## Abstract

Solid–solid and solid–fluid impacts and bouncing are the concern here. A theoretical study is presented on fluid–body interaction in which the motion of the body and the fluid influence each other nonlinearly. There could also be many bodies involved. The clashing refers to solid–solid impacts arising from fluid–body interaction in a channel, while the skimming refers to another area where a thin body impacts obliquely upon a fluid surface. Bouncing usually then follows in both areas. The main new contribution concerns the influences of thickness and camber which lead to a different and more general form of clashing and hence bouncing.

## 1. Introduction

The present motivation, apart from much intrinsic interest in the subject of free body movement within a fluid in general, is from industrial, biomedical and sports applications, and other arenas. The industrial applications are manifold including in particular ship slamming, sloshing and granular flows on chutes [1–20]. There is also a huge number of biomedical applications [21–31], while sports applications are such as in skeleton bobsleigh. The work is on fluid–body interaction [32]. The article is aimed mostly at analytical understanding and results.

Hydroelastic effects can be very important in practice during clashing, skimming and bouncing. Such effects are usually also rather subtle and challenging as regards calculation, depending on the context. The current focus, on the other hand, is on the influences of body shape in interactions with fluid motion, where the body can move but preserves its shape, as explained in detail within the current section. This is with a view to gaining more insight into the subtleties of fluid–body interplay, which can themselves be very challenging indeed. An in-depth follow-up study of hydroelastic features seems called for based on the properties found here.

The present contribution is based on two recent findings and one new piece of work, with bouncing or rebounding being a common feature. The first is Smith & Ellis [32] where interactions between a finite number of moving bodies and the surrounding fluid in a channel are investigated. The bodies or grains are thin, straight and free to move in a nearly parallel configuration in quasi-inviscid fluid, the combined motion being assumed to be planar for modelling purposes. Numerical and analytical aspects are presented. Three major features appear, namely an instability about the uniform state, a solid-to-solid clashing (body–wall or body–body) within a finite time, and a continuum limit for many bodies. The finding of most interest here is that of clashing. Second is the theoretical investigation by Hicks & Smith [33] of the evolution for the combined solid–fluid motion when a solid body undergoes a skimming impact with and possibly rebounds from a shallow liquid layer, e.g. shallow-water skipping. Depending on the parameter range involved, the induced lift on the body owing to flow pressure is sufficient to entirely retard the incident downward motion before causing the body to exit from the layer within a finite scaled time. The work with almost exactly the same interaction structure as in the first study thus yields skimming, lift-off from the water and subsequent bouncing. Third, the new piece here is on the unsteady interactions between a body and the fluid flowing past it, inside a channel again but with a more realistic body. Indeed, the study by Smith & Ellis [32] is for one or more infinitesimally thin plates in inviscid fluid, whereas the current investigation is also for a (quasi-) inviscid incompressible fluid but the contained moving body has thickness and camber. In the parameter study presented below, these turn out to be important features.

Section 2 below describes the interactive body–fluid configuration, formulation and main properties of the numerical solutions obtained for the nonlinear setting with a single body having thickness and/or camber. All cases are found to yield a solid–solid clash within a finite scaled time. This can be contrasted and compared with the interesting findings of Yih [34], Korobkin [35], Korobkin & Ohkusu [36], Schonberg & Hinch [19], Newman [37], Tuck [38], Fortes *et al*. [39], Huang *et al*. [40], Yang *et al*. [41] and Ardekani *et al*. [42]. The typical present clash occurs mid-body instead of at the leading edge as in Smith & Ellis [32] and this new form is examined in detail in §§3 and 4. Comparisons between the numerical results and the clash analysis are in §5. Section 6 presents final comments.

## 2. Configuration and solution properties

The focus is on a single body as in figure 1. In terms of the formulation and methodology, the body–fluid interaction takes place inside a slender channel in a two-dimensional *x*–*y* plane where *x* and *y* are Cartesian coordinates non-dimensionalized with respect to a representative longitudinal length *L**. The body or grain is thin but with non-zero thickness or camber (or both) and with its angles of inclination during motion being of the same small order *b*, say, as those of the containing channel. The typical flow speed over the time scales of concern is *U**, with the (*x*,*y*) velocity components (*u*,*v*), pressure variation *p* and time *t* being based on *U**, *ρ***U*^{*2} and *L**/*U**, respectively, where *ρ** is the fluid density. The representative length scale for *x* is of order unity by definition, whereas that for *y* is *b*; similarly, *u* is of *O*(1), whereas *v* is of *O*(*b*) by continuity and *p* and *t* are typically of *O*(1). For convenience, we write *y*=*bY* and *v*=*bV* , so that *Y* lies between 0 and 1. Also the quantity *Reb*^{2} is supposed to be large, meaning that viscous effects are nominally negligible, with *Re* being the appropriate Reynolds number, while the effective Froude number is likewise taken to be large such that gravity effects are negligible. Two-way nonlinear interaction takes place simply because the fluid dynamical forces cause body movement, which in turn affects the fluid motion.

The scales involved imply the governing equations
2.1a
and
2.1b
for each *u*=*u*_{n}, *V* =*V* _{n}, *p*=*p*_{n} owing to continuity and streamwise momentum in the two thin layers present as in figure 1. The normal momentum balance yields ∂*p*/∂*Y* being zero at this level, so that *p*(*x*,*t*) is independent of *Y* in each layer, in effect. The subscripts *n*=1,2 here and below refer to values within the two fluid layers on either side of the body. The range of *x* is from 0 to 1. The lateral boundary conditions are the kinematic ones
2.2
for each solid moving boundary at scaled position *Y* =*f* (determined by a central position, a scaled thickness effect *T* and a scaled camber *C* as specified later), acting together with the tangential flow conditions at the straight fixed walls,
2.3

The oncoming flow has (*u*,*V*,*p*)=(1,0,0) ahead of the leading-edge position *x*=0, and in order to satisfy the trailing-edge Kutta conditions downstream, a local leading-edge region is generated which leads to the jump requirements
2.4a
and
2.4b
from the Bernoulli equation and conservation of vorticity in each gap as in the study of Smith & Ellis [32]. The fluid flow equations in each gap now yield the shallow-water relations
2.5a
and
2.5b
in view of equations (2.1)–(2.4), where the two unknown *H* values denote the scaled gap widths in the form
2.6a
and
2.6b
for definiteness, where camber and thickness are defined in figure 1. The total mass balance then requires
2.7
Also at the trailing edge *x*=1, a Kutta condition holds, imposing on the fluid flow the constraint
2.8
with an unknown pressure value *π*_{e}, which depends on *t*.

Along with the above, the body itself moves mainly in response to the fluid dynamical pressure forces acting on the body surfaces. Thus, since the unknown body position is that implied by equations (2.6*a*,*b*), the balances of lateral and angular motion require
2.9a
and
2.9b
respectively, where a dot represents differentiation with respect to *t*. Here, to clarify, *h* is the *Y* -position of the body centre of mass (the *x*-position being close to 1/2), *θ* is the angle of inclination of the body, while *M* and *I* are the scaled mass and moment of inertia in turn. All numerical results presented herein take *M*=1=*I* for illustrative purposes, although we note that the studies of Smith & Ellis [32] and Hicks & Smith [33] show that the thin-straight-plate system is sensitive in quantitative terms to the values of *M* and *I*. The streamwise motion is dominated by the requirement of constant streamwise velocity of the body as the fluid dynamical forces are comparatively small in that direction.

The system of interest is therefore equations (2.4a), (2.5)–(2.9). Numerical solutions of this nonlinear system were obtained by finite differencing and iteration as in the study of Smith & Ellis [32] with analytical treatments used as checks. An integrated form of equation (2.5a) proves advantageous here, giving
2.10a
and
2.10b
with the temporal functions of integration being constrained by
2.11
because of equation (2.7). Results are shown in figures 2 and 3 for a number of particular cases. When the camber is zero but thickness varies, figure 2*a* shows that the early motion is dominated by a linear increase in *θ*. This is followed by a continuing growth of *θ* accompanied by motion towards the lower wall. The added-mass effect for larger variations in thickness accelerates the motion towards the wall. Figure 2*b* shows that flow reversal can occur in the closing gap between the body and the wall, and that the flows in both gaps show Bernoulli-like increases towards the centre of the body, where the thickness is greatest. The clash in the *C*(*x*)=0=*T*(*x*) case is occurring at the leading edge, whereas in the remaining cases it appears to occur away from the leading and trailing edges in general, also in keeping with figure 4. When the thickness is zero but the camber varies, figure 3*a* suggests that the resulting lift-induced migration to the wall and angular acceleration affect the flow more strongly than in the zero camber cases. Figure 3*b* shows that, once again, flow reversal can occur in the closing gap, and Bernoulli-like effects are again seen as the concave-downwards body migrates to the wall. Again, clashes are occurring in general away from the leading and trailing edges. In all cases, leading- and trailing-edge velocity appears to approach a linear asymptote before the clash. Similar effects are to be expected for cases with camber and thickness both present.

Since the cases for which *C*(*x*)=0=*T*(*x*) agree with those of the study of Smith & Ellis [32], similar properties are to be expected for many bodies present, as in Smith & Ellis [32]. Clashing within a finite time is clearly suggested with the minimum of one of the gap widths approaching zero at an *x*-station away from the leading and trailing edges in general.

## 3. Finite-time clashes and local effects

The task now is to explain the mathematical structure of the fluid–body interaction as the clashing occurs, and in particular to elucidate the contributions of the local and non-local dynamics, which turn out to have similar degrees of importance. Also, this analysis is in preparation for future study of the post-clash phase, i.e. the start of the rebound, subsequent interactive motion, and so on, where an analogous solution structure is to be expected. The finite-time clashing is found analytically here to have a solution structure different from that of Smith & Ellis [32], which corresponded to perhaps rather gentle clashes happening at leading edges. Instead, the typical clash is now centred mid-chord, as follows, with both local and global effects having significant roles.

Supposing the clash is centred near some *x*=*x*_{0} with 0<*x*_{0}<1, and that the gap width *H*_{1} first tends to zero there at a time *t*=*t*_{0}, we have locally *H*_{1} to be of order (*t*_{0}−*t*)^{n}, say, with the unknown power *n* being positive. The smooth local body shape then indicates that typically *Y* is of order (*x*−*x*_{0})^{2} and so the spatial scalings (*x*−*x*_{0})∼(*t*_{0}−*t*)^{n/2}, *Y* ∼*H*_{1}∼(*t*_{0}−*t*)^{n} are inferred. An argument based on the consequent fluid velocities and hence pressure then leads through the added mass concept to the conclusion that *n* must be zero or unity on account of the equations of body motions (2.9*a*,*b*). The former value is contradictory; hence *n*=1. The resulting asymptotic description therefore takes the form
3.1a
3.1b
3.1c
3.1d
for . Here, the coefficients multiplying the powers of (*t*_{0}−*t*) are of *O*(1), while *F*_{1}=*T*/2−*C*, *F*_{2}=*T*/2+*C* are functions of *x* only. The 3/2 and 1/2 powers above follow from a line of reasoning similar to that employed earlier. Also, the appropriate scales spatially are given by *x*−*x*_{0}=(*t*_{0}−*t*)^{1/2}*ξ* with the coordinate *ξ* being of order unity, and *H*_{1}∼(*t*_{0}−*t*), specifically
3.2
where by definition of a clash; *F*_{11}=*θ*_{10} as slopes cancel; and . The constants *A* and *B* are positive. Comparisons with numerical work are shown in figure 5 and these tend to support the trends of equations (3.1*a–d*) and (3.2) and their immediate consequences.

Substitution into equation (2.10a) shows that for consistency the relation
3.3
must hold, together with the expansion
3.4
of the velocity in gap 1. By contrast, the velocity in gap 2 remains finite. The numerical findings in figure 5 tend to support the implication (3.3). Furthermore, from equation (2.10a) the velocity coefficient here is given by
3.5
with *α*=*c*_{12}+(3/2)(*x*_{0}−1/2)*h*_{12}+(3/4)(*x*_{0}−1/2)^{2}*θ*_{12} and *β*=*h*_{11}+(*x*_{0}−1/2)*θ*_{11}=*A*, the approach speed of the body, which is given by d*H*_{1}/d*t* evaluated from equation (3.2) and is generally positive. From equations (3.4) and (2.5b), we infer that the local pressure expansion in gap 1 must have
3.6
whereas in gap 2, the pressure *p*=*p*_{2} is substantially smaller. Hence, the governing equation
3.7
applies owing to equation (2.5b). In addition, the body movement according to equations (2.9*a*,*b*) requires
3.8a
and
3.8b
formally, as long as the local contributions to the integrals here are the dominant ones—this needs to be checked later. The major point about equations (3.8*a*,*b*) at this stage is simply that they appear to fit together in a self-consistent manner.

The scaled pressure term *P*_{1} is from the integration of equation (3.7), with the constant of integration *D*_{1} being determined as −*A*/(2*B*) to ensure decay at large positive or negative *ξ*. Hence, the local scaled velocity and pressure solutions can be written in the normalized form
3.9
where and
3.10a
and
3.10b
These are plotted in figure 5 for various values of *α*_{b}. Clearly, if *α*_{b} is zero then the pressure *P*_{1b} is even in *ξ*_{b}, *U*_{1b} is odd in *ξ*_{b}, and the pressure decay is relatively fast (like *ξ*^{−4}_{b}) at large *ξ*_{b}; also the integral appearing in equations (3.8*a*,*b*) is then convergent. If, on the other hand, *α*_{b} is non-zero, then evenness and oddness of the solutions do not hold, the pressure decay is slower (like *ξ*^{−1}_{b}) and the integral in equations (3.8*a*,*b*) is logarithmically divergent.

The most broadly applying case has *α*_{b} being non-zero (and this case is supported by the comparisons presented later in §5). That corresponds to *α* being non-zero in general and so the asymptotes acting at large *ξ*_{b} values,
3.11a
and
3.11b
when combined with the local expressions in equations (3.4) and (3.6), indicate that the velocity and pressure must have the underlying formation
3.12a
and
3.12b
outside of the local region, i.e. where *x* is of order unity along the majority of the body. These predictions lead to the non-local effects studied in §4. We observe also here that the contributions from the leading-edge jump conditions and the trailing-edge smoothness requirements (2.4a) and (2.8), respectively, have been negligible in the local analysis so far in contrast with their effects that emerge globally as described in §4. The background issue of whether the body and fluid velocities are mostly finite at a clash has also still to be addressed.

## 4. Non-local effects

The non-local or global effects operating over the majority of the body as hinted towards the end of the preceding section contribute more or less equally to the major finite-time clashing behaviour. Their description is distinct generally from the local one above but the two descriptions have to match exactly where overlap occurs.

The appropriate expansions for height, orientation and flux, with the typical *x* being of order unity, again have the patterns (3.1*a–c*), whereas the global *F*_{1} here is simply the smooth prescribed shape function *F*_{1}(*x*)=*T*/2−*C*, while *F*_{2}(*x*)=*T*/2+*C*. Hence, for *x* not equal to *x*_{0} now the gap widths *H*_{1} and *H*_{2} develop according to
4.1a
and
4.1b
with the dominant terms being
4.2a
and
4.2b
which give the two scaled gap widths at the clashing time *t*=*t*_{0}. The scaled velocity components in the corresponding two gaps are then of the form
4.3a
and
4.3b
where, from substitution into equations (2.10*a*,*b*), the leading contributions satisfy
4.4a
and
4.4b
and equations governing the higher order corrections can also be written down readily. Mass conservation as represented by equation (2.11) requires here that
4.5a
and
4.5b
and so on for successive terms.

The induced gap pressures as anticipated at the end of §3 are therefore described by
4.6a
and
4.6b
The controlling equation (2.5b) thus yields direct balances between the temporal acceleration forces and the pressure gradients,
4.7a
and
4.7b
at leading order. The high pressure *p*_{1} produced within gap 1 is just as expected from the conclusion to §3, but the similarly high pressure *p*_{2} induced within gap 2 is perhaps surprising at first sight but is necessary for consistency on the global scale. The leading- and trailing-edge constraints (2.4a) and (2.8) here require, respectively,
4.8a
and
4.8b

Finally, here the body-movement relations (2.9*a*,*b*) give the global contributions
4.9a
and
4.9b
to be viewed in conjunction with those in equations (3.8*a*,*b*). Comment is postponed to §6.

## 5. Comparisons

The comparisons in question are between the analysis of §§3 and 4 and the numerical results that were presented in §2. The aim is to shed extra light where possible on the clashing process. The comparisons are shown in figures 6–9.

Figure 6 shows a check on prediction (3.3) of the terminal analysis just prior to a clash. Here, gap 1 is that in which the clash occurs or is implied by the numerical results. The quantity tested numerically in the figure as regards gap 1 seems clearly to be passing through zero within a finite time, and in line with equation (3.3), while the streamwise position *x* of the zero point is near *x*=*x*_{0}=0.4, a value agreeing quite well with the results in the earlier figures, for example, figure 4.

Velocity profiles *u*_{1}, *u*_{2} obtained numerically near termination are highlighted in figure 7*a*. Here, the *u*_{2} profiles have values very much smaller than those of the *u*_{1} velocity profiles, again immediately prior to a clashing taking place. The trend is as predicted in the analysis specifically in §§3 and 4. Corresponding pressure profiles *p*_{1}, *p*_{2} computed for the two gaps are shown in figure 7*b*. Again, the variations associated with the *p*_{2} profiles are considerably smaller in general than for *p*_{1}, in keeping with the overall argument in §§3 and 4.

Figures 8 and 9 show other quantities of prime interest for comparison with the touch-down analysis, namely . The message appears to be reasonably affirmative as in the previous results. Here, the touch-down analysis of §§3 and 4 suggests that three square roots and two inverse-square roots should emerge asymptotically near the termination of the computations as pointed out in the figures, and indeed the numerical results appear to support this conclusion.

The above comparisons of analytical and computational results seem fairly encouraging altogether for the terminal description put forward in §§3 and 4, for the clashing process arising at a finite scaled time.

## 6. Further comments

Solid–solid impacts and the approach towards such impacts in the presence of surrounding fluid have provided the major focus of the present study, as opposed to the solid–fluid interfacial impacts of Hicks & Smith [33]. Physical and mathematical understanding of the reasons for approach and the dynamics of the clashing seem inherent in the descriptions in §§2–5. The clash structure fits together at the algebraic level subject to logarithmic effects being implied by equations (3.8*a*,*b*) with equations (4.9*a*,*b*). We believe that these effects correspond to additional terms proportional to multiplied by the fractional powers in equations (3.1*a–c*). Increased understanding of subsequent rebounds or bounces from the impact and then quite possibly further multiple clashes is also in prospect depending on the specific contexts and follow-up. On the one hand, there is the detailed question of whether the clash in the presence of containing fluid leads simply to a reversal of the approach velocity or not and, on the other hand, there is the clear suggestion shown by Smith & Ellis [32] that owing to interactive instabilities continual clashing is inevitable in the current settings. The times, scales and orientations at which such clashes continue to take place are governed by the system and typical solutions investigated in the paper. What happens here in the longer run as clashes abound is of much wider concern.

Only a single body has been included so far with thickness or camber or both being significant in the fluid–body interactions. The approach, however, is readily extendable to more bodies, in fact to any finite number of them and then onward to the large-number (continuum) limit, which was addressed in detail in Smith & Ellis [32] for that particular scenario. This aspect would be interesting to pursue in further work. The issue also of precisely what effect clashes followed by bounces act as or lead on to within the continuum limit is an intriguing one.

Several miscellaneous matters of relevance here also arise, as detailed in the following points. (i) We would expect linear instability to be present as in the earlier Smith–Ellis scenario say for a uniform configuration but this has not yet been investigated in detail. (ii) The extension of the theory to three spatial dimensions would mark a great step forward in understanding. (iii) The parameter range associated with viscous–inviscid interactions as distinct from the inviscid interactions studied herein also poses a significant challenge to the theoreticians. (iv) Another issue surrounds whether anything general can be taken from the results in the sense of clashes, rebounds and then multiple rebounds that involve smooth surfaces; this likewise has still to be addressed fully. (v) The extreme of nearly ballistic motion of the contained body or bodies has been investigated in preliminary work by F.T.S. and Amy Brookes–Beighton. (vi) Interaction results obtained for increasing body thicknesses suggest that a subtle changeover can arise at comparatively small values of the thickness in relation to the zero-thickness case of Smith & Ellis [32]; this is because of the delay in thickness effects influencing the leading-edge region that dominates in the zero-thickness case. (vii) Essentially the same physics and issues arise for the related context of skimming as in Hicks & Smith [33]; this indicates that there may well be a similar more general description of lift-off holding for the skimming context when rebounds are produced; such solid–fluid impacts lead on, of course, to multiple subsequent skimmings and rebounds.

## Acknowledgements

Thanks are due to Roger Gent, Jim Oliver and Sasha Korobkin for their interest and comments at various stages, and to a referee for helpful comments.

## Footnotes

One contribution of 13 to a Theme Issue ‘The mathematical challenges and modelling of hydroelasticity’.

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