Flow–structure interaction of a starting jet through a flexible circular nozzle

Abstract

In the present study, the flow–structure interaction of a starting jet through a flexible nozzle is experimentally investigated, with a focus on the optimal flexibility for thrust generation. Water slug is impulsively accelerated through a cylindrical nozzle, fabricated with silicone rubber of varying flexibility. In general, the flexible nozzle modifies the vortical structure of the jet and augments the thrust of the starting jet. The measurement of nozzle surface deformation revealed that a back-and-forth wave propagation on the nozzle surface is responsible for the jet-vortex evolution augmenting the thrust generation. Combining the hydrodynamic conservation equations and the linearized shell theory, we also formulated the governing equations, dominated by two relevant dimensionless parameters: the effective acceleration time of the jet ($\varPi _0$) and the effective nozzle stiffness ($\varPi _1$). Asymptotic analysis of the equation showed that the dimensionless wave speed ($\hat {c}$) is expressed as $\hat {c}=(\varPi _{0}^{2}\varPi _{1}/2)^{0.5}$, and the jet momentum is maximized at $\hat {c}=\hat {c}_{crit}$ ($\simeq 3.0$), the condition at which the release of elastic energy stored during nozzle contraction to the jet is synchronized with the instant of termination of jet acceleration. While $\hat {c}=\hat {c}_{crit}$, the achievable maximum jet velocity decreases with the effective acceleration time of the jet ($\varPi _0$), which is attributed to the reduced speed of the surface wave by the flow inside the nozzle.

1. Introduction

A starting jet is ubiquitous in diverse fluid applications that require the transport of fluid energy to fulfil various purposes, such as the cleaning process of bag filters (Li et al. 2016) and pulsed-jet propulsion of aircraft (Garnier 2015), underwater vehicles (Linden 2011; Renda et al. 2015) and natural organisms such as squid (Packard 1969; Gosline & DeMont 1985; Weymouth & Triantafyllou 2013), salp (Sutherland & Madin 2010) and jellyfish (Park et al. 2014; Fang et al. 2017). In the biomedical field, blood influx to the atrium through the heart valve (Mittal 2018) and the transmission of viral droplets through sneezing (Bourouiba, Dehandschoewercker & Bush 2014; Mittal, Ni & Seo 2020) can be understood through the dynamics of a starting jet.

The flow induced by a starting jet can be characterized in terms of the hydrodynamic impulse, contributed by fluid inertia and pressure difference, because it is directly related to not only thrust generation but also the entrainment (mixing) of the ambient fluid (Krueger & Gharib 2003; Krieg & Mohseni 2013). Because the momentum flux is an integration of $\rho _{f} u_{e}^{2}$ (where $u_e$ is the jet-exit velocity and $\rho _f$ is the fluid density) on the jet-exit plane during a specified time duration, it is generally affected by the jet generator (e.g. piston and impeller), in particular for a rigid nozzle or steady jetting condition. On the other hand, the contribution of the pressure difference ($p-p_\infty$, where $p_\infty$ is the ambient pressure) is affected by various factors (Krueger & Gharib 2003; Krieg & Mohseni 2013; Gao et al. 2020). Because the jet-exit pressure is closely related to the pinch-off dynamics of the vortex ring and the subsequent entrainment process, previous studies used the formation number ($F$), defined as the ratio of piston stroke to nozzle diameter, to understand it. Considering the formation number as a characteristic time scale, it was suggested that the fully grown primary vortex ring pinches off the nozzle tip at a critical value of $F \simeq 4.0$, which entrains the ambient fluid and increases the pressure rise at the exit, thereby maximizing the total thrust normalized by the momentum flux (Gharib, Rambod & Shariff 1998; Krueger & Gharib 2003). At the same formation number, on the other hand, a faster acceleration of the jet results in a higher thrust production (Krueger & Gharib 2003), and Gao et al. (2020) mentioned that the pressure rise is altered by transient jet evolution such as the growth of the primary vortex and the deceleration process of the jet. Meanwhile, Krieg & Mohseni (2013) measured the increase in the impulse from a converging nozzle by $70\,\%$ compared with a circular cylinder nozzle and explained that the radial velocity contributes to the rise in pressure at the jet exit using the potential flow theory. Meanwhile, Limbourg & Nedić (2021a,b) proposed the modified slug-flow model, which can account for flow contraction at the nozzle exit.

In terms of a jet-propelled vehicle (or robot), it was reported that pulsed-jet propulsion is more efficient than that by a steady jet because of the added mass and entrainment effect (Siekmann 1963; Krueger & Gharib 2003; Dabiri 2009; Bujard, Giorgio-Serchi & Weymouth 2021), which depends on the formation number (Moslemi & Krueger 2010; Nichols & Krueger 2012; Whittlesey & Dabiri 2013), jet acceleration (or deceleration) (Krieg & Mohseni 2015) and Reynolds number (Moslemi & Krueger 2011). Despite these achievements, studies concerned with maximizing thrust are quite lacking. When the nozzle through which a propulsive jet flow is issued is rigid, the generated impulse is directly determined by the piston movement and nozzle geometry; however, if the nozzle material is sufficiently flexible so that it can interact with the jet, the fluid–structure interaction will significantly affect the generated impulse as well as the efficiency.

As a method to control the hydrodynamics of a moving body, on the other hand, many nature-inspired researches have investigated the fluid–structure interaction in the flow over a flexible (compliant) surface (Triantafyllou, Triantafyllou & Grosenbaugh 1993; Kang et al. 2011; Marais et al. 2012; Park et al. 2012, 2016; Dewey et al. 2013; Quinn, Lauder & Smits 2014, 2015; David, Govardhan & Arakeri 2017; Medina & Kang 2018). In common, they reported that a certain level of flexibility enhances thrust generation or power efficiency of a flapping wing and fin model, which has been explained through the alignment of shed vortices with surface deformation. Dewey et al. (2013) showed that thrust generation and propulsive efficiency of a flexible pitching panel are maximized when the dimensionless pitching frequency (Strouhal number, $St$) is in the range $0.2< St<0.25$ (where $St = f_{p} A_{p}/u_\infty$, in which $f_p$ is the pitching frequency, $A_p$ is the pitching amplitude and $u_\infty$ is the free-stream velocity) and matches the natural (structural) frequency of the panel. Park et al. (2012, 2016) explained the optimal flexibility for thrust generation in terms of the phase lag ($= {\rm \pi}/2$) between the pitching angle, i.e. the condition of vortex shedding, and the rotation of the trailing edge owing to the flexibility. Li, Jaiman & Khoo (2021) reported that the flow-induced oscillation of a flexible membrane enhances lift generation by extending the leading-edge vortices, and a transition of vibrational modes appears owing to the frequency lock-in between the natural frequency of the membrane and vortex-shedding frequency. In addition, a flexible flapping foil is known to suppress asymmetric vortex shedding, inhibiting deflection of the propulsive jet (Marais et al. 2012).

Despite the advances in understanding the role of flexible surfaces in flow control, its application is confined to undulating appendages like wing, fin and flag. Furthermore, the effect of flexibility on jet flow requires a more fundamental investigation. For example, Dabiri & Gharib (2005) reported that the circulation of the leading vortex ring is enhanced by forcing the active shrinkage of a circular nozzle exit during the operation of the starting jet, which decreases the kinetic energy of the vortex and thereby sustains the vorticity supply. Das, Govardhan & Arakeri (2018) showed that a flexible flap attached to the tip of a two-dimensional slit increases the jet impulse two times by enhancing the circulation of jet vortices. Also, Jung, Song & Kim (2021) recently showed that the eversion of an elastic sheet installed at the nozzle exit enhances the impulse of a starting jet, which increases with bending rigidity. Dabiri (2009) suggested that a living structure (such as the human heart and jetting marine animals) operates under specific optimal conditions in terms of vortex formation. However, the governing parameter (i.e. formation number only) obtained for the ideal case (i.e. piston experiment) has a limitation in terms of explaining the different strategies used universally. Furthermore, a physical model for optimal flexibility, beyond a qualitative description, needs to be developed based on fluid–structure interaction analysis of unsteady jet-flow evolution. Indeed, most previous studies tend to simply conclude that a flexible surface performs better than a rigid one without a systematic quantification of the relevant parameters.

It would be helpful to briefly summarize the previous studies of flow inside a flexible tube (Lin & Morgan 1956; Kraus 1967; Païdoussis & Denise 1972; Chen & Rosenberg 1974; Païdoussis 1998), which has gained more attention than jet flow. The natural frequency and vibration mode of an immersed thin flexible tube can be predicted using the thin elastic shell theory, together with hydrodynamic equations (Lin & Morgan 1956; Kraus 1967). While the inertia and acoustic couplings between the flexible tube and viscous fluid flow significantly affect the frequency and wavenumber of the resulting wave on the tube (dispersion relation), each contribution can be characterized by the mass ($\chi$) and speed ratio ($c_{sr}$). The mass ratio is defined as $\chi = \rho _{f} R/(\rho _{s} h)$, where $\rho _s$, $h$ and $R$ are the tube density, wall thickness and tube radius, respectively, and the speed ratio ($c_{f}/c_{s}$) is the ratio of the speed of sound in fluid ($c_f$) to the wave speed ($c_s$) on the tube surface (Lin & Morgan 1956; Chen & Rosenberg 1974; Païdoussis 1998). Furthermore, the flow inside the flexible tube could lead to self-excited oscillation of various modes and frequencies (Bertram, Raymond & Pedley 1991; Heil 1997; Kumaran 1998; Heil & Waters 2006; Heil & Boyle 2010; Whittaker et al. 2010; Paak, Païdoussis & Misra 2013; Siviglia & Toffolon 2014; Stewart 2017; Zhang, Luo & Cai 2018; Abdelbaki, Paidoussis & Misra 2020; Podoprosvetova & Vedeneev 2022). When one end of the tube is clamped and the other is free, similar to the present condition, steady flow with a bulk velocity of $u_b$ excites periodic, multimode or chaotic vibration at a specific velocity ratio, $u_{r}=u_{b}/c_{s}$ (Paak et al. 2013). Thus, it is reasonable to assume that there should exist a specific interaction between starting jet and flexible tube, which we do not understand clearly yet.

Therefore, in this study, we experimentally investigated the evolution of a starting jet through a circular nozzle, the flexibility of which is varied, with a focus on the relation between fluid–structure interaction and thrust generation (the hydrodynamic impulse and total circulation). The conditions of liquid flow developing through the nozzle are controlled by varying the acceleration duration ($T_{acc}$) and the terminal jet centreline velocity ($u_t$), such that the jet Reynolds number (defined as $Re_{j}=2\rho _{f} u_{t} R/\mu _f$, where $\mu _f$ is the dynamic viscosity of water) and the effective acceleration time (defined as $\varPi _{0}=T_{acc} u_{t}/L$, where $L$ denotes the nozzle length) vary as $3290\unicode{x2013}12\,500$ and $1.05\unicode{x2013}2.33$, respectively. To understand the phenomenon better, we simultaneously measured the deformation of the nozzle surface and flow fields (both inside and outside of the nozzle) using high-speed particle image velocimetry (PIV). Based on the analysis, we suggest the optimal condition for maximizing the thrust generation by analytically combining the shell theory and principle of conservation (of mass and momentum). Asymptotic analysis and energy balance were also used to formulate the model. We believe that the present results will significantly enhance our understanding of the fluid–structure interaction occurring in diverse flow geometries found in nature and in industry.

The remainder of this paper is organized as follows. We explain the experimental set-up and characterize the flexible nozzle used in § 2. In § 3, we discuss the effects of flexibility and flow conditions on jet evolution and the resultant thrust. In § 4, the analytical derivation of the optimal condition for maximum thrust is explained and validated. Finally, a summary and outlook of our work are given in § 5.

2. Experimental set-up and procedure

2.1. Facility for generating a starting jet

We established a piston–motor system to create a starting jet in a transparent acrylic chamber ($300 \times 300 \times 700$ mm$^{3}$) filled with tap water at room temperature (figure 1a). Actuated by a servomotor (ELDM6020, LeadShine), an aluminium piston (with a diameter of $50$ mm) inside the acrylic cylinder instantly moves downward guided by a linear stage placed above the chamber. The cylindrical passage of the piston is connected to a smaller-diameter pipe (with an inner diameter of $15$ mm) through a smooth contraction part (contraction ratio of $0.09$). A motor driver (ELD5-400, LeadShine) and built-in control algorithm enable precise control of the motor to follow a designated velocity profile with respect to the time at the jet exit (for the rigid nozzle). We observed that there is neither mechanical vibration nor acoustic noise during the actuation, which may originate from failure in optimizing the motor–driver system.

Figure 1. (a) Schematic diagram for the jet-flow generating facility and the detailed installation of the flexible nozzle on the piston system. (b) Image of the manufactured flexible nozzle with its support.

The motor is controlled to perform impulsive acceleration of the piston (i.e. flow inside the pipe and subsequent jet flow) with time duration $T_{acc} = 0.12\unicode{x2013}0.43$ s (figure 2a) to reach the targeted terminal velocity, $u_{t} = 0.20\unicode{x2013}0.84$ m s$^{-1}$ (figure 2b). Here, the acceleration time ($T_{acc}$) denotes the time when the jet-exit velocity at the nozzle centre ($u_e$) reaches $90\,\%$ of the terminal velocity ($u_{e} = 0.9u_t$) with the rigid nozzle. Notably, the exit velocities ($u_e$ and $u_t$) are measured at $(x/D, y/D) = (0.0, -0.2)$ to avoid interference with the nozzle wall. The Reynolds number of the jet is considered to be $Re_{j} = 3290\unicode{x2013}12\,500$. As the jet flow becomes as fast as $u_{t} = 0.64\unicode{x2013}0.84$ m s$^{-1}$, it tends to weakly oscillate (figure 2b); for example, the oscillation amplitude of $u_t$ is approximately $0.05$ m s$^{-1}$ ($6.8\,\%$ of $u_t$) at $u_{t} = 0.84$ m s$^{-1}$. This periodic fluctuation originates from the feedback control implemented in the motor driver; however, we confirmed that the nozzle deformation is dominated by the rapid acceleration of the jet, and the slight oscillation occurring at $t>T_{acc}$ does not affect the results. Before initiating each motor operation, the piston is precisely aligned with the cylindrical container by adjusting the location of the linear guide on the uppermost plate of the chamber to avoid physical contact with the inner wall of the container. At $u_{t} = 0.22$ m s$^{-1}$, the deviation of the jet velocity at the nozzle exit is approximately $0.03u_t$. Figure 2(c) shows the temporal variation of the jet-velocity profile at the nozzle exit for the rigid nozzle, which shows the symmetry against the nozzle centreline. Initially, the jet flow develops with two peaks near the nozzle wall ($t/T_{acc} = 0.5$), which is attributed to the formation of a vortex ring (Didden 1979), and gradually evolves into a top-hat profile because of the boundary layer that develops inside the nozzle ($t/T_{acc} > 1.5$). Figure 2(d) shows the piston velocity compared with the designated and actual jet-exit velocities. Piston velocity was measured using a high-speed camera (SpeedSense M310, Dantec Dynamics), on which the oil pattern was marked to be tracked by a correlation-based algorithm. To compensate the effect of contraction geometry (see figure 1a), the effective piston velocity ($u_{p,eff}$) is defined based on the contraction ratio ($\simeq 0.09$), expressed as $u_{p,eff}=u_{p}/0.09$. We measured five sets of piston motion and their corresponding jet-exit velocities to obtain the averaged values and ensure repeatability. It is confirmed that $u_e$ follows the planned velocity profile accurately. As shown, they have the same acceleration time ($0.2$ s) and target velocity ($-0.2$ m s$^{-1}$); however, there is a slight deviation between $u_{p,eff}$ and $u_e$ in the accelerating stage. This deviation would be caused by the cylinder geometry (e.g. shape and length), piston setting (e.g. sealing and stiffness of O-ring rubber) and coefficient of motor feedback control.

Figure 2. Time history of jet-exit velocity ($u_e$) with varying (a) acceleration time ($T_{acc}$) under the same terminal velocity of $0.2$ m s$^{-1}$ and (b) terminal velocity ($u_t$) with $T_{acc}=0.12$–$0.43$ s. (c) Velocity profile in radial direction with $u_{t} = 0.2$ m s$^{-1}$ and $T_{acc} = 0.2$ s. (d) Comparison of $u_e$ with the piston and targeted velocities. The jet velocities were measured at $y/D = -0.2$.

Notably, the starting jet in the present study refers to the initiation of a continuous jet, indicating that the bulk velocity at the inlet of the nozzle is steady throughout the entire process, rather than a pulsed jet. In previous studies, a pulsed jet was employed to make all the ejected fluid slug contribute to the formation of the leading vortex ring, i.e. without the trailing jet, which was mainly characterized by the formation number ($F$) (Gharib et al. 1998; Krueger & Gharib 2003). In contrast, we are mainly interested in the role of the interplay between the flexible nozzle and the accelerating jet in thrust generation. Thus, a relatively longer jet (i.e. $F = \infty$) was chosen, which is more suitable for revealing the fundamental mechanism of the fluid–structure interaction in the jet flow.

2.2. Preparation and installation of the flexible nozzle

The flexible nozzle in the present study needs to satisfy two conditions: (i) it has to be thin enough to interact with the jet flow (below $200\ \mathrm {\mu }$m for the present condition) and (ii) it must be optically accessible (i.e. transparent) to allow measurement of the fluid flow inside the nozzle. To achieve this, the dip-mould method was used to make a transparent cylindrical nozzle with a wall of thickness of $\textbf {\textit {O}}(10\unicode{x2013}10^{2})\ \mathrm {\mu }$m. First, liquid-state silicone rubber (SortaClear 40A, SmoothOn) was poured onto an acrylic cylinder rod (mould) with a diameter of $15$ mm, mounted vertically on a support. The silicone flows down along the sidewall of the cylinder and gradually forms a thin film during curing. The silicone nozzle was then cut to the desired length and was carefully removed from the mould, yielding a thin-walled and transparent nozzle (figure 1b). By varying the mass fraction ($9.1\unicode{x2013}32.4$ wt%) of the thinner (Silicone Thinner, SmoothOn) to the liquid-state silicone rubber, it was possible to control the mean thickness ($h$) of the wall from $170$ to $70\ \mathrm {\mu }$m, and the corresponding Young's modulus ($E$) from $270$ to $100$ kPa, with a constant Poisson's ratio of $0.4$. The Young's modulus and Poisson's ratio were measured by the two-dimensional digital image correlation (DIC) technique. First, specimens of $1$ mm in thickness of silicone rubber with a mass fraction of $9.1\unicode{x2013}32.4$ wt% were prepared for tensile testing, on which a speckle pattern was printed using oil-based ink. Next, the tensile test machine extended the silicone rubber specimen while the high-resolution camera captured the deformed speckle pattern on the rubber surface. Based on this, a strain–stress curve is obtained, the slope of which corresponds to the Young's modulus, and the Poisson's ratio also comes out from the two-dimensional strain field calculated by digital image correlation (for details, see the supplementary material available at https://doi.org/10.1017/jfm.2022.781).

The uniformity of the thickness distribution was also verified. Along the azimuthal direction, we confirmed that the variation was negligible (a maximum deviation of $11.1\,\%$ at a mass fraction of $9.1\unicode{x2013}23.1$ wt%) by measuring the cross-sections at multiple streamwise locations using a microscope. However, the nozzle wall becomes thicker at a rate of $12.5\,\%$ per nozzle diameter ($= 15$ mm) along the longitudinal direction, which is inevitable for the present manufacturing process. For a nozzle of $45$ mm in length, the wall thickness at the nozzle tip ($= 51.1\ \mathrm {\mu }$m) is approximately $27\,\%$ thinner than that at the support ($= 70\ \mathrm {\mu }$m) for a mass fraction of $9.1$ wt%. Considering this variation, the averaged thickness ($h$) is used as a characteristic thickness of the nozzle and, as we will discuss, the theoretical analysis (see § 4) based on the assumption of a constant wall thickness agrees well with the experimental results. On the other hand, the uncertainty in the repeatability of the manufactured nozzle is found to be within $6.6\,\%$ (in terms of the wall thickness) from six independent trials (for details, see the supplementary material available at https://doi.org/10.1017/jfm.2022.781).

In total, we tested three flexible nozzles with nozzle structural stiffnesses of $Eh = 7.0$, $14.4$ and $43.2$ N m$^{-1}$. The length ($L$) and inner diameter ($D$) of the cylinder were $41.1$ and $15.0$ mm, respectively, corresponding to an aspect ratio ($L/D$) of $2.74$ (figure 1a). Because a higher strain rate is concentrated at the junction between the flexible nozzle and the rigid support as the nozzle is deformed by fluid forces, which can damage the silicone wall, it is necessary to mount the flexible nozzle in a robust manner. To cope with this, we manufactured a soft support together with the silicone nozzle (figure 1a), having a smooth strain gradient at the junction of the thin nozzle and support. In addition to flexible nozzles, we also tested a rigid nozzle as a reference, which is an aluminium tube with a wall thickness of $0.5$ mm. The length and inner diameter were the same as those of the flexible nozzles. In table 1, we summarize the conditions of jet flow and the nozzle morphology considered in the present experiments.

Table 1. Summary of the geometric and material properties of the nozzle and the jet condition considered in the present study, with the dimensionless parameters governing the starting jet–nozzle interaction.

2.3. Measurement of flow velocity and nozzle deformation

Considering the nominally axisymmetric nozzle geometry and resultant flow, we performed two-dimensional PIV to measure the flow fields inside and outside the nozzle. Likewise, previous studies of jet flow from a circular nozzle focused on the velocity fields measured on the centre plane (Dabiri & Gharib 2005; Gao et al. 2020). As tracer particles, we used glass particles (HSG-10, Dantec Dynamics) of nominal diameter of $50\ \mathrm {\mu }$m. A $5$ W Nd:YAG laser (RayPower 5000, Dantec Dynamics) generated a green-coloured laser sheet of wavelength $532$ nm and illuminated the seeding particles on the measurement ($x\unicode{x2013}y$) plane (figure 1a). A high-speed camera (with a resolution of $1000 \times 800$ pixels) captured images of the particles with a frequency ranging from $1000$ to $2000$ Hz depending on the targeted jet velocity. Acquisition of the image is triggered by the initiation of the piston–motor system by directly hardwiring the motor driver and camera adaptor. The time delay between the motor start and image acquisition is within $1.0\ \mathrm {\mu }$s. To capture both the development of the jet structure and the nozzle deformation, the field of view has a width of $3D$ ($D$ being the inner diameter of the nozzle) and vertically ranges over $8D$ from the nozzle support in the jet direction.

To measure the unsteady deformation of the nozzle wall and velocity fields together, an additional image-processing step is applied to the raw images, shown in figure 3(a). As intended, the particles are clearly observed inside the nozzle. First, we apply a median filter to remove the particle images (figure 3b). In the figure, the boundary of the nozzle is detected as the position where the horizontal gradient of the grey-level intensity becomes a maximum (figure 3c). To enhance the detection resolution as much as the subpixel displacement ($\sim 0.1$ pixels or $0.04\,\%$ of $D$), we used Gaussian interpolation. By subtracting the nozzle image from the original image, we obtained an image of particles only (figure 3d), which is used to measure the flow velocities. The velocity vectors were evaluated by a custom-made PIV algorithm adapting the subpixel offset-window scheme (Scarano 2001). In detail, the interrogation window ($32 \times 32$ pixels) is offset to the subpixel displacement, pre-calculated from the larger-sized ($64 \times 64$ pixels) interrogation window. The resultant velocity distribution has a spatial resolution of $0.066D$ ($= 0.99$ mm).

Figure 3. Image-processing procedure to measure the nozzle shape and particle image: (a) raw image including the image of nozzle and particles; (b) nozzle image; (c) detected nozzle shape; (d) image of seeding particles for PIV.

Based on error propagation analysis (Clifford 1973), the uncertainty of the flow velocity obtained by PIV can be expressed as $\delta (u_{piv})=\sqrt {\delta (MF)^{2}+\delta ({\rm \Delta} s)^{2}+\delta ({\rm \Delta} t)^{2}}$ (Lawson et al. 1999; Choi & Park 2018; Choi, Lee & Park 2019; Maeng & Park 2021). Here, $\delta (x)$ denotes the percentage error while measuring $x$, $MF$ is the pixel magnification factor, ${\rm \Delta} t$ is the time difference between two particle images and ${\rm \Delta} s$ is the displacement of the particle through ${\rm \Delta} t$. In the present set-up, the percentage error involved in determining the magnification factor is measured as $0.41\,\%$ with $MF = 61.0\ \mathrm {\mu }$m pixel$^{-1}$. The error in the time interval (${\rm \Delta} t$) is estimated as $0.1\,\%$, given that the maximum uncertainty is within $1.0\ \mathrm {\mu }$s whereas the time interval between the image pair is $1.0$ ms. Lastly, the uncertainty of ${\rm \Delta} s$ is calculated as $3.8\,\%$ with an average particle displacement of $3.0$ pixels (Scarano 2001). Combining all contributions, the overall uncertainty for the velocity measurement is approximately $3.8\,\%$ for the present experiments.

2.4. Dimensionless parameters

As we will discuss, the analysis of the fluid–structure interaction for jet flow through the deforming flexible nozzle is governed by a combination of the jet-flow conditions (terminal velocity and acceleration time) and the morphological properties of the nozzle (Young's modulus, aspect ratio and wall thickness). Here, we propose two dimensionless parameters to characterize the jet condition and the nozzle stiffness: the effective acceleration time ($\varPi _{0}=T_{acc} u_{t}/L$) and the effective nozzle stiffness ($\varPi _{1}=Eh/(\rho _{f} u_{t}^{2} R$)), where $R$ is the inner radius of the nozzle ($= D/2$). Here, $\varPi _0$ is defined as the ratio of jet acceleration time ($T_{acc}$) to the characteristic time scale of the starting jet ($L/u_t$), which is further understood as the inverse of the Strouhal number ($St$), $\varPi _{0} = u_{t} T_{acc}/L = (f_{acc}L/u_{t})^{-1}={St}^{-1}$, by introducing $f_{acc}=1/T_{acc}$. In previous studies of thrust generation by a flexible flapping surface, the Strouhal number was considered to explain the interaction between the geometric deformation and surrounding flow structure (Kang et al. 2011; Marais et al. 2012).

The second parameter ($\varPi _1$) is the ratio of the elastic force released during the expansion of the nozzle ($\sim EhR$; see Kraus 1967) to the inertia of the jet flow ($\sim \rho _{f}u_{t}^{2}R^{2}$). A similar non-dimensional parameter ($\varPi '_{1}$) was adopted in the study of the interaction between a thin flexible body and flow around it, which was derived as $\varPi '_{1}=E/(\rho _{f}u_{c}^{2})\cdot(h/l_{c})^{3}$ from the Euler–Bernoulli beam equation, where $u_c$ and $l_c$ denote the characteristic velocity and length of geometry, respectively (Shapiro 1977; Kang et al. 2011; Dewey et al. 2013; Siviglia & Toffolon 2014; Medina & Kang 2018). In those cases, the elastic body exchanges momentum with the fluid flow through the bending force ($\sim Eh^{3}/l_{c}$, if the magnitude of the beam deflection is comparable to the characteristic length) and the inertia of flow is scaled as $\rho _{f}u_{t}^{2}l_{c}^{2}$. For the present problem, nozzle expansion is dominant over bending, and thus it is reasonable to consider the expansion force ($\sim EhR$) as the relevant dimensionless parameter. From the perspective of discussing the balance between elastic and fluid forces, we understand that the present parameter ($\varPi _1$) is consistent with $\varPi '_{1}$. In the following sections, we discuss how the behaviour of the flexible nozzle and modulation of the jet-flow structure are governed by these parameters. In § 4, their roles are emphasized by analytically showing that they appear in the governing equations describing the internal flow through the flexible nozzle and its shell deformation.

3. Experimental results and discussion

3.1. Change in the jet characteristics with nozzle stiffness

First, we discuss the evolution of a starting jet depending on the structural stiffness ($Eh$) of the nozzle (figure 4). Figure 4(a) shows the time history of the jet vortical structure from the rigid nozzle, for the case of $Re_{j} = 3286$ and $T_{acc} = 0.2$ s. As the jet starts, a pair of counter-rotating vortices representing the cross-section of the vortex ring evolves gradually (numbered as ‘1’ in figure 4a-ii), which pinches off from the nozzle exit (figure 4a-iii) at $t/T_{acc} \simeq 0.76$. Following the primary vortex, a trailing shear layer develops and rolls up into the secondary vortex ring (numbered as ‘2’ in figure 4a-iv). As the flow develops, the lateral distance between counter-rotating vortices, i.e. size of the vortex ring, increases, and the secondary vortex is entrained and merged into the primary vortex (figure 4a-v,a-vi), whereas the third and fourth vortices (numbered as ‘3’ and ‘4’ in the figure) show up successively along with the jet shear layer. Overall, this temporal evolution of the vortical structure of the starting jet from a circular nozzle is consistent with that observed in a previous study (Zhao, Frankel & Mongeau 2000).

Figure 4. Evolution of jet flow (velocity vectors and vorticity ($\hat {\omega }=\omega D/u_{t}$) contour) depending on the structural stiffness ($Eh$) of the nozzle at $Re_{j} = 3286$ and $T_{acc} = 0.2$ s: (a) $Eh = \infty$ ($\varPi _{0}=1.05$ and $\varPi _{1}=\infty$); (b) $43.2$ N m$^{-1}$ ($1.05$ and $120.02$); (c) $14.4$ N m$^{-1}$ ($1.05$ and $40.0$); (d) $7.0$ N m$^{-1}$ ($1.05$ and $19.45$). The profile of the deformed nozzle is drawn as black lines, and the numbered index in each box indicates the birth order of each vortex.

Under the same flow conditions, the change in the jet vortical structures from flexible nozzles with different structural stiffness ($Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$) is shown in figure 4(b–d), together with the nozzle profile. With flexible nozzles, similar flow structures are retained such that the primary vortex develops (figure 4b-ii–d-ii) and pinches off (figure 4b-iii–d-iii) as the jet flow accelerates. However, the pinch-off instant of the primary vortex from the shear layer is affected by the flexibility of the nozzle. It occurs at $t/T_{acc}= 0.35$, $0.6$ and $0.77$ for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively, while it is at $t/T_{acc} = 0.76$ for the rigid nozzle ($Eh = \infty$). Thus, it is understood that the promotion and delay of the vortex pinch-off for the relatively stiffer ($Eh = 43.2$ and $14.4$ N m$^{-1}$) and softer ($Eh = 7.0$ N m$^{-1}$) nozzles, respectively, are closely associated with the expansion and contraction behaviour of the deformable nozzle, which is discussed in the following sections. Away from the nozzle exit, the primary vortex merges with the secondary vortex (figure 4b-v–d-v) while moving downward. For $Eh = 43.2$ N m$^{-1}$, interestingly, the merging of the first and second vortex rings is less organized than that in other cases, which is judged from the scattered small-scale eddies around the merged vortex (figure 4b-vi). In addition, the radius of the merged vortex ring ($\sim 0.9D$) is larger than others ($\sim 0.8D$). This is related to the early pinch-off ($t/T_{acc} \simeq 0.35$) of the primary vortex; the strength of the secondary vortex ring is comparable (figure 4b-iv) with the primary vortex ring with an insufficient feeding of the vorticity from the shear layer. This results in a more vigorous interaction between the vortex rings, which does not occur for other nozzles (figure 4a-iv,c-iv,d-iv). After merging with the secondary vortex ring, the primary vortex is convected downward faster as the nozzle becomes more flexible, indicated by the longer distance travelled from the nozzle exit: at $t/T_{acc} = 2.66$, it is measured to be $3.7D$ ($Eh= 43.2$ N m$^{-1}$), $4.5D$ ($14.4$ N m$^{-1}$) and $5.0D$ ($7.0$ N m$^{-1}$) (figure 4b-vi–d-vi). The convection speed of the vortex is proportional to its circulation ($\varGamma _{vor}$), the growth rate of which is associated with the jet-exit velocity ($u_e$), expressed as ${\rm d}\varGamma _{vor}/{\rm d}t=u_{e}^{2}/2$, if the horizontal velocity is absent at the jet exit (Saffman 1995; Krieg & Mohseni 2013). For the present set-up, the pinch-off time and the jet-exit velocity increase (see figure 7b) with decreasing $Eh$, which results in faster migration of the vortex ring.

Because the amplitude of nozzle deformation is less than $5\,\%$ of the nozzle diameter, which is not sufficiently large to make a quantitative analysis (figure 4), we additionally performed subpixel image processing (Scarano 2001) to extract the unsteady variation of radial displacement ($w_n$) of the nozzle wall from processed images (figure 2c) with a resolution of approximately $0.0004D$. The measured $w_n(y,t)$ is further normalized with the maximum radial displacement ($w_{n,max}$) as $w_{n}^{*}=w_{n}/w_{n,max}$. Note that the nozzle deformation was not averaged circumferentially but was measured on one centre plane considering the axisymmetric expansion and contraction of the nozzle. Figure 5 shows the spatio-temporal contour of $w_{n}^{*}$ with varying $Eh$ of the nozzle for the same flow conditions as those of figure 4. As shown, the nozzle initially expands axisymmetrically and reaches the maximum expansion, which is delayed with decreasing $Eh$: $w_{n,max} = 0.0040D$ (at $t/T_{acc} = 0.15$), $0.012D$ (at $0.20$) and $0.024D$ (at $0.35$) for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively. Once the expansion wave is transmitted to the nozzle exit ($y/D = 0$), it returns to the original position with almost the same speed (note that the slopes of the solid and dashed lines drawn around the first expansion wave are equivalent). That is, the nozzle contracts towards its original shape ($w_{n} = 0$) without a delay. After one cyclic transmission of the expansion wave, the nozzle subsequently undergoes irregular collapses (buckling that causes flow asymmetry inside the nozzle in figure 4d-iv) because of the negative pressure inside the nozzle induced by the accelerated fluid jetting out of the nozzle (at $t/T_{acc}> 0.25$, $0.4$ and $0.60$ for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively) (figure 5). At this stage, optical edge detection shows its limitation as the nozzle wall is folded and wrinkled, over-reflecting light, and we removed the corresponding data in figure 5 to avoid any misunderstanding. Notably, the jet characteristics responsible for thrust generation are mainly relevant to the stage of expansion–contraction by the first surface wave (i.e. $t/T_{acc}< 0.25$, $0.4$ and $0.60$ for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively). Thus, in the following sections, we focus on the initial stage to discuss the optimal fluid–structure interaction condition to achieve enhanced thrust generation.

Figure 5. Spatio-temporal contour of the normalized nozzle displacement ($\hat {w}^{*} = w_{n}/w_{n,max}$) at $Re_{j} = 3286$ and $T_{acc} = 0.2$ s, depending on the structural stiffness: (a) $Eh= 43.2$; (b) $14.4$; (c) $7.0$ N m$^{-1}$. Here, the maximum displacement $w_{n,max}/D$ is measured as (a) $0.010$, (b) $0.022$ and (c) $0.036$.

As highlighted with arrows in figure 5, expansion–contraction of the nozzle results in the surface wave propagating with constant speed (the edge of displacement contour is fitted linearly) between the nozzle support ($y/D \simeq 2.5$) and tip ($y/D = 0$). Through this surface deformation, the elastic energy stored during expansion is transported to the ejecting jet, increasing the impulse of the jet flow. The first cycle ($T_c$) of expansion–contraction gets longer as the nozzle becomes more flexible; for example, $T_{c}/T_{acc} = 0.3$, $0.4$ and $0.6$ for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively. Interestingly, this matches the instant when the primary vortex ring pinches off: $0.35T_{acc}$, $0.6T_{acc}$ and $0.77T_{acc}$ (for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively), indicating the close relationship between nozzle deformation and vortex evolution, which is further analysed in the following sections.

The visualization shown in figure 6 describes the mechanism of vortex formation (figure 6a–d) and nozzle deformation along the azimuthal direction (figure 6e–h) for the case of $Eh = 7.0$ N m$^{-1}$ corresponding to the most flexible condition tested. Near the end of jet acceleration ($t/T_{acc} = 0.7$), the primary vortex is developing at the nozzle exit (figure 6a), which subsequently pinches off from the shear layer and convects downstream. At this time, the nozzle tip maintains its circular shape (figure 6e). After acceleration of the jet is completed ($t/T_{acc} = 1.7$), the jet shear layer following the primary vortex becomes unstable (indicated by the arrow in figure 6b), which rolls up into the secondary vortex. As can be expected from figure 4, the nozzle tip deforms (shrinks) beyond its original size, driven by the strong jet inertia, resulting in a wrinkled structure (figure 6f), which is known to have lower energy compared with the axisymmetric shape under the same pressure distribution (Hibbeler 2013). For a flexible nozzle, the roll-up and formation of subsequent (third, fourth and so on) vortex rings occur at the nozzle tip (figure 4b-v–d-v) driven by its lateral flapping movement, rather than the growth of the shear-layer instability, which is the leading cause in the jet from the rigid nozzle (figure 4a-vi). Figures 6(c) and 6(d) show the formation of the third and fourth vortices triggered by the movement of the nozzle tip (noted with the solid and dotted arrows, respectively, in the figures). During this process, flapping motion of the nozzle tip results in a triangular shape, the orientation of which is synchronized with sequential vortex shedding (figure 6g,h). Interestingly, the nozzle deforms in an organized shape after the transient (shedding of the second vortex ring) stage of irregular wobbling. For the case of $Eh = 7.0$ N m$^{-1}$, the wavenumber (i.e. the number of waves on the circumference) along the circumferential direction corresponding to this organized deformation is $3.0$ (triangle), and it varies depending on the stiffness of the nozzle and the jet conditions. Although this is not the primary concern of the present study and requires a detailed structural analysis in the future, we performed a brief modal analysis to understand this behaviour (see Appendix A).

Figure 6. (a–d) Temporal variation of dye visualization of vortex formation for the case of $Eh = 7.0$ N m$^{-1}$ and (e–h) the deformation of the nozzle tip viewed from the bottom at the same instant: (a,e) $t/T_{acc} = 0.7$; (b,f) $1.7$; (c,g) $2.5$; (d,h) $2.8$. The flow conditions are $Re_{j} = 3286$ and $T_{acc} = 0.2$ s. In (e–h) the laser sheet is positioned at $y/D = 0.1$ to highlight the profile of the nozzle tip.

On the other hand, the inherent azimuthal instability of a vortex ring (Widnall & Sullivan 1973; Widnall, Bliss & Tsai 1974; Hattori & Fukumoto 2003; Fukumoto & Hattori 2005; Dazin, Dupont & Stanislas 2006) can be a potential cause of the circumferential deformation of the nozzle. This instability originates from the resonance between azimuthal perturbations with two different wavenumbers and simultaneously amplifies the disturbance in a linearly developing stage. It was reported that the growth of azimuthal instability in a vortex ring incurs a marked decrease in both the circulation and translational velocity of a vortex ring because of the generation of secondary vortex and subsequent nonlinear processes (Dazin et al. 2006; Bergdorf, Koumoutsakos & Leonard 2007). However, as we discuss below (see figure 8), the vortex ring tends to grow by being fed continuously from a shear layer or merging with following vortices. Therefore, the waviness of the nozzle tip (figure 6f) seems to be irrelevant to the instability of a vortex ring; rather, it is related to the fluid–structure interaction of the elastic nozzle. On the other hand, the present flexible nozzle can be stretched (figure 6a,e) or bent (figure 6b–d,f,g). Through stretching, the nozzle stores and releases elastic energy to the jet flow until the expansion–contraction cycle ends (figures 5 and 8). Bending occurs after this stage (figure 6f–h); however, the increment to the hydrodynamic impulse is found to be negligible. As we discuss in relation to figure 7(c), bending does not produce additional thrust but contributes to the generation of successive vortex rings (figure 6c,d).

Figure 7. Temporal evolution of the jet-flow characteristics: (a) dimensionless total circulation, $\hat {\varGamma }=\varGamma /(u_{t}D)$; (b) dimensionless jet-exit velocity, $\hat {u}_{e}=u_{e}/u_{t}$; (c) relative hydrodynamic impulse, $\hat {I}_{h}=I_{h}/I_{h,rigid}|_{t/T_{acc}=1.0}$. For the cases plotted, $Re_{j} = 3286$ and $T_{acc} = 0.2$ s. The solid arrow in each panel denotes the direction of increasing $Eh$.

Das et al. (2018) reported that a flexible flap installed at the outlet of a slit (plane) nozzle is deformed by a starting jet approximately twice the nozzle diameter ($\sim 2D$) under similar conditions ($Re_{j} = 2700$, $T_{acc} = 0.1$ s, $D = 15$ mm and $L/D = 3$), which is larger than that in the current study ($< 0.2D$). For the two-dimensional flap, the fluid pressure inside the nozzle due to jet acceleration balances with the bending moment rather than the tensile force, while the latter dominates in the flow through the circular flexible nozzle. A brief scale analysis is given as follows. When the amplitude of nozzle deflection is $\delta _b$, the bending force can be scaled as $Eh^{3}\delta _{b}/l_{c}^{2}$ (Shapiro 1977; Kang et al. 2011; Dewey et al. 2013; Siviglia & Toffolon 2014; Medina & Kang 2018), whereas the tensile force is associated with $Eh\delta _t$, where $\delta _t$ denotes the displacement of the axisymmetric expansion (Kraus 1967). If the nozzle is very thin (i.e. $h/l_{c} \ll 1.0$), $\delta _{b}/\delta _{t} \sim (l_{c}/h)^{2} \gg 1.0$, indicating that the bending amplitude is much greater than that by expansion.

Figure 7(a) shows the temporal variation of the total circulation ($\varGamma$) depending on nozzle stiffness, which is calculated as $\varGamma = 0.5\int _{C}{|\omega _{\theta }|}\, {\rm d}A$, where the integration area $C$ ranges between $-2 \le x/D \le 2$ and $-5.5 \le y/D \le 0$ on the measurement plane, excluding the area inside the nozzle located at $y/D>0$. It is further non-dimensionalized as $\hat {\varGamma }=\varGamma /(u_{t}D)$. Consistent with the jet-flow development in figure 4, the total circulation at $t/T_{acc}=1.0$ (when the acceleration ends) is maximum for the case of $Eh=7.0$ N m$^{-1}$, and it decreases with increasing $Eh$. When we consider the trends closely, an initial time lag (up to $0.2T_{acc}$) is observed before $\hat {\varGamma }$ starts to rapidly increase, which is attributed to the time required for the fluid to be accumulated inside the nozzle sufficiently to radially expand the nozzle. The fluid is then released during the contraction phase through the nozzle exit, which increases the circulation (during $t/T_{acc} \simeq 0.2\unicode{x2013}1.0$). As the nozzle becomes more flexible (smaller $Eh$), $\hat {\varGamma }$ increases sharply for a longer duration because the contraction (as well as the expansion) time of the nozzle is increased, as shown in figure 5. While recovering the original shape ($t/T_{acc} > 1.0$), the growth rate of the total circulation is reduced, which gradually converges to that for the rigid nozzle. This can be understood from the conservation of the circulation according to the Helmholtz theorem. Because all nozzles are supplied with the same influx of vorticity through the same movement of the piston–motor system, the time history of the total circulation contained in the control volume has to be identical. After jet acceleration ends and the nozzle recovers its original shape, the influx from different nozzles becomes identical and the total circulation converges.

Figure 7(b) shows the jet velocity at the nozzle exit normalized by the terminal jet velocity, $\hat {u}_{e}=u_{e}/u_{t}$. As the flow inside the flexible nozzle is accelerated, $\hat {u}_{e}$ abruptly increases after a short time lag owing to nozzle expansion and reaches the maximum velocity faster than the terminal velocity ($u_t$), i.e. $|\hat {u}_{e}| \ge 1.0$. Clearly, the maximum jet-exit velocity becomes higher as the nozzle becomes more flexible, which is responsible for the rapid increase of $\varGamma$ (figure 7a). The time ($t_{max}$) at which the jet reaches the maximum velocity is delayed as the nozzle becomes flexible; it is $t_{max}/T_{acc} = 0.2$, $0.3$ and $0.6$ for $Eh = 43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively. It coincides with the instants at which the nozzle contraction ends (figure 5) and the vortex ring pinches off (figure 4), indicating that the fluid–structure interaction associated with nozzle deformation governs the jet characteristics. It is meaningful to mention that the agility of the nozzle system as well as the thrust (i.e. the maximum velocity) are markedly enhanced using the flexible surface. For a flexible nozzle of $Eh = 43.2$ N m$^{-1}$, for example, only $0.17T_{acc}$ is required to reach $90\,\%$ of terminal velocity. This implies that the elastic appendage can increase the accelerating performance of the system. In physiology and biomimetic engineering, a flexible actuator quickly releases the stored force in movements such as jumping, capturing prey and escaping, and it is even advantageous in overcoming physiological and neurological limits (Higham & Irschick 2013; Haldane et al. 2016).

To quantify the effect of nozzle flexibility, the hydrodynamic impulse ($I_h$) is calculated, which has been used as a measure of thrust generation (Krueger & Gharib 2003). It is expressed as follows (Saffman 1995):

(3.1)\begin{equation} I_h = \frac{1}{2}\rho_{f}\int_{C}{\boldsymbol{x}\times\boldsymbol{\omega}}\, {\rm d}V.\end{equation}

Here, $\boldsymbol {x}=(r,\theta,y)$ is the position vector whose origin is located at the centre of the nozzle exit (figure 1b) and the vorticity vector is given as $\boldsymbol {\omega }=(\omega _{r},\omega _{\theta },\omega _{y})$. Considering that the flow is axisymmetric ($\omega _{r}=\omega _{y}=0$) and ${\rm d}V=2{\rm \pi} r\,{\rm d}A$, hydrodynamic impulse can be calculated as $I_{h}=0.5{\rm \pi} \rho _{f}\int _{C}{r^{2}|\omega _{\theta }(r,y)|}\, {{\rm d}}A$, where the control surface $C$ is the same as for the calculation of $\varGamma$ (figure 7a) and $\omega _{\theta }$ denotes the component of vorticity perpendicular to the measurement plane. In figure 7(c), we compare the hydrodynamic impulse from the flexible nozzle with that from the rigid nozzle, which is defined as $\hat {I}_{h}=I_{h}/I_{h,rigid}|_{t/T_{acc}=1.0}$. As expected, the relative hydrodynamic impulse becomes stronger as the nozzle becomes more flexible. When the jet acceleration ends ($t/T_{acc} = 1.0$), $\hat {I}_h$ becomes $1.44$ (for $Eh = 43.2$ N m$^{-1}$), $1.46$ ($14.4$) and $1.9$ ($7.0$). Thus, the flexible nozzle with $Eh = 7.0$ N m$^{-1}$ (the most flexible case among those considered) can produce a thrust $90\,\%$ larger than that produced by the rigid nozzle instantly.

Considering the potential application of a flexible nozzle in a jet-propelled robot, we estimated the hydrodynamic impulses ($I_h$) in figure 7(c) in a physical unit, which is $I_{h} = 0.93$, $1.25$, $1.39$ and $1.79\ {\rm mN}\,{\rm s}$ at $t/T_{acc} = 1$ for $Eh = \infty$, $43.2$, $14.4$ and $7.0$ N m$^{-1}$, respectively. Krueger & Gharib (2003) reported an impulse of a starting jet of around $6.0\unicode{x2013}6.6\ {\rm mN}\,{\rm s}$, when $F = 3.0$ (for the present study, the instantaneous formation number ($F_i$) defined by Limbourg & Nedić (2021a) is calculated as $F_{i} = 2.7$ at $t/T_{acc} = 1$). Since the contribution of the pressure to the impulse is less than $10\,\%$ of the total impulse (Gao et al. 2020), the total impulse can be roughly estimated by the momentum contribution ($\sim \rho _{f} u_{e}^{2} A$, where $u_e$ and $A$ denote the velocity and cross-section at the nozzle exit, respectively). By comparing the jet velocity and the exit area, it is estimated that the jet impulse from Krueger & Gharib (2003) would be $4.5$ times larger than ours, which is consistent with the measurement.

As we have shown, the increase in the jet impulse is associated with delayed vortex pinch-off, and previous studies tried to achieve this by controlling the exit diameter of the nozzle (Dabiri & Gharib 2005) and by inducing a radial velocity of the jet using the converging circular nozzle (Krieg & Mohseni 2013), by which the jet thrust was enhanced by $35\,\%$ and $70\,\%\unicode{x2013}75\,\%$, respectively. In the present study, compared with active control (Dabiri & Gharib 2005) requiring external energy input, the hydrodynamic impulse is improved by a passive fluid–structure interaction, and the magnitude of the increment is larger. On the other hand, the increment of the impulse may be contributed by either an increase in the fluid momentum (i.e. the accelerated jet exit velocity) or a pressure rise at the exit, which is attributed to the radial velocity at the exit (Krieg & Mohseni 2013). Because the radial component of the velocity is negligible compared with the streamwise one in the present configuration, the augmentation of the jet impulse is caused by the increased fluid momentum (evidenced in figure 7b), and it is closely related to the expansion–contraction of the nozzle (an analytical analysis of this is given in § 4).

As shown in figure 4, on the other hand, the evolution of a leading vortex ring from the nozzles is also affected by nozzle flexibility. To investigate this, the horizontal ($x_v$) and vertical ($y_v$) locations of vortex ring are detected based on the maximum vorticity ($\omega _{max}$). Then, its translational velocity ($V_v$) is calculated as $V_{v}={\rm d}y_{v}/{\rm d}t$, and the vortex radius ($R_v$) is obtained from $x_v$. Finally, the core radius ($a_v$) of a vortex ring is defined as the equivalent radius of the closed loop at which the vorticity corresponds to $0.9\omega _{max}$ (Das, Bansal & Manghnani 2017). Das et al. (2017) showed that the core radius obtained based on vorticity is comparable with the velocity-based one (Arakeri et al. 2004) as well as the theoretical prediction of $a_{v} = \sqrt {4\nu T_{s}}$ (where $T_s$ is the stroke time during vortex ring generation), which assumes that the vorticity in a core follows a Gaussian distribution (Saffman 1995). To measure the circulation of vortex ring ($\varGamma _v$), the inner region of a vortex on the two-dimensional flow field should be specified, on which the vorticity is integrated. We used the swirling strength ($\lambda$) to define that region with a closed curve (Zhang & Rival 2020). That is, based on the maximum swirling strength (centre of the vortex ring), the isocurve of $0.1\lambda _{max}$ is drawn and the vorticity is integrated to obtain $\varGamma _v$. Considering the symmetric vortex ring, the variables mentioned above were obtained for both sides ($x > 0$ and $x < 0$) and averaged, which were normalized with the nozzle diameter ($D$) and jet terminal velocity ($u_t$).

Figure 8 shows the temporal variation of the characteristics of a leading vortex ring depending on $Eh$. Figure 8(a) shows that the vortex ring from a flexible nozzle rapidly accelerates and saturates (at $t/T_{acc} = 0.5\unicode{x2013}1.0$) to a specific velocity that increases with decreasing $Eh$, whereas the vortex from the rigid nozzle gradually accelerates. At $t/T_{acc} > 2.0$, vortex velocity slightly decreases for flexible nozzles of $Eh = 14.4$ and $7.0$ N m$^{-1}$, while that from the stiffer nozzle ($Eh = 43.2$ N m$^{-1}$) marginally increases at $t/T_{acc} \simeq 1.5$. The trend of circulation (figure 8b) is similar to that of $V_v$. However, the circulation tends to remain constant for $Eh = 14.4$ and $7.0$ N m$^{-1}$, and slightly increases for $Eh = 7.0$ N m$^{-1}$. Assuming that the vortex radius ($R_v$) is much larger than core radius ($a_v$), and the vorticity follows the Gaussian distribution inside a core, translational velocity ($V_v$) of a vortex ring is expressed as $V_{v} = \varGamma _{v}/4{\rm \pi} R_{v}(\ln (8R_{v}/a_{v})-0.552)$ (Lamb 1945; Saffman 1970; Sullivan et al. 2008). This relation explains the similarity between the translational velocity (figure 8a) and circulation (figure 8b) of a vortex ring because $R_v$ and $a_v$ are quite similar for all nozzles (figure 8c,d). Interestingly, the difference in both radii is barely noticeable between nozzles despite the strong nozzle–jet interaction (figure 6), except for $R_v$ of $Eh = 43.2$ N m$^{-1}$. In the present configuration, the piston supplies momentum for entire duration of measurement (thus, the stroke time is the same for all cases), and the vorticity is sufficiently fed to vortex ring for all nozzles. Therefore, considering the theoretical model of Saffman (1995), the core radius is almost the same for all nozzles (figure 8c). Meanwhile, for $Eh = 43.2$ N m$^{-1}$, the vortex radius becomes larger than others at $t/T_{acc} > 1$. This is attributed to the vortex interaction, shown in figure 4b-iv; that is, the early pinch-off of a leading vortex results in coalescence (at $t/T_{acc} \simeq 1.5$) with subsequent vortices, enlarging the vortex radius of the leading vortex ring, and also slightly increasing the translational velocity (figure 8a) and circulation (figure 8b). Vortex merging also happens for $Eh = 14.4$ (figure 4c-v) and $7.0$ (figure 4d-v,vi) but circulation and radius are not affected, unlike the case of $Eh = 43.2$ N m$^{-1}$. This is because the trailing vortex ring is not sufficiently large to incur the vortex interaction but is passively entrained by the leading vortex. In brief, the nozzle flexibility alters the jet vortices not only for the generation and growth of a leading vortex but also for the birth of trailing vortices (figure 6c,d,g,h).

Figure 8. Temporal variation of the characteristics of a leading vortex ring depending on $Eh$: (a) translational velocity of the vortex ($\hat {V}_{v}=V_{v}/u_{t}$); (b) vortex circulation ($\hat {\varGamma }_{v}=\varGamma _{v}/(u_{t}D$)); (c) vortex radius ($\hat {R}_{v}=R_{v}/D$); (d) core radius ($\hat {a}_{v}=a_{v}/D$). For the cases plotted, $Re_{j}= 12\,555$ and $T_{acc} = 0.12$ s.

3.2. Effect of flow condition on jet characteristics

Figure 9 shows the temporal evolution of the jet characteristics when $Re_j$ increases to $12\,555$ and $T_{acc}$ is shortened to $0.12$ s ($\varPi _{0} = 2.33$ and $\varPi _{1}=1.33\unicode{x2013}8.22$). For the cases shown in figure 7, $Re_{j} = 3286$ and the acceleration time is $0.2$ s, corresponding to $\varPi _{0} = 1.05$ and $\varPi _{1} = 19.45\unicode{x2013}120.02$. Interestingly, the total circulation for the intermediate flexibility ($Eh = 14.4$ N m$^{-1}$) is larger than that for the more flexible nozzle ($Eh = 7.0$ N m$^{-1}$) at $t/T_{acc} = 1.0$ (figure 9a), contrary to the result in figure 7(a), in which the maximum total circulation (at $t/T_{acc}=1.0$) is achieved with the minimum stiffness ($Eh=7.0$ N m$^{-1}$). Similarly, reversed trends in the jet exit velocity (figure 9b) and hydrodynamic impulse (figure 9c) are observed. The maximum $\hat {u}_{e}$ and $\hat {I}_h$ achieved with $Eh = 14.4$ N m$^{-1}$ at $t/T_{acc}=1.0$ are slightly larger than those for $Eh = 7.0$ N m$^{-1}$. This implies that enhanced flexibility does not always augment the jet thrust, whereas many previous studies tended to conclude that the aerodynamic (or hydrodynamic) performance of a body is enhanced monotonically as its surface becomes more flexible (Kim & Gharib 2011; Marais et al. 2012; Das et al. 2018). Rather, there should be an optimal flexibility for maximizing the thrust, and it is not only determined by the stiffness but also depends on the flow condition (such as the acceleration time and the terminal velocity for the present problem). Meanwhile, the optimal condition for enhancing thrust (or efficiency) of a flexible flapping wing (panel) is closely associated with the phase lag in the angular displacements of the root and the deflected tip (Kang et al. 2011; Dewey et al. 2013; Quinn et al. 2014; Park et al. 2016). The physical significance of the phase lag implies that the time-scale ratio of the structural natural frequency to the undulation (flapping) frequency governs thrust generation and vortical structures emanating from the flexible body. Similarly, for the present flexible nozzle, we need a similar approach or a parameter that can compare the jet-flow time scale with the structural response time scale (material and geometrical properties of the nozzle).

Figure 9. Temporal evolution of jet-flow characteristics: (a) dimensionless total circulation, $\hat {\varGamma }=\varGamma /(u_{t}D)$; (b) dimensionless jet-exit velocity, $\hat {u}_{e}=u_{e}/u_{t}$; (c) relative hydrodynamic impulse, $\hat {I}_{h}=I_{h}/I_{h,rigid}|_{t/T_{acc}=1.0}$. For the cases plotted, $Re_{j} = 12\,555$ and $T_{acc} = 0.12$ s.

In figure 10, we show the temporal variation of the averaged nozzle displacement along the radial direction, defined as $w_{avg}^{*} = (w_{left}^{*}+w_{right}^{*})/2$, where $w_{left}^{*}$ and $w_{right}^{*}$ correspond to the normalized radial displacement at $x < 0$ and $x > 0$, respectively. The operation condition is the same as that for figure 9. Note that the contour range is deliberately biased to positive value to effectively show the expansion–contraction behaviour of the nozzle. Similar to the case of lower $Re_{j} = 3285$ (figure 5), the flexible nozzle radially expands and contracts as the jet accelerates, and the period of expansion–contraction ($t_{cycle}$) becomes longer as the stiffness of the nozzle decreases. For the stiffest case ($Eh = 43.2$ N m$^{-1}$), nozzle contraction is completed at $\sim 0.6T_{acc}$ when the growth rate of the exit velocity is significantly reduced (figure 9b), which is attributed to the early end of the supply of elastic energy from the nozzle to the jet flow. Notably, for $Eh = 14.4$ N m$^{-1}$, the expansion–contraction cycle corresponds to the time for jet acceleration ($t_{cycle} \simeq 1.4T_{acc}$), indicating that elastic energy stored in the nozzle is fully transferred to the flow during acceleration. For a nozzle with $Eh = 7.0$ N m$^{-1}$, contraction persists up to $t/T_{acc}>1.5$, implying that the elastic energy does not entirely add fluid momentum even after jet acceleration ends. That is, the response of the nozzle lags behind the acceleration of the jet.

Figure 10. Spatio-temporal contour of the averaged nozzle displacement ($w_{avg}^{*} = (w_{left}^{*}+w_{right}^{*})/2$) at $Re_{j} = 12\,555$ and $T_{acc} = 0.12$ s, depending on the structural stiffness: (a) $Eh = 43.2$; (b) $14.4$; (c) $7.0$ N m$^{-1}$. Here, the maximum displacement, $w_{n,max}/D$, is given as (a) $0.025$, (b) $0.062$ and (c) $0.14$.

Based on our analyses so far, the expansion–contraction cycle ($t_{cycle}$) should be comparable to the acceleration time of the jet, $t_{cycle} \simeq T_{acc}$, to enhance thrust generation. Precisely, on the other hand, the nozzle may further release energy at $t>t_{cycle}$ until the nozzle fully contracts at $t=t_{fc}$. Therefore, the effective time required to maximize thrust generation must cover the duration for complete nozzle shrinkage ($t_{fc}$). Based on the present measurements (figure 5a,b), the time span of $t_{cycle}< t< t_{fc}$ is comparable to half of the expansion–contraction cycle; thus, the optimal jet acceleration time ($T_{acc}$) should match $t_{fc} \simeq 1.5t_{cycle}$. Hence, it is expected that the elastic energy stored in the flexible nozzle is fully transferred to the kinetic energy of the ejecting jet when satisfying $t_{fc} = T_{acc}$, and this would incur the highest jet-exit velocity (figure 9b). Otherwise, for $t_{fc} < T_{acc}$ or $t_{fc} > T_{acc}$, the maximum jet-exit velocity decreases due to the early (figure 10a) or retarded (figure 10c) contraction of the nozzle, respectively.

As noted, the acceleration time ($T_{acc}$) is generally predetermined by the configuration of the motor–piston system; however, the instant of full contraction ($t_{fc}$) is determined by the jet–nozzle interaction. Because $t_{cycle}$ can be understood as the time during which the travelling wave on the nozzle (which originally starts from the support of the nozzle at $y/D = 2.74$) comes back to its original position, it is expressed as $t_{cycle}=2L/c$, where $c$ and $L$ denote the wave speed and the nozzle length, respectively. Thus, the above optimal condition can be written as $T_{acc}(=t_{fc})=3L/c$.

Before discussing theoretical aspects, notably, the measured data already indicate the strong dependency of the wave speed on the nozzle stiffness. Figure 11 shows the variation of $w_{avg}^{*}$ depending on the acceleration time ($T_{acc}=0.20$ and $0.22$ s in figures 11a and 11b, respectively) and the terminal velocity of the jet (corresponding to $Re_{j}=3286$ and $12\,555$ in figures 11c and 11d, respectively), while fixing the structural stiffness as $Eh = 7.0$ N m$^{-1}$. Note that the time in the figure is dimensional. During the expansion ($t < 0.05$ s) stage, the wave on the nozzle surface travels from the support to the tip (denoted as solid arrows in the figure), and, interestingly, its speed is almost independent of both $T_{acc}$ and $Re_j$. This is contrary to the cases shown in figures 5 and 10, such that the wave speed significantly varies with nozzle stiffness; however, the only invariant parameter in figure 11 is the structural stiffness. Therefore, it can be speculated that the wave speed is determined by the material property rather than by the flow condition. To support this, quantitative measurements of the wave speed and its analytical analysis are provided in the following sections.

Figure 11. Spatio-temporal contour of the averaged nozzle displacement ($w_{avg}^{*} = (w_{left}^{*}+w_{right}^{*})/2$) with acceleration time (a,b) and jet Reynolds number: (a) $T_{acc} = 0.20$ s; (b) $0.22$ s; (c) $Re_{j}=6360$; (d) $12\,555$. For (a,b), $Re_{j} = 3286$ and $T_{acc} = 0.12\unicode{x2013}0.13$ s for (c,d). For all cases, the structural stiffness is fixed as $Eh = 7.0$ N m$^{-1}$. Here, the maximum displacement $w_{n,max}/D$ is given as (a) $0.036$, (b) $0.031$, (c) $0.081$ and (d) $0.14$.

4. Analytical formulation of the optimal flexibility condition

4.1. Derivation of governing equations for nozzle deformation and jet velocity

To construct equations for the fluid–structure interaction occurring in the present flexible circular nozzle, we defined the problem using the schematics of the deformed flexible nozzle in figure 12. Initially (without external forces), the nozzle has the shape of a circular cylinder (dashed line in figure 12a) with diameter $D$ ($= 2R$), length $L$ and thickness $h$. Here, the boundaries $B_1$ and $B_2$, respectively, correspond to the support and tip of the nozzle at the initial state. Global coordinates ($r,\theta,y$) are defined with the origin at the centre of the nozzle tip. During deformation, the shape and location of support ($B_1$) remain fixed, but are not necessarily fixed for the tip ($B_2$) that is a free end. For the present set-up, the vertical location ($y$) of the nozzle tip changes slightly (maximum deviation of $0.2D$). Thus, we assume that the nozzle length is constant at $L$ (the plane of $B_2$ and $B'_2$ is the same in figure 12a). As a result of deformation owing to the accelerated flow inside, we assume that the arbitrary material curve $C_m$ on the nozzle is distorted to the curve $C'_m$; the point on curve $C_m$ (denoted as point $p_m$) translates to point $p'_m$ on $C'_m$ (figure 12a). The displacement of $p_m$ can be expressed as $\boldsymbol {w}=(w_{y},w_{s},w_{n})$ (figure 12b,c). Here, the $n$ and $s$ axes are defined to be normal to the surface of the initial geometry and parallel to the $\theta$ axis, respectively (figure 12b). The radial direction outward from the nozzle is set to be positive, and the $y$ axis is parallel to the direction of the nozzle length (figure 12c). To characterize the fluid flow inside the nozzle, the vector field of $\boldsymbol {v}(r,\theta )=(v_{r},v_{\theta },v_{y})$ is considered on the arbitrary cross-sectional (normal to the $y$ axis) planes of the deformed nozzle.

Figure 12. Schematic diagram of the deformed cylindrical shell: (a) isometric view; (b) top view; (c) view with respect to the plane ($A\unicode{x2013}A'$) involving both the centre plane ($r = 0$) and the point $p_m$. The dashed lines denote the original circular cylinder before deformation. During deformation, the material curve $C_m$ and the point $p_m$ (on $C_m$) translate to the curve $C'_{m}$ and point $p'_{m}$ (on $C'_{m}$) with a displacement of $\boldsymbol {w}=(w_{y},w_{s},w_{n})$. Note that the curve $C'_{m}$ is not necessarily parallel to the $r\unicode{x2013}\theta$ plane.

Because the material of the nozzle is impermeable, fluid flow inside the nozzle should satisfy mass conservation, and momentum conservation can also be formulated from the pressure distribution and surface force acting inside the nozzle. As we have shown, the nozzle deformation is assumed to be nominally axisymmetric ($w_{s}\simeq 0$) and both the radial and azimuthal velocity components are negligibly small, i.e. $\boldsymbol {v} \simeq (v_{y},0,0)$. For the characteristic vertical velocity, on the other hand, the bulk velocity ($u$) is used, defined as $u=\int _{S}v_{y}\, {{\rm d}}A/S=Q/S$, where $S$ and $Q$ denote the arbitrary cross-sectional area normal to the $y$ axis and the volume flux passing through $S$, respectively. Therefore, the conservation law for mass and momentum is expressed as follows (Shapiro 1977; Siviglia & Toffolon 2014):

(4.1)\begin{gather} \frac{\partial Q}{\partial y} + \frac{\partial S}{\partial t} = 0, \end{gather}

(4.2)\begin{gather}\frac{\partial Q}{\partial t} + \frac{\partial(uQ)}{\partial y} +\frac{S}{\rho_f}\frac{\partial p}{\partial y} ={-}Bu. \end{gather}

Here, $t$, $\rho _f$ and $p$ denote the time, fluid density and pressure difference between the outer and inner wall of the nozzle, respectively. The term on the right-hand side of (4.2) corresponds to the viscous friction on the inner wall of the nozzle, modelled as being linearly proportional to the bulk velocity with dimensional coefficient $B$. The cross-sectional area ($S$) and fluid flux ($Q$) can be calculated as $S={\rm \pi} (R+w_{n})^{2}$ and $Q=uS=u{\rm \pi} (R+w_{n})^{2}$, respectively. Hence, (4.1) and (4.2) are written in terms of $u$ and $w_n$ ((4.3) and (4.4)):

(4.3)\begin{gather} \frac{\partial w_n}{\partial t} + \frac{R+w_n}{2}\frac{\partial u}{\partial y} + u\frac{\partial w_n}{\partial y}=0, \end{gather}

(4.4)\begin{gather}\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial y} + \frac{1}{\rho_f}\frac{\partial p}{\partial y} ={-}\frac{B}{{\rm \pi}(R+w_{n})^{2}}u. \end{gather}

The above equations can be solved if pressure $p$ is determined based on the physical relation to the nozzle deformation and fluid flow, which is known as the ‘tube law’ (Siviglia & Toffolon 2014). Meanwhile, based on the assumption of axisymmetric deformation of the nozzle, the pressure can be analytically derived from the shell theory (detailed derivation is provided in Appendix B) as follows:

(4.5)\begin{equation} p = \frac{Eh}{R^{2}}w_n.\end{equation}

Equation (4.5) states that the pressure is linearly proportional to the radial displacement ($w_n$) with structural stiffness ($Eh/R^{2}$) as a proportional rate, resembling Hooke's law. When the jet accelerates, the pressure ($p$) at $y=y_p$ pushes the fluid volume at $0 < y < y_p$, which is accelerated with acceleration $a$ towards the nozzle exit. Considering the control volume surrounded by the inner walls of the nozzle and the planes $y=y_p$ and $y=0$, the change in the vertical momentum is determined by the difference in the pressure on the planes $y=0$ and $y=y_p$, and the viscous friction ($\tau _w$) on the inner wall, expressed as $\rho _{f}({\rm \pi} R^{2}y_{p})a=({\rm \pi} R^{2}){\rm \Delta} p - \tau _{w}(2{\rm \pi} R y_{p})$. For accelerating pipe flow, the unsteady wall-shear stress was obtained as $\tau _{w}=\rho _{f}aR(4\eta ^{0.5}+3\eta )/2$, where $\eta = t \nu _{f}/R^{2}$ ($\nu _f$, kinematic viscosity of the fluid; $t$, time) (Annus et al. 2013). Thus, the ratio of inertia to pressure rise is expressed as $f_{in}=\rho _{f}({\rm \pi} R^{2}y_{p})a/({\rm \pi} R^{2}{\rm \Delta} p)=1/(1+2\tau _{w}/(\rho _faR))=1/(1+4\eta ^{0.5}+3\eta )$. This indicates that inertia solely determines the pressure rise initially, i.e. $f_{in} = 1.0$ at $t = 0$. As the jet accelerates, $f_{in}$ slightly decreases to $0.85$ in the middle of acceleration (at $t=0.5T_{acc}$) when the frictional loss would be the largest. Thus, we may ignore the viscous stress for the whole process, and the expression for the pressure difference is reduced to ${\rm \Delta} p=\rho _{f} y_{p} a$. Assuming that the reference pressure at the nozzle exit is zero, (4.5) is rewritten as $\rho _{f} a y_{p}=Ehw_{n}/R^{2}$. Here, the flow acceleration can be approximated as $a \simeq u_{t}/T_{acc}$. Because the maximum displacement ($w_{n,max}$) occurs near the support of the nozzle at $y_{p}\simeq L$ (see figure 5), $w_{n,max}/D$ can be obtained as follows:

(4.6)\begin{equation} \hat{w}_{n,max} = \frac{w_{n,max}}{D} = \frac{\rho_{f}u_{j}LD}{4EhT_{acc}}.\end{equation}

In figure 13, we compare the measured $w_{n,max}$ with that predicted by (4.6), which shows that the above analytical derivation is quite robust.

Figure 13. Comparison of the measured maximum radial displacement ($\hat {w}_{n,max} = w_{n,max}/D$) of the nozzle diameter with the prediction of (4.6).

Confirming the reliability of (4.5), it is plugged into (4.3) and (4.4), and the resulting governing equations for mass and momentum conservation are derived as (4.7) and (4.8), respectively:

(4.7)\begin{gather} \frac{1}{\varPi_0}\frac{\partial \hat{w}_y}{\partial \hat{t}} + \hat{u}\frac{\partial \hat{w}_n}{\partial \hat{y}} ={-} \frac{1+\hat{w}_n}{2}\frac{\partial \hat{u}}{\partial \hat{y}}, \end{gather}

(4.8)\begin{gather}\frac{1}{\varPi_0}\frac{\partial \hat{u}}{\partial \hat{t}} + \frac{\hat{u}}{(1+\hat{w}_{n})^{2}}\frac{\partial \hat{u}}{\partial \hat{y}} ={-}\varPi_{1}\frac{\partial \hat{w}_{n}}{\partial \hat{y}}. \end{gather}

Here, the variables were normalized as $\hat {y}=y/L$, $\hat {t}=t/T_{acc}$, $\hat {w}_n=w_{n}/R$ and $\hat {u}=u/u_{t}$, and the friction was neglected ($B = 0$). Equations (4.7) and (4.8) explain how nozzle displacement and bulk velocity of the fluid are correlated. Interestingly, two dimensionless parameters ($\varPi _{0}=T_{acc} u_{t}/L$ and $\varPi _{1}=Eh/(\rho _{f} u_{j}^{2}R)$) appear, the physical meanings of which (the effective acceleration time and the effective nozzle stiffness, respectively) were discussed in § 2.4. This supports our understanding that $\varPi _0$ and $\varPi _1$ dictate the fluid–structure interaction between the starting jet and deformable nozzle. For both equations, the two terms on the left-hand side are similar to the material derivative ($\mathcal {D} = (1/\varPi _{0})\partial/\partial \hat {t}+\hat {u}\partial / \partial \hat {y}$) because $\hat {w}_n<0.1$ and $(1+\hat {w}_n)^{2}\simeq 1.0$, as shown in figure 13. Here, the effective acceleration time ($\varPi _0$) determines the relative contribution of the local acceleration ($\partial /\partial \hat {t}$) to the convection acceleration ($\hat {u}\,\partial /\partial \hat {y}$). This understanding is extended such that $\varPi _0$ plays a similar role to the Strouhal number characterizing the unsteadiness of the flow structures. Equations (4.7) and (4.8) explain that the rate of change in $\hat {w}_y$ and $\hat {u}$ ($\mathcal {D}\hat {w}_y$ and $\mathcal {D}\hat {u}$, respectively) is determined by the spatial gradient of the counterpart variables, $\partial \hat {u}/\partial \hat {y}$ and $\partial \hat {w}_{y}/\partial \hat {y}$, respectively. For example, a positive $\partial \hat {u}/\partial \hat {y}$ indicates that the nozzle should contract in diameter, i.e. $\mathcal {D}\hat {w}_{y} < 0$, to satisfy the mass conservation, which is consistent with (4.7). In contrast, in (4.8), a positive $\partial \hat {w}_{y}/\partial \hat {y}$ corresponds to $\partial p/\partial y>0$ following (4.5). The favourable pressure gradient will accelerate the bulk velocity, which agrees with the equation. Notably, the acceleration of the flow inside the nozzle ($\mathcal {D}\hat {u}$) is correlated with the nozzle deformation ($\partial \hat {w}_{y}/\partial \hat {y}$) with the proportional coefficient of $\varPi _1$, which originates from the relation between the displacement and pressure in the flow.

4.2. Description of optimal flexibility for maximum thrust generation

To determine the optimal condition ($T_{acc} \simeq 3L/c$) for maximizing thrust generation, it is necessary to find the speed ($c$) of the wave travelling on the nozzle surface. However, the governing equations (4.7) and (4.8) are nonlinear partial differential equations that can not be easily solved analytically. Rather than calculating the exact solution for the wave speed, we found that the asymptotic approach can successfully estimate the wave speed in a reasonable manner. At the early stage of jet acceleration ($t/T_{acc} \ll 1.0$), the bulk velocity is smaller than the terminal velocity ($\hat {u}=u/u_{t} \ll 1.0$), and the nozzle deformation is also negligible ($\hat {w}_{n}=w_{n}/R<0.2$). Therefore, the second terms on the left-hand sides of (4.7) and (4.8) can be ignored, resulting in

(4.9)\begin{gather} \frac{1}{\varPi_0}\frac{\partial \hat{w}_y}{\partial \hat{t}} + \frac{1}{2}\frac{\partial\hat{u}}{\partial\hat{y}} = 0, \end{gather}

(4.10)\begin{gather}\frac{1}{\varPi_0}\frac{\partial \hat{u}}{\partial \hat{t}} + \varPi_{1}\frac{\partial \hat{w}_{n}}{\partial \hat{y}} = 0. \end{gather}

Subtraction between (4.9) and (4.10), after they are differentiated in terms of $\hat {t}$ and $\hat {y}$, respectively, provides the equation for $\hat {w}_n$ (equation (4.11)), and a similar procedure can be performed to obtain the equation for $\hat {u}$ (equation (4.12)):

(4.11)\begin{gather} \frac{\partial^{2}\hat{w}_{n}}{\partial \hat{t}^{2}} - \frac{\varPi_{1}\varPi_{0}^{2}}{2}\frac{\partial^{2} \hat{w}_{n}}{\partial\hat{y}^{2}}=0, \end{gather}

(4.12)\begin{gather}\frac{\partial^{2}\hat{u}}{\partial \hat{t}^{2}} - \frac{\varPi_{1}\varPi_{0}^{2}}{2}\frac{\partial^{2} \hat{u}}{\partial\hat{y}^{2}}=0. \end{gather}

The resulting equations have the form of a one-dimensional wave equation; both $\hat {w}_n$ and $\hat {u}$ have identical relations with the same coefficient of $\varPi _{1} \varPi _{0}^{2}/2$. To extract the wave speed from the equation, a normal mode of the perturbation of $\hat {w}_n$ and $\hat {u}$ (expressed as $\hat {u}, \hat {w}_{n} \sim \exp [i(\hat {k}\hat {y}- \hat {\omega }\hat {t})]$, where $\hat {k}$ and $\hat {\omega }$ correspond to the normalized frequency and the wavenumber of the perturbation, respectively) is replaced in (4.11) and (4.12), resulting in $i^{2}\hat {\omega }^{2}-(\varPi _{1}\varPi _{0}^{2}/2)i^{2}\hat {k}^{2}=0$. Therefore, the dimensionless wave speed is as follows:

(4.13)\begin{equation} \hat{c} = \frac{\hat{\omega}}{\hat{k}} ={\pm}\sqrt{\frac{\varPi_{1}\varPi_{0}^{2}}{2}} ={\pm}\sqrt{\frac{EhT_{acc}^{2}}{\rho_{f}DL^{2}}}.\end{equation}

The dimensionless wave speed is determined by two dimensionless parameters that we derived from the governing equations (4.7) and (4.8). The effective acceleration time ($\varPi _{0}=u_{t} T_{acc}/L$) is the non-dimensional jet velocity ($u_t$) normalized by the characteristic velocity ($L/T_{acc}$). In the meantime, $\varPi _{1}^{0.5}$ can be understood as the dimensionless wave speed such that $\varPi _{1}^{0.5}=\sqrt {Eh/(\rho _{f}u_{t}^{2}R)}=(c_{s}/u_{t})\chi ^{0.5}$, where $c_s$ and $\chi$ denote the wave velocity of the elastic material (Païdoussis 1998; Paak et al. 2013) and the mass ratio between the nozzle wall and fluid ($\chi = \rho _{s}h/(\rho _{f}R$)). Therefore, $\varPi _{0}\varPi _{1}^{0.5}$ physically indicates the ratio of the surface wave speed to the characteristic flow velocity, as $\hat {c}=c/(LT_{acc}^{-1})$.

In dimensional form, the wave speed can be rewritten as follows:

(4.14)\begin{equation} c = \hat{c}\frac{L}{T_{acc}} ={\pm}\sqrt{\frac{Eh}{\rho_{f}D}}. \end{equation}

This implies that the wave speed is independent of the jet velocity and the acceleration time, which is consistent with the observation in figure 11. The positive and negative signs, respectively, denote that the surface wave may propagate either downward ($-y$) or upward ($+y$). As the nozzle stiffness ($Eh$) increases and the fluid inertia ($\rho _{f}D$) decreases, the wave speed increases. Equation (4.14) can also be deduced as one of the limiting cases from the acoustic interaction between an elastic cylinder and contained fluid (Lin & Morgan 1956). If the sound speed ($c_f$) and fluid inertia ($\rho _{f}R$) are much greater than the wave speed on the nozzle surface ($c$) and the nozzle inertia ($\rho _{s}h$), two solutions of the wave with mode $n = 0$ (axisymmetric) can be expressed as $c_{1}^{2} = E(1+\nu ^{2}/(2\chi ))/(\rho _{s}(1-\nu ^{2}))$ and $c_{2}^{2} = Eh/(2\rho _{f}R(1+\nu ^{2}/(2\chi )))$, where $\rho _s$ and $\nu$ correspond to the nozzle density and Poisson's ratio, respectively, obtained from the linearized equation of a thin circular cylindrical shell and two equations regarding the wave propagation (Laplace equation) and the momentum conservation. The former solution ($c_1$) denotes the axial vibration due to the negligible fluid viscosity and the latter ($c_2$) corresponds to the radial vibration that is related to the present point of interest. For all cases in the present study, we found that $\chi > 46.9$ and $\nu ^{2} = 0.16$, so the latter solution is directly simplified to (4.14) since $\nu ^{2}/(2\chi ) \ll 1.0$.

To further validate our analysis, we directly measured the wave speed from the result of nozzle deformation (figures 5, 10 and 11); the time delay ($T_{wave}$) between the instants when the enlarged diameter reaches $50\,\%$ of its maximum at two locations of $y_{1}/D = 2.5$ and $y_{2}/D = 1.5$ is measured, and the wave speed is calculated as $c = (y_{1}-y_{2})/T_{wave}$. Figure 14 compares the measured wave speed with the prediction (4.14), which shows remarkable agreement. This supports our analysis such that the surface wave speed solely depends on the material property of the nozzle rather than on the jet conditions. Additionally, we observed that the wave is faster than the terminal velocity of the jet ($u_{t} = 0.20\unicode{x2013}0.84$ m s$^{-1}$), implying that there exist two characteristic velocities for the starting jet issuing from a flexible nozzle.

Figure 14. Validation of the wave speed ($c$) on the nozzle wall with the prediction of (4.14).

Now, the optimal condition ($T_{acc} \simeq 3L/c$) for maximum thrust can be expressed with two non-dimensional parameters ($\varPi _0$ and $\varPi _1$) representing the flow and nozzle conditions, respectively, by substituting the relation for surface wave speed. The resulting relation is given as

(4.15)\begin{equation} \hat{c} = c\frac{T_{acc}}{L} = \sqrt{\frac{\varPi_{1} \varPi_{0}^{2}}{2}} \simeq 3.0.\end{equation}

As noted, the optimized value of the dimensionless wave speed is given together, which is ${\sim}3.0$ for the present configuration. To confirm the validity of condition (4.15), we have plotted all the considered cases in the parameter space of $\varPi _0$ and $\varPi _1$ in figure 15(a) (optimal condition (4.15) is shown as a dashed line in the figure). Clearly, the cases with a larger thrust lie on the line $\hat {c} = 3.0$. As we analysed above, for $\hat {c} > 3.0$ (above the dashed line in figure 15a), the wave speed is relatively fast so that the full contraction is completed before jet acceleration ends (see figure 10). Because the nozzle releases its elastic energy before the jet reaches the maximum velocity, the timing mismatch weakens thrust generation (figure 9c). On the other hand, for $\hat {c} < 3.0$, the response of the nozzle to the acceleration is too slow, forcing the nozzle to contract after the acceleration finishes, which also produces weaker thrust.

Figure 15. (a) Regime map ($\varPi _{0}$–$\varPi _{1}$) and (b) impulse generation depending on the dimensionless wave speed ($\hat {c}$) for all cases tested in the present study. The dashed lines correspond to $\hat {c}=3.0$, and the symbols with cross marks denote the cases near the optimal condition.

Figure 15(b) shows the relative impulse ($\hat {I}_h$) depending on the dimensionless wave speed ($\hat {c}$) for the same dataset shown in figure 15(a). The impulse is measured at $t/T_{acc}=1.0$ and normalized with the value from the rigid nozzle, expressed as $\hat {I}_{h}|_{t=T_{acc}} = I_{h}/I_{h,rigid}|_{t=T_{acc}}$. Clearly, the impulse increases as $\hat {c}$ approaches $3.0$ from both regimes of $\hat {c} > 3.0$ and $\hat {c} < 3.0$. Interestingly, for the cases of $\hat {c} \simeq 3.0$ (denoted with symbols with cross marks), the amplification of the relative impulse is not identical but tends to increase with increasing effective stiffness, $\varPi _1$ (or tends to decrease with increasing effective acceleration time, $\varPi _0$): e.g. $\hat {I}_{h}|_{t=T_{acc}} = 1.35$ (for $\varPi _{1} = 2.74$), $1.47$ ($4.72$), $1.77$ ($10.68$) and $2.07$ ($19.45$).

4.3. Scaling law for optimal flexibility

The enhancement in thrust (figure 15b) comes from the elastic potential energy ($P$) transferred to the kinetic energy of the fluid flow inside the nozzle during contraction. The elastic potential energy is expressed in terms of the deformation amplitude ($w_n$) as (Kraus 1967)

(4.16)\begin{equation} P = \int_{0}^{L}\int_{0}^{2{\rm \pi}}{\frac{1}{2} N_{ss} w_{n}}\, {\rm d}\theta \, {{\rm d} y}.\end{equation}

Here, $N_{ss}$ corresponds to the tensile stress in the azimuthal direction (see the detailed definition and its derivation in Appendix C). Considering that (i) the nozzle expands predominantly in the radial direction ($\hat {w}_{n} \gg \hat {w}_{y}$), (ii) the deformation is axisymmetric and (iii) the relation between pressure and radial displacement (4.6) holds, the elastic energy can be rewritten as (the detailed procedure of derivation is given in Appendix C)

(4.17)\begin{equation} P \simeq \frac{{\rm \pi} R^{3} a^{2} \rho_{f}^{2} L^{3}}{3(1-\nu^{2})Eh}.\end{equation}

The acceleration is approximated as $a \simeq u/T_{acc}$, and $\nu$ denotes Poisson's ratio of the nozzle material. Meanwhile, the kinetic energy ($M$) of the jet is calculated as $M = \rho _{f}({\rm \pi} R^{2}L)u^{2}/2$, and its increment (${\rm \Delta} M$) corresponding to the change in jet velocity (${\rm \Delta} u$) is obtained as ${\rm \Delta} M = {\rm \Delta} (\rho _{f} {\rm \pi}R^{2} Lu^{2}/2) = \rho _{f} {\rm \pi}R^{2} L(u{\rm \Delta} u)$.

Using the experimental data, it is possible to examine the transfer of elastic potential energy of the nozzle to jet kinetic energy. The elastic potential energy ($P$) of the nozzle deforming axisymmetrically with a normal displacement ($w_n$) is calculated as $P = {\rm \pi}EhR/(1-\nu ^{2})\int _{0}^{L}(w_{n}/R)^{2}{{\rm d} y}$ ((C4) in Appendix C), which is further non-dimensionalized as $\hat {P} = P/({\rm \pi} EhRL)$. Meanwhile, the kinetic energy ($E_k$) of the axisymmetric and non-swirling flow can be calculated as $E_{k} = 0.5{\rm \pi} \rho _{f}\int _{C}r(u^{2}+v^{2})\, {{\rm d}}A$ (Saffman 1995; Limbourg & Nedić 2021a), where the control surface ($C$) is the same as that used for calculating $\varGamma$ (figure 7a), which corresponds to the flow domain below the nozzle exit ($y < 0$). Here, $u$ and $v$ denote the horizontal and vertical velocity, respectively. The normalized kinetic energy is defined as $\hat {E}_{k} = E_{k}/(\rho _{f} u_{t}^{2} D^{3})$. Figure 16 shows the time history of $\hat {E}_k$ and $\hat {P}$ depending on the structural stiffness ($Eh$). For the rigid nozzle ($Eh = \infty$), the kinetic energy starts to increase gradually from $t = 0$; however, for the flexible nozzles, it starts to increase at delayed instants of $t/T_{acc} = 0.16$ (for $Eh = 43.2$ N m$^{-1}$), $0.21$ ($14.4$ N m$^{-1}$) and $0.31$ ($7.0$ N m$^{-1}$), agreeing with the instants when the corresponding potential energy reaches its maximum value and decreases. This clearly indicates that the elastic potential energy is transferred to the jet kinetic energy during nozzle deformation. The stored elastic potential energy increases with decreasing $Eh$, implying that the optimally flexible nozzle can attain the largest amount of elastic energy from the accelerating jet. For the same reason, the growth rate of jet kinetic energy becomes steeper as $Eh$ decreases. If the stored potential (elastic) energy is fully converted to kinetic energy of the jet (i.e. ${\rm \Delta} M \sim P$) during the contraction of the nozzle, the increase in the jet velocity can be obtained by combining (4.17) and the relation for ${\rm \Delta} M$ (see the detailed derivation in Appendix D):

(4.18)\begin{equation} \frac{{\rm \Delta} u}{u} \sim \frac{\varPi_{0}^{2}}{\varPi_{1}} \sim \frac{1}{\hat{c}^{2}}.\end{equation}

Equation (4.18) implies that the change in the jet velocity is also related to the effective acceleration time ($\varPi _{0}=T_{acc} u_{t}/L$) and effective stiffness ($\varPi _{1}=Eh/\rho _{f} u_{j}^{2} R$). The increase in the jet velocity is further scaled as ${\rm \Delta} u/u \sim 1/\hat {c}^{2}$, indicating that the maximum velocity would remain unchanged if two dimensionless parameters satisfy the optimal condition ($\hat {c}= 3.0$). This is different from the result shown in figure 15(b), where the jet impulse varies although the dimensionless wave speed converges to a value of $3.0$.

Figure 16. Time history of jet kinetic energy ($\hat {E}_{k}=E_{k}/(0.5\rho _{f}u_{t}^{2}D^{3}$)) and elastic potential energy ($\hat {P}=P/({\rm \pi} EhLR$) of the flexible nozzle depending on $Eh$.

We believe that this discrepancy originates from the assumption that the entire elastic energy is transferred to the kinetic energy of the jet (${\rm \Delta} M \sim P$) during acceleration of the jet. In figures 10(b) and 10(c), we can see that the duration of nozzle contraction is longer than that of nozzle expansion. Other evidence is that the temporal variation of the averaged nozzle diameter for the three cases satisfying $\hat {c} \simeq 3.0$ (denoted as symbols with cross marks in figure 15) is shown in figure 17. The wave speeds along the expansion stage (denoted as dotted arrows in the figure) seem to be identical to each other because the dimensionless wave speed is very similar to that of the optimal value. In the contraction stage (denoted as solid arrows in the figure), however, the nozzle contracts more slowly as $\varPi _0$ increases (or $\varPi _1$ decreases). This is because the accelerated flow inside the nozzle involves the process of wave propagation (i.e. fluid–structure interaction). This does not occur when contraction ends far before acceleration ends ($t/T_{acc} \ll 1.0$). For this condition, the durations of both expansion and contraction are identical because the assumption for jet velocity ($u \simeq 0$) in (4.9) and (4.10) is valid, as shown in figure 5.

Figure 17. Spatio-temporal contour of the averaged nozzle displacement ($w_{avg}^{*} =(w_{left}^{*}+w_{right}^{*})/2$) for cases satisfying the optimal condition: (a) $\hat {c} = 3.08$ ($\varPi _{0} = 1.33$, $\varPi _{1} = 10.7$); (b) $2.73$ ($\varPi _{0} = 1.79$, $\varPi _{1} = 4.72$); (c) $2.70$ ($\varPi _{0} = 2.33$, $\varPi _{1} = 2.74$). Here, $Eh = 14.4$ N m$^{-1}$ and $T_{acc} = 0.12$–$0.13$ s for all cases. Here, the maximum displacement $w_{n,max}/D$ is given as (a) $0.042$, (b) $0.055$ and (c) $0.062$.

To model this decelerated wave speed, we revisit the governing equations that we derived for the nozzle deformation and jet speed. Let us consider the surface wave moving downward ($-y$) while the jet velocity saturates to $u=u_t$. A small perturbation on the jet velocity ($u_p$) is introduced as $\hat {u}=1+\hat {u}_p$ (note that a hat denotes normalization with $u_t$). Based on the observation that the radial displacement is small compared with the nozzle radius ($\hat {w}_{n}=w_{n}/R \ll 1$), (4.7) and (4.8) are reduced to

(4.19)\begin{gather} \mathcal{D}\hat{w}_{n} + \frac{1}{2}\frac{\partial \hat{u}_p}{\partial\hat{y}} = 0, \end{gather}

(4.20)\begin{gather}\mathcal{D}\hat{u}_{p} + \varPi_{1}\frac{\partial\hat{w}_n}{\partial\hat{y}} = 0. \end{gather}

Here, the differential operator $\mathcal {D}$ is defined as $\mathcal {D} = (1/\varPi _{0})\partial /\partial \hat {t}+\partial /\partial \hat {y}$, and the higher-order terms are ignored. Then, through similar procedures to construct single-variable equations, the above equations can be rearranged as follows:

(4.21)\begin{gather} \mathcal{D}^{2}\hat{w}_{n} - \frac{\varPi_1}{2} \frac{\partial^{2}\hat{w}_{n}}{\partial\hat{y}^{2}} = 0, \end{gather}

(4.22)\begin{gather}\mathcal{D}^{2}\hat{u}_{p} - \frac{\varPi_1}{2}\frac{\partial^{2}\hat{u}_{p}}{\partial\hat{y}^{2}} = 0. \end{gather}

As expected, $\hat {w}_{n}$ and $\hat {u}_{p}$ are the solutions of the same wave equation. By substituting the normal mode of perturbation defined as $\hat {u}_{p}, \hat {w}_{n} \sim \exp [i(\hat {k}\hat {y} - \hat {\omega }\hat {t})]$, the dispersion relation appears as $(1/\varPi _{0}^{2})\hat {\omega }^{2}-(2/\varPi _{0})\hat {k}\hat {\omega } + (1-\varPi _{1}/2)\hat {k}^{2}=0$. Using the quadratic formula for frequency ($\hat {\omega }$), the reduced wave speed ($\hat {c}_r$) is obtained in both dimensionless (4.23) and dimensional (4.24) forms, respectively,

(4.23)\begin{gather} \hat{c}_{r} = \frac{\hat{\omega}}{\hat{k}} = \varPi_{0}(1 \pm \sqrt{\varPi_{1}/2}), \end{gather}

(4.24)\begin{gather}c_{r} = \frac{\hat{c}_{r}L}{T_{acc}} = u_{t}(1 \pm \sqrt{\varPi_{1}/2}) = u_{t} \pm \sqrt{\frac{Eh}{\rho_{f}D}}= u_{t} \pm |c|. \end{gather}

The negative sign corresponds to the jet direction. According to (4.24), the wave speed is not only affected by the effective nozzle stiffness but is also augmented by the jet terminal velocity ($u_t$), unlike (4.13) and (4.14). A similar formulation was reported for the free vibrations of a fluid-conveying shell (Chen & Rosenberg 1974). From the shell equation without assuming a thin shell, i.e. Flügge equation, and the fluid momentum conservation equation, the water-hammer wave speed with vibration of $n = 0$ (axisymmetric mode) can be calculated as $u_{t} \pm c_{f}/\sqrt {1+2KR/(Eh)}$, where $K$ and $c_f$ denote the fluid's bulk modulus and wave speed, respectively. Because $K = \rho _{f} c_{f}^{2}$, this relation can be rewritten as $u_{t} \pm 1/\sqrt {(1/c_{f}^{2}+2\rho _{f}R/(Eh)} = u_{t} \pm 1/\sqrt {1/c_{f}^{2}+1/c^{2}}$, which is directly reduced to (4.24) under the condition of $c^{2}/c_{f}^{2} \ll 1$. Thus, relation (4.24) corresponds to the case in which acoustic waves decouple. Next, the wave speed in the upward direction is $c_{r,u} = u_{t} + |c|$ (note that $u_t$ has a negative sign). Therefore, the effective time duration corresponding to the release of the elastic energy stored in the expanded nozzle ($T_{eff}$) is calculated with the reduced wave speed as $T_{eff} = L/c_{r,u}$. Accordingly, the energy balance used to derive (4.18) can be modified such that the emitted potential (elastic) energy of the nozzle contributes to the generation of maximum thrust during the acceleration time ($T_{acc}$) only. This can be considered as follows:

(4.25)\begin{equation} {\rm \Delta} M \sim P \frac{T_{acc}}{T_{eff}}.\end{equation}

Substituting the relation for ${\rm \Delta} M$, $P$ and $T_{eff}$, the equation for the change in the jet velocity is rewritten as follows (the detailed derivation can be found in Appendix D):

(4.26)\begin{equation} \frac{{\rm \Delta} u}{u} \sim 1-\sqrt{\frac{2}{\varPi_1}} = J^{*}. \end{equation}

The scaling relation (4.26) indicates that if the nozzle material and the jet characteristics satisfy the optimal condition (i.e. $\hat {c}=3.0$), the maximum jet velocity increases with increasing effective stiffness ($\varPi _1$). While validating the modified argument on the transfer of elastic energy to the jet kinetic energy, figure 18 clearly shows that both the augmentation in jet velocity and the impulse for the cases around the optimal condition ($1.5 < \hat {c} < 4.5$) are well correlated with $J^{*}$. For the off-optimal condition ($\hat {c}>4.5$), flow variables related to thrust show poor correlation with $J^{*}$. Another implication that we can learn from $J^{*}$ is that there is an upper limit of $J^{*} \rightarrow 1.0$ as $\varPi _{1}\rightarrow \infty$. That is, increasing the nozzle stiffness will saturate the thrust to a certain value (under the condition that the jet and the nozzle satisfy the optimal condition). In figure 18(a), extrapolation of the optimal condition (denoted as dashed arrows) predicts that the augmented velocity will be approximately $131\,\%$ ($236\,\%$ for the hydrodynamic impulse) larger than that of the rigid nozzle. Unfortunately, the limitation in manufacturing thinner nozzles than those tested and the faster response time of the motor specification did not allow us to explore the corresponding regime of $J^{*}\simeq 1.0$. Together with (4.15), (4.26) raises the important implication that thrust generation must be treated for understanding the optimal condition (which considers the ratio between the structural response time and flow time scale) and the mechanism of the delayed release of the flexible nozzle (which stems from the slower surface wave speed with decreasing effective stiffness). Without this consideration, misinterpretations can occur; for example, reduced stiffness either enhances the impulse generation (as observed in figure 7) or, totally conversely, reduces it (as shown in figure 18b). This will be useful for understanding and predicting thrust generation from a flexible nozzle, which is prevalent in nature and industrial applications.

Figure 18. (a) The maximum increase of jet-exit velocity (${\rm \Delta} u_{e}/u_{t}|_{max}$) and (b) the relative hydrodynamic impulse at $t=T_{acc}$ depending on $J^{*}$: $\blacklozenge$, $1.5 < \hat {c} <4.5$; $\bullet$, $\hat {c}> 4.5$. The dashed lines are the extrapolation curves of $a_{1}J^{*} + a_{2}$, where ($a_{1},a_{2}$) = ($0.46, 0.15$) (a) and ($2.75, 0.38$) (b).

5. Concluding remarks

We conducted an experimental and analytical investigation of the evolution of a starting jet through a flexible circular nozzle. The flexible nozzles are manufactured with three different structural stiffnesses, and the terminal velocity of the jet and acceleration time are systematically adjusted using an in-house piston–motor system. To understand the dynamics of the jet structure in time and space, we performed two-dimensional PIV measurements, and measured the deformation of the nozzle surface together with subpixel image processing. The evolution of the flow field shows that the primary vortex shed from the nozzle exit convects faster as the nozzle is more flexible; consequently, the hydrodynamic thrust increases approximately $90\,\%$ greater than that of the rigid nozzle. Based on the quantification of nozzle deformation, we found that the expansion and contraction of the nozzle, respectively, accumulates the elastic potential energy and transfers it to the flow inside the jet, thereby increasing the jet velocity inside the nozzle. This is the key fluid–structure interaction mechanism responsible for maximizing thrust generation. To characterize the interaction between the flexible nozzle and the accelerating flow in it, the governing equations for the nozzle deformation and flow velocity are derived from the shell theory and conservation laws for mass and momentum, in which two dimensionless parameters, i.e. effective acceleration time ($\varPi _0$) and effective nozzle stiffness ($\varPi _1$), are used. The subsequent analytical investigation, supported by the measured flow field and nozzle deformation, indicates that the time when the contraction completely ends has to be comparable to the instance when the jet reaches maximum velocity, to produce maximum thrust. This optimal condition is expressed as $\hat {c}=\sqrt {\varPi _{0}^{2}\varPi _{1}/2} \simeq 3.0$, where $\hat {c}$ is the dimensionless surface wave speed. While satisfying this condition, furthermore, the maximum jet velocity of the flexible nozzle depends on the effective nozzle stiffness ($\varPi _1$), which is attributed to the reduced wave speed drawn by asymptotic analysis. The augmentation in the jet-exit velocity is found to increase with effective stiffness, which is scaled as ${\rm \Delta} u/u \sim 1-\sqrt {2/\varPi _1}$.

Therefore, we understand that the flexibility of the nozzle has the potential not only for thrust improvement in jet flow but also for the agility of the system. Numerous organisms and industrial applications make use of a flexible nozzle, which ejects an impulsively starting jet. Thus, it will be fascinating to examine if jet conditions (i.e. the terminal velocity and the acceleration time) and the material property of the nozzle (i.e. Young's modulus and thickness) follow the optimal condition derived in the present study. In addition, the dimensionless surface wave speed (it was approximately 3.0 for the circular nozzle) must be quantified, depending on the nozzle geometry such as a cone-shaped nozzle.