# Initial stage of plate lifting from a water surface

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## Abstract

This study deals with the flow induced by a rigid flat plate of finite length, initially touching a horizontal water surface, when it starts to move upwards with constant acceleration. In the present model, negative hydrodynamic pressures on the lower (wetted) surface of the plate are allowed, and thus, the water follows the plate due to the resulting suction force. The acceleration of the plate and the plate length are such that gravity, surface tension and viscous effects can be neglected during the early stages of the motion. Under these assumptions, the initial two-dimensional, potential flow caused by the plate lifting is obtained by using the small-time expansion of the velocity potential. This small-time solution is not valid close to the plate edges, as it predicts there singular flow velocities and unbounded displacements of the water-free surface. It is shown that close to the plate edges the flow is nonlinear and self-similar to leading order. This nonlinear flow is computed by the boundary-element method combined with a time-marching scheme. The numerical time-dependent solution approaches the self-similar local solution with time.

### Keywords

Free-surface flows Matched asymptotics Numerical methods Water exit### Mathematics Subject Classification

76B10## 1 Introduction

The unsteady two-dimensional flow caused by the impulsive vertical motion of a floating rigid plate is considered. Initially, the liquid occupies the lower half-plane and is at rest. The draft of the floating plate is much smaller that the plate length, 2*L*, and is neglected in the present analysis. The body starts to move suddenly upwards from the liquid at a constant acceleration *a*, which is assumed to be much greater than the gravitational acceleration *g*. The liquid is assumed to be ideal and incompressible with negligible surface tension. Consequently, the generated flow can be modelled as symmetric and potential. We shall determine the flow during the early stages of the motion.

The present study is motivated by the experiments performed by Reis et al. [1] with a glass disc representing the tip of a cat’s tongue. The disc was initially placed on the still water surface, with a negligible draft, and then pulled vertically upwards. The motion of the disc reproduced the motion of the cat’s tongue recorded with a high-speed camera, which followed the time-dependent acceleration. In [1], the authors claimed that “the fluid dynamics of lapping is governed by inertia and gravity, whereas viscous and capillary forces are negligible”.

A linearized model of water exit was developed in [2] by means of the results and ideas from [1]. In this model, the hydrodynamic equations and boundary conditions were linearized, which is acceptable during the initial stage of the flow, and the unknown size of the wetted surface was suggested to be determined by using the condition that the speed of the wetted area shrinking is proportional to the local velocity of the flow at its periphery. Comparing the analytical results provided by the model from [2] with the numerical results by Piro and Maki [3], it was found that the coefficient of proportionality is equal to two. This value of the coefficient was used in [4, 5] to calculate the hydrodynamic forces acting on two-dimensional and axisymmetric bodies exiting from a water surface and to compare the theoretical results with the numerical ones computed by fully nonlinear numerical simulations. Note that this value of the coefficient has not been justified. More research is needed to explain this value and conditions of exit leading to this value. The theoretical model from [2] does not account for the shape of the body, gravity, surface tension or liquid viscosity. Nevertheless, its predictions compare well with CFD results, using the same value of the coefficient of proportionality for all the cases.

The theoretical model requires the solution of a nonlinear and singular integral equation with respect to the time-dependent size of the wetted area. It was discovered in [2] that both the solution and the convergence of the algorithm used for its computation are strongly dependent on an initial condition that must be obtained by examining the initial stages of the flow. Specifically, the integral equation (2.25) from [2] could not be solved numerically on its own without the asymptotic formula (3.11). The initial motion of the lifting body is also important for lapping, which is possible only for specific lifting-velocity laws. If the lifting is too slow, gravity could preclude the liquid to be lifted up to a required height. In contrast, if the body motion is too fast, cavitation may occur at the wetted surface, and the body may separate from the liquid at early stages. A successful law of a disc motion, when the liquid is lifted by the disc up to a significant height, can be found in [1].

As a consequence of its relevance to the naval and aeronautical industry, as well as to understand some fundamental questions in natural sciences (such as the cat lapping problem), the water-exit problem has been explored extensively in the past. Greenhow [6] and Greenhow and Moyo [7] studied the exit of initially submerged bodies by numerical methods. Oliver [8] was concerned with the stability of the advancing liquid–solid contact line. Baarholm [9], Scolan et al. [10] and Faltinsen et al. [11] considered the problem of wave impact from below on a platform placed just above the water surface. Piro and Maki [3, 12] analysed the combined entry–exit problem, using computational fluid mechanics with application to high-speed boats. Indeed, some of the analyses performed for the entry problem can be applied to the exit counterpart [13]. Finally, Tassin et al. [14] were concerned with the hydrodynamic loads in the rear part of the fuselage during aircraft ditching on water. See also the references in [3, 12, 14] about the relation of the exit problem to the naval and aeronautical fields.

The configuration of the problem is such that the plate motion upwards does not generate a flow if there is no adhesion force between the fluid and the surface of the plate. This configuration is different from those studied in [15, 16], where an initially submerged circular cylinder exits from the liquid piercing its free surface. We consider a plate motion with large acceleration such that the gravity acceleration can be neglected. For slow motions of the plate, gravity plays an important role and the present model is not applicable. The results of the present paper show that the hydrodynamic pressure is below the atmospheric pressure at any point of the contact region. This explains the shrinking of the contact region in time. This phenomenon is known as ventilation. However, separation of the liquid from the lower surface of the plate does not occur instantly. The speed of the contact region shrinking depends on the physical characteristics of the liquid, the plate surface and the plate motion. These physical properties are important close to the plate surface but can be neglected far from the surface, as was argued in [1].

The analysis of the present paper is restricted to the initial stage of the flow, when the plate displacement is small compared with the plate length. Asymptotic methods are used to obtain a uniformly valid initial asymptotic solution. The approach is similar to that developed by Iafrati and Korobkin [17, 18] for the problem of initial flow caused by sudden motion of a floating body. In this approach, the flow region is divided into the main flow region, where the solution can be obtained by the small-time expansion method, and the immediate vicinities of the plate edges, where the inner leading-order solution is obtained numerically. This inner solution is then matched to the small-time solution in the main region. The matching procedure provides the far-field condition for the inner solution.

The problem is formulated in Sect. 2 with respect to the velocity potential and the shape of the liquid free surface in non-dimensional variables. Conditions on the parameters of the problem, which allow us to neglect viscosity of the liquid, surface tension on the liquid free surface and the gravity are derived at the end of Sect. 2. The second-order small-time solution of the problem is obtained in Sect. 3 together with the corresponding eigen solution. The equations and boundary conditions governing the flow close to the plate edge during the initial stage are derived in Sect. 4. Theoretical analysis of the inner solution is provided. The numerical solution of the inner-boundary problem is presented in Sect. 5. Finally, the conclusions are drawn and future work is discussed in Sect. 6.

## 2 Formulation of the problem

The two-dimensional unsteady problem of a plate lifting from the water surface is formulated in Cartesian coordinates and in non-dimensional variables. Initially, a plate of length 2*L* touches the still and horizontal water surface. Then the plate starts to move suddenly upwards at a constant acceleration *a*. The half-width of the plate *L* is taken as the length scale, and the time scale *T* is such that the displacement of the plate, which is of order \(aT^2\), is much smaller than the length scale of the problem, \(aT^2\ll L\). The velocity scale is *aT*, and the pressure scale is \(\rho a L\), where \(\rho \) is the liquid density. The ratio \(\varepsilon =aT^2/L\) plays the role of a small parameter of the problem. Gravity and surface tension are not included in the present model.

*xy*. The position of the plate at time

*t*is given by \(y=\frac{1}{2} \varepsilon t^2\), \(|x|<1\). Assuming that the wetted part of the plate is shrinking in time, we introduce a function

*c*(

*t*) such that the interval \(|x|<c(t, \varepsilon )\), \(c(0,\varepsilon )=1\), corresponds to the wetted part of the plate. The flow is assumed two-dimensional, symmetric with respect to the vertical line \(x=0\), and potential with velocity potential \(\varphi (x,y,t)\). The scale of this velocity potential is

*aTL*. The hydrodynamic pressure

*p*(

*x*,

*y*,

*t*) is given by the Bernoulli equation

In the formulation (2)–(6), there is still no equation which determines the motion of the contact points \(x=\pm c(t, \varepsilon )\). It is not clear at present which condition should be imposed at the contact point. Baarholm [9] in his numerical analysis of wave impacts used the condition that the speeds of the contact points are equal to the local velocities of the flow at these points; that is, the contact points are material points. In the linearized model of water exit [2], the speeds of the contact points were assumed to be proportional to the local velocities of the flow at these points. The coefficient of proportionality was chosen as 2 using the CFD results from [3]. In previous studies, it was noticed that the speeds of the contact points are small during the initial stage of exit, \(c^{\prime }(t)=O(t^{1/3})\) as \(t\rightarrow 0\) (see [2]). This result allows us to neglect the motion of the contact points during the early stage of the plate lifting in the calculation of the flow far from the plate edges. However, close to the edges, the motions of the contact points should be taken into account as it will be shown later. We will investigate the effect of the motion of the contact point on the flow close to the plate edge.

In the present study, we are concerned with the asymptotic solution of the problem (2)–(6) accurate up to \(o(\varepsilon )\) as \(\varepsilon \rightarrow 0\) in the main flow region. The viscous term \(\nu \nabla ^2\vec {v}\) in the Navier–Stokes equation, where \(\nu \) is the kinematic viscosity of the liquid, can be neglected compared with the inertia term \(\vec {v}_t\) and the convective term \((\vec {v}\cdot \nabla )\vec {v}\) if \(\nu T/L^2\ll \varepsilon \), which gives \(T\gg \nu /aL\). For \(\nu \approx 10^{-6}\) m s\(^{-1}\) (water at 20 \(^{\circ }\)C), \(L=5\) cm and \(a=20\) m s\(^{-2}\), the inequality gives \(T\gg 10^{-6}\) s. In the immediate vicinities of the plate edges, the viscous effects are expected to be more important than in the main flow region. However, it can be shown that the liquid viscosity can be neglected at leading order, even in the immediate vicinities when \(T\gg 10^{-5}\) s for the same conditions. Therefore, the viscous effects are important only during the very early stages of the flow, the duration of which can be estimated as \(10^{-5}\) s.

It can be shown that the surface tension can be neglected in the main part of the free surface with accuracy \(o(\varepsilon )\) and in the immediate vicinities of the plate edges at leading order after a very short initial stage of duration \(10^{-3}\) s. Indeed, in the main part of the flow region, the pressure is of the order of \(\rho a L\), whereas the capillary pressure is of order \(\sigma /L\), where \(\sigma \) is the surface tension coefficient. Thus, the ratio \( \sigma /L\ll \varepsilon \) when \(T^2\gg \sigma /(\rho a^2 L)\). This inequality gives \(T\gg 1.9 \times 10^{-3}\) s for water of density \(\rho =1000\) kg m\(^{-3}\) and surface tension coefficient \(\sigma \approx 72\times 10^{-3}\) N m\(^{-1}\). As will be shown here, the pressure is of order of \(\rho a L \varepsilon ^{1/3}\) in the vicinity of the plate edge, the size of which is of order \(L\varepsilon ^{2/3}\), where the capillary pressure is of order of \(\sigma /(L\varepsilon ^{2/3})\). Then the ratio \(\left( \sigma /(L\varepsilon ^{2/3})\right) /\big (\rho a L \varepsilon ^{1/3}\big )= \sigma /(\rho a L^2 \varepsilon )\) is much smaller than one when \(T\gg 1.9\times 10^{-3}\) s. We conclude that viscous and surface tension effects can be neglected in the analysis of the plate-lifting problem after a short early stage, when the plate velocity is small, both in the main flow region and in the immediate vicinities of the plate edges. Note that the typical acceleration \(a=20\) m s\(^{-2}\) , which comes from the experimental results in [3], was used in the above estimates. This value was chosen to satisfy another limitation of the present study, namely, that gravity can be neglected compared with the acceleration of the plate. For higher accelerations of the plate, the duration of the early stage, when viscous and surface tension effects should be included in the analysis, is smaller than in the reference case shown above.

## 3 Second-order solution in the main flow region

*q*and the function \(C_e(t)\) in (20)–(23) are undetermined and will be obtained using the matching condition between the second-order outer solution (7) with corrections (21)–(23) and the leading-order inner solution, which will be studied in the next section. The small-time solution derived in this section is not valid near the plate edges, where an inner solution is needed to correct the flow and the pressure distribution.

## 4 The inner flow close to the plate edges

*k*is selected in such a way that the quadratic term in the Bernoulli equation (1),

*q*and the function \(C_\mathrm{e}(t)\) in (23) are obtained by matching the eigen value term (21) along the plate to the second term with \(C_*\) in (39). The result is \(q=2/3\) and \(C_\mathrm{e}(t)=-\sqrt{2} C_* t^{7/3}\). Then the correction (23) to the pressure distribution (19) reads

*C*and \(C_*\). These constants are obtained by matching the outer solution with the inner solution near the plate edges. In particular, the pressure at the centre of the plate is given by

## 5 Numerical analysis of the inner flow

The potential nonlinear problem (27)–(29), where the position of the free surface is unknown *a priori*, is solved in a numerical domain bounded by the free surface, \(\Sigma _{{\mathrm {FS}}}(t)\), a semi-circular far-field boundary, \(\Sigma _\infty \), with radius \(R_\infty \), where the potential satisfies equation (29), and a flat plate lying on \(\eta =0\), where the no-flux condition (27) is imposed. A sketch of the domain and coordinate system is shown in Fig. 1.

The slow convergence near the corner is compatible with the existence of a discontinuity in the slope of the free surface at that point. In fact, Needham and co-workers showed that in an analogous system such singularities do exist [21]. In their particular problem, the flow sets in motion by an inclined plate that accelerates at a constant rate from rest, a corner is formed depending on the inclination angle of the plate, which precluded the numerical computation of the solution from that point on. In our case, the time-marching procedure allows us to infer the shape of the free surface with a reasonable accuracy, except obviously at the singular point and, as will be shown below, at the contact point.

Regarding the conditions for the formation of the singularity, although it would be tempting to think that, similar to what Needham and co-authors found in [21], there could exist a critical value of the contact angle between the plate and the free surface, \(\alpha _{{\mathrm {c}}}\), that would yield a solution that is regular everywhere. However, this is not the case here. Indeed, Fig. 7 shows the final free surface, normal velocity of the modified potential, \(S_n\), and dimensionless tangential velocity, \(\Phi _\tau \), in the vicinities of the plate. It can be seen how, except very close to it, the structure of the flow is not affected by the choice of this contact angle. In particular, the corner found near \(\tau = 1.75\) is found for all the values of \(\alpha \) explored. This points out that using plates with different wettability properties (hydrophobic or hydrophilic surface, for instance) would not affect significantly the structure of the flow described in this work, except, naturally, very close to the surface

Presumably, the effect of the singular corner has also an effect on the contact point. Figure 4 displays the tangential velocity, \(\Phi _\tau \), along the free surface. It can be seen how this velocity shows a peak close to the contact point, \(\xi =c_0(t)\), \(\eta =0\), before becoming zero at the plate. Consequently, the convergence of the solution near the plate is also very slow. Figure 5 shows the evolution of the rescaled *x* coordinate of the contact point, \(c_0(t)\), and the potential at that point, \(\phi _0(t) = \phi (c_0(t), 0, t)\) rescaled according to the proposed self-similar structure of the problem, i.e. \(c_0(t)/t^{4/3}\) and \(\phi _0(t)/t^{5/3}\). Although both magnitudes follow the self-similar scaling for \(t > 20\), the potential does not coincide with the theoretical value, exhibiting a relative error of about \(5 \%\). In that figure, the grey solid line (red in the online version) is computed using the first equation (41), taking the value of \(\lambda _c\) corresponding to the last time step. Conversely, the black solid line has been calculated with the second equation (41), using the value of \(\Phi _\lambda (\lambda _c, 0)\) of the final integration time.

Finally, it is interesting to show the evolution of both the velocity, \(\phi _x\), and pressure distribution, \(p = -\phi _t - \frac{1}{2} \phi _x^2\), along the plate (see Fig. 6). Whereas the velocity converges very quickly to the self-similar distribution, the pressure near the contact takes longer to do so, specially close to the contact point. Most likely, this is an effect of the influence of the corner on this region discussed in the previous paragraph. Despite this, the pressure converges to the far-field solution, \(p/t^{2/3} \sim -\sqrt{2\lambda }\) (black dashed line).

## 6 Conclusions

The numerical solution of the problem exhibits the self-similar scaling proposed by the asymptotic analysis, which proves the validity of the theory to describe the flow close to the contact point between the plate and the free surface. Interestingly, some discrepancies arise, although they might be associated to the numerical error introduced by the existence of a singularity in the mathematical solution, very much like the one found by Needham et al. [21]. Although our numerical method, based on time-marching the flow in a pseudo-time, is at least able to describe the overall structure of the flow, it clearly fails to accurately accommodate the existence of the singularity. However, in contrast to the flow considered in reference [21], the corner found here seems to be insensitive to the particular choice of the contact angle, \(\alpha \), as is illustrated in Fig. 7.

## Notes

### Acknowledgments

This work has been supported by the NICOP research grant “Fundamental Analysis of the Water Exit Problem” N62909-13-1-N274, through Dr. Woei-Min Lin. The authors wish to acknowledge the support of the Spanish Ministry of Economy and Competitiveness through grant DPI2014-59292-C3-1-P. The authors also acknowledge the support of the London Mathematical Society that funded a research stay of J. R-R at the University of East Anglia in the summer of 2014. The authors are grateful to the referees for their careful readings and comments, which helped to improve an earlier version of this paper.

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