2.1 Stiffness of the deformable flaps

We used flaps with three different flexural rigidity ( $EI$ ) values, where $E$ is the Young’s modulus and $I$ is the area moment of inertia of the flap. The flexural rigidity or stiffness of the flaps per unit span length and other details are listed in table 1. The flexural rigidity is an important parameter in our study, and we shall refer to the three values shown in table 1 often and hence in the rest of this paper we shall refer to the three $EI$ cases as least, intermediate and most stiff flaps. The span of the flaps used in our experiments was held constant at 30 cm that is equal to the span of the rectangular channel through which the starting jet ensues. The rectangular channel had a width ( $d$ ) of 1.5 cm. We used three different flap lengths ( $L_{f}$ ) of 3 cm, 6 cm and 9 cm, corresponding to ( $L_{f}/d$ ) values of 2, 4 and 6, respectively.

Table 1.

The details of the flap materials used in this study.

The flaps used were heavier than the fluid medium with specific gravities ranging from 1.3 to 1.6, and were mounted in a vertical configuration (figure 1). This implies that the net downward gravitational force (after subtracting buoyancy) experienced by the flaps would cause an additional restoring moment as the flaps open outward, in addition to the restoring moment due to the flexural rigidity ( $EI$ ) of the flaps. The net gravitational force acts as a uniformly distributed load on the flap (beam) trying to bring it back to its initial position, and a precise solution to such a problem is quite complicated and involves integro-differential equations. However, we can get a reasonable estimate for the increase in $EI$ caused by the net downward gravitational force by considering tip deflection of the flap modelled as a rigid beam with a torsional stiffness at the hinge point representing the restoring effect of flexural rigidity. With this simplification, it is easy to see that the net downward gravitational force acting at the mid-point of the flap would cause an increased restoring moment, which can in turn be written in terms of a higher equivalent flexural rigidity ( $EI_{eq}$ ) given by $EI_{eq}=EI(1+F_{g}{L_{f}}^{2}/(6EI))$ , where $F_{g}$ is the net downward gravitational force per unit span length of the flap after subtracting the buoyancy force. The values of the ratio of this equivalent flexural rigidity ( $EI_{eq}$ ) to the actual $EI$ are listed in table 1 for all the different $EI$ and flap length ( $L_{f}/d$ ) cases studied. It is clear from the table that the variations in $EI_{eq}$ from $EI$ are very small for the most and intermediate stiff flaps, and are significant only for the least stiff flaps, and in particular for the longer ( $L_{f}/d=4$ and 6) least stiff flap cases.

The parameters involved in the study with deformable flaps can be noted from figure 1. The channel width ( $d=1.5~\text{cm}$ ) was kept constant throughout this work. The flap length ( $L_{f}$ ) is non-dimensionalized with the channel exit width ( $L_{f}/d$ ) and three different lengths, $L_{f}/d=2$ , 4, and 6 were studied. The instantaneous exit width at the flap tips is measured as $d_{e}$ . The flexural rigidity of the flaps ( $EI$ ) is a parameter, and we studied three flaps, each with a different $EI$ value (table 1). The fluid slug length ( $L/d$ ) is defined as the normalized streamwise distance moved by the fluid if the fluid moved as a slug. We have kept the final fluid slug length ( $L/d$ ) to be a constant at 15 in all the cases studied. The piston velocity program used was an impulsively started piston motion. For our experiments, we have used five different piston speeds between $10$ and $25~\text{cm}~\text{s}^{-1}$ giving us a Reynolds number ( $Re=(U_{p}d)/\unicode[STIX]{x1D708}$ ) range of 1500–3750. The dimensional time at which the piston stops is referred to as $\unicode[STIX]{x1D70F}$ hereafter, and the value of $\unicode[STIX]{x1D70F}$ varies with piston speed to keep $L/d$ constant at 15.

3.1 Flap kinematics for a sample case

We begin by presenting in figure 3, a typical sequence of the flap kinematics visualized with time corresponding to the least stiff flap with $L_{f}/d=4$ and a piston velocity ( $U_{p}$ ) of $18~\text{cm}~\text{s}^{-1}$ . As the flow starts, the flaps bulge outwards (in a) due to an over-pressure that is developed in the flap region to accelerate the fluid slug contained between the flaps. This was also responsible for the opening out of the hinged rigid flaps in the study of Das et al. (Reference Das, Govardhan and Arakeri2013). However, since the flaps are deformable here, this excess pressure results in a local bulge as in figure 3(b). With time, this bulge is seen to grow and move downstream with the flow, leading to the opening outwards of the flap tip (figure 3 c) and ultimately resulting in the flap tips opening to a maximum value/extent in figure 3(d). Thereafter, the flaps gradually return to their initial position (figure 3 f). It should be noted that the piston accelerates rapidly from rest and reaches a steady speed before the maximum bulge is seen in figure 3(b). The inward motion of the flaps after their maximum opening can be attributed to two broad reasons, namely, the stiffness ( $EI$ ) of the flaps and to the lower pressure created in the flap region due to the ‘velocity head’ developed, the latter being similar to the reasoning made in Das et al. (Reference Das, Govardhan and Arakeri2013). Immediately after the piston is stopped ( $t=\unicode[STIX]{x1D70F}$ ), corresponding to a fluid slug length of 15, an under-pressure is developed within the flap region leading to the inward deflection of the flaps seen in figure 3(g) at $t/\unicode[STIX]{x1D70F}\approx 1.03$ . This under-pressure within the flaps is the opposite effect of the over-pressure seen at flow initiation, and is caused by the rapid deceleration of the flow between the flaps. This under-pressure leads to the inward deflection of the flaps that grows with time leading to the two flaps approaching each other (figure 3 h) and ultimately to the closure of the flaps with the tips touching each other (figure 3 i). This closure of the flaps by the under-pressure is similar to the observation of Lee & Talbot (Reference Lee and Talbot1979), who showed that an adverse pressure gradient due to fluid deceleration was the dominant reason behind closure of heart valves. As time progresses further, the flaps gradually open out to their initial position (figure 3 j).

Figure 3.

Sequence of images showing the kinematics of the least stiff flaps with $L_{f}/d=4$ . The piston velocity program is impulsively started with $U_{p}=18~\text{cm}~\text{s}^{-1}$ , and sets up the flow that is from left to right. The piston stops at $\unicode[STIX]{x1D70F}\approx 1.3~\text{s}$ and the corresponding fluid slug length reached is $L/d=15$ . The instances shown in the figure are at $t/\unicode[STIX]{x1D70F}=$ (a) 0 (b) 0.1 (c) 0.19 (d) 0.28 (e) 0.59 (f) 0.97 (g) 1.03 (h) 1.1 (i) 1.16 (j) 4.8.

Figure 4.

Plot of exit width with time for the least stiff flap and $L_{f}/d=4$ . The instances (b–i) are the same as shown in figure 3 with all other parameters being the same. The solid line is a trend line drawn for clarity and uncertainty estimates are shown at every alternate measurement point.

Figure 5.

Plot of exit width with $L/L_{f}$ for varying piston speeds ( $U_{p}$ ). In all cases, the flap length ( $L_{f}/d$ ) is 4, the flaps are of least stiffness and the piston stops when the fluid slug length reaches $L/d=15$ .

In Das et al. (Reference Das, Govardhan and Arakeri2013), we used the rigid leaflet’s angle with respect to the channel walls to compare different cases and to study the leaflet kinematics. In the cases with deformable leaflets, however, there is no clear angle because the flaps deform. We therefore start with plotting the exit width based on flap tips ( $d_{e}$ from figure 1) to look at the flap kinematics as a function of time. We present time variation of the exit width ( $d_{e}$ ) at the flap tips normalized by the channel width ( $d$ ) in figure 4, for the same case as in figure 3. The piston motion is nearly impulsively started to $18~\text{cm}~\text{s}^{-1}$ . The different instances (b–i) marked in the figure are the same instances for which the flap shapes are shown in figure 3. The uncertainty estimates in every alternate data point are shown in the figure, and are obtained from multiple runs of the same case. The initial streamwise central bulging of the flaps results in a reduction of the exit width ( $d_{e}$ ) as seen in the plot in figure 3(b). Thereafter the flap tips move out to reach the maximum exit width as shown in figure 3(d). The collapse of the flaps after the piston stops is reflected by a rise in the exit width (figure 3 h) as the tips move outwards, followed by a relatively fast coming together of the flaps resulting in the exit width reaching zero (figure 3 i).

We next show the kinematics for different piston speeds (figure 5) keeping the flap length constant at $L_{f}/d=4$ and employing the flaps of least stiffness. We have thus far presented the kinematics as a function of a simple non-dimensional time ( $t/\unicode[STIX]{x1D70F}$ ), where $\unicode[STIX]{x1D70F}$ is the time when the piston stops. This choice was deliberately made to permit discussions of flap kinematics even after the piston stops ( $t/\unicode[STIX]{x1D70F}>1$ ). The conventional time scale in such starting jet studies is the fluid slug length ( $L$ ), which is more relevant while the fluid is being pushed out, but remains constant once the piston stops, and hence is not convenient to discuss flap kinematics after the piston stops. However, hereafter we will be predominantly looking at the kinematics while the piston is in motion, and so we shall use the conventional fluid slug length ( $L$ ) as the non-dimensional time. In the study with hinged leaflets (Das et al. Reference Das, Govardhan and Arakeri2013), we found that a time scale based on the fluid slug length and flap length ( $L/L_{f}$ ) collapsed the flap kinematics across different piston speeds ( $U_{p}$ ) and flap lengths ( $L_{f}/d$ ), where $L$ is the streamwise distance moved by the fluid if it moved as a slug. We therefore plot the kinematics for the different $U_{p}$ cases from this study using $L/L_{f}$ as the time scale in figure 5. In the present case of flexible flaps, we can clearly see changes in the exit width ( $d_{e}$ ) with $U_{p}$ , noting that the observed changes are significant compared to the uncertainty in $d_{e}$ shown in figure 4. This suggests that unlike in the hinged rigid flap study, the piston speed ( $U_{p}$ ) does influence the flap kinematics significantly. We shall address this aspect later in the section by defining an appropriate non-dimensional flexural rigidity ( $EI^{\ast }$ ) using both the flap stiffness ( $EI$ ) and $U_{p}$ , to capture this effect. We shall see that to a reasonable extent increasing values of $U_{p}$ may be seen as leading to a decrease in $EI^{\ast }$ , and hence an increase in the maximum observed flap opening. We shall explore this further after presenting results for varying flap lengths and stiffnesses in the following subsections.

Broadly, across all flap stiffness and lengths studied, we observe the features described above. However, the time scales and the extent of opening and closing can be very different as we vary the stiffness and length. This also implies that the flow dynamics (to be discussed in § 4) is different for each case. We shall now look at the flap kinematics for all the different cases systematically. We will make comparisons across different cases and try to explain the underlying reasons behind the observations.

3.2 Effect of flap stiffness and length on flap kinematics

We have used flaps with three different stiffness values ( $EI$ ) with three different flap lengths ( $L_{f}/d$ ) for each flap stiffness. We shall first present measured flap shapes as a function of time for different flap $EI$ values followed by different $L_{f}/d$ values. In all cases shown we keep the piston speed $U_{p}$ fixed at $18~\text{cm}~\text{s}^{-1}$ , as in the sample case shown earlier, for ease of understanding. The effects of $U_{p}$ will be presented separately later in the section.

We first show the time sequence of flap kinematics for different stiffness values ( $EI$ ) in figure 6 keeping the flap length ( $L_{f}/d$ ) fixed at 4, as in the sample case already discussed. In each of the three stiffness cases shown, the instances correspond to flaps at the maximum bulge (minimum $d_{e}$ at flow initiation), maximum opening of the flap tips, just before the piston stops, immediately after the piston stops and the minimum $d_{e}$ reached after the piston stops. We chose these physical events, rather than a specific non-dimensional time, as none of the common non-dimensional time scales based on the piston velocity profile such as $t/\unicode[STIX]{x1D70F}$ or $L/L_{f}$ collapse these physical events across the different flap stiffness and length cases studied. This is contrary to our previous study (Das et al. Reference Das, Govardhan and Arakeri2013) without stiffness, where the piston velocity based time scale, $L/L_{f}$ , was sufficient to collapse the data across different flap lengths. In addition, we have also tried to collapse the data using non-dimensional time based on the flap stiffness ( $t/\sqrt{(\unicode[STIX]{x1D70C}L_{f}^{5}/EI)}$ ) (Das Reference Das2016), this being the relaxation time scale for a flap with flexural rigidity ( $EI$ ) and with inertia that is dominated by fluid inertia (negligible mass flap as in the present case). However, even this flap stiffness based time scale did not collapse the flap kinematic data as shown in Das (Reference Das2016), highlighting the fact that there are competing time scales in the present problem, which makes the dynamics richer than in our previous study (Das et al. Reference Das, Govardhan and Arakeri2013) that was in the absence of stiffness. As noted above, the instances corresponding to these physical events occur at different non-dimensional times (say $t/\unicode[STIX]{x1D70F}$ ) for each of the cases, and the corresponding values of ( $t/\unicode[STIX]{x1D70F}$ ) are given in the figure. We can see from the figure that as the flap stiffness is increased, as one would expect, the maximum flap tip displacements reduce, and the changes in deflections are more gradual along the length of the flaps. The latter are especially clear at flow initiation (first image in each sequence) and immediately after the piston stops (fourth image in sequence).

The effect of flap length ( $L_{f}/d$ ) on the flap kinematics is shown in figure 7 with the flap stiffness being maintained at the lowest value in all cases. The physical events or instances that we have selected to show are the same as in the previous figure. Although the $EI$ values are the same, we can see that larger flap lengths decrease the effective stiffness of the flap with longer flaps showing larger flap tip displacements, and larger variations in deformations along the flap length. As in the previous figure, the latter point is clear in the second and fourth images of the sequence for each case that shows the flap deformation at flow initiation and just after the piston stops. In particular, when the piston stops, the longer flap case ( $L_{f}/d=6$ ) shows a clear coming together of the two flaps at approximately 30 % of the flap length. Subsequently, the two flaps collapse and touch each other along a large fraction of the flap length as seen in the final image of the sequence for the longest flap case. This clearly highlights the fact that the total amount of fluid ejected in forming the starting jet can be significantly more than the case with no flaps. This is because some of the fluid from the flap region is expelled in addition to the amount of fluid pushed by the piston, the former effect being absent if there were no flaps. Thus, the net mass of fluid forming the jet is larger in the flap cases, although the same amount of fluid is pushed by the piston. We shall discuss this point in more detail in the next section. An overview of the flap deformation shapes for all the different flap stiffness values and lengths studied at four key instances in time corresponding to the maximum bulge, maximum opening, just after piston stops and when the flaps close fully is shown in the Appendix.

Figure 6.

Flap deformation profiles with time for different flap stiffness values corresponding to (a) least stiff, (b) intermediate stiffness and (c) stiffest. In all cases, $L_{f}/d=4$ , $U_{p}=18~\text{cm}~\text{s}^{-1}$ and the piston stops at $\unicode[STIX]{x1D70F}=1.3~\text{s}$ corresponding to a fluid slug length ( $L_{d}$ ) of 15. The five time instances shown are the maximum bulge, maximum opening of the flap tips, just before the piston stops, immediately after piston stops and the final closing of the flap tips.

Figure 7.

Flap deformation profiles with time for different flap length values corresponding to (a) $L_{f}/d=2$ , (b) $L_{f}/d=4$ and (c) $L_{f}/d=6$ . In all cases, the flaps are of least stiffness, $U_{p}=18~\text{cm}~\text{s}^{-1}$ , and the piston stops at $\unicode[STIX]{x1D70F}=1.3~\text{s}$ corresponding to a fluid slug length ( $L_{d}$ ) of 15. The five time instances shown are the maximum bulge, maximum opening of the flap tips, just before the piston stops, immediately after piston stops and the final closing of the flap tips.

Figure 8.

Plot of exit width ( $d_{e}/d$ ) with $L/L_{f}$ . In (a), the stiffness ( $EI$ ) of the flap is varied while keeping the flap length fixed at $L_{f}/d=4$ , while in (b), the flap length ( $L_{f}/d$ ) is varied while keeping $EI$ fixed at the least stiffness value. In both cases, the piston velocity ( $U_{p}$ ) is $18~\text{cm}~\text{s}^{-1}$ , and the piston is stopped when the fluid slug length reaches $L/d=15$ .

In figure 8, we plot the exit width as a function of ( $L/L_{f}$ ) for the different flap stiffness and flap length cases shown in the preceding two figures. Figure 8(a) shows the effect of variation in flap stiffness for the $L_{f}/d=4$ case, whereas figure 8(b) shows the effect of variation in flap length for the least stiff flap case. Both the plots show that there are significant variations in the flap kinematics with varying flap stiffness and flap length. It may also be noted that the changes due to these parameter variations are significantly larger than the uncertainties in measurement, which are shown at representative points in each plot. Further, as noted earlier, it is clear from these plots that physical events like maximum flap tip opening do not collapse when using the piston velocity profile based non-dimensional time scale $L/L_{f}$ . Other time scales, such as a flap stiffness based time scale ( $t/\sqrt{(\unicode[STIX]{x1D70C}L_{f}^{5}/EI)}$ ), also do not show collapse of the data (Das Reference Das2016). It is also clear, as shown earlier (figure 5), that the flap kinematics is also significantly affected by the piston velocity ( $U_{p}$ ). This is contrary to what was observed in the hinged leaflet cases of Das et al. (Reference Das, Govardhan and Arakeri2013), where a time scale based on ( $L/L_{f}$ ) collapsed the kinematics across different flap lengths and piston speeds. The differences in the flap motion observed in these figures clearly indicates that the flexural rigidity of the flaps ( $EI$ ), flap length ( $L_{f}/d$ ) and piston speed ( $U_{p}$ ) are all important in determining the kinematics of the flaps. This naturally raises the question of finding alternate non-dimensional parameters for the flap kinematics in the present case that should necessarily include $EI$ besides $L_{f}/d$ and $U_{p}$ .

With a view to finding alternate non-dimensional parameters involving $EI$ , we look at non-dimensionalized forms of flexural rigidity ( $EI$ ) used in the literature. Non-dimensionalized forms of flexural rigidity ( $EI$ ) have been used in other studies related to flag flutter (Connell & Yue Reference Connell and Yue2007), flexible flaps in the wake of a cylinder (Shukla, Govardhan & Arakeri Reference Shukla, Govardhan and Arakeri2013) or flapping foils (Shinde & Arakeri Reference Shinde and Arakeri2014). In our problem, the energy of the fluid coming out of the channel ( $1/2\unicode[STIX]{x1D70C}dL_{f}U_{p}^{2}$ ) can be expected to be of the same order as that of the energy stored by the flaps in the form of its bending stiffness ( ${\sim}\unicode[STIX]{x1D6FF}^{2}EI/L_{f}^{3}$ , where $\unicode[STIX]{x1D6FF}$ is the flap tip deflection due to fluid forces). If we define a non-dimensional flexural rigidity as the ratio $L_{f}^{2}/\unicode[STIX]{x1D6FF}^{2}$ , then we get

(3.1) $$\begin{eqnarray}\displaystyle & EI^{\ast }={\displaystyle \frac{EI_{eq}}{{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}U_{p}^{2}L_{f}^{2}d}}, & \displaystyle\end{eqnarray}$$

where we have used $EI_{eq}$ , the equivalent flexural rigidity defined previously to take the self-weight of the flaps into account. The $EI^{\ast }$ defined here is slightly different from the previous studies of Connell & Yue (Reference Connell and Yue2007), Shukla et al. (Reference Shukla, Govardhan and Arakeri2013) and Shinde & Arakeri (Reference Shinde and Arakeri2014) that use ( ${L_{f}}^{3}$ ) in the denominator, while in the present definition we also have the channel width ( $d$ ) in the denominator which is a relevant parameter here.

Figure 9.

Plot of flap kinematic parameters from all experiments in the present study as a function of non-dimensional stiffness ( $EI^{\ast }$ ). The data shown here correspond to studies with different flap lengths ( $L_{f}/d$ ), flap stiffness values ( $EI$ ) and piston velocities ( $U_{p}$ ). In (a), the maximum tip based angle ( $\unicode[STIX]{x1D703}_{max}$ ) is shown, while in (b), the non-dimensional time ( $L/L_{f}$ ) required to reach the maximum $\unicode[STIX]{x1D703}_{max}$ is shown. The piston velocity program is impulsively started and the fluid slug length reaches $L/d=15$ in each case. The legends are: ○, $L_{f}/d=2$ ; ▫, $L_{f}/d=4$ ; ♢, $L_{f}/d=6$ ; with open symbols for least stiff, half-filled for intermediate and filled for stiffest flaps. The arrows in both the panels represent the values for the hinged-rigid flap case of Das et al. (Reference Das, Govardhan and Arakeri2013), where the stiffness was close to zero.

We show that $EI^{\ast }$ is a useful non-dimensionalization in our experiments by directly plotting some overview parameters representing the flap kinematics against this non-dimensional parameter. Two such overview parameters representing flap kinematics could be the maximum flap tip opening and the time taken by the flap tips to reach that maximum opening. As noted before, in the present case, we do not have a unique angle because the flaps deform. However, we can define an angle $\unicode[STIX]{x1D703}$ based on a line joining the flap tip to the rigid end where the flap is fixed (as shown in figure 1). This angle $\unicode[STIX]{x1D703}$ is useful in making a broad comparison with the hinged-rigid cases, while also capturing the exit width ( $d_{e}$ ) as it is directly related to $d_{e}$ ( $\unicode[STIX]{x1D703}=\tan ^{-1}(0.5(d_{e}-d)/L_{f})$ ). We show in figure 9, the maximum flap opening ( $\unicode[STIX]{x1D703}_{max}$ ) and the time taken to reach $\unicode[STIX]{x1D703}_{max}$ , for all the different experiments that we have done, which include variations in $EI$ , $L_{f}/d$ and $U_{p}$ , plotted with $EI^{\ast }$ . In figure 9(a), we show the maximum flap tip opening in terms of the angle ( $\unicode[STIX]{x1D703}_{max}$ ) defined above. It is clear that the data from all the different cases fall onto an approximately linear trend line (on the log plot) indicating a reasonably good collapse of the data with $EI^{\ast }$ . It should be noted here that the collapsed data include variations in $U_{p}$ and $L_{f}/d$ apart from $EI$ , and hence, indicate that as far as the maximum flap opening is concerned, increases in $U_{p}$ and $L_{f}/d$ may be interpreted as equivalent to a reduced $EI^{\ast }$ . Further, the figure shows that as $EI^{\ast }$ is increased, the maximum flap tip angle ( $\unicode[STIX]{x1D703}_{max}$ ) decreases from approximately $20^{\circ }$ at the lowest values of $EI^{\ast }$ and drops to approximately $5^{\circ }$ at the higher end of the $EI^{\ast }$ regime studied. It may be noted here that the largest $\unicode[STIX]{x1D703}_{max}$ values seen here (approximately $20^{\circ }$ ) are comparable to, but slightly higher than, the maximum angles seen for the hinged-rigid flap case studied in Das et al. (Reference Das, Govardhan and Arakeri2013) (shown by the arrow in the figure) where there was no hinge stiffness. In the present case, we see that at low values of $EI^{\ast }$ the angles are comparable to that case, with the angles decreasing from those values as $EI^{\ast }$ is increased. This plot thus gives a reasonable idea of the extent of flap opening across a range of $EI$ , $L_{f}/d$ and $U_{p}$ values. As $EI^{\ast }$ is further decreased towards zero, one would expect that at some value of $EI^{\ast }$ , the maximum flap tip angle would saturate to a value, with further reductions in $EI^{\ast }$ not leading to an increase in flap opening. From the plot, it is not clear if this saturation has been reached. Figure 9(b) shows the time taken by the flaps in terms of ( $L/L_{f}$ ) to reach the maximum angle as a function of $EI^{\ast }$ . Although the collapse is not as good as in figure 9(a), we find that the data from all cases can be reasonably well represented by a single trend line in the plot. It may be noted that the variations seen in figure 9(b) around the broad trend line are not due to experimental uncertainties, which are small, as shown by the uncertainty bars in the figure at representative points. In the plot, we have indicated data for different flap stiffness cases with open/half-filled/filled symbols, while each flap length is indicated by a different symbol. A closer inspection of the data indicates that $EI^{\ast }$ is able to reasonably well collapse the data across different flap stiffnesses and piston speeds for each of the three flap lengths. However, the collapse of the data for different flap lengths is not as good, with the longer $L_{f}/d=6$ flap data showing a small but distinctly different trend from the smaller two flap lengths data. This is likely related to the presence of more complex flap mode shapes with large bending close to the flap tips in the longer $L_{f}/d=6$ flap case, as shown in figure 7. However, a broad trend line is shown for all the three flap length data sets to give an overview of the data across all flap lengths, stiffnesses and piston speeds. At low values of $EI^{\ast }$ , as in figure 9(b), we find a reasonably good agreement between the present results and the data for the hinged-rigid flap case studied in Das et al. (Reference Das, Govardhan and Arakeri2013), both showing $L/L_{f}$ values of approximately 1.75–2. As the $EI^{\ast }$ value is increased, the $L/L_{f}$ values corresponding to maximum $\unicode[STIX]{x1D703}_{max}$ drop down to approximately 0.5 at the highest $EI^{\ast }$ value studied of approximately 1. The reasonable collapse of the flap kinematics data with $EI^{\ast }$ indicates that $EI^{\ast }$ is a good normalization for flap stiffness, and can be used to broadly collapse data from a wide range of independent experiments covering variations in $EI$ , $L_{f}/d$ and $U_{p}$ .

4.1 Vortex dynamics for a sample case

In figure 10, we show the vorticity contours for the sample case of least stiff flaps with $L_{f}/d=4$ (same as that in figure 3). At flow initiation in figure 10(a), the flaps bulge outwards in the centre due to an over-pressure, after which the bulge gradually moves outwards resulting in the flap tips opening outwards. In figure 10(b), we can see formation of two vortices ( $\text{C1}$ and $\text{C2}$ ) that are due to the outward motion of the flaps, which results in separation of the boundary layer formed on the outer surface of the flaps at the flap tips. In Das et al. (Reference Das, Govardhan and Arakeri2013), we saw similar vortices being formed when the rigid flaps moved outwards. As in that case, we will refer to these vortices as $\text{C}$ vortices, which are of opposite sense to the vortices formed if the flaps did not move. As the flaps open out fully and start returning to their initial position, the ensuing fluid starts separating from the flap tips (figure 10 d). These vortices ( $\text{A}$ ) result from the separating boundary layer on the inside surface of the flaps. The $\text{A}$ vortices are the ones which would form if the flaps were absent, or if the flaps did not move. Since the flaps open out to sufficiently large angles, there is also flow separation in the flap region due to a diffuser kind of flow, resulting in the formation of $\text{B}$ vortices, which can be seen forming in figure 10(d) and that come out of the flaps in figure 10(e). These $\text{B}$ vortices are also of the same sense as the $\text{A}$ vortices. However, the $\text{B}$ vortices not only form due to flow separation in the flap region, but also get a contribution from the flow separation at the flap tips at later times (as seen in figure 10 e). As the piston stops, the flaps collapse inwards (figure 10 f) resulting in the formation of $\text{D}$ vortices, which are distinctly visible in figure 10(h). Thus, with the introduction of flexible flaps at the exit, we observe the evolution of four distinct kinds of vortices as opposed to a single vortex pair. The relative strength of these vortices depend on the flap length and the stiffness of the flaps besides $U_{p}$ . As discussed in the previous section, all the flaps display qualitatively similar kinematics, hence we expect the formation of all these vortices in all cases, although their individual strengths may vary.

Figure 10.

Time sequence of vorticity contours for $L_{f}/d=4$ , least stiffness flap case with $EI^{\ast }=0.034$ . The piston velocity ( $U_{p}$ ) is $18~\text{cm}~\text{s}^{-1}$ , and the piston stops at $\unicode[STIX]{x1D70F}\approx 1.3~\text{s}$ corresponding to a final fluid slug length reached of $L/d=15$ . In each plot, 20 equally spaced contour levels are shown between ( $\pm$ ) $\unicode[STIX]{x1D714}d/U_{p}$ , with the value of $\unicode[STIX]{x1D714}d/U_{p}$ being in (a) 1.36, (b) 1.33, (c) 1.35, (d) 1.86, (e) 1.83, (f) 2.38, (g) 2.21, (h) 2.53. The instances shown in the figure are at $t/\unicode[STIX]{x1D70F}=$ (a) 0.1, (b) 0.19, (c) 0.28, (d) 0.59, (e) 0.97, (f) 1.03, (g) 1.1, (h) 1.16.

Figure 11.

Plot of circulation as a function of time for the sample case shown in figure 10. (a) Circulation of individual vortices as a function of time. (b) The net circulation ejected as a function of time. In both cases, uncertainty in circulation values are shown at representative points.

Figure 11 shows the plot of circulation as a function of time for this sample case. The circulation for each of the individual vortices described before is plotted in figure 11(a) as a function of time. The text $t_{a}$ , $t_{b}$ , $t_{c}$ , $t_{d}$ refers to the time at which vortices $\text{A}$ , $\text{B}$ , $\text{C}$ and $\text{D}$ are first observed respectively. To compute the value of circulation for, say the $\text{A}$ vortices, we measure the circulation of $\text{A1}$ and $\text{A2}$ separately and then take the average of their absolute values. The uncertainty in the evaluation of circulation obtained from different realizations is also shown on the plot at chosen representative points, indicating that the uncertainties are small compared to the observed variations. The figure shows that the $\text{C}$ vortices are formed first at time $t_{c}$ and thereafter remain almost constant in strength. The $\text{A}$ vortices (at time $t_{a}$ ) form second and start gaining strength as more and more fluid separates at the flap tips. The $\text{A}$ vortices stop growing and the circulation reaches a saturation point when the $\text{B}$ vortices form due to separation in the flap region ( $t_{b}$ ). The subsequent vorticity from the boundary layer is then fed to these vortices which start gaining in strength. Once the piston stops at $t=1.3~\text{s}$ , fluid from the flap region is pushed out resulting in further growth of these $\text{B}$ vortices and making their circulation larger than all other vortices. The rapid closure of the flap tips results in the formation of $\text{D}$ vortices at $t_{d}$ . The circulations of $\text{A}$ , $\text{B}$ and $\text{D}$ vortices are similar in value whereas the $\text{C}$ vortices are the weakest. It should be noted that the relative strengths of these vortices will depend on the kinematics of the flaps for each flap stiffness and length case.

Figure 11(b) plots the total circulation ejected from the flap region. As the fluid gets pushed out of the flap region, the circulation increases continuously with time. Once the piston stops at $t/\unicode[STIX]{x1D70F}=1$ , we see that the circulation still keeps increasing because of more fluid being pushed out of the flap region in the form of $\text{D}$ vortices (at $t_{d}$ ). Thereafter, the net circulation reaches a constant final value once the flaps close.

We next look at the fluid impulse of the flow coming out of the nozzle. It has been shown that the impulse ( $\boldsymbol{I}$ ) of a flow given by a velocity field $\boldsymbol{u}$ is defined as the time integral of the impulsive force $\boldsymbol{F}$ that has to be applied to the flow to generate the velocity field (Saffman Reference Saffman1993; Voropayev & Afanasyev Reference Voropayev and Afanasyev1994; Lighthill Reference Lighthill1996), with $\boldsymbol{I}$ being given by:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{I}=\int _{0}^{\unicode[STIX]{x1D70F}}\boldsymbol{F}\,\text{d}t, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ is the time over which the force is applied. This impulse can also be formulated using the vorticity ( $\unicode[STIX]{x1D74E}$ ). This is of interest because the rate of change of fluid impulse is linked to the total non-conservative body forces acting on the fluid. Noca (Reference Noca1997) and Epps (Reference Epps2010) have studied this in detail and provide many expressions for evaluating the time-dependent forces on a body in a fluid, which can be used to estimate the forces on a body from measurements of the flow field around it. DeVoria & Ringuette (Reference DeVoria and Ringuette2013) for example have used this method to evaluate the force on a rotating plate by evaluating the fluid impulse from the measured velocity field. To calculate the fluid impulse in our case, consider a control volume which extends to infinity ( $V_{\infty }$ ) that contains all the fluid which is outside of a control volume $V_{b}$ containing the piston cylinder and the flaps. Now, the control volume $V_{\infty }$ is large enough to encompass all of the vorticity coming out of $V_{b}$ , and the velocity and vorticity become negligible in the limits of its boundaries at $x_{\infty }$ and $y_{\infty }$ . Since the control volume chosen is such that the volume $V_{b}$ (which generates the flow) does not move, we can evaluate the impulse ( $\boldsymbol{I}$ ) of the fluid contained in $V_{\infty }$ using the expression:

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{I}=\unicode[STIX]{x1D70C}\int _{V_{\infty }}(\boldsymbol{x}\times \unicode[STIX]{x1D74E})\,\text{d}{\mathcal{V}}. & \displaystyle\end{eqnarray}$$

We are interested in the impulse in the $x$ direction since the impulse in the $y$ direction will be zero because of flow symmetry about the $x$ axis. For a two-dimensional flow field, equation (4.2) simplifies to

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle I_{x}=\unicode[STIX]{x1D70C}\int _{A}y\,\unicode[STIX]{x1D714}_{z}\,\text{d}A, & \displaystyle\end{eqnarray}$$

where $A$ is a large enough area which includes all the vorticity and $I_{x}$ is evaluated per unit depth.

Figure 12.

Plot of impulse as a function of time for the sample case shown in figure 10 along with a rigid exit case. All the other parameters for the two cases are the same. The times $t_{a}$ , $t_{b}$ , $t_{c}$ , $t_{d}$ at which the vortices $\text{A},~\text{B},~\text{C}$ and $\text{D}$ are first observed respectively are also marked in the plot for the flap case. Uncertainty in impulse values are shown at representative points for both cases. The piston stops at the dotted line along the $x$ axis when the fluid slug length $(L/d)$ reached is 15.

From our experimental data, this impulse is evaluated by taking a large flow field which encompasses all the vorticity that has come out of the flap region. We do this by masking the flaps along with the region contained between the flaps and then processing the images to get the flow field. A large enough field of view ensured that all the vorticity was contained within the control volume and that the fluid velocities were sufficiently small on the outer boundaries for the duration studied. The $x$ component (streamwise) of the impulse is plotted in figure 12 for the rigid exit case and for the sample flap case. Although not shown here, the $y$ component is approximately zero because of flow symmetry. We notice that by introducing the flaps at the exit of the channel, the fluid impulse changes dramatically in comparison to the rigid exit case, where the impulse increases almost linearly until the piston stops (similar to figures 18 and 19 in Krieg & Mohseni (Reference Krieg and Mohseni2013)). In the rigid case, after the piston stops, the impulse of the flow becomes nearly a constant value ( $I_{xr}$ ).

In the flaps case, the times $t_{a}$ , $t_{b}$ , $t_{c}$ , $t_{d}$ at which the vortices $\text{A},~\text{B},~\text{C}$ and $\text{D}$ are, respectively, first observed, are also marked in the plot. Initially, as the flaps bulge, the impulse is close to zero as all the vorticity is present only within the flap region. As the flap tip opens outwards, the $\text{C}$ vortices that are of opposite sign to the other vortices start to form at time $t_{c}$ outside the flap region, and we find that the impulse of the fluid becomes negative due to the opposite sign of $\text{C}$ vortices. Subsequently as the flaps close, the $\text{A}$ vortices begin to form at $t_{a}$ with their strengths being much larger and of opposite sign to the $\text{C}$ vortices. This leads to a very rapid increase in the impulse, with the rate of increase of impulse being considerably larger than the rigid exit case. The increase in impulse continues at approximately the same rate as $\text{B}$ vortices are formed and add to the total impulse. The impulse data in the flap case are approximately linear between $t/\unicode[STIX]{x1D70F}\approx 0.4$ and 1, with a slope of approximately 1.8 times that of the slope of the impulse data for the rigid exit case. This indicates that the net force is significantly higher for the flap case over a reasonable amount of time. Once the piston stops, unlike in the rigid exit case, the impulse of the fluid continues to increase, but at a slower rate than before. This is due to the fact that after the piston stops, the flaps close rapidly but over some finite time, during which they continue to push out more fluid and vorticity resulting in a larger total impulse in the end. The final impulse of the sample flap case shown is approximately 60 % higher compared to the final impulse of the rigid case ( $I_{xr}$ ).

The final impulse can also be understood as the product of the final circulation ( $\unicode[STIX]{x1D6E4}=\int \,\unicode[STIX]{x1D714}_{z}\,\text{d}A$ ) and an effective lateral spacing ( $y_{eff}$ ) between the two oppositely signed regions of vorticity. We can therefore determine $y_{eff}$ from (4.3) as:

(4.4) $$\begin{eqnarray}\displaystyle & y_{eff}={\displaystyle \frac{\displaystyle \int y\,\unicode[STIX]{x1D714}_{z}\,\text{d}A}{\unicode[STIX]{x1D6E4}}}. & \displaystyle\end{eqnarray}$$

This effective lateral separation distance can be thought of as the distance between the centroid of the vorticity on each side of the line of symmetry ( $x$ axis). Knowing the final fluid impulse and the final circulation, we can then find the corresponding $y_{eff}$ . Therefore, for the same circulation, a higher impulse would imply a higher $y_{eff}$ . The results thus suggest that in the flap case, the opening of the flaps results in larger $y_{eff}$ due to vorticity redistribution, thereby increasing the final fluid impulse. We shall see in the following subsections that this is broadly true for all the flap cases, with the final impulse and $y_{eff}$ being larger for nearly all the flap cases compared to the rigid exit case.

Figure 13.

Instantaneous streamlines along with velocity magnitude contours for the sample case with least flap stiffness and $L_{f}/d=4$ . The colour variation shows the absolute velocity magnitude of the fluid normalized with the piston speed. All the parameters are the same as in figure 10.

We next plot the streamlines for the sample case in figure 13 along with colour contours of the fluid velocity magnitude. This combination of streamlines and velocity magnitude normalized by the piston speed gives a broad overview of the fluid motion in the vicinity of the flaps at different times. Due to the initial bulge (figure 13 a), the ambient fluid is displaced creating a flow resembling potential flow outside the flap region. As the flaps open outwards, the streamlines close to the flaps follow the flap deformation profile. Until the instance shown in figure 13(c), the fluid velocity outside of the flap region is very small suggesting that most of the fluid has until now only been accommodated in the extra volume created by the flaps opening out. At instance (d), the rolling up of the streamlines shows the formation of $\text{A}$ vortices at the flap tips. At instance (e), we see large fluid velocities outside of the flap region and in between the vortices that are due to two reasons. The first reason is that the flaps moving inwards pushed fluid out of the flap region adding to the fluid being pushed by the piston, resulting in larger fluid velocities. The second reason is the induced flow created by the vortices on either side of this jet of fluid. At the instance seen in figure 13(f), the collapsing flaps result in the fluid being dragged in to fill the space created on the outer side of the flaps. Once the flaps begin to close the fluid starts getting pushed out of the flap region. This results in large fluid velocities in between the vortices as seen in figure 13(g). In instance (h) the flaps have closed fully and we see that the fluid velocity in some regions has reached almost 1.35 times the piston speed ( $U_{p}$ ). The streamlines here show that the fluid velocities are large and concentrated in the region between the vortices.

4.2 Effect of flap stiffness and length on vortex dynamics

We shall now present time sequences of the flow field for variations in flap stiffness and flap length along with the corresponding circulation and impulse values. Figure 14 shows the vorticity contours for three different flap stiffness cases ( $EI^{\ast }$ ), all with $L_{f}/d=6$ . The instances shown in each case, as before, are at the maximum bulge, at the maximum opening, just before piston stops, after piston stops and after all the fluid has been ejected out. The contour levels which are plotted are kept the same for all the cases. It is clear that although the flow fields are qualitatively similar, there are important differences as well. For example, distinct $\text{B}$ vortices are formed in the least stiff case in figure 14(a), while in the stiffest case in figure 14(c), the $\text{B}$ vortices can hardly be seen. This difference may be attributed to the flaps considerable opening out in figure 14(a), which leads to flow separation within the flap region that leads to formation of $\text{B}$ vortices, while the flaps do not open out as much in the stiffer case in figure 14(c), preventing flow separation. In addition to the larger opening in figure 14(a), it should also be noted that the least stiff flaps also open more gradually giving larger times for the flow to separate as compared to the stiff flap cases. Differences can also be seen with respect to the formation of $\text{C}$ vortices formed during the outward motion of the flap tips. These $\text{C}$ vortices can be seen clearly in figure 14(a) and (b). In the stiffest flap case in figure 14(c), the initial bulge and subsequent opening of the flap tips happens very rapidly, resulting in a to and fro motion of the flap tips. This results in the formation of a vortex pair near each flap tip seen in the second image (of figure 14 c), which then quickly shoot out in the transverse direction, thus removing the $\text{C}$ vortices from the flap tip region. These differences lead to significant changes in the final vorticity configuration in the three cases indicating that the stiffness of the flaps is an important parameter in determining the flow.

Figure 14.

Plot showing time sequence of vorticity contours for (a) least stiff flaps, $EI^{\ast }=0.028$ (b) intermediate stiffness flaps, $EI^{\ast }=0.096$ and (c) stiffest flaps, $EI^{\ast }=0.49$ . The flap length is $L_{f}/d=6$ and the increment between successive vorticity ( $\unicode[STIX]{x1D714}d/U_{p}$ ) contour levels is approximately 0.14 in each case. The piston speed $U_{p}\approx 18~\text{cm}~\text{s}^{-1}$ , and the piston stops when the fluid slug length reached is $L/d=15$ . The different times for each flap stiffness are the time taken for maximum bulge, maximum opening, the time just before piston stops, just after piston stops and the time taken for complete closure of the flaps respectively from top to bottom of the plot. The instantaneous time ( $t/\unicode[STIX]{x1D70F}$ ) for each instance is shown in the figure.

Figure 15.

Plot of (a) impulse as a function of time and (b) circulation as a function of time, for different flap stiffness cases along with the rigid exit data. Uncertainty estimates are shown at a few representative points. In all cases, the flap length ( $L_{f}/d$ ) is 6, the piston speed $U_{p}\approx 18~\text{cm}~\text{s}^{-1}$ , and the final fluid slug length $(L/d)$ reached is 15. The piston stops at the dotted line along the $x$ axis, where the fluid slug length $(L/d)$ is 15.

In figure 15, we plot the impulse (figure 15 a) and the circulation (figure 15 b) for the three flap stiffness cases discussed above along with the reference rigid exit case. Both the impulse ( $I_{x}$ ) and circulation ( $\unicode[STIX]{x1D6E4}$ ) values are normalized by the final impulse ( $I_{xr}$ ) and circulation ( $\unicode[STIX]{x1D6E4}_{r}$ ) values of the rigid exit case. With respect to the impulse, the most important point to note is that the final impulse of the flow is higher in all the flap cases when compared to the rigid exit case, with the final impulse being about 1.5 to 1.75 times higher than the rigid exit case. In all three cases, we can see that there are some common characteristics of the impulse variation with time, while there are also some distinct differences. In all flap cases, the impulse does not immediately increase after the flow starts due to the bulging of the flaps, which leads to very little vorticity ejection from the flap region. Subsequently, the impulse increases in all cases with the rise being more rapid in the least and stiffest cases, compared to the intermediate stiffness case, although the final impulse values are all reasonably similar. The differences with stiffness may be better understood by simultaneously looking at the circulation values for the three cases. In the stiffest flap case, the flaps do not open out as much and hence vorticity is ejected from the flap region relatively early and therefore shows up as an earlier rise in circulation in the plot compared to the intermediate and least stiff flaps. In these latter two cases, the flaps open out more, and hence the vorticity takes longer to be ejected from the flap region, and hence the circulation rise is delayed. We find that the final circulation values reached are similar for the most and least stiff flap cases both of which are also close to the rigid exit case. This suggests that in these cases, the net circulation is about the same but is redistributed to have a higher impulse, which is linked to the larger lateral separation ( $y_{eff}$ ) between the two opposite regions of vorticity as discussed in the previous subsection. In the intermediate stiffness case, we find that the final circulation values are a little lower (approximately 15 %), which is likely linked to cancellation of opposite signed vorticity within the flap region. The reason for the larger vorticity cancellation in this intermediate stiffness case may be related to the larger time taken for opposite signed vorticity within the flap region to be ejected in this case, compared to both the least and most stiff flap cases. This can be seen in figure 14 and especially figure 15(b), where the intermediate and least stiff flap cases have nearly the same circulation ejected until the instant when the piston stops ( $t/\unicode[STIX]{x1D70F}=1$ ). Thereafter, in the least stiff flap case, the circulation rapidly increases as the flaps swiftly deform and eject vorticity, while in the intermediate stiffness case, a much more gradual increase in circulation is seen related to the slower ejection of vorticity. In contrast to these, the stiffest flap has a circulation variation with time that is much more rapid right from the start of piston motion and closer to that of the rigid case, as the flaps do not deflect as much as the least or intermediate stiffness cases. Related to this, one can also notice a more diffuse vorticity field in the intermediate stiffness case. In general, more diffuse vortices are seen to form when the motions of the flap as a whole are more gradual, and the flap has no rapid local deformations near the flap tip. On the other hand, more tightly rolled up vortices are seen to form when there are rapid local deformations near the flap tip (least stiff flap case) and when the flap as a whole has more rapid motions (stiffest flap case).

Figure 16.

Plot showing time sequence of vorticity contours for (a) $L_{f}/d=2$ , (b) $L_{f}/d=4$ and (c) $L_{f}/d=6$ . The flap stiffness ( $EI$ ) is the least in each case, and the piston speed $U_{p}\approx 18~\text{cm}~\text{s}^{-1}$ , thus giving $EI^{\ast }$ values of 0.097, 0.035 and 0.028, respectively, in the three cases. The different times for each flap length case shown are the time taken for maximum bulge, maximum opening, the time just before piston stops, just after piston stops and the time taken for complete closure of the flaps respectively from top to bottom. The time instances are the same as shown in figure 7 and the increment between successive vorticity contour levels is approximately 0.14 in each case.

We now look at the effect of flap length $L_{f}/d$ while keeping the flap material and hence the flap stiffness ( $EI$ ) constant. Figure 16 shows the vorticity contours for the least stiff flap case ( $EI=2\times 10^{-5}~\text{Nm}$ ). We choose this flap stiffness because it undergoes larger deformations and therefore shows more pronounced differences in flow features. The flap lengths used are $L_{f}/d=2$ (figure 16 a), $L_{f}/d=4$ (figure 16 b) and $L_{f}/d=6$ (figure 16 c). The instances shown are the same as in figure 7 with the vorticity contour levels shown being the same in all cases. It is clear that there are important changes in the vorticity dynamics as the flap length is varied. For the longest flap case ( $L_{f}/d=6$ ), when the flaps are at their maximum opening, there is a very clear flow separation in the flap region due to the large angle created by the flaps. This results in the formation of $\text{B}$ vortices and these can be clearly seen at this instance. In contrast, the $\text{B}$ vortices are not as strong in the $L_{f}/d=4$ case, and are completely absent in the $L_{f}/d=2$ case, where the flap opening is relatively small. In this latter case ( $L_{f}/d=2$ ), both the $\text{B}$ and $\text{C}$ vortices cannot be seen and only strong $\text{A}$ vortices and relatively weaker $\text{D}$ vortices corresponding to flap closing can be seen. In the $L_{f}/d=4$ and $L_{f}/d=6$ cases, we can see all the different types of vortices $\text{A}$ , $\text{B}$ , $\text{C}$ and $\text{D}$ , although their relative strengths are different. The final vorticity configuration in the three cases shows significant differences, as seen in the final image of each sequence, highlighting the importance of flap length on the flow.

Figure 17.

(a) Impulse as a function of time and (b) circulation as a function of time, for different flap length cases along with the rigid exit data. In all cases, the flap stiffness ( $EI$ ) is the least, the piston speed $U_{p}\approx 18~\text{cm}~\text{s}^{-1}$ and the final fluid slug length $(L/d)$ reached is 15. The piston stops at the dotted line along the $x$ axis, where the fluid slug length $(L/d)$ is 15.

Figure 18.

Plot of the normalized volume flow rate out of the flap region as a function of time for (a) the $L_{f}/d=6$ case with variations in flap stiffness, and (b) for the least stiff ( $EI$ ) flap case with variations in flap length. The filled symbols in each case show the time instance when the maximum flap tip opening occurs. The rigid exit case is shown by the solid line in both plots. In each case, the piston velocity $U_{p}\approx 18~\text{cm}~\text{s}^{-1}$ , and the final fluid slug length $(L/d)$ reached is 15.

The impulse and circulation values as a function of time for the three different flap length cases are shown in figure 17(a) and (b), respectively. As before, the final impulse is higher in all the flap cases compared to the rigid exit case with the largest value being for the shortest flap ( $L_{f}/d=2$ ) case that has a final impulse that is approximately twice the value of the rigid exit case. Among the different flap length cases, the $L_{f}/d=2$ case is strikingly different. One can see a clear rise in impulse within a relatively small time after flow initiation, as circulation is relatively rapidly ejected from the flap region for this short flap case, as shown in figure 17(b). The impulse increases approximately linearly until slightly after the piston stops, with the slope of this line being roughly twice that of the rigid exit case. We will see later in § 4.3 that this is related to the overall configuration of vorticity, more precisely to the extent of lateral motion of the centroid of vorticity that we shall see is the largest for this case. Compared to the $L_{f}/d=2$ case, the longer flaps take larger times after flow initiation before they show a significant rise in impulse as the vorticity takes longer to emerge from the flap region, consistent with the measurement of circulation shown in figure 17(b). The time taken for the circulation to rise from around zero is longest for the $L_{f}/d=6$ case, as might be expected, with the final circulation however being nearly the same in all three cases and equal to the rigid case, as observed before. It may be noted here that the non-dimensional flexural rigidity or stiffness ( $EI^{\ast }$ ) value for the $L_{f}/d=2$ case (figure 16 a) is approximately 0.097, which is nearly the same as the intermediate stiffness case (with $L_{f}/d=6$ ) shown in figure 14(b). Clearly, although the two have nearly the same values of $EI^{\ast }$ , the vorticity distribution and the time variation of the impulses are very different indicating that $EI^{\ast }$ alone is not sufficient to determine the vorticity dynamics, although it does give a reasonable collapse of the flap kinematics as shown in the previous section.

We shall now highlight the effect that flap deformation has on the net volume flow rate out of the flexible flap exit ( $\dot{Q}$ ). The volume flow rate out of the rigid exit is completely determined in our incompressible case by the piston speed ( $U_{p}$ ), as is shown by the solid line in figure 18, which is representative of the near impulsive piston velocity program used in our study. We define $\dot{Q}_{r}$ as the flow rate associated with the constant speed portion of this velocity program for the rigid exit case, and use this to normalize the flow rate for all the flap cases shown in the figure, with effects of variations in $EI$ of the flap being shown in figure 18(a) and effects of variations in flap length ( $L_{f}/d$ ) shown in figure 18(b). In the flexible flap cases, the flap deformations change with time, leading to the possibility of transient storage and expulsion of fluid within the flap region, and hence, the flow rate out of the flap region can vary significantly from the piston velocity program. In these cases, we have another term that is dependent on the rate of change of volume contained within the two (deforming) flaps, which we evaluate from the recorded flap deformation profiles at different instances in time. Initially, as the flaps bulge out, the volume flow rate out of the flexible exit is less as the flow pushed out by the piston is taken up to fill the increasing volume between the two flaps, with this effect being more pronounced for the longer and lower stiffness cases. Subsequently, as the region between the flaps stops expanding, the flow rate out ( $\dot{Q}$ ) equals $\dot{Q}_{r}$ that is pushed by the piston. It may be noted that the time instance where the maximum flap tip opening occurs, shown by the filled symbols in the plot, does not correspond to $\dot{Q}$ equals $\dot{Q}_{r}$ , as the latter condition corresponds to the case where the volume of the region between the flaps is maximum. As the volume of the region between the flaps begins to reduce, the fluid between the flaps is pushed out in addition to the fluid pushed out by the piston leading to substantially higher volume flux than $\dot{Q}_{r}$ , reaching values of approximately $2\dot{Q}_{r}$ . One can also see that the volume flux out continues to be non-zero even after the piston stops, this being caused by the collapsing flaps that continue to push fluid out from the flap region. These effects can be seen in all the different flap stiffness and length cases shown in figure 18, although the extent of these effects vary, with the more pronounced effects being seen for the less stiff (in figure 18 a) and the longer flap cases (in figure 18 b). In the smallest flap case ( $L_{f}/d=2$ ) in figure 18(b), the changes in volume between the flaps is relatively small, and the volume flow rate is reasonably close to the flow rate generated by the piston motion alone, except during the initial times and after the piston stops.

4.3 Final vorticity configuration

We now present in figure 19 the final vorticity configuration, after all of the fluid has left the flap region, for the different flap stiffness ( $EI$ ) and flap length ( $L_{f}/d$ ) cases studied. The time at which this occurs is different for each case, but it gives an overall picture of the final flow configuration. In the figure, the flap length increases as we go from top to bottom, and the stiffness increases as we go from left to right. In each case, the different kinds of vortices ( $\text{A}$ , $\text{B}$ , $\text{C}$ and $\text{D}$ ) are labelled and one can see the significant changes with stiffness and flap length. For the shortest flap length of $L_{f}/d=2$ , only the $\text{A}$ and $\text{D}$ vortices are present. This is because the time in which the flaps open and return to their initial position is small and therefore the flow does not separate in the flap region to form the $\text{B}$ vortices. In the other two flap length cases, all these vortices are present. The stiffest flaps however, irrespective of the flap length, do not clearly show the formation of $\text{B}$ and $\text{C}$ vortices. The $\text{D}$ vortices in all the $L_{f}/d=4$ and $L_{f}/d=6$ cases move reasonably fast compared to the other vortices and often either merge with the $\text{B}$ (and/or $\text{A}$ ) vortices or overtake them (as in the $L_{f}/d=6$ , intermediate stiffness case). This can be attributed to the fact that the $\text{D}$ vortices are generated by a rapid inward motion of the flaps after the piston stops. This results in the formation of vortices which are laterally less spread out compared to the $\text{A}$ , $\text{B}$ and $\text{C}$ vortices. This would mean that the self-induced velocity of this pair would be higher as observed in the experiments. The final configuration of the vorticity out of the flap region is thus dramatically different with the introduction of flexible flaps and varies with the flexural rigidity and length of the flaps. However, in almost all the cases, the total circulation ejected from the flap region is nearly the same as the rigid exit case, indicating that the total circulation is shared among the different vortices seen in each case.

Figure 19.

Vorticity contours for the flap cases at the instance when all the fluid has been ejected from the flap region after the piston stops. The flap length increases from $L_{f}/d=2$ to $L_{f}/d=6$ from the top to bottom image, and the stiffness ( $EI$ ) increases from least to most as we go from left to right. The $EI^{\ast }$ values for (a) ( $L_{f}/d=2$ ) from left to right are 0.097, 0.69, 3.84 respectively; for (b) ( $L_{f}/d=4$ ) from left to right are 0.035, 0.18, 1.01 respectively; for (c) ( $L_{f}/d=6$ ) from left to right are 0.028, 0.096, 0.49 respectively. In all cases, the piston speed $U_{p}$ is $18~\text{cm}~\text{s}^{-1}$ , and the piston stops when the fluid slug length reached is 15. The increment between successive vorticity contour levels is approximately 0.14 in each case.

Figure 20.

The final vorticity contours for (a) rigid exit case, (b) hinged-rigid flap cases ( $L_{f}/d=2$ & 4), (c) least stiff flexible flap cases ( $L_{f}/d=2$ & 4). The instance chosen is when all the fluid has been ejected after the piston stops. The piston velocity program is impulsively started in each case with the piston speed being $18~\text{cm}~\text{s}^{-1}$ in the rigid and flexible flap cases, and $5.5~\text{cm}~\text{s}^{-1}$ for the hinged-rigid flap cases (Das et al. Reference Das, Govardhan and Arakeri2013), with the final fluid slug reached in all cases being close to 15. The increment between successive vorticity contour levels is approximately 0.14 in each case.

The present study has been done with deformable flexible flaps having a flexural rigidity or stiffness ( $EI$ ). In a previous study, Das et al. (Reference Das, Govardhan and Arakeri2013) studied the case of hinged-rigid flaps with negligible hinge stiffness, and a comparison of the final vorticity configurations between the two studies is shown in figure 20 along with the reference rigid exit case. Since the study of Das et al. (Reference Das, Govardhan and Arakeri2013) was for the negligibly small hinge stiffness case, we choose for the comparison, low stiffness cases from our present study, and in particular show two flap length cases $L_{f}/d=2$ and $L_{f}/d=4$ . The first observation is that qualitatively the vorticity field is very different from the rigid exit case for all flap cases. Comparing the hinged-rigid flaps and the deformable flap cases, we can see that there are some striking similarities and some distinct differences. There is a clear similarity across both flap types for the $L_{f}/d=2$ and $L_{f}/d=4$ cases, with the shorter flap showing just a relatively simple configuration in both cases, while the longer flap cases have a more complex distribution of vorticity with the formation of many different types of vortices. A distinct difference between the two flap types is the clear formation of separate vortices $\text{A}$ , $\text{B}$ , $\text{C}$ and $\text{D}$ in the deformable flap case with $L_{f}/d=4$ , in contrast to the hinged-rigid case where the different vortices are present, but are not as clear. The same is true for the $L_{f}/d=2$ case, where the $\text{D}$ vortices are somewhat clearer in the deformable flap case as compared to the hinged-rigid flap case. These differences appear to be linked to the fact that in the flexible flap cases, the deformations can be local and rapid (say near the flap tips) and hence permits clear formation of distinct vortices, while in the hinged-rigid flap case, the entire flap would respond more slowly as it has to move as a whole in response to the fluid forces.

Figure 21.

Figure showing the plot of final (a) impulse (b) circulation and (c) $y_{eff}$ for all the flap cases as a function of $EI^{\ast }$ . The impulse and circulation values are normalized by the corresponding rigid exit case values. Symbols are: ○, $L_{f}/d=2$ ; ▫, $L_{f}/d=4$ ; ♢, $L_{f}/d=6$ ; with open symbols for least stiff, half-filled for intermediate and filled for stiffest flaps. The open symbols with a dot inside are for the corresponding hinged leaflet cases from Das et al. (Reference Das, Govardhan and Arakeri2013). In (a) and (b) the values are normalized by the corresponding rigid exit case. In (c), for reference, $y_{eff}/d\approx 1$ for the rigid exit case.

Figure 21 shows final fluid impulse (figure 21 a), final circulation (figure 21 b) and $y_{eff}/d$ (figure 21 c) for all the flap cases plotted against the non-dimensionalized flexural rigidity ( $EI^{\ast }$ ). In addition to the present data, the hinged-rigid data from Das et al. (Reference Das, Govardhan and Arakeri2013) corresponding to nearly zero stiffness are also shown in each plot of the figure. The impulse and circulation values have been normalized with the corresponding rigid exit values, with both plots showing that the data are relatively insensitive to $EI^{\ast }$ over the range of $EI^{\ast }$ values studied from nearly zero to approximately 10, although there is some variation. The uncertainties in both impulse and circulation are shown by uncertainty bars for all the cases in figure 21. While these experimental uncertainties are present, the impulse values are still significantly higher compared to the rigid exit values whereas the net ejected circulation values are close to the rigid exit value. For the impulse values, the variations across different stiffness cases also appears to be real and not because of measurement uncertainties. As noted before, the flap deformations in these different cases can vary significantly (as shown in § 3), and $EI^{\ast }$ only broadly collapses the maximum flap tip opening data. This implies that considerable differences in vorticity distribution can result from the variations in flap deformation, even for the same $EI^{\ast }$ , as highlighted by the differences in vorticity distribution between figures 14(b) and 16(a) (both having the same $EI^{\ast }$ ). Hence, variations, such as those seen in figure 21, between the different flexible flap cases are not surprising, although it is not easy to explain them. Despite these complexities, a broad overall result is that the final fluid impulse in the cases with flaps ( $I_{x}$ ) is approximately twice the impulse values in the rigid exit case ( $I_{xr}$ ), as shown by the dashed line in figure 21(a) over the range of $EI^{\ast }$ values studied.

As $EI^{\ast }$ increases, we expect the normalized impulse to reduce gradually to the rigid exit value of 1.0. The circulation plot shows that the final circulation in all cases is about the same as the rigid exit case, although there are some variations. The increase in impulse is thus mainly related to the changes in the lateral spacing of the generated vortices ( $y_{eff}/d$ ). This is reflected in the $y_{eff}/d$ plot, which is qualitatively similar to the impulse plot, with $y_{eff}/d$ being approximately about 2 over the range of $EI^{\ast }$ values studied. We mark this distance in figure 19 by drawing two dotted lines in each vorticity plot, the distance between them being $y_{eff}/d$ . At higher $EI^{\ast }$ values, $y_{eff}/d$ is expected to reduce gradually to the rigid case value of approximately 1.0 as $EI^{\ast }$ tends to large values. It may also be seen that the data for the hinged-rigid flap cases (Das et al. Reference Das, Govardhan and Arakeri2013), with nearly zero stiffness, are broadly in agreement with the present deformable flap cases in each of the three plots. Overall the motion of the flaps, as discussed in detail before, leads to a redistribution of vorticity, which in turn results in higher final impulse values.

In this section, we have looked at the effect of flap stiffness ( $EI$ ) and flap length ( $L_{f}/d$ ) on the vorticity dynamics. The results show that even when the non-dimensional flexural rigidity ( $EI^{\ast }$ ) is maintained (as in figures 14(b) and 16 a), the detailed vorticity dynamics varies with flap length ( $L_{f}/d$ ). This indicates that $EI^{\ast }$ alone is not sufficient to determine the detailed vorticity dynamics, although it does give a reasonable collapse of the flap kinematics and integrated quantities such as the total final impulse. The observed differences for example in the final normalized impulse about the mean value of 2 appear to be related to these observed differences in the vorticity dynamics. In general, the formation of up to four kinds of vortices were observed with the introduction of flaps at the exit, which we refer to as $\text{A}$ , $\text{B}$ , $\text{C}$ and $\text{D}$ , along with significant changes in the final impulse of the flow that can be more than 100 % higher than the rigid exit case. The rate at which impulse increases, which is linked to force, was also found to be substantially higher over a significant time for the flexible cases in comparison to the rigid exit case.