2.1. Non-dimensional parameters

We consider the 2D problem described in figure 1, consisting of a horizontal cylinder of radius $a=1$ vertically lifted out of a bath of liquid at a velocity $U$ , released from a depth  $d$ below the free surface, in a tank of lateral dimension $2W$ and depth $H$ . In the inviscid limit, the water exit dynamics is known to be governed by inertia and gravity (Telste Reference Telste1987; Liju et al. Reference Liju, Machane and Cartellier2001); their relative effect can be expressed as the Froude number $\textit{Fr} = U^2 / g a$ , where $g$ is the acceleration of gravity. When the cylinder is initially close to the interface, the next parameter by order of importance is the starting depth $d$ . It is commonly assumed that transient effects do not impact the exit dynamics if $d$ is sufficiently large, whereas gravitational waves are triggered at the interface when $d$ is small. In the latter case, they can have noticeable effects on the bulge height and the hydrodynamic force (Tyvand & Miloh Reference Tyvand and Miloh1995a ; Telste Reference Telste1987). Wake characteristics are controlled by the Reynolds number, defined as $\textit{Re} = \rho (2a) U / \mu$ , where $\rho$ is the liquid density and $\mu$ its dynamic viscosity. For large Reynolds numbers, we expect viscosity not only to affect the wake development (growth and stability), but also to play a hindering role in the drainage phase.

Figure 1.

(a) Problem description in dimensionless parameters: a horizontal circular cylinder of radius $a=1$ is lifted out of a liquid bath of initial height $H$ and width $2W$ at a constant velocity $U=1$ , starting from rest at a depth $d$ below the free surface. A bulge is formed at short times. (b) As the cylinder crosses the interface, a liquid film of quasi-uniform thickness wraps around the cylinder while draining. A symmetric wake is formed below the cylinder. Here $y_c$ is the cylinder vertical position, while $y_s$ and $h$ are the free-surface elevation and film thickness directly above the cylinder ( $y_s = y_c + 1 + h$ ).

All lengths in figure 1 are non-dimensionalised by the cylinder radius $a$ and the fluid velocity by the nominal cylinder velocity $U$ . Consistently, the time is normalised by the inertial time scale $a/U$ and the force by the inertial force scale $({1}/{2}) \rho S U^2$ , where $S = (2 a) L$ is the cross-sectional area of the cylinder in the direction of motion and $L$ is the cylinder length. Note that in this 2D problem, $L$ is irrelevant. In what follows, all quantities (variables and parameters) are dimensionless unless otherwise stated. The range of non-dimensional numbers considered in our work is shown in table 1, along with typical dimensional equivalents for reference.

Table 1.

Typical ranges of dimensional parameters (see Ashraf et al. (Reference Ashraf, Vincent, Falla, Terrapon, Scheid and Dorbolo2024)) and dimensionless numbers considered in the present work.

The effect of surface tension is disregarded in this work. This choice is supported by both physical and numerical arguments. Our work is geared towards a comparison with future in-house experiments and focuses on exit cases of macroscopic (centimetre-sized) cylinders from water and 50-cSt silicone oil at speeds between 0.1–1.3 m s⁻¹. Taking $a = 25$ mm and typical parameters for water (density $\rho = 1000$ kg m $^{-3}$ , surface tension $\sigma = 72$ mN m⁻¹), we can verify that the object is significantly larger than the capillary length $\sqrt {\sigma / \rho g} = 2.7$ mm. Moreover, the balance between surface tension and inertia can be written in terms of a Weber number, $W\!e = \rho a U^2 / \sigma$ , which varies between $4$ and $714$ with the above velocity range. This result suggests that, while low-velocity cases may be affected by the effect of surface tension, moderate- and high-velocity cases should not. Finally, we confirmed our order-of-magnitude approach by running simulations with and without surface tension for specific cases. We quantified and checked that the effects of surface tension on the variables of interest were negligibly small compared with all other effects already taken into account. Due to curvature calculations, these simulations with surface tension are significantly slower, right from the start, and even more so when the cylinder reaches the interface and the free surface is subjected to strong deformations, which would have effectively prevented the systematic parameter screening we propose in this work.

We anticipate, however, that surface tension may play a role in the later stages of water exit, e.g. stages that involve large local surface curvature or where volumes of fluid detach from the free surface. These events are beyond the scope of this work and inaccessible with our current numerical method that does not handle an intersecting or folding interface. To allow exploring beyond surface folding events and the later stages of exit, we developed a significantly more resource-intensive in-house method that handles topological changes. This more robust method is used to validate our present numerical method; see Appendix A.

2.2. Governing equations

The flow around the rising cylinder can be modelled with the continuity and the Navier–Stokes equations, which have been rendered dimensionless using the characteristic length  $a$ , the characteristic velocity $U$ and characteristic stress $\rho U^2$ as

(2.1) \begin{align} \boldsymbol{\nabla } \boldsymbol{\cdot }\boldsymbol{v} = 0, \quad \partial _t \boldsymbol{v} +\left ( \boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla }\right ) \boldsymbol{v} = \boldsymbol{\nabla } \boldsymbol{\cdot }\tau - \frac {1}{\textit{Fr}} \boldsymbol{e}_{\!y} \quad \mbox{in} \quad \mathcal V , \end{align}

where $\tau =-p \mathcal I + 2 \textit{Re}^{-1} [ \boldsymbol{\nabla } \boldsymbol{v} + (\boldsymbol{\nabla } \boldsymbol{v})^T ]$ is the stress tensor, $\boldsymbol{v}=(u,v)$ the velocity and  $p$ the pressure (the factor 2 stands for the fact that the length scale is the cylinder radius whereas the Reynolds number is based on the cylinder diameter). These equations hold at any position, $\boldsymbol{x}$ , in the deformable liquid domain $\mathcal{V}(t)$ , initially of width $2W$ and uniform depth $H$ .

Let $\varSigma _{w\textit{all}}$ be the lateral and bottom boundaries of the container, $\varSigma _{\textit{cyl}}$ the interface between the 2D cylinder and the liquid, and $\varSigma$ the free surface. The boundary conditions on each of these surfaces are the no-penetration conditions at the tank bottom and slide walls, the no-slip condition at the cylinder surface and the stress-free condition at the free surface:

(2.2a) \begin{align} \tau \boldsymbol{\cdot }\boldsymbol{n} = \frac {1}{\textit{Fr}} y \, \boldsymbol{n} \, &\quad \mbox{at} \quad \varSigma _{w\textit{all}} , \end{align}

(2.2b) \begin{align} \boldsymbol{v} = \, \boldsymbol{e}_{\!y} \, &\quad \mbox{at} \quad \varSigma _{\textit{cyl}}(t) , \end{align}

(2.2c) \begin{align} \tau \boldsymbol{\cdot }\boldsymbol{n} = 0 \, &\quad \mbox{at} \quad \varSigma (t) . \end{align}

Here $\boldsymbol{n}$ is the unit vector normal to the surface and pointing into the liquid domain. As an initial condition, the system is considered at rest, i.e.

(2.3) \begin{align} \boldsymbol{v} &= \boldsymbol{0} \quad \mbox{in} \quad \mathcal{V}(0). \end{align}

Additionally, simulations with free slip at the cylinder surface offer a reference scenario that will alternatively be used in this work to probe the influence of the wake. This free-slip condition substitutes to the no-slip condition (2.2b ) and writes

(2.4a) \begin{align} \tau \boldsymbol{\cdot }\boldsymbol{n} = -p_s \boldsymbol{n} &\quad \mbox{at}\quad \varSigma _{\textit{cyl}}(t) , \end{align}

(2.4b) \begin{align} \boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{n} = \boldsymbol{e}_{\!y} \boldsymbol{\cdot }\boldsymbol{n} &\quad \mbox{at}\quad \varSigma _{\textit{cyl}}(t), \end{align}

where $p_s$ is the pressure at the cylinder surface necessary to prevent liquid penetration and, therefore, acts as a Lagrangian multiplier.

To deal with the deformable domain, the arbitrary Lagrangian–Eulerian method is used. In this method, any point $\boldsymbol{X}$ of the initial domain $\mathcal{V}(0)$ is transformed into one in the actual domain, such that $\boldsymbol{x}=\boldsymbol{x}(\boldsymbol{X},t) \in \mathcal{V}(t)$ and $\boldsymbol{X}=\boldsymbol{x}(\boldsymbol{X},0) \in \mathcal{V}(0)$ . We choose the mesh displacement $ \boldsymbol{q} = \boldsymbol{x}-\boldsymbol{X}$ to fulfil

(2.5) \begin{align} {\nabla }^2 \boldsymbol{q} = 0 \quad \mbox{in} \quad \mathcal V(t) , \end{align}

for which boundary conditions must be imposed on each of the domain boundaries, i.e. no deformation of the tank walls, specified velocity at the cylinder surface and a kinematic condition at the free surface:

(2.6a) \begin{align} \boldsymbol{q} &= \boldsymbol{0} \qquad\, \mbox{at} \quad \varSigma _{w\textit{all}}, \end{align}

(2.6b) \begin{align} \boldsymbol{q} &= \,t\, \boldsymbol{e}_{\!y} \quad \mbox{at}\quad \varSigma _{\textit{cyl}}(t), \end{align}

(2.6c) \begin{align} \partial _t \boldsymbol{q} &= v_n \boldsymbol{n} \quad\, \mbox{at} \quad \varSigma (t) . \end{align}

Here $v_n=\boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{n}$ is the normal component of the velocity. The initial condition writes

(2.7) \begin{equation} \boldsymbol {q} = \textbf {0} \quad \mbox{in } \quad \mathcal{V}(0). \end{equation}

2.3. Numerical method

The system of equations (2.1)–(2.7) is solved by means of the finite element software Comsol Multiphysics, with the moving mesh and ‘weak form PDE’ modules. Linear Lagrangian interpolants are used for the pressure and quadratic interpolants for the velocity and geometry. For the moving mesh, we use an arbitrary Lagrangian–Eulerian (ALE) formulation with a specific mesh treatment at the boundary, named the BALE method (Rivero-Rodríguez et al. Reference Rivero-Rodríguez, Pérez-Saborid and Scheid2021). The mesh is locally refined in the vicinity of the cylinder and on curved free-surface sections by setting a curvature-based criterion, leading to a uniform mesh size $\Delta x/a$ around the cylinder. This minimum mesh size is coupled with a growth factor $m$ from one layer of elements to the next. Typical parameters are $\Delta x/a=0.01$ and $m=1.04$ for the no-slip cases, and $\Delta x/a=0.05$ and $m=1.08$ for the free-slip cases at $\varSigma _{\textit{cyl}}$ . In addition, we set a minimum mesh size of $0.75a$ on the free surface to avoid excessive smoothing of the free surface when the cylinder is far (which may affect surface wave propagation). The good quality of the mesh is maintained by periodically remeshing when the mesh distortion exceeds a certain threshold ( $20\,\%$ ), resulting in a typical remeshing frequency of one event per radius travelled by the cylinder. The time stepping is done via the adaptive backward differentiation formula, with minimum order $2$ and maximum order $5$ . We performed a convergence study and checked that the results were independent of the mesh parameters, element size and time stepping, for every set of physical parameters. For the highest Reynolds number tested, $\textit{Re}=10\,000$ , converged results were obtained for a mesh size $\Delta x/a=5 \times 10^{-3}$ and a time step $\Delta t=10^{-3}$ . Simulations are run until topological changes (e.g. interface folding) are detected at the free surface. A single run typically takes a few hours on a standard laptop computer.

Finally, we chose to constrain the domain to a half-plane ( $x\gt 0$ ) and use the mirror condition in x = 0. This choice improves computational efficiency and allows us to explore more of the parameter space; besides, it is consistent with our focus on symmetric, transient wakes. However, it implies that the faithfulness of our simulations will degrade when, at longer times, the real wake is no longer symmetric (vortex shedding). This transition to an asymmetric flow is sensitive to physical parameters such as Reynolds number and pre-existing turbulence in the tank. For cylinders, it happens typically after 10–20 radii travelled, with evidence showings symmetric transient wakes up to 36 travelled radii (Ashraf et al. Reference Ashraf, Vincent, Falla, Terrapon, Scheid and Dorbolo2024).

3.1. Impulsively started cylinder

In this first set of comparisons, we seek to validate our code’s performance against well-studied cases of impulsively started cylinders in an infinite fluid medium. In the absence of the interface, the problem is set by three non-dimensional numbers, namely the Reynolds number $\textit{Re} = (2 a) U / \nu$ , the tank width $W$ and the initial distance from the cylinder to the bottom wall, that is, $H{-}d$ with our notation. We ran a preliminary analysis showing that this last parameter did not matter if greater than 5, and we kept that condition true for all simulations. In contrast, the tank width and Reynolds number do have a noticeable effect, and we matched them exactly when their values were available.

3.1.1. Comparison with experiments

Figure 2.

Transient wake after impulsive start, shown by the streamlines in the cylinder’s frame of reference (left and right) and coloured vorticity contour (left). Left: our work; right: experiments from Lin et al. (Reference Lin, Hsieh, Lin and Raikar2012). $\textit{Re} = 500$ , $W =28$ .

We start with a moderate/low Reynolds number ( $\textit{Re} = 500$ ) comparison, representative of a slow water exit situation. Figure 2 compares our work (left panels) to experiments by Lin et al. (Reference Lin, Hsieh, Lin and Raikar2012) (right panels), for several time snapshots up to $t = 11.2$ (that is, when the cylinder travelled 11.2 radii). In spite of the eventual and inevitable loss of symmetry observed in experiments, the simulation-to-experiment agreement is excellent throughout the wake development. In particular, the bottom stagnation point in the wake, the locii of the vortex centres and the wake width all compare quantitatively well for all three time snapshots.

For larger Reynolds numbers characteristic of fast water exits, figures 3 and 4 compare our work (left panels) to long-exposure experimental snapshots from the seminal work of Bouard & Coutanceau (Reference Bouard and Coutanceau1980) (right panels), for several time snapshots up to $t = 5$ . Here again, the simulation-to-experiment agreement is excellent for all snapshots. We observe the faithful reproduction of even the smallest features in the wake close to the cylinder wall. It should be emphasised that the experiments of Bouard & Coutanceau (Reference Bouard and Coutanceau1980) were performed in a relatively narrow tank, i.e. $W=8$ , a width for which the lateral bounding has a sizeable influence on the wake development, namely a quicker wake development than the infinite-medium situation. Thus, exact matching of $W$ was proven necessary for relevant comparison. We note that no experimental snapshot was provided for $t \gt 5$ ( $\textit{Re} = 5000$ ) and $t \gt 3$ ( $\textit{Re} = 9500$ ). A plausible reason for this omission is the inevitable symmetry breaking of the initial 2D wake by 3D destabilisation mechanisms for these longer times.

Figure 3.

Transient wake after the impulsive start, shown by the streamlines in the cylinder’s frame of reference (left and right) and vorticity contour (left). Left: our work; right: experiments from Bouard & Coutanceau (Reference Bouard and Coutanceau1980). Here $\textit{Re} = 5000$ , $W =8$ . The thin white circle in the left panel marks the initial position of the cylinder.

Figure 4.

Transient wake after the impulsive start, shown by the streamlines in the cylinder’s frame of reference (left and right) and vorticity contour (left). Left: our work; right: experiments from Bouard & Coutanceau (Reference Bouard and Coutanceau1980). Here $\textit{Re} = 9500$ , $t = 3$ , $W =8$ .

3.1.2. Comparison with simulations

There exists a wide array of impulsive start simulations (in an infinite medium) with sufficient details to allow quantitative comparisons. For the sake of brevity, we selected for this comparison the work of Wu et al. (Reference Wu, Carvalho and Kumar2019b ), but stress that this work is a good representation of the state-of-the-art in terms of computational development for impulsive wakes; most notably, it compares favourably with the seminal computational work of Koumoutsakos & Leonard (Reference Koumoutsakos and Leonard1995). The comparison between Wu et al. (Reference Wu, Carvalho and Kumar2019b ) and our simulations shows in figure 5 a near-perfect agreement for both Reynolds numbers, 550 and 3000.

This first set of comparisons demonstrates that our numerical code faithfully reproduces the wake development following an impulsive start of a cylinder, and provides a benchmark for the wake-related effects (including forces) that will be represented henceforth.

Figure 5.

Transient wake after impulsive start, shown by the streamlines in the cylinder’s frame of reference. Left: our work; right: simulations from Wu et al. (Reference Wu, Carvalho and Kumar2019b ) for (a) $\textit{Re} = 550$ and (b) $\textit{Re} = 3000$ , and $W =28$ , $t = 5$ .

3.2. Water exit

In order to validate our free-surface simulations, we now compare our numerical results to the less proficient literature of water exit.

Several studies in the literature feature the computationally lighter free slip case, i.e. with no shear stress at the surface of the cylinder. We present the three most relevant comparisons in the first part of this section.

In contrast, a very limited amount of literature is available with no-slip conditions at the cylinder surface. Among this limited amount, and restricting to forced motion studies (constant velocity or acceleration), no publication provided enough information or detailed enough figures to allow direct comparison. Therefore, we used instead an in-house solver (Cerquaglia et al. Reference Cerquaglia, Deliége, Boman, Terrapon and Ponthot2017; Falla et al. Reference Falla, Bobach, Boman, Ponthot and Terrapon2023) based on the PFEM (Oñate et al. Reference Oñate, Idelsohn, Del Pin and Aubry2004; Cremonesi et al. Reference Cremonesi, Franci, Idelsohn and Oñate2020) that was developed to resolve the full Navier–Stokes equations in the presence of a deformable free surface that can undergo topological changes.

For Comsol simulations, and unless otherwise specified, we used the following values for the bath width and height: W = 28 and H = 48. These values are large enough to effectively mimic an infinite bath for the sake of comparison.

3.2.1. Free slip

For these comparisons, we also use the free-slip boundary condition on the cylinder (2.4), and the largest value of Reynolds number for which we have validated the numerical convergence, $\textit{Re} = 10\,000$ , in order to approach the inviscid limit. Note however that, for our simulations, the viscosity, albeit small, is not zero in the bulk.

Figure 6.

Validation of water exit with the free-slip boundary condition at the cylinder surface. Generalised vortex method simulations by Telste (Reference Telste1987) (a) and our work (b) coloured by vertical velocity; streamlines are drawn in thin grey lines. Results are shown for (a) $\textit{Fr} = 0.2$ and (b) $\textit{Fr} = 2$ . The initial depth is $d = 2$ .

The first comparison is with the reference work of Telste (Reference Telste1987), who studied the water exit of a 2D cylinder at a constant velocity by means of a generalised vortex method for four Froude numbers from 1/100 to 2. We selected the two largest Froude numbers, $\textit{Fr} = 0.2$ and $2$ , that both show large surface deformations. As can be seen in figure 6, our results compare very well with Telste’s computations, only showing minor differences in streamlines and free-surface location close to the points of strongest curvature and for the latest time snapshot available.

Figure 7.

Validation of water exit with the free-slip boundary condition at the cylinder surface. Boundary integral method simulations by Gorlov (Reference Gorlov2000) (a) and our work (b), coloured by vertical velocity, superimposed on the right half of each panel. Results are shown for $\textit{Fr} = 0.2$ (a) and $\textit{Fr} = 2$ (b), for $d = 5$ .

The second comparison shown in figure 7 is with the work of Gorlov (Reference Gorlov2000), slightly posterior to Telste’s computations, based on a boundary integral method at the same conditions (except for $d=5$ ) and pushing the simulations further in time. There again, the comparison with our results holds remarkably well up to the last available snapshots provided in Gorlov’s work, for $\textit{Fr} = 0.2$ and 2.

Figure 8.

Validation of water exit with the free-slip boundary condition at the cylinder surface. Boundary integral method simulations by Greenhow (Reference Greenhow1988) superimposed with our work coloured by the vertical velocity at $y_c=0$ (left) and $y_c=0.5$ (right). Results are shown for $\textit{Fr} = 0.51$ and $d = 2$ .

The last comparison is with the work of Greenhow (Reference Greenhow1988) for the test case most relevant to our problem of interest, namely $\textit{Fr} = 0.51$ and $d = 2$ . Two snapshots in time ( $y_c = 0$ and $y_c = 0.5$ ) are shown in figure 8. Again, the agreement is overall very good. However, Greenhow’s zoomed-in figure provides an opportunity to focus on the small differences occurring at the point of largest curvature, a common feature seen in all three free-slip comparisons previously described. More specifically, the fluid neck directly below the base of the cylinder predicted by our simulations, at later times and large $\textit{Fr}$ , is always a little bit thinner in our results. Two possible reasons can be put forward to explain this difference: the presence of non-zero (albeit small) viscosity in our numerical simulations and the difference in resolution. Note that the surface tension is not taken into account in any of these simulations. The region in which the difference occurs presents large gradients and small scales, which suggests a high spatial resolution needed to resolve it. This second reason is supported by Greenhow’s own account of computational blowup occurring just after the last snapshot he presented. By design, our numerical approach allows a very fine meshing in this area (due to the proximity of the cylinder), which gives us confidence in the higher accuracy.

5.1. Initial surface dynamics: conclusion

In this section we characterised the free-surface dynamics that follows the cylinder motion inception. We found that the dynamics is primarily conditioned by the tank width $W$ , and we identified two regimes: a regime of strong lateral bonding (depth-to-width ratio $\gt$ 0.5) and an ‘unbounded’ regime (depth-to-width ratio $\lt$ 0.5). Then, we identified regime-dependent scaling laws for two macroscopic observables: the initial surge velocity $c^*$ and the maximum surge amplitude $b^*$ . The maximum surge amplitude occurs at a time $\tau$ after the motion inception, which we predicted based on wave theory.

We expect the initial surface dynamics to have a limited impact on the free-surface behaviour if the cylinder’s starting depth is large enough, giving time for the impulsively started surface waves to settle and damp. In contrast, the initial surge is expected to have a sensible effect for a shallow starting position (small $d$ ) or a fast motion (large $\textit{Fr}$ ), for which an excess or deficit of bulge height is expected when the cylinder reaches the free surface. This aspect is analysed further in the next section.

6.1. Reference case: free-slip, unbounded domain

Figure 14.

Characteristic bulge height $h^*=h(y_c=-1)$ for the free-slip case and an unbounded domain and $\textit{Re} = 10^4$ . For large values of $d$ (black dashed line), $h^*$ follows an algebraic trend for $\textit{Fr} \leqslant 0.2$ and a logarithmic trend for $\textit{Fr}\gt 0.2$ . Surface colour indicates the distance to the best logarithmic fit for large $d$ (red line). The ridge betrays transient dynamics; see § 6.2. The thick faded lines mark the approximate boundaries of areas where $h^*$ becomes independent of either $d$ (continuous line) or $\textit{Fr}$ (dashed line). The solid line on the crest corresponds to the relation (6.2). Insets show snapshots colour coded with upwards velocity, with the initial condition d = 36.

Figure 14 shows the dependence of $h^*$ on the Froude number $\textit{Fr}$ (exit velocity) and starting depth $d$ . Here, in order to show the most general results, we present data using the free-slip condition and for an impulsive start in a simulated unbounded domain ( $W=\infty$ ) by using permeable side and bottom boundaries set at hydrostatic pressure (respectively placed at $W=28$ and $H=48$ ). Several trends can be observed.

First, as anticipated, $h^*$ monotonically increases with $\textit{Fr}$ regardless of $d$ over the entire parameter space. For $0.2 \leqslant \textit{Fr} \leqslant 10$ and a large enough $d$ that we specify further below, the value of $h^*$ matches the logarithmic fit $h^*_{\textit{sf,log}} = 0.17 \log (\textit{Fr}) + 0.55$ within $\pm 0.02$ . The ‘sf’ subscript denotes free-slip condition at the cylinder’s surface. The corresponding region is coloured teal in figure 14. For $\textit{Fr} \lt 0.2$ , however, the logarithmic trend is lost and the dependence becomes instead algebraic: a reasonable fit is $h^*_{\textit{sf, po}w} = 0.6 \textit{Fr}^{0.45}$ . Figure 14 also reveals that $h^*$ is a non-monotonic function of $d$ , increasing from $d=1$ (a limit value for which the layer above the cylinder is non-existent), and reaching a plateau for $d=36$ at all $\textit{Fr}$ , indicating that the effect of transient dynamics has faded out.

We first focus on the small- $d$ (shallow start) region. First, the rise of $h^*$ for small $d$ is a strong function of $d$ , but is initially independent of the Froude number. However, segregation by $\textit{Fr}$ soon follows: for $\textit{Fr} \geqslant 10$ , the rise is seemingly monotonic until the large-depth plateau is reached, whereas for $\textit{Fr} \leqslant 1$ , the sharp rise is followed by a local maximum and a subsequent dip. This pattern is stretched to larger $d$ values as $\textit{Fr}$ is increased while becoming less prominent both in relative and absolute intensity. This non-monotonic behaviour reflects the presence of gravity-induced wave dynamics triggered by the impulsive start, which we have explained in more detail in § 5. Taking a step further, we propose a prediction of the locus of the local maxima. We go with the intuitive idea that, for a fixed $\textit{Fr}$ , the maximum value of $h^*$ occurs when the cylinder’s top reaches the free surface at the same time as the surface surge hits its maximum elevation, that is,

(6.1) \begin{equation} d-1 = \tau ^*. \end{equation}

The underlying assumption allowing us to write (6.1) is that the rise of the cylinder does not modify appreciably the surface wave behaviour, even as the cylinder approaches the surface. Combining (6.1) with (5.5) yields a quadratic equation; solving it produces two roots, of which only one makes physical sense ( $d\gt 1$ ):

(6.2) \begin{equation} d = m + \sqrt { m^2 - 1} ,\quad m = \frac {\pi }{4} \textit{Fr} + 1. \end{equation}

This relationship, plotted in figure 14 as a solid line, is very close to the observed crest, suggesting that this approach provides a good estimate of the starting depth that would lead to the maximum $h^*$ .

After reaching a local maximum and a local minimum, the value of $h^*$ eventually reaches a plateau as $d$ increases. It is useful to define the boundary separating the region of $d$ impacted by transient effects from the ‘deep-enough’ region where initial dynamics no longer matter. We follow the intuitive idea that the ridge-and-trough pattern (and its damping in time or, equivalently, in starting depth) scales with $\tau ^*$ , and we propose that the boundary lays at a value of $d$ that is simply twice (6.2), that is, $d=2(m+\sqrt {m^2-1})$ . This boundary is represented in figure 14 with a thick faded line and seems to satisfactorily separate the two regions. Beyond this limit, we expect that surface transient effects no longer impact $h^*$ . It is worth noting, however, that the threshold value of $d$ increases steadily with $\textit{Fr}$ ; the gain is particularly pronounced for larger-than-unity $\textit{Fr}$ values for which $d \simeq ({\pi }/{2}) \textit{Fr}$ according to (6.2): as a result, the ‘deep-enough’ limit drifts to unexpectedly large values of $d$ for high-speed exits.

In conclusion, we have characterised the bulge height $h^*$ above the cylinder in a canonical setting (unbounded domain and free-slip boundary condition), and we proposed predictions of $h^*(\textit{Fr})$ in both low- $\textit{Fr}$ (algebraic) and moderate-to-high- $\textit{Fr}$ (logarithmic) regimes. We also rationalised the existence of a local maximum when considering increasing starting depths based on transient surface dynamics, and proposed predictions of the starting depth $d$ leading to this maximum, as well as the minimum value of $d$ needed to reach the ‘deep-enough’ behaviour for which transient wave dynamics no longer matters. Because these results have been obtained in a simplified idealised setting, we now delve into the influence of several important parameters on $h^*$ : lateral bounding (§ 6.2) and wake strength (§ 6.3)

6.2. Influence of lateral bounding

Figure 15.

Effect of lateral bounding on the bulge height $h^*$ using the free-slip boundary condition. (a) Raw bulge height, (b) bulge height corrected for the effect of lateral bounding using $\delta _0$ from relation (6.4), (c) schematic, and (d) computed free surface for $y_c=-1$ , renormalised by $h^*$ in the $x$ direction. In a finite-size tank, volume conservation implies that the bulge (magenta area) is accompanied by a level drop $\delta _W$ , which in turn causes a reduction of $h^*$ by an amount $\delta _0 \approx 0.4 \delta _W$ . The bulge width is $w^* \approx 3 h^*$ for all $\textit{Fr}$ .

In § 6.1 results were obtained in an unbounded domain. However, the bulge height is affected by the lateral bounding, as figure 15(a) shows by reporting $h^*$ as a function of the starting depth $d$ for several Froude numbers and decreasing tank widths $W$ , from $W=\infty$ (unbounded) to $W=7$ (strong bounding). The results show that the bounding almost always plays a moderating role: a decrease of $W$ leads to a reduction of $h^*$ for almost all values of $\textit{Fr}$ and $d$ , even though marginal gains are observed in some narrow range of parameters due to transient effects. The intensity of the global reduction is shown to increase sharply with $\textit{Fr}$ : its effect is almost imperceptible for $\textit{Fr} = 0.1$ , noticeable for $\textit{Fr} = 1$ and strong for $\textit{Fr} = 10$ with up to a $15 \,\%$ drop at $W=7$ compared with $W=\infty$ .

We rationalise the drop in bulge height in the presence of lateral bounding and predict its magnitude $\delta _0$ using an argument based on volume conservation. We follow a two-step process: we first estimate the level drop $\delta _W$ in the bath, then we identify and use the relationship between $\delta _W$ and the decrease in bulge height $\delta _0$ . The creation of the bulge implies the elevation of a fluid volume directly above the cylinder. Due to volume conservation, this elevated volume is balanced by an average drop in level away from the bulge. The volume of fluid pushed above the initial fluid surface, assuming a roughly triangular shape, is approximated by $({1}/{2}) w^* h^*$ , where $w^*$ is the width of the bulge taken at $y=0$ . Making the additional assumption that the level change $\delta _W$ is uniform everywhere away from the bulge, the volume conservation reads

(6.3) \begin{equation} \frac {1}{2} w^* h^* = \delta _W (W-w^*). \end{equation}

This relationship involves the unknown bulge width $w^*$ . Free-surface profiles for various Froude numbers and various $W$ , shown in figure 15(d), suggest that there exists a proportionality between $w^*$ and $h^*$ , namely $w^* \approx 3 h^*$ . To close the reasoning, we need a relationship between the level drop $\delta _0$ and the level drop in the bulk $\delta _W$ (see schematic in figure 15 c). Measurements suggest that they are simply proportional, i.e. $\delta _0 \approx 0.4 \delta _W$ , and we can now rewrite (6.3) in its final form:

(6.4) \begin{equation} \delta _0 \approx 0.6 h^{*2} / (W-3h^*). \end{equation}

To check the validity of this prediction, we plot in figure 15(b) the quantity $h^* + \delta _0 = h^* [ 1 + 0.6 h^{*2} / (W-3h^*) ]$ as a function of $d$ . We observe a satisfying collapse of the data for the various lateral widths $W$ at each $\textit{Fr}$ , suggesting (6.4) provides a reliable way to predict the effect of the tank width, and hence, the lateral bounding, on bulge height.

The quadratic rise of $\delta _0$ with $h^*$ implies that the effect of $W$ can quickly become important as $\textit{Fr}$ is raised. Consequently, as the exit velocity (and $h^*$ ) becomes larger, the minimum tank width needed for the system to behave as ‘infinitely wide’ increases quickly as well. At $\textit{Fr} = 10$ , a tank at least $13$ cylinder radii wide is necessary for the effect of lateral bounding to drop below $5\,\%$ , while $30$ radii are needed to ensure a deviation smaller than $2\,\%$ compared with the ‘infinitely wide’ reference value.

6.3. Influence of wake

Figure 16.

Wake-induced surge excess. (a) Characteristic bulge height $h^*$ as a function of the Froude number for various starting depths, with and without wake. (b) Wake-induced surge normalised by $\sqrt {j^*}$ , where $j^*$ is the characteristic jet area. A deeper starting depth induces a larger wake at the time of crossing, which in turn promotes additional swelling $\delta _{{wake}}$ of the water layer above the cylinder. Here $W=28$ , $H=48$ , $\textit{Re}=10\,000$ .

Until now, all simulations have been performed using the free-slip boundary condition. However, in reality, the friction at the cylinder surface creates a wake (absent in the previous results), which alters the surge dynamics. The effect of the wake is thus now investigated by considering simulations with no-slip boundary conditions at the cylinder surface. Figure 16 shows the bulge height $h^*$ as a function of the Froude number for varying starting depths $d$ . When $d$ increases, we observe for all Froude numbers an increasing increment $\delta _{w\textit{ake}}$ of the bulge height with respect to the free-slip situation (red-edged symbols and red line), suggesting that the presence of the wake increases the bulge height.

Larger starting depths correlate with a larger wake size: because the cylinder starts from a lower position, the wake is given additional time to develop. Figure 17(a) shows the time evolution of the ‘jet area’ $j$ , a representative measurement of wake size defined as the area where the vertical velocity $v$ equals or exceeds that of the cylinder ( $v/U \geqslant 1$ ).

Figure 17.

Wake size and jet area (no-slip). (a) Time evolution of the jet area $j$ for various $\textit{Fr}$ (colour bar), $d$ and two values of $\textit{Re}$ , and (b) characteristic value of the jet area at the crossing, $j^*=j(y_c=-1)$ , as a function of $\textit{Fr}$ for various $d$ and $\textit{Re}$ . The jet area, proportional to the wake size, undergoes a quasi-algebraic $\textit{Fr}$ -independent (but $\textit{Re}$ -dependent) growth until the cylinder crosses the interface, followed by a quick $\textit{Fr}$ -dependent decay. Both the growth and the characteristic wake size exhibit a slight dependence on $\textit{Re}$ .

Our measurements show that during the cylinder’s submerged phase, the jet area monotonically increases with time, just as we would expect for a submerged cylinder. It is, remarkably, independent of $\textit{Fr}$ . We characterised the growth by fitting a $\textit{Re}$ -dependent algebraic relationship of the form $j \propto (t-\delta (\textit{Re}))^\alpha$ , where $\delta (\textit{Re})$ is a small $\textit{Re}$ -related delay. This delay reflects the fact that the wake does not immediately start growing as the cylinder starts to move. Indeed, during a short-time period, there exist a potential-like flow pattern around the cylinder; then the boundary layers on the bottom part of the cylinder gradually grow and give way to the main counter-rotating vortices Koumoutsakos & Leonard (Reference Koumoutsakos and Leonard1995).

The initial growth is substantially quicker for $\textit{Re}=500$ (exponent $\approx 0.77$ ) than for $\textit{Re}=2500$ (exponent $\approx 0.68$ ). Data for $\textit{Re}=10\,000$ , not shown in figure 17(a), closely follows the trend for $\textit{Re}=2500$ ; however, it appears significantly noisier due to the larger influence of small vortical structures in the wake at that Reynolds number.

As a result, the characteristic jet area $j^*$ (the jet area at $y_c = -1$ ) does not depend on $\textit{Fr}$ either, but instead bears a strong dependence on $d$ and almost no dependence on $\textit{Re}$ as figure 17(b) shows.

Following the growth, shortly after the cylinder reaches the interface ( $y_c=-1$ ), the wake growth turns into a rapid $\textit{Fr}$ -dependent decay, fuelled by the rapid draining of the water around the cylinder.

Taken together, these results suggest that the free surface does not influence the wake until after the cylinder reaches the interface. On the other hand, the influence of the wake on the free surface is significant: the wake bends the streamlines in the vicinity of the cylinder, both below and above, and interferes with the bulge formation. We pushed the correlation further and identified a relationship between the characteristic jet size $j^*$ and the surge increment $\delta _{w\textit{ake}}$ . Normalising the latter by $\sqrt {j^*}$ , we obtain an excellent collapse of the data for Froude numbers $\textit{Fr} \lesssim d/3$ , as illustrated in figure 16(b). The master curve is a logarithmic function of $\textit{Fr}$ , suggesting that the free surface is virtually insensitive to the wake size for a small exit velocity, whereas it is very sensitive to it for large exit velocities (figure 16 a). To put this result in perspective, for $\textit{Fr}=10$ and $d=36$ , up to $30\,\%$ of the bulge height is due to the cylinder’s wake (figure 16 a). For $\textit{Fr} \gtrsim d/3$ however, the data deviates significantly from the master curve and seems to reach a plateau at large $\textit{Fr}$ ( $\textit{Fr}$ independent), whose level increases with $d$ .

In the numerical results presented above, the wake, symmetric by construction, stays attached to the cylinder regardless of the distance travelled (up to $36a$ ). We find it important to point out that in typical experimental conditions, a stable symmetric wake is only observed for up to approximately 10–15 radii travelled. Beyond this limit, one vortex usually grows at the expense of the other (Bouard & Coutanceau Reference Bouard and Coutanceau1980; Lin et al. Reference Lin, Hsieh, Lin and Raikar2012), followed by its shedding, and the eventual transition to an alternating vortex street (von Karman vortices). Nonetheless, we believe that our results beyond this typical $15R$ limit are still valuable, for two reasons: (i) for large initial depths, the effect of the wake size is stronger, which allows us to verify correlations over a larger range of $d$ (see figure 16 b); and (ii) because this symmetrically growing wake is representative of what one would observe in the absence of finite perturbations. One could imagine that very careful experiments could maintain that symmetry state up to large values of $d$ , especially at moderate Reynolds numbers where viscous effects damp symmetry-breaking perturbations.

6.4. Film spreading and entrained volume (slip condition, no wake)

Figure 18.

(a) Raw and (b) renormalised angular extent $\alpha$ of the liquid film, from the inertial bulging phase to the drainage phase. A transition occurs at $h/\textit{Fr}^{0.25}\sim 1$ , after which (i.e. for lower $h$ ) the evolution converges to a self-similar spreading phase.

We now shift the focus to the later stage of the exit and aim to characterise with global characteristics the dynamics of the quasi-uniform liquid film spreading and thinning on the cylinder. In particular, we look for a renormalisation able to provide a global picture of the coating process. We define the liquid film as the region comprised between the cylinder apex and the point of curvature change at the base.

In figure 18 we first look at the angular extent of the film $\alpha$ for Froude numbers comprised between 0.02 and 2. Panel (a) shows the raw data as a function of the film thickness at the apex $h$ . We use the film thickness $h$ instead of $y_c$ as a measurement of drainage progression; as a result, time goes from right to left. Far ahead of the crossing, when $h \sim 2{-}3$ , all $\alpha$ are varying little in time and seemingly converge to a single value around $0.5$ rad. A constant angle is a synonym of a bulge whose lateral dimensions are proportional to the object’s depth, consistent with the literature.

As time progresses, the general trend is an accelerating spreading of the liquid film; that acceleration is progressive and earlier for large Froude numbers (top curves) and much sharper for small Froude numbers (bottom curves), consistent with the rapid free-surface modifications observed when the cylinder approaches the free surface at small $\textit{Fr}$ . Renormalising both the film thickness and the angular extent by $\textit{Fr}^{1/4}$ reveals a self-similar spreading behaviour in the later phase (figure 19 b). Moreover, we observe for all $\textit{Fr}$ a sharp transition from a slow progressive spreading (earlier times) to a quickly accelerating spreading (later times), happening around $h/\textit{Fr}^{0.25} \simeq 1$ for all $\textit{Fr}$ . We interpret this behavioural shift as a transition between a bulging regime and a drainage/spreading regime.

Figure 19.

(a) Raw and (b) renormalised non-dimensional volume $\varOmega$ of the film during the drainage phase. The renormalisation allows a prediction of the film volume eventually deposited on the cylinder for all Froude numbers: $\varOmega ^* \approx 0.22 \textit{Fr}^{1/2}$ .

To gain further insight into the changes occurring in the liquid film, figure 19 shows the evolution of the volume $\varOmega$ contained in the liquid film, measured as the integration of $h(\alpha )$ , from $\alpha = 0$ to $\alpha _C$ , where $\alpha _C$ is the angular position of the curvature change.

Raw results of $\varOmega$ vs $h$ are shown in (figure 19 a) for Froude numbers comprised between 0.02 and 2. We observe that, for a given $h$ , increasing the Froude number leads to a moderate increase of $\varOmega$ , suggesting a larger Froude is associated with a thicker film.

Renormalising both the film thickness and the film volume by $\textit{Fr}^{1/4}$ and $\textit{Fr}^{1/2}$ , respectively, we uncover a self-similar drainage behaviour (figure 19 b). Extrapolating the master curve to $h = 0$ leads to a single value of $\varOmega / \textit{Fr}^{1/2} \approx 0.22$ , which can be interpreted as an approximation of the volume coated on the cylinder; strictly speaking, the latter would be $\varOmega (\alpha = \pi )$ , which would correspond to a slightly different $h$ for each $\textit{Fr}$ .

As a final note, let us note that these results were obtained with the free-slip condition on the cylinder. We made no attempt to extend these results to no-slip cases, and we expect the results to be noticeably different due to the importance of viscous effects in thin layers.

6.5. Free-surface shape during crossing: conclusion

In this section we characterised the bulge height and interface shape during interface crossing. First, we laid out a map of the characteristic bulge height $h^*$ as a function of the Froude number $\textit{Fr}$ and initial depth $d$ , which we rationalised by making use of findings from § 5. This map allows us to extract scaling laws and discriminate a window of $\textit{Fr}$ , $d$ where initial dynamics still affect the free-surface dynamics when the cylinder reaches it. We also quantified and modelled the influence of the bounding $W$ , which induces a measurable liquid depletion above the cylinder. Most importantly, we identified the leading role of wake size in setting the characteristic film thickness above the cylinder, characterised its growth (rise and fall) and assessed its effect. Finally, we analysed the spreading of the liquid film during interface crossing, and we were able to propose a prediction of the volume of liquid coating the cylinder after crossing.

7.1. Free slip

Figure 21.

(a) Evolution of the force acting on the cylinder at $\textit{Fr} = 0.1$ , 0.2, 1, 5 and 10, and (b,c) typical pressure fields for $\textit{Fr} = 0.1$ and 10 during interface crossing under slip conditions, all other parameters being identical to figure 20. The flat-interface (loss of) buoyancy (7.4) is represented as solid thin lines for reference for each $\textit{Fr}$ . Results from `no interface’ simulations (simulations where the cylinder stays submerged and far from the free surface) are also represented with a thick line for reference. The inset in (a) presents renormalised data using the buoyant force as scale instead of the dynamic force. Squares indicate occurrences of free-surface folding and halting computations. Arrows denote the intensity of the net downward (blue) or upward (green) force felt by the cylinder.

In this section, tangential viscous stress and, thus, skin friction drag are zero as a result of the free-slip condition on the cylinder.

Consequently, we choose to follow a completely inviscid approach and simply define the vertical force $F$ as the integral of the dynamic pressure, $p_{\textit{dyn}} = p + \rho y / \textit{Fr}$ . Making use of the left–right flow symmetry of the free-slip condition and following the convention that a resistive force (i.e. a force acting vertically downwards) is positive, the force per unit length in the transverse direction reads (Telste Reference Telste1987)

(7.3) \begin{equation} F = 2 \int _{0}^{\pi } p(\theta ) \cos {(\theta )}\, {\rm d}\theta + \frac {\pi }{\textit{Fr}}. \end{equation}

Using this definition, a submerged cylinder at rest experiences no force ( $F=0$ ), while a dry cylinder outside the bath is subjected to a downward force equal to the buoyancy of the immersed cylinder, that is, $F = \pi / \textit{Fr}$ in our non-dimensional framework. Now, assuming that the cylinder is pulled out of the bath in a quasi-steady fashion and that the interface remains flat and pinned at $y_c = 0$ , the resistive force felt by the cylinder during interface crossing can be estimated as the complement of the buoyancy on the wet area:

(7.4) \begin{align} F_b (y_c) = \frac {\pi }{\textit{Fr}}\! \left [ 1 - \frac {1}{\pi }\! \left ( \arccos (y_c)-y_c \sqrt {2(1-y_c)-(1-y_c)^2} \right ) \right ]\!, \quad -1 \leqslant y_c \leqslant 1. \end{align}

This baseline is represented by thin solid lines in figure 21(a). It corresponds to the force evolution we expect for an infinitely slow cylinder crossing the undisturbed interface (no liquid entrained by the cylinder, neither below or above).

Figure 21(a) presents the computed force evolution for five representative cases of cylinder exit at finite speed, from $\textit{Fr} = 0.1$ to $10$ . As expected, the force $F$ is vanishingly small when the cylinder is deep enough. Moreover, when the cylinder is outside the bath, well past the interface crossing, the force eventually settles to the expected value of $\pi / \textit{Fr}$ for a dry cylinder (see inset), at least for computations that were able to run that far (interface folding halts the runs at smaller Froude numbers; see figure 21 b). Also note that in our simulations the cylinder is always covered by liquid (at the least by a very thin layer), in contrast to the underlying assumption behind (7.4).

Between these two extreme points in time, figure 21(a) immediately makes clear that both small and large Froude cases show substantial differences to the hydrostatic reference (7.4): small Froude values are associated with a relative excess of resistive force at a given $y_c$ , while large Froude values show instead a deficit during crossing. This is further exemplified by renormalising the force by $\pi /\textit{Fr}$ , which sets the buoyancy as the reference force (inset of figure 21 a). A second general remark is that, while the terminal value of $\pi /\textit{Fr}$ is met starting at $\textit{Fr} \simeq 1$ , larger-than-unity $\textit{Fr}$ cases reach this terminal value with a delay that appears to increase with $\textit{Fr}$ . We now take a closer look and lay out the physical mechanisms behind these variations.

For small Froude numbers ( $\textit{Fr} \leqslant 1$ ), the force is nearly always larger than the hydrostatic reference (7.4). We rationalise this overall force excess by invoking two distinct contributions. Firstly, in the earlier stages of the exit, the free surface resists deformation due to gravity (dominant for that $\textit{Fr}$ range) and pressure builds up in the region between the cylinder and the surface (figure 21 b), inducing an overall increase of resistance prior to the cylinder reaching the interface. This is analogous to the case of a cylinder approaching a slip wall. This effect was previously identified by Telste (Reference Telste1987); see figure 9 and accompanying text in that reference. Secondly, during the late stages of the exit, inertial effects draw the free surface downwards on both sides of the cylinder, which only leaves a thin, dynamically inert film on the cylinder (last panel of figure 21 b). The loss of buoyancy thus occurs comparatively earlier (at a smaller $y_c$ ) than for a quasi-static exit, and contributes to a comparatively larger force in these late stages. Past the simulation breakdown point, we expect the force level to reconnect with the ‘dry’ reference with no or little overshoot, just as we see for the $\textit{Fr} = 1$ case.

For larger Froude numbers, the force evolution also deviates from the hydrostatic reference (7.4). Perhaps counter-intuitively, the resistance decreases as $\textit{Fr}$ or velocity increases. We attribute this striking change of behaviour to a favourable net top-to-bottom pressure difference setting around the cylinder at a high Froude number. This pressure gradient is driven in part by a relative pressure drop ahead of the cylinder that we impute to a thinning layer: as the amount of fluid pushed forward by the cylinder decreases, so does the pressure needed to sustain the fluid’s outward motion. This gradient is also fuelled by the pressure increase below the cylinder, as highlighted in figure 21(c). This boosting effect becomes very strong for large Froude numbers: around $\textit{Fr} = 5$ , it cancels out the buoyancy loss during the crossing phase (zero average force from $y_c = -1$ to $y_c = 1$ ), and at $\textit{Fr} = 10$ the cylinder now experiences an assisting force of the same order as the buoyancy loss!

To our knowledge, this favourable effect has never been identified. We hypothesise that this is due to the limited range of Froude numbers considered in previous controlled studies of water exit or to a lack of appropriate renormalisation of the force data. Indeed, renormalising force data based on the inertial force (such as in the main plot in figure 21 a), makes the high- $\textit{Fr}$ force changes look deceptively small compared with the low- $\textit{Fr}$ ones. Additionally, the effect can be partially obscured by wake-induced drag, as we show in § 7.2.

7.2. No-slip

Figure 22.

(a) Evolution of the force acting on the cylinder at $\textit{Fr} = 0.08$ , 0.33, 1.3, 5.22 and 10, and (b,c) typical pressure fields for $\textit{Fr} = 0.08$ and 10 during interface crossing under a no-slip condition at $\textit{Re} = 500$ , all other conditions being identical to figure 20. See figure 21 for full caption.

We now characterise the force during interface crossing for no-slip simulations. The major difference in this case is the development of a prominent wake downstream of the cylinder, as emphasised by the side-to-side comparison of figure 20. The presence of this strong wake significantly affects the pressure acting on the cylinder’s surface, and thus, induces important changes for the vertical force. Most notably, the pair of large symmetric vortices directly in the cylinder’s wake creates a low-pressure area below the cylinder (shown in figure 22 b,c). As a result, the net resistive force quickly rises, reaching a first maximum (around $y_c=-4$ for all $\textit{Fr}$ values) before plummeting to smaller values (figure 22 a), in stark contrast to the flat start of the ‘free-slip’ behaviour. A second contribution to drag is the skin friction arising from the no-slip boundary condition at the cylinder’s surface; this contribution is smaller, but nonetheless present over the range of Reynolds numbers tested in this work ( $\textit{Re} = 500 {-} 10\,000$ ).

Closer to the interface crossing, several phenomena take place simultaneously. First, in a trend similar to the free-slip simulations (see § 7.1), the force evolution for small Froude numbers ( $\textit{Fr} = 0.08$ and $0.33$ ) loosely follows the loss of buoyancy occurring around interface crossing, with a sharp rise of resistive force starting shortly before $y_c=-1$ . Conversely, for larger Froude numbers, the loss of buoyancy is barely noticeable in the force evolution: indeed, for the $\textit{Fr} = 5.22$ and $10$ cases, the cylinder experiences a large and steady decrease of force before and across the $y_c = 0$ region (inset). At first glance, this decrease is not necessarily the result of interface crossing. For a submerged cylinder, as the wake grows, the low-pressure area is slowly pushed away from the cylinder’s surface, weakening its contribution to drag. Yet, this explanation is insufficient. In figure 22(a) we plotted in faded lines the drag corresponding to simulations for which the free surface is far enough to not play any role in the force evolution, simulating a submerged cylinder. Looking at the inset of figure 23(a) whose renormalisation by $\pi /\textit{Fr}$ better separates various $\textit{Fr}$ , we notice that indeed the natural evolution of the wake is only responsible for a moderate force drop, compared with the steep total drop in the presence of an interface. The mechanism at play is suggested by wake size measurements of figure 17(c): as the cylinder approaches the free surface, the wake growth stalls and sharply reverses, and the low-pressure areas are pushed further away (not shown): as a consequence, the drag sharply decreases. Lastly, the inertial and free-slip mechanism described in § 7.1 is still expected to play a role, and one could wonder about its relative contribution to the total force drop.

Figure 23.

Net downward force excess due to interface crossing under the no-slip condition for $\textit{Fr} = 0.08$ , 0.33, 1.3, 5.22 and 10, and compared with the corresponding free-slip case for $\textit{Fr} = 5.22$ and $10$ (inset). All other parameters are identical to figure 22. The force reference is that of a submerged cylinder moving at the same speed. Wake weakening shortly before and during the interface crossing is responsible for an important drop of force during interface crossing.

In order to gauge more precisely the net effect of the interface crossing, and compare it to the free-slip case, we represent in figure 23 the next force excess $\Delta F$ between a submerged case and an interface-crossing case, both in no-slip conditions. As anticipated, the interface only starts to have an impact a few radii before the cylinder reaches it, in a fashion similar to the free-slip case for small Froude numbers (see figure 21): a sharp rise loosely matching the theoretical buoyancy loss variation. Large-Froude-numbers cases also share the similarity that the force is decreasing significantly during crossing; however, the comparison stops there.

The inset figure presents the renormalised force excess $\Delta F / (\pi / \textit{Fr} )$ , as well as the force evolution for the free-slip cases from figure 21 in faded lines (only shown for $\textit{Fr} = 5.22$ and $10$ for clarity). We emphasise that this comparison makes more physical sense for $y_c\lt 1$ , when the free-surface location is comparable between a free-slip (vanishingly small wake) and a no-slip situation. Regardless, the results show that the contribution of wake-related effects is large, overpowering that of the free-slip effect identified in § 7.1.

Finally, to support the generality of these observations, we present in figure 24 the force evolution for various $\textit{Re}$ from $500$ to $10\,000$ and for two values of $\textit{Fr}$ , $1.3$ and $10$ . We also choose a deeper start for these simulations ( $d = 14$ vs $d = 7$ ) in order to capture more details during the immersed phase.

Figure 24.

Evolution of the force acting on the cylinder for various $\textit{Re}$ , two representative Froude numbers $\textit{Fr} = 1.3, 10$ and $d = 14$ , all other parameters are identical to figure 22. Data for different $\textit{Re}$ at a given $\textit{Fr}$ show similar variations, but large oscillations are present for $\textit{Re} = 10\,000$ due to wake instabilities. Inset shows the dynamic pressure fields near the cylinder for one representative pair of successive maximum and minimum downward forces; note the growing distance from the main low-pressure system to the cylinder surface (green arrows). Like for shallower start cases, the interface starts to play a role around $y_c = -5$ , after which segregation by $\textit{Fr}$ occurs.

Reynolds number $500$ and $2500$ cases have very similar time variations, the $\textit{Re} = 500$ case being smoother (due to viscous damping) and with substantially larger values (due to larger skin friction). Reynolds number $10\,000$ cases, however, show large, virtually undamped oscillations, evidence of a chaotic wake. Yet, seeing through these oscillations, one can read an underlying evolution close to the $\textit{Re}=2500$ case, characterised by a local maximum a few radii after start-up followed by a steady drop that accelerates around $y_c = 0$ . Segregation by $\textit{Fr}$ occurs around 3–5 radii before interface crossing for all cases.

Taken together, these observations suggest that the mechanisms leading to force change, identified and rationalised for $\textit{Re} = 500$ and $d=7$ , are relevant throughout the entire range of parameters explored in our work.

7.3. Force evolution through interface crossing: conclusion and applications

In this section we analysed the resistive force felt by the cylinder as it rises, approaches and crosses the interface, starting with the free-slip cases (§ 7.1) and then the no-slip cases (§ 7.2). Low and intermediate Froude cases generally follow loss-of-buoyancy pattern, with a sharp rise followed by a plateau. For large Froude numbers, we observe a radically different picture: a pronounced decrease of resistive force during interface crossing, present in both free-slip and no-slip conditions, but stronger in no-slip conditions due to the wake being stripped away during crossing. This conclusion is applicable to the entire parameter range.

The striking force boost during interface crossing is foreseen to be fairly general. Specifically, it is expected to hold (at various degrees) for any bluff or streamlined object and should play an important role in situations where the wake (pressure) drag is relatively small. Such a situation can arise, for instance, when the wake is only given little time to develop (e.g. for objects starting close enough to the interface) or when the object is travelling at Reynolds numbers past the drag crisis threshold ( $\textit{Re} \approx 3\times 10^5$ for a sphere, Van Dyke Reference Van Dyke2008). This positive contribution is expected to play a role in various fast exit situations, for instance, the pop-up of buoyant spheres (Bourrier, Guyon & Jorre Reference Bourrier, Guyon and Jorre1984; Truscott et al. Reference Truscott, Epps and Munns2016). In this situation, we expect interface crossing at large Froude numbers to show unusually large pop-up heights, which is precisely what the authors observe (Truscott et al. Reference Truscott, Epps and Munns2016). We hypothesise that the effect contributes to this anomaly.