3. Rotor and vortex dynamics

As a rotor approached the water–air interface from below, it experienced abrupt, yet predictable, changes in frequency, lift and torque. Consider, as a demonstration, the dynamics of rotor 2 at median conditions, $\dot {z}_{{R}}=100\,$ mm s⁻¹ and $16\,\%$ throttle (figure 2). Far below the water, the rotor’s frequency, lift and torque converged to equilibrium values, and the free surface became depressed, presumably due to the low-pressure core of the vortex produced by the rotor. In air, the rotor’s frequency, lift and torque converged to different equilibrium values, as expected. However, at a critical transition depth $z_{{R}}=z_{{R,c}}$ , well before the rotor passed the steady-state free surface, the frequency began to increase, while the lift and torque began to decrease. Video stills demonstrate that this early transition corresponds to the rotor reaching the depressed free surface ( $z_{{R}} = \eta (0)$ ) before reaching the steady-state free surface ( $z_{{R}}=0$ ) (figure 2, insets). Across trials with the same input kinematics, the critical transition depth $z_{{R,c}}$ was largely unchanged.

Figure 2. Representative vortex and rotor dynamics during water-to-air transition. Rotor 2; $\dot {z}_{{R}}= 100$ mm s⁻¹, throttle $= 16\,\%$ . (a) Rotor frequency increases after $z_{{R}}\gt z_{{R,c}}$ . (b) Rotor lift decreases after $z_{{R}}\gt z_{{R,c}}$ . (c) Rotor torque follows lift trends. Insets: key moments of the highlighted trial (darker curve; Trial 1 in Movie 1): (1) far underwater, $z_{{R}}/a=-1.76$ ; (2) surface depression about to contact rotor, $z_{{R}}/a=-0.68$ ; (3) rotor–surface contact and subsequent aeration, $z_{{R}}/a=-0.28$ ; (4) breach, $z_{{R}}/a=0$ ; (5) far above water, $z_{{R}}/a=+1.5$ .

Another notable feature of the vortex/rotor dynamics is the aeration caused by rotor–surface contact. Soon after the rotor reached the depressed surface, it often experienced a blip or inflection in frequency, lift and torque (figure 2). The inflection was more pronounced at lower ascent speeds. Consider, for example, a proxy for quantifying inflection during the transition: the tortuosity of the $L(z_{{R}})$ data, i.e. the ratio of arc length to distance between end points for $z_{{R,c}}\lt z_{{R}} \lt 0$ , $\int _{-z_{{R,c}}}^0 \sqrt {1+L'(z_{{R}})^2}\mathrm{d}z_{{R}}\,/\,\sqrt {(L(z_{{R,c}})-L(0))^2+z_{{R,c}}^2}$ . Averaged across all rotors, throttles and trials, the tortuosity was 1.272 $\pm$ 0.081 for $\dot {z}_{{R}}=20\,\mathrm{mm\, s^{- 1}}$ and 1.003 $\pm$ 0.005 for $\dot {z}_{{R}}=180\,\mathrm{mm\, s^{-1}}$ . Video stills demonstrate that this inflection corresponded to the onset of aeration (figure 2, inset (3)). Beyond that point, but before the rotor exited the water, air was entrained into a bubble-rich turbulent wake beneath the rotor. This state is similar to the one induced purposefully in stirred tanks, where aeration enhances the mixing effects of a submerged impeller (Nagata Reference Nagata1975).

The spinup time had no noticeable effect on the transition depth. With the shorter spinup time (0.075 s), the frequency had time to converge before the rotor began its ascent, whereas the lift and torque did not (figure 2). For all cases, frequency, lift and torque converged to within 2 % of their steady-state values underwater in less than 1.075 s. Therefore, with the longer spinup time (1.075 s), frequency, lift and torque all converged before the ascent. The trials with longer spinup times are presumably more representative of water-to-air trajectories starting at deeper depths ( $z_R/a\lt -2$ ), where secondary flows have time to evolve well before the transition. However, it seems that the initial conditions of these secondary flows have little effect on the transition depth.

The ascent speed $\dot {z}_{{R}}$ also appeared to have little effect on the critical depth at which the rotor contacted the free surface ( $z_{{R,c}}$ ) (figure 3 a,b). The depth of the surface depression therefore tended to be similar at a given rotor throttle and rotor depth across different ascent speeds (figure 3 a,b, insets). This similarity is presumably what led to the similarity in frequency and lift curves beneath the water (above the water, the ascent speed $\dot {z}_{{R}}$ of the rotor led to different $f(z_R)$ curves; see figure 3 a). The fact that ascent speed had little effect on the timing of rotor–surface contact suggests that the dynamics of the surface were largely quasi-steady. At higher advance ratios, unsteady effects would likely appear, but for the advance ratios considered here (0.01–0.17), ascent speed had no measurable effect on $z_{{R,c}}$ .

Figure 3. Vortex and rotor dynamics at differing ascent speeds and throttles. (a,b) Rotor frequency (a) and lift (b) begin transitioning at similar depths, $z_{{R,c}}$ , regardless of ascent speed. (c,d) Rotor frequency (c) and lift (d) begin transitioning at lower depths as throttle increases. Insets: video snapshots at $z_{{R}}/a=-0.47$ in the highlighted trials (darker curves; Trial 2 in Movie 1).

In contrast, throttle did have an effect on the transition depth ( $z_{{R,c}}$ ). As throttle increased, $z_{{R,c}}$ decreased, i.e. the stronger free vortex led to an earlier growth of the surface depression and subsequent contact with the rotor (figure 3 c, d). Therefore, at a given rotor depth, the free surface was often in a qualitatively different stage depending on throttle. For example, when rotor 2 was at $z_{{R}}/a=-0.47$ , the rotor was either below the free surface, just reaching the free surface or mid-aeration, depending on whether the throttle was 10 %, 16 % or 20 % (figure 3 c, d, insets). This dependence on throttle is consistent with studies of stirred tanks, where higher impeller speeds lead to stronger free vortices and deeper surface depressions (Halász et al. Reference Halász, Gyüre, Jánosi, Szabó and Tél2007; Busciglio et al. Reference Busciglio, Caputo and Scargiali2013; Kim & Kim Reference Kim and Kim2021). In the case of transitioning rotors, this dependence means that higher throttles cause frequency to rise and lift to drop at lower depths.

All six rotors that we tested had similar frequency, lift and torque trends as those of rotor 2 (figures 2 and 3). The only clear difference was that the transition depth ( $z_{{R,c}}$ ) was lower when more blades were present. An effect of blade number is perhaps expected given that blade geometry also influences the effectiveness of impellers in stirred tanks (Pincovschi, Dragomirescu & Modrogan Reference Pincovschi, Dragomirescu and Modrogan2025). Rather than replotting frequency and lift curves, we will consider the effect of blade number/geometry when examining the scaling of $z_{{R,c}}$ in the following section.

4. Scaling the transition depth

Motivated by the links between the free vortex and the rotor’s performance, we explored how transition depth ( $z_{{R,c}}$ ) might scale with rotor geometry and kinematics. For a given rotor type, we propose that $z_{{R,c}}$ must be a function of the variables present: the acceleration due to gravity $g$ , the rotor radius $a$ , the rotor frequency $f$ , the rotor ascent speed $\dot {z}_{{R}}$ and the density $\rho$ , viscosity $\mu$ and surface tension $\gamma$ of the surrounding water. Dimensional analysis reveals that the scaled transition depth must be some function $\phi$ of four dimensionless groups, e.g.

(4.1) \begin{equation} \frac {z_{{R,c}}}{a}=\phi \left (\frac {\dot {z}_{{R}}}{2af},\frac {\rho g a^2}{\gamma },\frac {\rho f a^2}{\mu },\frac {fa}{\sqrt {ga}}\right )\equiv \phi (J,Bo,{Re},Fr). \end{equation}

The first dimensionless group in $\phi$ is the advance ratio $J$ . The other three are various ratios of gravity forces, surface tension forces, viscous forces and rotor-imposed inertial forces, i.e. they are forms of a Bond number $Bo$ , Reynolds number ${Re}$ and Froude number $Fr$ .

Our results suggest no dependence on the advance ratio over the range considered here (figure 3 a,b). The Bond number, a ratio of gravity to surface tension forces, was high throughout our study ( $\approx 100$ to $2700$ ), and the rounded shape of the free surface – which differs from the needle-like profiles of vortices dominated by surface tension (Andersen et al. Reference Andersen, Bohr, Stenum, Rasmussen and Lautrup2003) – suggests that $Bo$ is not a key parameter here. The Reynolds number was also high throughout our study ( $\approx 28\,000$ to $84\,000$ ), and studies of stirred tanks have found inviscid vortex models to work well for ${Re} \gtrapprox 10^4$ (Nagata Reference Nagata1975; Deshpande et al. Reference Deshpande, Kar, Walker, Pressler and Su2017). In contrast, the Froude number based on rotor diameter, $Fr$ , was $O(1)$ throughout our study ( $\approx$ 0.7–1.5). In stirred tank literature, the Froude number based on impeller diameter has been described as the ‘key dimensionless parameter for describing the vortex depth’ (Deshpande et al. Reference Deshpande, Kar, Walker, Pressler and Su2017). As we will show, the same conclusion can be drawn from our data.

Plotting transition depth against Froude number (figure 4) supports the hypothesis that $z_{{R,c}}/a$ is primarily a function of $Fr$ . Not only does Froude number do well to collapse the transition depths for different rotor radii, but also the relationship is largely linear within each blade type. To test the linear scaling’s robustness, we also produced nonlinear fits $z_{{R,c}}/a = \alpha Fr^\beta$ for the 2-, 3-,and 4-blade rotor data, with fit coefficients $\alpha$ and $\beta$ . The fitted $\beta$ values were near unity (0.93, 0.84 and 0.89) with residuals comparable to the purely linear fit ( $R^2=0.78$ , $0.82$ , $0.92$ ), supporting the claim that $z_{{R,c}}/a \propto -Fr$ .

Figure 4. Scaled transition depth versus Froude number. The depth at which lift begins to drop ( $z_{{R,c}})$ , as calculated using derivative thresholding on $L(z_{{R}})$ data, generally scales with Froude number for the 2-, 3- and 4-blade rotors.

5. Vortex modelling

We wondered if the observed scaling of $z_{{R,c}}$ could be better understood using a Burgers vortex, a steady solution to the Navier–Stokes equations in which an inward radial flow and an outward diffusion of vorticity balance to form a stable vortex (Drazin & Riley Reference Drazin and Riley2006). The Burgers vortex has been successfully used to model a variety of vortex-related phenomena, including the free vortex in stirred tanks (Nagata Reference Nagata1975; Lugt Reference Lugt1983; Halász et al. Reference Halász, Gyüre, Jánosi, Szabó and Tél2007). In a Burgers vortex, a Gaussian distribution of vorticity centred at the origin leads to an azimuthal flow speed

(5.1) \begin{equation} u_\theta =\frac {\varGamma }{2\pi r} \left (1-\mathrm{e}^{-\frac {r^2}{c^2}}\right ), \end{equation}

where $\varGamma$ is circulation and $c$ is the vortex core radius. Far outside the core, the vortex converges to irrotational flow ( $\lim _{r\to \infty } u_\theta = \varGamma /(2\pi r)$ ); deep in the core, the vortex converges to a solid body rotation ( $\lim _{r\to 0} u_\theta = \varGamma r/(2\pi c^2)$ ).

Equation (5.1) can be used to model the depression of a free surface above the vortex. Following others (e.g. Halász et al. Reference Halász, Gyüre, Jánosi, Szabó and Tél2007), we equate the radial and vertical pressure gradients with the centrifugal force and gravity,

(5.2) \begin{equation} \frac {\partial p}{\partial r}=\frac {\rho u_\theta ^2}{r}\quad \mathrm{and}\quad \frac {\partial p}{\partial z}=-\rho g, \end{equation}

leading to a differential equation for the free surface height $\eta (r)$

(5.3) \begin{equation} \frac {\partial \eta }{\partial r}=-\frac {\partial p/\partial r}{\partial p /\partial z}=\frac {u_\theta ^2}{rg}. \end{equation}

The height at the centre of the vortex (with $\eta (\infty )\equiv 0$ ; figure 5) follows from integration

(5.4) \begin{equation} \eta (0)=\int _\infty ^0 \frac {u_\theta ^2}{rg} = -\frac {\mathrm{ln}2}{4\pi ^2}\frac {\varGamma ^2}{c^2g}. \end{equation}

Figure 5. Model set-up. (a) As $z_{{R}}$ rises, the surface depression grows. (b) As $z_{{R}}$ rises to the transition depth, the rotor contacts the free surface ( $z_{{R}}=z_{{R,c}}=\eta (0)$ ).

We expect the vortex core radius to scale with rotor radius ( $c\sim a$ ) (Halász et al. Reference Halász, Gyüre, Jánosi, Szabó and Tél2007, see also free surface profiles in the Supplementary Materials). We expect the circulation to be primarily set by the rotor radius and tip speed (e.g. $\varGamma \sim f a^2$ , Abedi Reference Abedi2011); however, studies of stirred tanks have found that vortex depth also includes a dependence on impeller height. For example, Halász et al. (Reference Halász, Gyüre, Jánosi, Szabó and Tél2007) found that vortex strength $\varGamma$ included a $1/\sqrt {z_{{R}}/a}$ dependence. What differs in our system is that the rotor rises and thus the vortex ultimately reaches the rotor in every case (figure 5 b). We therefore expect $\eta (0)\sim z_{R,c}$ throughout much of the water-to-air transition phase. With $\eta (0)\sim z_{R,c}$ , $c\sim a$ and $\varGamma \sim f a^2 \boldsymbol{\cdot }(1/\sqrt {z_{R,c}/a})$ in (5.4) leads to

(5.5) \begin{equation} \frac {\eta (0)}{a}\sim -\frac {f a}{\sqrt {ga}}=-Fr. \end{equation}

The prediction that scaled vortex height would scale with $Fr$ during the transition phase offers some physical rationale for the linear relation between transition depth and Froude number (figure 4). Curiously, in stirred tank studies, vortex depth is often found to scale with $f^2$ , with additional geometric factors such as impeller depth also playing a role (Nagata Reference Nagata1975; Smit & During Reference Smit and During1991; Halász et al. Reference Halász, Gyüre, Jánosi, Szabó and Tél2007; Busciglio et al. Reference Busciglio, Caputo and Scargiali2013; Deshpande et al. Reference Deshpande, Kar, Walker, Pressler and Su2017). In our case, because the vortex depth and the rotor/impeller depth are inherently linked, it appears that the transition depth ultimately scales with $f$ (5.5, figure 4).