\langle T_{{avg}} \rangle$ values are 37 % and 22 % higher than at $Re_f=200$ , respectively. The merged vorticity near the leading edge is stronger in the non-periodic case than in the periodic case, and hence more capable of bringing cold fluid towards the plate.

4.2.2. Oblique oncoming flow: $\gamma =45^\circ$

Figure 12. The cases that optimise global heat transfer ( $\langle \overline {Nu} \rangle$ and $\langle T_{{avg}} \rangle$ ) at $\gamma =45^\circ$ for $A/|\boldsymbol{U}_{\infty }|=0.2$ (a) and 0.3 (b). The supplementary movies for panel (a) and (b) are named ‘movie 7’ and ‘movie 8’, respectively, and can be found here. In each panel, the four snapshots of the vorticity and temperature field correspond to times marked with red dots on the graphs of $ \overline {Nu}$ and $\langle T_{{max}} \rangle$ , with the red arrows indicating the instantaneous direction of plate motion. The bottom three panels show the cycle-averaged $\langle Nu \rangle$ of the top and bottom walls for the isothermal plate, and the cycle-averaged $\langle T_{{plate}} \rangle$ with fixed heat flux. The first 5 cycles are omitted to avoid transient effects. The black dashed line shows the time-averaged results for the non-oscillating plate in the final periodic state for $\gamma =45^\circ$ .

The previous section considered the case of the background flow along the plane of the plate, $\gamma = 0^\circ$ . We now consider heat transfer in the same system with $\gamma$ increased to $45^\circ$ , an oblique background flow. For any $\gamma$ significantly different from $0^\circ$ , the wake oscillates periodically (for a range of $Re_U$ near the present value of 100) even for a steady plate, as shown in § 3. When we oscillate the plate also but at a different period, the dynamics can be almost periodic on a longer time scale (i.e. the flow period locks to a multiple of the plate period) or clearly non-periodic, as indicated by the open and filled circles respectively in figure 6 for $\gamma = 45^\circ$ and 90 $^\circ$ . Almost-periodic cases are the most common at $A/|\boldsymbol{U}_{\infty }|=0.2$ , while non-periodic cases are approximately equally common at $A/|\boldsymbol{U}_{\infty }|=0.3$ , and more common with transverse oscillation, when the shed vorticity is stronger. In almost-periodic cases, vortex merging occurs over the same number of plate oscillation cycles and at the same location each time. In non-periodic cases, the locations and dynamics of vortex merging are somewhat irregular. In general, the almost-periodic and non-periodic cases are distinguished by the regularity of the vortex dynamics, but not heat transfer performance. Generally, we see larger heat transfer with stronger vortices, which are also associated with non-periodic flows. But we do not see a direct mechanism by which non-periodicity increases heat transfer relative to almost-periodic flows with vortices of roughly the same strength. If we compare almost-periodic and non-periodic cases with the same $A/|\boldsymbol{U}_{\infty }|$ (i.e. the same oscillation velocity), for example, $\gamma =45^\circ$ and $\alpha =120^\circ$ at $A/|\boldsymbol{U}_{\infty }|=0.3$ in figures 6 and 7, we see similar global heat transfer for the open and filled circles.

In figure 12, we show the cases with the best global heat transfer ( $\langle \overline {Nu} \rangle$ ) when $\gamma = 45^\circ$ , for $A/|\boldsymbol{U}_{\infty }|=0.2$ (panel (a)) and 0.3 (panel (b)). Both occur with transverse oscillation ( $\alpha =90^\circ$ ), which is also optimal at $\gamma = 0^\circ$ and 90 $^\circ$ . Both optima occur at an intermediate $Re_f$ of 150, but other $Re_f$ give similar $\langle \overline {Nu} \rangle$ and $\langle T_{{avg}}\rangle$ . As indicated by the time series of the global quantities in the first row of each panel, panel (a) is nearly periodic with a period of 17 plate oscillation cycles, while panel (b) is non-periodic. In both cases a dipole sheds alternately from the left and right edge every half-cycle; the stronger vortex of the dipole pair merges with the cloud of previously shed vortices of the same sign and the weaker vortex diffuses. Over many cycles, negative vortices group at the left edge, detach and then positive vortices group at the right edge, detach and the process repeats. (Supplementary movies of these cases, named ‘movie 7’ and ‘movie 8’, can be found here.) During this time, the graphs of the global quantities $\overline {\textit{Nu}}$ , $T_{\textit{avg}}$ and $T_{{max}}$ show oscillations on the time scale of the plate oscillation and on much longer time scales. At the bottom of each panel, the graphs of the local quantities $\langle Nu_{{top}} \rangle$ , $\langle Nu_{{bot}} \rangle$ and $\langle T_{{plate}} \rangle$ also oscillate, between distributions with maxima on the left and right alternately for the first and third quantities. These oscillations are correlated with the vortex dynamics shown in the snapshots in the middle of each panel.

In the first vorticity snapshot of panel (a), a region of negative merged vortices lies above the left edge, which makes $\langle Nu_{{top}} \rangle$ low on the left and high on the right and gives a maximum of $\langle T_{{plate}} \rangle$ near the left edge. The top subpanel shows $\overline {Nu}$ is generally increasing at this time. In the second vorticity snapshot, a positive vortex group forms and repels the negative vortex group. The maximum of $\langle Nu_{{top}} \rangle$ shifts from the right towards the left. The second red dot on the $\overline {Nu}$ time series above shows that the global heat transfer reaches a maximum at this time. In the third snapshot, the positive vortex group detaches, while $\overline {Nu}$ decreases, reaching a minimum at the time of the fourth snapshot, when the negative vortex group begins to grow again. The graphs of $T_{{avg}}$ and $T_{{max}}$ below show nearly the inverse behaviour for the fixed-flux case, as expected. During this entire cycle $\langle Nu_{{bot}} \rangle$ , which is less influenced by the vortex wake dynamics, has much smaller oscillations about a mean distribution slightly above that of the steady case (black dashed line).

A few differences are seen in panel (b) ( $A/|\boldsymbol{U}_{\infty }|=0.3$ ). The vortex groups form over more oscillation cycles than at $A/|\boldsymbol{U}_{\infty }|=0.2$ . In the first snapshot of panel (b), the negative vortex group is more diffuse than in panel (a), with a longer chain of dipoles emanating from the right edge. Meanwhile, the global heat transfer quantities ( $\overline {Nu}$ , $T_{{avg}}$ and $T_{{max}}$ ) are nearly steady. In the second snapshot, the positive vortices from the right edge have formed a loose group that repels the negative vortex group, and $\overline {Nu}$ drops to a local minimum. In the third snapshot the positive vortices have formed a more compact group attached to the right edge of the plate, during a long period of increase of $\overline {Nu}$ . In the fourth snapshot, the positive vortex group has detached, the negative vortex group is reforming and $\overline {Nu}$ drops to another local minimum. Meanwhile $\langle Nu_{{top}} \rangle$ , $\langle Nu_{{bot}} \rangle$ and $\langle T_{{plate}} \rangle$ at the bottom oscillate between distributions similar to those in panel (a), but more erratically and with larger oscillation amplitudes.

Figure 13. Vorticity fields and cycle-averaged local heat transfer quantities for an oscillating plate in an oblique flow ( $\gamma =45^\circ$ ) with $A/|\boldsymbol{U}_{\infty }| = 0.2$ and $\alpha \neq 90^\circ$ , i.e. oscillation directions other than transverse. The red arrow in each vorticity panel indicates the instantaneous plate velocity direction. Supplementary movies are available at this link for the left case of panel (a) (‘movie 9’), the two cases in panel (b) (‘movie 10’ and ‘movie 11’) and the two cases in panel (c) (‘movie 12’ and ‘movie 13’).

Figure 13 shows vorticity field snapshots and local heat transfer quantities for oblique flow ( $\gamma =45^\circ$ ) and the remaining seven plate oscillation directions, i.e. $\alpha \neq 90^\circ$ . We only discuss $A/|\boldsymbol{U}_{\infty }|=0.2$ because the phenomena are generally similar at $A/|\boldsymbol{U}_{\infty }|=0.3$ . In figure 13, we show the variations in vorticity and local heat transfer distributions with a simplified presentation. In each row, two cases are positioned side by side and compared. Two representative snapshots of the vorticity field are selected for each case to indicate the motions of the vortices. We also provide supplementary movies of each case (at the link in the figure caption). The three subpanels to the right of each pair of vorticity fields show the cycle-averaged local quantities. We have removed the axis labels to make the figure compact, but all the subpanels for the cycle-averaged local $Nu$ and plate temperature share the same y axes, respectively, and the non-oscillating-plate values (the black dashed curve) can be taken as a reference. (Besides the cycle-averaged local quantities, we also show the long-time-averaged local $Nu$ in figure 28 in Appendix E.)

The top row (panel (a)) shows in-plane oscillation ( $\alpha =0^\circ$ ), for which the largest heat transfer enhancement occurs at $Re_f=50$ (shown on the left side), and mainly on the top side of the plate near the left edge. As $Re_f$ increases and the amplitude decreases, the enhancement drops. The flow and local heat transfer quantities converge to those of the steady plate, as shown on the right side of panel (a), neglecting the dark blue graphs of the initial transient period before the steady state. With in-plane oscillations, each edge mainly sheds vorticity of a single sign, as for the static plate. Without strong interactions of oppositely signed vorticity, there is relatively weak fluid mixing near the plate.

The remaining rows, panels (b)–(d), show $\alpha = 30^\circ , 45^\circ $ and $60^\circ$ on the left paired with the supplementary angles $150^\circ , 135^\circ $ and $120^\circ$ on the right. For each pair, the transverse velocity component is the same, ranging from smaller (more in plane) at $\alpha = 30^\circ$ and $150^\circ$ (panel (b)) to larger (more transverse) at $\alpha = 60^\circ$ and $120^\circ$ (panel (d)). But the in-plane velocity components are reversed, so the left side ( $\alpha \lt 90^\circ$ ) oscillates more along the oncoming flow direction while the right side ( $\alpha \gt 90^\circ$ ) oscillates more across it, as seen from the plate velocity vectors (red arrows) in each panel. In figure 7 we saw that oscillating along the flow is better, i.e. within each pair $\alpha \lt 90^\circ$ outperforms $\alpha \gt 90^\circ$ , and the difference between the two becomes smaller as $Re_f$ increases. This is also seen in the local heat transfer graphs of figure 13, for which $\langle Nu_{{top/bot}} \rangle$ is generally higher and $\langle T_{{plate}} \rangle$ is lower relative to the steady reference case (black dashed) on the left side, oscillating along the flow.

Previously, we mentioned that stronger vorticity is found near the plate for $\alpha \lt 90^\circ$ than $\alpha \gt 90^\circ$ (as shown in figure 25 in Appendix D), which might be a short explanation for the better heat transfer achieved with $\alpha \lt 90^\circ$ . We now give a more detailed explanation by showing the differences in vortex dynamics for $\alpha \lt 90^\circ$ and $\alpha \gt 90^\circ$ . With $A/|\boldsymbol{U}_{\infty }|$ = 0.2, the ratio of the maximum plate velocity to the oncoming flow velocity is $2\pi \cdot 0.2 = 1.26$ , so the plate oscillation is strong enough to reverse the oncoming flow velocity. This occurs for $\alpha = 45^\circ$ , when the plate oscillates along the flow, and at nearby values, i.e. over most of the range $0^\circ \lt \alpha \lt 90^\circ$ . In this case, oppositely signed vortices are shed from each edge on each cycle, and they have relatively strong interactions and mix the fluid close to the plate, as can be seen in the vorticity snapshots on the left sides of panels (b)–(d). When $\alpha = 135^\circ$ , the plate oscillates across the flow, and there is always a positive flow component in the oncoming flow direction. The same is true over most of the range $90^\circ \lt \alpha \lt 180^\circ$ . In this case, the vorticity shed from each edge is predominantly (but not completely) single signed, similarly to the in-plane oscillation of panel (a). Here there are weaker interactions between the vortices near the plate, before they are advected away by the oncoming flow, as seen in the vorticity snapshots on the right sides of panels (b)–(d). These differences may underlie some of the common features of the local heat transfer quantities on the left and right. For example, the right subpanels but not the left subpanels show a dip in $\langle Nu_{{bot}} \rangle$ , and a rise in $\langle T_{{plate}} \rangle$ , at the left edge relative to the steady graph (black dashed line). The difference between the two cases diminishes as $Re_f$ increases, because the vorticity fields become more similar. This is seen in the second rows of panels (c) and (d), where the vorticity snapshots are more similar on the left and right sides than in the first rows, at lower $Re_f$ .

Figure 14. The cases that optimise global heat transfer ( $\langle \overline {Nu} \rangle$ and $\langle T_{{avg}} \rangle$ ) at $\gamma =90^\circ$ for $A/|\boldsymbol{U}_{\infty }|=0.2$ (a) and 0.3 (b), with supplementary movies named ‘14‘ and ‘15’, respectively, linked here. In each panel, the four snapshots of the vorticity and temperature field correspond to times marked with red dots on the graphs of $ \overline {Nu}$ and $\langle T_{{max}} \rangle$ , with the red arrows indicating the instantaneous direction of plate motion. The bottom three panels show the cycle-averaged $\langle Nu \rangle$ of the top and bottom walls for the isothermal plate, and the cycle-averaged $\langle T_{{plate}} \rangle$ with fixed heat flux. The first 5 cycles are omitted to avoid transient effects. The black dashed line shows the time-averaged results for the non-oscillating plate in the final periodic state for $\gamma =90^\circ$ , after the symmetry breaking.

4.2.3. Transverse oncoming flow: $\gamma =90^\circ$

Having discussed in-plane and oblique oncoming flows, $\gamma =0^\circ$ and $45^\circ$ , we now consider transverse oncoming flow, $\gamma =90^\circ$ . Due to symmetry, we can restrict ourselves to $0^\circ \leqslant \alpha \leqslant 90^\circ$ . As before, the best global heat transfer occurs with transverse plate oscillations ( $\alpha =90^\circ$ ). At these particular $\alpha$ and $\gamma$ , as for the steady plate at $\gamma =90^\circ$ , a uniform initial oncoming flow with small asymmetric perturbations reaches the asymmetric steady state slowly, after ${\approx}300$ time units. To reach the steady state faster, we instead initialise the flow and temperature fields to be those at $t=400$ for the steady plate with $\gamma =90^\circ$ , similar to the cases shown at the latest times in figure 5, which have a von Kármán vortex wake.

The optimal flows for heat transfer at $A/|\boldsymbol{U}_{\infty }|=0.2$ and 0.3 are shown in figures 14(a) and 14(b), respectively, using a layout similar to figure 12, with links to corresponding movies in the caption. With the combination of a plate oscillation frequency and the bluff-body wake shedding frequency, the vortex dynamics and resulting heat transfer for $\gamma =90^\circ$ are generally similar to those for $\gamma =45^\circ$ in a few ways. First, in both cases complex vortex-shedding patterns occur in the wake, and greatly enhance heat transfer on the top of the plate (the side facing the wake). Second, shed vortices merge into large regions of single-signed vorticity over many oscillation periods, and then detach, alternately at one edge and then the other. Third, in most cases at $A/|\boldsymbol{U}_{\infty }|=0.2$ the vortex dynamics is close to periodic and locked to a multiple of the plate oscillation period (10 in panel (a)), while at $A/|\boldsymbol{U}_{\infty }|=0.3$ more cases are non-periodic, particularly near $\alpha = 90^\circ$ . One difference is the bilateral symmetry of the $\gamma =\alpha = 90^\circ$ cases, e.g. in figure 14, so that the positive and negative vortex groups are essentially mirror images in the two leftmost and rightmost snapshots of the periodic case in panel (a), and almost mirror images in the non-periodic case of panel (b). A corresponding left-right symmetry is seen in the ensemble of cycle-averaged $\langle Nu_{{top}} \rangle$ , $\langle Nu_{{bot}} \rangle$ and $\langle T_{{plate}} \rangle$ distributions in panel (a), and to a lesser extent in panel (b). Key features of the graphs of the global heat transfer quantities are correlated with events in the vortex dynamics, similarly to $\gamma = 45^\circ$ . Troughs in $\overline {Nu}$ and peaks in $T_{{avg}}$ and $T_{{max}}$ in panels (a) and (b) occur when a vortex group detaches from one edge and starts to form at the other edge, as shown in the second and fourth snapshots of each panel. During the process of vortex merging, the global heat transfer quantities first increase and then oscillate about fairly constant levels, as occurs during the first and third snapshot of panel (b).

Figure 15. Vorticity fields and cycle-averaged local heat transfer quantities for an oscillating plate in a transverse flow ( $\gamma =90^\circ$ ) with $A/|\boldsymbol{U}_{\infty }| = 0.2$ (a) and 0.3 (b) and $\alpha \neq 90^\circ$ , i.e. oscillation directions other than transverse. The red arrow in each vorticity panel indicates the instantaneous plate velocity direction. The red dash-dotted curves show instantaneous distributions of cycle-averaged local heat transfer quantities that are discussed in the main text. Supplementary movies named ‘movie 16’ and ‘movie 17’ are available here for panels ( $a1$ ) and ( $b3$ ), respectively.

Examples of flows and heat transfer quantities for the other plate oscillation directions are shown in figure 15, for $A/|\boldsymbol{U}_{\infty }|=0.2$ (panel (a)) and 0.3 (panel (b)). For each case shown, the first vorticity snapshot shows the negative vortex/vortex group at the left edge, while the second vorticity snapshot shows the positive vortex/vortex group at the right edge. To the right of the two snapshots, the cycle-averaged local quantities are shown with the same $y$ -axis scales. Similar to figure 13, the axis labels are removed to make the figure compact but the result for the non-oscillating plate (black dashed curve) can be taken as a reference. The second vorticity snapshots in the last three rows show typical vorticity patterns for $\gamma =90^\circ$ , with the corresponding distributions of the cycle-averaged local quantities highlighted with red dash-dotted curves. The caption lists links to two supplementary movies that show the dynamics of the vortices and local heat transfer distributions. As a supplement, the time-averaged local Nusselt number is shown in figure 29 in Appendix E.

Subpanels (a1) and (b1) show the in-plane-oscillation case ( $\alpha =0^\circ$ ). As occurred with $\gamma = 45^\circ$ , the most noticeable heat transfer enhancement relative to the steady case occurs at this value of $Re_f$ , the lowest (see figure 7). The graphs of $\langle Nu_{{top}} \rangle$ and $\langle T_{{plate}} \rangle$ show that the local heat transfer is enhanced mainly at the edges and less enhanced or even worsened in the middle. The overall enhancement in both subpanels (a1) and (b1) is very limited. In these subpanels the cycle-averaged local heat transfer quantities have a left–right symmetry that is broken for the $\alpha$ in the following subpanels. Here, we have vortex groupings at both edges, with the positive vortex group (above the right edge) more prominent. It usually lasts longer, is more compact and intense and is closer to the top wall than the negative vortex group. In subpanels ( $a2$ ), ( $b2$ ), ( $a3$ ) and ( $b3$ ), the negative vortex group is partially or fully separated into individual vortices, while the positive vortex group is a single compact region at the right edge. When the positive vortex group grows to a certain size, it is repelled by the negative vortex group and detaches from the plate. At higher $Re_f$ , the negative vortex group also forms into a compact group attached to the plate, as shown in subpanels ( $a4$ ) and ( $b4$ ), but the positive vortex group still lasts longer. The second snapshots in the last three rows of subpanels show an instant when the positive vortex group is detaching from the plate and dipoles have been shed from the left edge. At these times, $\langle Nu_{{top}} \rangle$ has a local maximum near the left edge and a local minimum near the right edge, with the inverse pattern in $\langle T_{{plate}} \rangle$ , as shown by the red dash-dotted lines.