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Wedge entry into initially calm water
Wedge entry into initially calm water MARTIN GREENHOW The Norwegtan Marine Technology Research Institute, PO Box 4125 - Valentinlyst, N- 7001 Trondheim, Norway
This paper presents results of calculations based on the Cauchy's theorem method of Vmje and Brevig1 for the two-dtmensional entry of wedges of various angles into initially calm water. The problem has a long history winch is briefly reviewed in the introduction, and significant progress has been made with both linear theories (valid for low entry speed) and with theories which treat the free surface conditions exactly but with the assumptions of zero gravity and constant speed of entry. This simplifies the problem to one which 1s self-similar in dimensionless space variables ~ = x/vt and = y/vt and this has a number of consequences. For wedges with half-angles up to about 45 ° and with high entry speeds, the numerical approach, winch Includes gravity, validates these assumptions and the agreement between both free surface displacements and pressure distributions on the wetted wedge surface is excellent except in the region of the jet of fluid which rises up the side of the wedge Because the potential flow initial value problem is singular at the Intersection of the free surface and wedge surface, exact numerical resolution of the jet is not possible. Nevertheless, the rest of the fluid motion Is insensltwe to the treatment of the jet, which itself may be calculated quite realistically. Of particular interest (but little practical relevance) is the pressure on the upper part of the wedge surface (in the jet region) wluch according to self-similar theories is very small but positive, but which is calculated to be small but negative by the numerical scheme. This effect, which is enhanced when gravity is included, is insensltwe to the numerical resolution of the jet and suggests that the jet may separate from the wedge surface, the new intersection point being where the pressure vanishes on the wedge surface. A modified numerical scheme allows this to happen and the results are an quahtatlve agreement with the experiments by Greenhow and Lln.2 The numerical method presented here is extremely versatile and a number of other effects may be explored. Examples of transient motion, non-constant speed of entry, oblique entry and complete penetration of the surface so that a cavity as formed behind the wedge are presented.
1. INTRODUCTION Thls work seeks to quantify the pressure distributions and forces on a body when it enters initially calm water, and the resulting fluid motion. It is motivated by the slamming of ship bows as they re-enter the water in heavy seas, and by the engulfing of oil rig cross-members by large and possibly breaking waves. For ships the magmtude and dlstnbutton of the pressure ts sometmaes sufficient to deform the bow area quite severely (see Yamamoto et al.3), whale for rigs the wave impacts may be a serious source of fatigue (see Attfield4). These difficult problems may be simplified by assuming that they are equivalent to the entry of a body into calm water with velocity equal to the relative velocity of the body and the moving free surface. Such an assumption has been justified for fixed cyhnders being engulfed m waves of sufficiently long wavelength by Rldley s and is expected to hold when the body velocity greatly exceeds the fluid velocities within the wave, as is the case for free fall lifeboats for example. In any case, the present work is regarded as an essential prerequisite to the full problem of a body entering a wave. These comments reflect the importance of the study of water entry and the need for practical methods of solution. In view of the significant progress Accepted December 1986. Discussion closes December 1987.
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made recently and the fact that Moran's (1965) 11 extensive review of earher work is not widely available, it seems worthwhile to briefly present the basic ideas and approaches below. For entry into water at speeds low enough to regard the free surface displacements as small, one can linearlse the free surface condition while treating the body boundary conditions exactly (i.e. the boundary condition is applied on the actual body surface). Papers in this category include Chapman,6 Yim 7 and Doctors.8 Alternatively, if the wedge half-angle a (see Fig. 1) is small, then the free surface displacements wall be small even for high entry speed. This slenderness assumption has been used by Mackle, 9'1° Moran 11 and Borg x2 and we make a comparison with the numerical results for a = 9 °. Of interest IS Mackie's 9 closed form solution for the free surface elevation. r/ .
In.
+.
.2~ tan -1 1
2
(1)
rr
where ~ = x/vt and 7? = y/vt with v being the wedge entry velocity. For the entry of non-slender wedges at high speed, the usual approach ts to equate the force on the body with the rate of change of added mass of the body as it enters the water times ItS velocity (see, e.g. yon Karman,~3Wagner,~4
0141-1187/87/040214-10 $2.00 © 1987 Computational Mechanics Pubhcations
Wedge entry into initially calm water: 214. Greenhow
t
Figure I.
L
Definition sketch
Pierson Is and Fabula16). This approach is fairly simple and has gained widespread use for a variety of body shapes (for a cylinder see Faltinsen et al., 17 for a sphere see Miloh, TM and for a cone see Shiffman and Spencer*9). It is also possible to take into account the free surface rise, albmt approximately, by using a method described by Wagner) 4 In that paper Wagner also models the wedge entry problem by an expanding flat plate of width equal to the instantaneous (elevated) waterline of the body during penetration, and this gives the pressure distribution on the wetted wedge surface as p ( x ) = ~Ov2
[
tan6(1 --x2/L2) 1/2
2y
- - - - (L 2 --x2) a/= u2
1 --x2/L 2
1
(and still less the contact angles) is possxble even when this jet separation is not allowed, because of the numerical difficulties of resolving the jet. When considering splash height and contact angles, an analytical description of the fluid motion would be highly destrable. Unfortunately, even with the assumptions necessary for self-similar flow, no complete solution has been found yet, but significant analytical progress has been made by Garabedian, 2s Mackle 26 and Johnstone and Mackie? 7 Garabedlan shows that the arc length along the surface between any two free surface particles remains constant with time and since the present numerical method follows free surface particles, this is a particularly useful result. (This was first shown by Wagner.14 Garabedlan also solves a wedge oblique entry problem subject to somewhat unphysical boundary conditions.) Mackie 26 discusses the condition of convexity to the fluid of the free surface, which Is guaranteed If the pressure Is positive on the wetted wedge surface. Assuming this to be true, Mackle places bounds on the contact angle: (i) For all c~, 0 rr/4, fl < rr/2 -and considers the curvature of the free surface at the contact point, which if zero implies that fl < rr/6. By considering the problem with Lagranglan variables, Johnstone and Mackie show that' cot fl + 2fl = rr + 2a
(2)
Here j~ is the wedge acceleration and other symbols are defined in Fig. 1. At the ends of the plate, x = + L, the fluid velocity is infinite and so equation (2) cannot be valid there, but it does predict the maxima in pressure some way away from the wedge vertex observed by Chuang 2° and others for small deadrise angles 8. For intermediate 8, however, approximating the body by a fiat plate IS clearly very crude: for 8 = 30 °, Fig. 7 shows disagreement with more exact approaches and with Borg'sal approxtmate analysis which is also simple but more accurate over most of the wedge surface. The problem of constant-velocity entry of a wedge with general angle and fully non-linear free surface conditions, but with gravity neglected, may be simplified by using the self-similarity variables ~ and ~7 (see Gurevich 22 for the more general result for non-constant velocities). Solutions based on self-similarity have been given by Borg la using a relaxation method, and Dobvrovol'skayaas and Hughes 24 using the Wagner h-function (see Gurevich22). Although these solutions are hmited to zero gravity self-similar flows, they do yield much information about the flow: the spray root singularity position and order, splash height and con. tact angle with the wedge, shape of the free surface and pressure distributions on the wedge m~¢ all be calculated at least for half-angles up to about 60~. Hughes presents experiments which clearly validate the self-similarity assumption when the entry speed is high except m the jet region where gravity (and other effects) may play a part. Thus these self-similar solutions provide and rigorous check on the numerical method presented here. As will be seen, the agreement between the free surface profiles and pressure dlstribuUons is good except m the jet region where the effect of gravaty enhances a negative pressure region on part of the wedge surface and this is thought to lead to jet separation. No direct comparison between the splash heights
(4)
While this relation satisfies condition (31) above (m fact fl hes between 23.2 ° and 9.5 ° for a varying between 0 and 90°), condition (3il) is violated if a exceeds approximately 79 °. This lmphes that the pressure within a region of fluid winch contains part of the wetted wedge surface must be negative, an interesting result in view of the present results which show a similar effect for even smaller wedge angles especially when gravity is included. It will have been noticed that practical apphcation of the above theories is lmalted to low speed of entry, slender body, zero gravity or self-similar muaUons. To avoid these restrictions and to provide good resolution of the fluid motion, Greenhow and Lm as and Yim 29 have used the Vinje and BrevigI approach based on Cauchy's theorem. (Another possibility is the method presented m Nichols and Hirt a° but this reqmres solving equations in the entire fluid domain, rather than at the boundary only.) Ylm discusses the singular nature of the initial and boundary value problem, by analogy with the potential flow perpendicular to a plate, which has infinite velocity at its ends as mentioned before. This has the undesirable consequence that good resolution of the fluid near the intersection of free and wedge surfaces causes very large velocities in the jet which are difficult to handle numerically and may lead to breakdown. However, Greenhow and Lin show that resolving the jet rather more poorly avoids this problem and does not lead to sigmficant errors m the free profile away from the jet or in the pressures calculated on the wetted wedge surface. The present paper considerably extends these catculatmns. 2. SOME COMMENTS ON THE METHOD The method of Vmje and Brevlg, which is based upon the approach of Longuet-Higgins and Cokelet,al is well described in Vinje and Brevig 1 and Greenhow et al. a2 and we therefore only describe here recent developments concerning the
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Wedgc entry into blittally cabn water 34. Green~tow treatment of the intersection points of the body and free surfaces. The problem arises because of the need to satisfy two boundary conditions at this point, namely the body boundary condition and the free surface boundary condition. In linear steady state wavemaker theou¢ it is well known that imposing both boundary conditions can result m a singularity in the complex potential ~(z) = q~ + t~, see particularly Kravtchenko 33 Here ~bis the velocity potential, IS the stream function and z = x + 0'. For time dependent problems, Peregnne 34 has analysed tire Impulsive start-up to velocity U of a wave maker in water of depth h. His calculations predict a logarithmically singular free surface profile at the intersection point x = 0 given by ). . . .
In tanh 7r I_ k4h 7A
(5)
which is m close agreement with the experiments of Greenhow and Lm 28 except at the intersection point where a jet is ejected almost perpendicularly to the wavemaker. In order to deal with this and other wavemaker motions, and as a prototype calculation for general surface-piercing bodies, Lln, Newman and Yue 3s modified the Wnje and Brevlg method as described below. Since /3(z) is analytic within the fluid domain, then Cauchy's theorem holds. : 0
(6)
Z --Z 0
c with z0 outside the contour of integration C, which is taken to include the free surface and body surface. C is divided into two parts, C¢ on which ~ is known and C¢ on which is known. Letting z o move onto the contour of mtegranon C, and taking real or Imaginary parts of equation (5) as appropriate yields.
Oo~(X,y,t)+Ref¢+t~dz=O
for Zo on C¢
(7)
Z --Z 0
and
tt
Oo¢(x,y, t) + Re i
= dZ
--g o
o
)
Here 0o is the angle between two tangents of C at Zo (mathematically equal to zr for any smooth part of C, but generally different from 7r numerically, especially in regions of sharp free surface curvature). Numerically equations (7) and (8) are dlscretlsed at N points around the boundary and this results in an N × N matrix equation of the form Ax = b where x is a vector of unknowns corresponding to the unknown part of/3(z), (either ¢ or 4). Similar equations and comments to the time derivatives Odp/bt and ~ / ~ t needed to step the system forward in time and to evaluate pressures on the body. It is not clear how to apply equations (7) and (8) when the point Zo is at the intersections of the free and body surfaces since both q~ and ~b are known there. Lm, Newman and Yue 3s specify both q~ and ff here and remove these equations completely from the system, resulting i n n being replaced by N - - 2 in the numerical method. This appears to be completely satisfactory for practical calculations giving results for the wavemaker problem which are almost indistinguishable from equation (5). Furthermore Greenhow ~ also uses this algorithm to calculate extreme wave profiles with fixed or moving cylinders m the free surface, giving reahstlc results.
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o
/ Figure 2. Comparison o f sell-similar flow with the intersection point retained on the wedge surface (shown by o) with the results of Dobrovol'skaya (shown dotted). The solid hne shows the flow when the let ts allowed to leave the wedge surface, a = 30 °
In contrast to this, Ylm 29 presents results of alternative formulations which specify either 4~ or ~b (but not both) at the intersection points This reults in unreasonable free surface elevations for the wedge entry problem and he concludes, along with Greenhow and Lin 2s that the above algorithm is preferable. A related concern of considerable nnportance in the numerical lmplementanon of equations (7) and (8) is how well one should try to resolve the behavlour of the free surface near the possibly singular intersection points. For the wedge entry problem, good resolution around this point leads to a very high velocity jet which almost immediately leads to numerical breakdown. On the other hand, poor resolution here leads to a violation of the conservation of fluid mass as experienced by Yim. 29 We here present numerical evidence to show that between these two extremes there IS a broad range of resolutions where practical calculations can take place, at least for small and moderate values of a. Within this range the jet profiles calculated are thought to be fairly realistic in some examples, but in any case it is shown that the rest of the free surface as well as the pressure distributions on the wetted wedge surface xs insensitive to the treatment around the intersection points. For large values of a the calculations are less accurate since the jet must be inhibited (by resolving it poorly) m order to time step the numerical solution sufficiently many times to reach an approximately self-similar solution which is mdependent of the initial conditions. Most of the computations presented here were run on a VAX-785. Typically, with 84 points on the free surface and about 150 in total, the execution took about 50 s c.p.u per time step. 3. RESULTS - FLOWS WHICH A R E E S S E N T I A L L Y SELF-SIMILAR
For tugh speeds of entry it Is reasonable to expect selfsimilarity since locally near the wedge the fluid particle acceleration will be much greater than gravity, especially in the early stages of entry. As suggested by Mackie, 9 the effect of gravity vall be unimportant if t ' < v/g (since gravity introduces an extra variable gt/v which is neglected
Wedge entry into initially calm water," M. Greenhow m the self-similar assumption). The calculations here suggest that gravity may be ignored if t<<.v/2g, except perhaps in the jet region. Figure 2 shows the flow for a wedge of half-angle a of 30 ° entering the fluid at high speed. Starting from the mitml conditions with just the wedge vertex in calm water, the free surface quickly becomes self-similar and compares well with the results of Dobrovol'skaya.2a The particle at the jet t:p is treated as an ordinary free surface particle for the purposes for time-stepping the solution except that since in general the predicted position does not lie exactly on the wedge, the particle xs moved horizontally back onto the wedge surface at each time step from a posmon just outside the wedge envelope. This algorithm fulfills the condition, assumed by earher authors, that the particle imtially m contact with the wedge will always stay in contact, but results m flow with negatwe pressure on the upper
20
24-
V/ Figure 3. Development o f the flow when the ]et is allowed to leave the wedge surface. The point o f intersection o f the free and wedge surfaces is shown by e. Also shown on the arrowed curve are typical results from a computation with coarse spacing o f the free surface points, where although the let is poorly resolved, the rest o f the free surface is almost coincident with the more accurate calculation. c~= 30 °
ta¢
•
ca v~
K
I0
\
x "~
lo
lg
~o
3~;
Figure 4. Comparison o f the calculated pressure distribution (o) with that o f Dobrovol'skaya 2a shown as the solid line and terminating in o. The dashed lines are the pressure distributions for a wedge which is decelerating, a = 30 °. The wedge vertex is at dimensionless depth --1
,X
L
. xz(v"
F.
o., Lo
I
PRESSURE
COMPONENT
Figure 5. Development o f the pressure distribution .for a wedge which is decelerating. The numbers indicate the time step, o indicates the pressure distribution at time step 16 with constant wedge velocity and . . . . inidcates the pressure distribution for a decelerating wedge but with the jet retained on the wedge surface (note the small region o f negative pressure). The effect o f the initial conditions is dearly seen at time steps 4 and 8, a = 30 °
portion of the wetted wedge surface. Although the magnitude of the negative pressure is small compared with the pressures on the lower port:on of the wedge, the length of the wetted wedge surface where the pressure is negative is by no means small as can be seen in Fig. 5. Tins leads us to examine the physical conditions m the jet region more closely. Certainly the particle at the intersect:on point cannot move through the wedge surface. However, the pressure gradient within the very thin jet must be extremely small, and hence the fluid particles within the jet, and in particular the intersection point part:cle, will behave approxamately as free projectiles moving under graxaty alone. Clearly the numerically calculated pressure gradient within the jet is insufficient to keep it on the wedge surface, and doing so artlficxally, as in Fig. 2, may result m the negative pressure observed. In an effort to examine this effect further, some runs were made with :dentical numerical intervals in space and time, but with zero gravity. This resulted m the jet moving shghtly further up the wedge surface, and for the lower portion of the wetted wedge surface (below the undisturbed free surface level), the difference m the pressures was almost exactly hydrostat:c. Both of these effects were expected. In contrast, in the jet region, small negative pressures were again observed, with a magmtude only slightly smaller than for flows with gravity. This disagreement with the results of Dobrovol'skaya23 and Hughes ~ certainly indicates a weakness in the present scheme, probably assocxated with poor numerical resolution of the jet. With gravxty included, the negative pressure region may be a least partly physical but we can draw no firm conclusion from these numerical results. What happens if negative pressure actually occurs at the intersection point xs not clear. One possxbility lS that the free surface breaks up because of the pressure inversion across it as in Rayleigh-Taylor instability (see Taylora7). (The real situation would be further complicated by viscosity and surface tension which are neglected here .) This instability would eventually preclude further time stepping of the solution, and indeed calculations wath the intersection
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Wedge entry into imlmlh' eahn water M (;reenh~m'
Figure 6a
High speed entry ol a wedge l~'ltll c~ = 30 °
cultles in resolving the jet nuinertcally muat bc lccognlsed, since its position and profile ale very' senslnve to the spacing between particles and the tlnre step ~,hosen in the plogIam. The same comment applies (to a lesser extent) to the position of the (new) Intersection point. Physically we might have expected this since near the intersection point the jet leaves the wedge surface tangentially upwards and hence the pressure gradient along the wedge surface here must be very small. On the other hand, the rest of the free surface and" pressure dlstrlbuhon on the wetted wedge surface is almost entirely Insensitive both to the numerical spacings and to the treatment of the jet This is the main point here since it permits practical and accurate numerical calculation for the majotlty of the flow for many time steps by avmdmg the abovementioned calculation breakdown when the jet is retained on the wedge If the (small) hydrostatic component is subtracted from the pressme along the wedge, then the agreement with the results of the zero-gravity calculations and of Dobrovol'skaya 2a is excellent except m the jet region, see Fig. 4 It IS usual to define a force coefficient CF as (9)
CF = Flpvat = F/pvgD
whine F is the actual fmce on the body a n d D is the depth of the vertex. The results here are influenced by the initial conditions for the early stages of entry but appear to conveige from above to CF = 0 97 for later stages. Dobrovol'skaya gives CF = 0 925 including a factor of 1/2 noted by Hughes, 24 a difference of about 5%.
Figure 6b High speed ento' ot a wedge with a = 30 °. Oblique I,lew
point kept on the wedge surface do break down near the let some time after this pressure inversion. However, Fig. 6b, taken together with the numerical work below suggests another possibility, namely that the jet fails away from the wedge surface introducing new free surface between the tip and the (new) Intersection point. Such a mechanism would reqmre the material tame derivative of pressure to be zero on a (moving) zero pressure contour and this has been examined by Longuet-tttgglns as as a model of the rear face of a breaking wave For wedge entry such a (semi-Lagrangmn) approach would need to satisfy the wedge surface condition and have intersecting free surfaces at the jet tip, which does at least seem possible. To explore this possibility numerically Figs. 2 and 3 show results obtained by treating the intersection point particle as an ordinary free surface particle without keeping it on the wedge. The new mtersectmn point is then taken to be on the wedge surface where the pressure vanishes. We notice first that the flow outside the jet region is almost ennrely unaffected, see Fig. 2, as is the pressure distribution on most of the wetted wedge surface. The form of the jet, wlueh ts calculated by following the fluid particles and employs no explimt smoothing, is extremely suggestive, especmlly when compared with the photographs from Greenhow and Lin 2 shown in Fig. 6. However, the diffi-
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A p p h e d Ocean Research, 1987, Vol. 9, No 4
laa
-
,/,',,:.:/
g
,,,,
,
,
,
f
as°3o •
,s"
Figure 7. Dimensionless pressures (o) f o r various wedge angles compared with the results o f Dobrovol'skaya =a and Hughes. ~ Small deadrise angle approximate theo,~es are also shown f o r ~ = 60 °. Borg 2' .. and Wagner a4(equation 2 hereto) . . . . The wedge vertex is at dimensionless depth --1
Wedge entry into initially calm water: M. Greenhow Also shown in Fig. 4 are two dotted graphs of the pressure distribution when the wedge does not have constant velocity but is free to accelerate under gravity, hydrodynamic and hydrostatic forces. This condition is typical for drop tests, and here the mass of the wedge is chosen to correspond with the Greenhow and I_an experiments. We notice that although at time step 24 the position of the wedge and free surface is almost indistinguishable from that shown in Fig. 3, and the velocity has decreased by only 6%, the acceleration of the wedge Is almost 3 g upwards. As expected from equation (2) this has an important effect on the pressures whose development is shown in Fig. 5, and hence on the value of CF (for time step 24, CF = 0.725). These results underline the importance of keeping the wedge acceleration small during drop tests, preferably by driving it through the free surface rather than by letting it fall through, and may account at least in part for the wide scatter in data for the force coefficients, especially for cylinder slamming where the forces wtll be even larger than for wedges slamming. We note that in the above calculations, the fluid mass was always conserved to within 5% of the mass of fired displaced by the wedge. It appears that Yim's 29 difficulty m conserving the fluid mass was primarily due to poor numerical resolution, which could not be improved satisfactorily because the jet was retained on the wedge surface and this quickly led to numerical breakdown with good resolution. Fortunately this problem is somewhat alleviated
Figure 8. Development o f self-similar flow Jbr high speed wedge entry with ~ = 9 ° from initial conditions shown in solid black, o denotes results o f Dobrovol'skaya ~ whtch show good agreement with the present calculation in the early stages o f entry. Note that the wedge has penetrated the free surface entirely in the later stages, leaving a cavity behind it
Ftgure 9. Complete penetratton o f the free surface by a wedge with a = 9 °. Note the slight closmg o f the cavity behind the wedge
by letting the jet separate from the wedge as described above. It should also be noted that for some of the calculations the mass error was only 1% to 2%, and this is much less than the mass of fluid above the level of vertical tangency of the free surface defining the jet region (about 5%). Therefore on this basis we conclude that even an approximate calculation of the jet IS preferable to smoothmg it out or simply removing it during the calculations. Since the numerical method follows the free surface particles (Lagrangian description) it is easy to see that Garabedian's a9 result for the constancy for arc length along the free surface between any two particles is accurately satisfied for the bulk of the free surface. In the jet region, however, the particles tend to spread out (see Fig. 3). The above discussion and comments apply to wedge half-angles of a = 9 ° and 15 °, the second of which was treated at length by Greenhow and Lin. 28 Figures 7 and 8 show the pressure distributions and free surface profile for a = 9 °. For both a = 9 ° and a = 15 °, the ejected jet actually moves inside the wedge envelope, but for ct = 15 °, Greenhow and Lin 28 replace the calculated jet by a jet which assumes that the jet particles do indeed move as projectiles. In view of the difficulty of resolving the jets more accurately and the fact that the exact jet form has virtually no significance for the rest of the flow, this seems to be a practical approach. However, for the complete penetration of the free surface by the wedge shown in the later stages of Fig. 8, the cavity formed behind the wedge is free to collapse reside the wedge envelope. This effect is also seen experimentally as in Fig. 9. We note that in the calculation for ot = 9 ° the jet velocities are low enough to be able to continue the calculations almost indefinitely (here to over 100 tune steps) and it is gratifying that the flmd mass is conserved to within 2% even at these late stages. Figure 7 also shows that the results of Dobrovol'skaya are m good agreement with the present calculation, whilst Fig. 10 compares the small ct analysis of Mackie 9 (equation (1) m
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219
Wedge entry into tnitlall3' cabn water: M. Greenhow
•- ~
~-,..;'~"
~
.__,
Figure 10. Present results f o r a wedge with a = 9 ° compared with the small a s y m p t o t w forms o f Mackie 9 . . . . and Borg 21 .
.
.
needed to give more accurate results. Finally we note that the tbrce coefficients Cf for c~ = 9 ° and 15 ° ale also ill good agreement with Dobrovol'skaya, being 0 034 and 0.l 3 respectively. In contrast to the excellent results above for wedge angles c~~< 30 °, we now discuss the case with c~ = 45 ° and compare the results with Hughes a4 As mlgh( be expected from Fig 11, very severe difficulties are encountered when is as large as 45 ° because of the fast m o w n g l e t Starting from the mltml conditions of Fig 12, the free surface development may be calculated quite realistically, but the jet causes the numerical procedure to break down before self-slmdar flow IS attained (I.e. the results are stdl Influenced by the mltml conditlons). This is thought to be the cause of the high pressures on the wedge surface (about double those of Hughes). Another shortcoming of the results is that the conservatmn of fluid mass is only approximately satisfied (to about 12%). By placing free surface particles so that the jet IS less well resolved, the procedure may be time stepped untd an approximately self-slmdar flow is attained. The pressures along the wedge shown m Fig. 7 are still too large and this is almost certainly a consequence of the poor mass conservation, see Fig. 12. More seriously (and despite considerable effort and computer time), no systematic method for improving the resolution of flows which become self-similar has been found for large c~ whach simultaneously gwes accurate free surface profiles and pressure dxstrlbutlons on the wedge surface This places some restrictions on the sensible use of the program )'or example conect results for a = 60 ° could not be obtained even approxamately. In wew of this it seems maportant to compare the existing theories )'or the pressure distributions, see Fig. 7. We see that the results of Borg 21 and Dobrovol'skaya 2a are m quite close agreement especmlly in the region of lugh pressure occurring close to where the free surface overturns For Wagner's ~4 theory (see equation (2)) the agreement IS not so good, although experiments of Chuang a° and Bisphnghoff and Doherty 4° support its use except for very small deadrlse angles (~< 10 °).
Figure 11. The high speed entry o f a wedge with a = 45 ° N o t e the long thin j e t close to the wedge surface
this paper), and Borg. 21 Neither result is particularly good for wedge angles as large as 9 ° , and we conclude that the practical utthty of these small angle theories is very limited indeed (see also Dobrovol'skaya's comments). The agreement of the pressure distribution on the wedge with a = 9 ° between Dobrovol'skaya's results and those of the present calculations is excellent except in the jet region as expected, and at the wedge vertex where better resolution would be
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Applied Ocean Research, 198 7, Vol 9, No 4
Figure 12. Development o f the free surface f o r ~ = 45 ° f r o m the initial eonditions shown in solid black. --- -denotes the results o f Hughes (1972) while . . . . denotes results o f the numerical procedure with poor resolution needed to inhibit the l e t
Wedge entry into initially calm water: M. Greenhow
effects are important is shown in Fig. 13 where a displaced wedge oscillates about it equilibrium position generating waves. The wedge starts from rest at the initial position shown. The pressure is always positive on the wedge face and the solution may be continued almost indefimtely, the only limitation being reflections which would ulttmately return from the distant vertical boundaries. Clearly in the early stages shown here this has not yet happened and the calculations are accurate and realistxc. Of particular interest is the locus of the intersection point particle which moves in an almost elliptical path. For transient motion the mass of the wedge as well as its angle is an important parameter for determining the flow and, in particular, whether overturning occurs or not. Greenhow and Lin 28 give an example of a wedge with a = 15 ° but with four times the mass of the wedge in Fig. 13. This gives a wave which breaks some distance from the wedge, although the details of the overturning crest were not resolved• Figure 14 shows an example with a = 30 °
F •
"
"
~
•
Figure 13. Transtent motion o f a wedge wtth a 15 (a) The wedge initial position (solM black) is displaced f r o m its equilibrium position which has its vertex at x. The wedge moves downwards generating a wave. (b) The wedge moves up again. N o t e the locus o f the intersection point . . . . . From Greenhow and Lin 28
I
, /t[ A.Z'~, ~,..__ [ _ /'~,.Z__" Y ~ / " - . . ~ ¢ ~ ~-"~.~_.._ '/////// -
',"
....
r
,
4. SOME EXAMPLES OF NON-SELF-SIMILAR WEDGE MOTION This section presents results for the entry of a wedge for which the resulting flow is not self-similar, either because the wedge speed is slgmficantly variable, or because gravity is tmportant, or both. An interesting example where both
Figure 14. Transient motion and localised breaking f o r a wedge with a = 30 °
2o .1
1.......__._
0
20
}(
0
Figure 15. Transient motion o f a 60 ° wedge started f r o m rest below its equilibrium position with vertex position marked x. The numbers mdicate the time step and . . . . is the locus o f the intersection point
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Wedge ento' into mitzally cahn water M. Greenhow
", ,'\t -y Figure 16. Oblique entry o f a wedge at high constant speed. ~ = 7.5 ° and the wedge centreline is inclined 1 Z5 ° from the vertical The dotted lines and axes represent the non-dimensional pressure P/1/2pv 2 distrtbution along the wedge which is negative on the suction stde
where a wave is starting to form, but where the jet of fluid originally moving up the wedge side overturns and causes breaking (which stops the calculation). As expected this also happens with larger values of a, but ff the wedge is released from a position below its equilibrium position, this problem is postponed until the wedge begins to move downwards, see Fig. 15. Because of the large number of parameters involved m these flows where gravity is important (e.g. wedge angle and mass and lnmal conditions), exact criteria for the onset of breaking have not been established. It is noted, however, that the method is well suited to this type of problem, and other examples could be calculated without modification of the program. Finally an example of obhque entry at constant speed is shown m Fig. 16. This type of calculatmn is relevant to the problem of calculating the pressures on a partially submerged propellor where the blades enter (and exat) the water (see YimT). In contrast to the non-oblique entry cases, it is essential to remove the sharp point at the wedge vertex as shown in order to calculate realistic pressures. On the right hand, or suction side the pressures are negative which indicates that either cavitation or ventilation from the free surface will take place. More detailed study of this phenomenon should be possible using the present technique, but lies outside the scope of this paper.
5. CONCLUSIONS This paper has considered tune-dependent motions of wedges of various angles with both gravity and the nonlineanty of the boundary conditions on the wedge and free surfaces taken into account for the first time. The treatment of the intersection point due to Lin, Newman and Yue as has been extensively tested and shown to be satisfactory for a variety of transient motions capable of being treated by the versatile numerical approach. However, in cases where a jet is ejected, this intersection point treatment can result m negative pressures on the upper part of the wetted wedge surface. Since the jet is very thin, particles near the tip must behave very much as projectiles and may therefore be free to 'peel off' the wedge surface. The exact description of this phenomena is not yet clear but a possibility is the introduction of new free surface particles between the original mtersection point at the jet tip and the wedge surface - a somewhat unusual situation
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not encountered in, lot example, the trdnMent lnotloil cases where the intersection point is always the same fluid particle A modified numerical scheme suggests that tins occurs and achieves some success m simulating these flows. Howevm, it is shown that practical and accurate calculation of the rest of the fluid is possible without complete resolution ol the jet. The self-similarity assumed m earlier theories for high speed wedge entry has been validated very accurately for wedge half-angles ~ ~< 30 °, and approximately for c~ = 45 ° For larger wedge angles severe numerical problems were encountered.
ACKNOWLEDGEMENTS Some of the groundwork for this paper was done at the Dept. of Ocean Engineering M.I.T. where the photographs were taken with W.-M. Lan (see Greenhow and Lan~); the program was developed at the Dept of Mathematics, UmversIty of Manchester and the bulk of the computation was done at The Norwegmn Marine Technology Research Institute (MARINTEK) under an internal project 'Generation and Analysis of Waves and Currents m the Ocean Basin'. I wish to thank these restitutions as well as Prof. J. N. Newman, Dr W.-M. Lm and Dr T. Vlnje for thmr helpful suggesttons
REFERENCES 1 Vmje, T. and Brevlg, P Nonhnear two-dimensional ship monons, Proc 3rd Int ConJ Num. Shtp Hydro, Paris, June, pp 257-266, 1980). (See also 'Nonhnear two-dimensaonal ship motions, S I S Report ¥o 11281, NHL, Trondheim, for further detatls.) 2 Greenhow, M. and Lm, W.-M. Nonlmeaz free surface effects. Theory and experiment, M I T. Dep Ocean Eng Rep. 83-19 Unpubhshed report 3 Yamamoto, Y., hda, K., bukasawa, T., Murakaml, T., Aral, M and Ando, A. Structural damage analysis of a fast ship due to bow flare slamming, Int Shtpbulldtng Prog. 1985, 32 (369), 124 4 Attfleld, K. Gas platforms how designers underestimated the North Sea, Of]shore Engineering 1975, June, 19 5 Rldley, J. A A study of some theoretical aspects of slamming, N.M1 ReportNo R 158, 1982 6 Chapman, R. B Large amphtude motion ot two-dimensional floatmg bodms, J. Shzp Res 1979, 23 (1), 20 7 Ylm, B investigation of gravity and venttlatlon effects m water entry of thin foils, Proc 1UTAM Syrup on Nonsteady Flows at Htgh Speeds, Lemngrad, pp 471-489 8 Doctors, L J Solution of two-dxmenslonalslamming by means of grote pressure elements, 3rd hTt Symp Num Ship Hydro, June, Pans, pp 559-577 9 Mackie,A G. A hneansed theory of the water entry problem, Q. J Mech Appl. Maths. 1962, XV (2), 137 i0 Mackie, A G Gravity effects m the water entry problem, J Aust Maths Soc. 1965, 5,427 11 Moran, J P The vemcal water-exit and -entry of slender symmetric bodies, J. Aerospace Sctences October, 803 (see also Moran, J P On the hydrodynamic theory of water-exxt and -entry, TAR-TR 6501, Therm Advanced Research Inc, Ithaca, N Y , March 19651 12 Borg, S I Some contrxbutmns to the water-water entry problem, Proc. .elm. Soc Ctvtl b.)zg J Engng Mech Dry , Vol 83, No EM2, Paper 1214, pp. 1-28, 1957 13 Von Karman, T. The impact of seaplane floats during landing, 3/A CA TN321, October, Washington, 1929 14 Wagner, H Uber Stoss- und Gleatvorgange an der Oberflache yon Flusslgkelten, Z A MM 1932, 12, 193. (See also 'The Phenomena of Impact and Planning on Water", N A.C.A Translanon 1366) 15 Pierson, J D. On the vxrtual mass of water associated wxth an Immersing wedge, J Aeronautieal Sctences 1951, June, 430
Wedge entry into initially calm water: M. Greenhow 16 17
18 19 20 21 22 23 24 25 26 27 28
Fabula, A. G. Ellipse-fitting approximation of two-dimensional normal symmetric impact of rigid bodies on water, Proc. 5th Mid-Western Conf. Flutd Mech , pp 299-315, 1957 Faltmsen, O , Kjaerland, O., NCttveit, A and Vinje, T. Water impact loads and dynamic response of horizontal circular cylinders m offshore structures, OTC-paper 2741, Houston, Texas, 1977 Mlloh, T. Wave slam on a sphere penetratmg a free surface, J Eng. Maths 1981, 15 (3), 221 Shfffman, M. and Spencer, D.C The force of lmpact of a cone striking a water surface (vertical entry), Communs Pure Appl. Maths 1951, 4 , 3 7 9 Chuang, S.-L. Experiments on slamming of wedge-shaped bodies, J Shtp Res. 1967, 11 (3), 190 Borg, S. F. The maximum pressures and total forces on straightsided wedges with small deadrise, J Am. Soc Naval Engineers 1959, 71, August, 559 Gurevich, M. I. Theory of Jets in Ideal FluMs, Academic Press, pp. 435-446, 1965 Dobrovol'skaya, Z. On some problems of singularity flow of a fluid with a free surface, J. FluM Mech. 1969, 36 (4), 805 Hughes, O. F. Solution of the wedge entry problem by numerical conformal mapping, J FluidMech. 1972, 56 (1), 173 Garabedian, P. R. Obhque water entry of a wedge, Communs Pure Appl Math. 1953, V1, 157 Mackle, A. G. The water entry problem, Q. J Mech. Appl. Maths. 1968, XXII (1), 1 Johnstone, E. A. and Mackle, A. G. The use of Lagranglan coordinates in water entry and related problems, Proc. Comb. Phtl. Soc 1973, 74,529 Greenhow, M. and Lm, W.-M. Numerical simulation of nonhnear free surface flows generated by wedge entry and wavemaker motions, Proc. 4th Num. Conf Shtp Hydro, Washington DC, September, 1985
29 30 31 32 33 34 35 36 37 38 39 40
Yim, B. Numerical solution of two-dtmensional wedge slamming with a nonlinear free-surface condition, Proc. 4th lnt Conf. Num. Ship Hydro., Washington, September, 1985 Nichols, B. D. and Hirt, C. W. Nonlinear hydrodynamic forces on floating bodies, Proc. 2nd Conf. Num. Shtp Hydro., Berkeley, pp. 382-394, 1977 Longuet-Hlggms, M. S. and Cokelet, E. D. The deformation of steep surface waves on water. I. A numerical method of computation, Proc. Roy. Soc. Lond A 1976, 350, 1 Greenhow, M., Vmje, T., Brevig, P. and Taylor, J. A theoretical and expertmental study of the capsize of Salter's Duck m extreme waves, J. FlutdMech. 1982, 118,259 Kravtchenko, J Remarques sur le calcul des amphtudes de la houle hneaire engendree par un batteau, Proc. 5th Conf. Coastal Eng, Grenoble, September, pp. 50-61, 1954 Peregrine, D. H. Flow due to a vertical plate moving in a channel, Unpubhshed notes l_an, W-M, Newman, J. N. and Yue, D. K. Nonlinear forced motions of floating bodies, Proc 15th Syrup Naval Hydro, Hamburg, Germany, Session 1, pp. 1-15 Greenhow, M. Survival of surface piercing wave energy devices m extreme waves, Proc. IUTAM Con/ Wave Energy, Lisbon, Portugal, July, 1985 Taylor, G. I. The instability of liquid surfaces when accelerated m a direction perpendicular to thetr planes. I, Proc. Roy. Soc. London A 1950, 201,192 Longuet-Higgms, M S. Parametric solutions for breaking waves, J FluM Mechantcs 1982, 121,403 Garabedlan, P R Asymptotic description o1 a free boundary at the point of separation, Proc Symp. Appl. Maths. 1965, 17, 111, ed R Finn, Am. Math. Soc. Pubhcation Blsphnghoff, R L. and Doherty, C S. Some studies of the Impact of vee wedges on a water surface, J Frankhn lnt 1952, 253,547
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This paper presents results of calculations based on the Cauchy's theorem method of Vmje and Brevig1 for the two-dtmensional entry of wedges of various angles into initially calm water. The problem has a long history winch is briefly reviewed in the introduction, and significant progress has been made with both linear theories (valid for low entry speed) and with theories which treat the free surface conditions exactly but with the assumptions of zero gravity and constant speed of entry. This simplifies the problem to one which 1s self-similar in dimensionless space variables ~ = x/vt and = y/vt and this has a number of consequences. For wedges with half-angles up to about 45 ° and with high entry speeds, the numerical approach, winch Includes gravity, validates these assumptions and the agreement between both free surface displacements and pressure distributions on the wetted wedge surface is excellent except in the region of the jet of fluid which rises up the side of the wedge Because the potential flow initial value problem is singular at the Intersection of the free surface and wedge surface, exact numerical resolution of the jet is not possible. Nevertheless, the rest of the fluid motion Is insensltwe to the treatment of the jet, which itself may be calculated quite realistically. Of particular interest (but little practical relevance) is the pressure on the upper part of the wedge surface (in the jet region) wluch according to self-similar theories is very small but positive, but which is calculated to be small but negative by the numerical scheme. This effect, which is enhanced when gravity is included, is insensltwe to the numerical resolution of the jet and suggests that the jet may separate from the wedge surface, the new intersection point being where the pressure vanishes on the wedge surface. A modified numerical scheme allows this to happen and the results are an quahtatlve agreement with the experiments by Greenhow and Lln.2 The numerical method presented here is extremely versatile and a number of other effects may be explored. Examples of transient motion, non-constant speed of entry, oblique entry and complete penetration of the surface so that a cavity as formed behind the wedge are presented.
1. INTRODUCTION Thls work seeks to quantify the pressure distributions and forces on a body when it enters initially calm water, and the resulting fluid motion. It is motivated by the slamming of ship bows as they re-enter the water in heavy seas, and by the engulfing of oil rig cross-members by large and possibly breaking waves. For ships the magmtude and dlstnbutton of the pressure ts sometmaes sufficient to deform the bow area quite severely (see Yamamoto et al.3), whale for rigs the wave impacts may be a serious source of fatigue (see Attfield4). These difficult problems may be simplified by assuming that they are equivalent to the entry of a body into calm water with velocity equal to the relative velocity of the body and the moving free surface. Such an assumption has been justified for fixed cyhnders being engulfed m waves of sufficiently long wavelength by Rldley s and is expected to hold when the body velocity greatly exceeds the fluid velocities within the wave, as is the case for free fall lifeboats for example. In any case, the present work is regarded as an essential prerequisite to the full problem of a body entering a wave. These comments reflect the importance of the study of water entry and the need for practical methods of solution. In view of the significant progress Accepted December 1986. Discussion closes December 1987.
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made recently and the fact that Moran's (1965) 11 extensive review of earher work is not widely available, it seems worthwhile to briefly present the basic ideas and approaches below. For entry into water at speeds low enough to regard the free surface displacements as small, one can linearlse the free surface condition while treating the body boundary conditions exactly (i.e. the boundary condition is applied on the actual body surface). Papers in this category include Chapman,6 Yim 7 and Doctors.8 Alternatively, if the wedge half-angle a (see Fig. 1) is small, then the free surface displacements wall be small even for high entry speed. This slenderness assumption has been used by Mackle, 9'1° Moran 11 and Borg x2 and we make a comparison with the numerical results for a = 9 °. Of interest IS Mackie's 9 closed form solution for the free surface elevation. r/ .
In.
+.
.2~ tan -1 1
2
(1)
rr
where ~ = x/vt and 7? = y/vt with v being the wedge entry velocity. For the entry of non-slender wedges at high speed, the usual approach ts to equate the force on the body with the rate of change of added mass of the body as it enters the water times ItS velocity (see, e.g. yon Karman,~3Wagner,~4
0141-1187/87/040214-10 $2.00 © 1987 Computational Mechanics Pubhcations
Wedge entry into initially calm water: 214. Greenhow
t
Figure I.
L
Definition sketch
Pierson Is and Fabula16). This approach is fairly simple and has gained widespread use for a variety of body shapes (for a cylinder see Faltinsen et al., 17 for a sphere see Miloh, TM and for a cone see Shiffman and Spencer*9). It is also possible to take into account the free surface rise, albmt approximately, by using a method described by Wagner) 4 In that paper Wagner also models the wedge entry problem by an expanding flat plate of width equal to the instantaneous (elevated) waterline of the body during penetration, and this gives the pressure distribution on the wetted wedge surface as p ( x ) = ~Ov2
[
tan6(1 --x2/L2) 1/2
2y
- - - - (L 2 --x2) a/= u2
1 --x2/L 2
1
(and still less the contact angles) is possxble even when this jet separation is not allowed, because of the numerical difficulties of resolving the jet. When considering splash height and contact angles, an analytical description of the fluid motion would be highly destrable. Unfortunately, even with the assumptions necessary for self-similar flow, no complete solution has been found yet, but significant analytical progress has been made by Garabedian, 2s Mackle 26 and Johnstone and Mackie? 7 Garabedlan shows that the arc length along the surface between any two free surface particles remains constant with time and since the present numerical method follows free surface particles, this is a particularly useful result. (This was first shown by Wagner.14 Garabedlan also solves a wedge oblique entry problem subject to somewhat unphysical boundary conditions.) Mackie 26 discusses the condition of convexity to the fluid of the free surface, which Is guaranteed If the pressure Is positive on the wetted wedge surface. Assuming this to be true, Mackle places bounds on the contact angle: (i) For all c~, 0 rr/4, fl < rr/2 -and considers the curvature of the free surface at the contact point, which if zero implies that fl < rr/6. By considering the problem with Lagranglan variables, Johnstone and Mackie show that' cot fl + 2fl = rr + 2a
(2)
Here j~ is the wedge acceleration and other symbols are defined in Fig. 1. At the ends of the plate, x = + L, the fluid velocity is infinite and so equation (2) cannot be valid there, but it does predict the maxima in pressure some way away from the wedge vertex observed by Chuang 2° and others for small deadrise angles 8. For intermediate 8, however, approximating the body by a fiat plate IS clearly very crude: for 8 = 30 °, Fig. 7 shows disagreement with more exact approaches and with Borg'sal approxtmate analysis which is also simple but more accurate over most of the wedge surface. The problem of constant-velocity entry of a wedge with general angle and fully non-linear free surface conditions, but with gravity neglected, may be simplified by using the self-similarity variables ~ and ~7 (see Gurevich 22 for the more general result for non-constant velocities). Solutions based on self-similarity have been given by Borg la using a relaxation method, and Dobvrovol'skayaas and Hughes 24 using the Wagner h-function (see Gurevich22). Although these solutions are hmited to zero gravity self-similar flows, they do yield much information about the flow: the spray root singularity position and order, splash height and con. tact angle with the wedge, shape of the free surface and pressure distributions on the wedge m~¢ all be calculated at least for half-angles up to about 60~. Hughes presents experiments which clearly validate the self-similarity assumption when the entry speed is high except m the jet region where gravity (and other effects) may play a part. Thus these self-similar solutions provide and rigorous check on the numerical method presented here. As will be seen, the agreement between the free surface profiles and pressure dlstribuUons is good except m the jet region where the effect of gravaty enhances a negative pressure region on part of the wedge surface and this is thought to lead to jet separation. No direct comparison between the splash heights
(4)
While this relation satisfies condition (31) above (m fact fl hes between 23.2 ° and 9.5 ° for a varying between 0 and 90°), condition (3il) is violated if a exceeds approximately 79 °. This lmphes that the pressure within a region of fluid winch contains part of the wetted wedge surface must be negative, an interesting result in view of the present results which show a similar effect for even smaller wedge angles especially when gravity is included. It will have been noticed that practical apphcation of the above theories is lmalted to low speed of entry, slender body, zero gravity or self-similar muaUons. To avoid these restrictions and to provide good resolution of the fluid motion, Greenhow and Lm as and Yim 29 have used the Vinje and BrevigI approach based on Cauchy's theorem. (Another possibility is the method presented m Nichols and Hirt a° but this reqmres solving equations in the entire fluid domain, rather than at the boundary only.) Ylm discusses the singular nature of the initial and boundary value problem, by analogy with the potential flow perpendicular to a plate, which has infinite velocity at its ends as mentioned before. This has the undesirable consequence that good resolution of the fluid near the intersection of free and wedge surfaces causes very large velocities in the jet which are difficult to handle numerically and may lead to breakdown. However, Greenhow and Lin show that resolving the jet rather more poorly avoids this problem and does not lead to sigmficant errors m the free profile away from the jet or in the pressures calculated on the wetted wedge surface. The present paper considerably extends these catculatmns. 2. SOME COMMENTS ON THE METHOD The method of Vmje and Brevlg, which is based upon the approach of Longuet-Higgins and Cokelet,al is well described in Vinje and Brevig 1 and Greenhow et al. a2 and we therefore only describe here recent developments concerning the
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Wedgc entry into blittally cabn water 34. Green~tow treatment of the intersection points of the body and free surfaces. The problem arises because of the need to satisfy two boundary conditions at this point, namely the body boundary condition and the free surface boundary condition. In linear steady state wavemaker theou¢ it is well known that imposing both boundary conditions can result m a singularity in the complex potential ~(z) = q~ + t~, see particularly Kravtchenko 33 Here ~bis the velocity potential, IS the stream function and z = x + 0'. For time dependent problems, Peregnne 34 has analysed tire Impulsive start-up to velocity U of a wave maker in water of depth h. His calculations predict a logarithmically singular free surface profile at the intersection point x = 0 given by ). . . .
In tanh 7r I_ k4h 7A
(5)
which is m close agreement with the experiments of Greenhow and Lm 28 except at the intersection point where a jet is ejected almost perpendicularly to the wavemaker. In order to deal with this and other wavemaker motions, and as a prototype calculation for general surface-piercing bodies, Lln, Newman and Yue 3s modified the Wnje and Brevlg method as described below. Since /3(z) is analytic within the fluid domain, then Cauchy's theorem holds. : 0
(6)
Z --Z 0
c with z0 outside the contour of integration C, which is taken to include the free surface and body surface. C is divided into two parts, C¢ on which ~ is known and C¢ on which is known. Letting z o move onto the contour of mtegranon C, and taking real or Imaginary parts of equation (5) as appropriate yields.
Oo~(X,y,t)+Ref¢+t~dz=O
for Zo on C¢
(7)
Z --Z 0
and
tt
Oo¢(x,y, t) + Re i
= dZ
--g o
o
)
Here 0o is the angle between two tangents of C at Zo (mathematically equal to zr for any smooth part of C, but generally different from 7r numerically, especially in regions of sharp free surface curvature). Numerically equations (7) and (8) are dlscretlsed at N points around the boundary and this results in an N × N matrix equation of the form Ax = b where x is a vector of unknowns corresponding to the unknown part of/3(z), (either ¢ or 4). Similar equations and comments to the time derivatives Odp/bt and ~ / ~ t needed to step the system forward in time and to evaluate pressures on the body. It is not clear how to apply equations (7) and (8) when the point Zo is at the intersections of the free and body surfaces since both q~ and ~b are known there. Lm, Newman and Yue 3s specify both q~ and ff here and remove these equations completely from the system, resulting i n n being replaced by N - - 2 in the numerical method. This appears to be completely satisfactory for practical calculations giving results for the wavemaker problem which are almost indistinguishable from equation (5). Furthermore Greenhow ~ also uses this algorithm to calculate extreme wave profiles with fixed or moving cylinders m the free surface, giving reahstlc results.
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o
/ Figure 2. Comparison o f sell-similar flow with the intersection point retained on the wedge surface (shown by o) with the results of Dobrovol'skaya (shown dotted). The solid hne shows the flow when the let ts allowed to leave the wedge surface, a = 30 °
In contrast to this, Ylm 29 presents results of alternative formulations which specify either 4~ or ~b (but not both) at the intersection points This reults in unreasonable free surface elevations for the wedge entry problem and he concludes, along with Greenhow and Lin 2s that the above algorithm is preferable. A related concern of considerable nnportance in the numerical lmplementanon of equations (7) and (8) is how well one should try to resolve the behavlour of the free surface near the possibly singular intersection points. For the wedge entry problem, good resolution around this point leads to a very high velocity jet which almost immediately leads to numerical breakdown. On the other hand, poor resolution here leads to a violation of the conservation of fluid mass as experienced by Yim. 29 We here present numerical evidence to show that between these two extremes there IS a broad range of resolutions where practical calculations can take place, at least for small and moderate values of a. Within this range the jet profiles calculated are thought to be fairly realistic in some examples, but in any case it is shown that the rest of the free surface as well as the pressure distributions on the wetted wedge surface xs insensitive to the treatment around the intersection points. For large values of a the calculations are less accurate since the jet must be inhibited (by resolving it poorly) m order to time step the numerical solution sufficiently many times to reach an approximately self-similar solution which is mdependent of the initial conditions. Most of the computations presented here were run on a VAX-785. Typically, with 84 points on the free surface and about 150 in total, the execution took about 50 s c.p.u per time step. 3. RESULTS - FLOWS WHICH A R E E S S E N T I A L L Y SELF-SIMILAR
For tugh speeds of entry it Is reasonable to expect selfsimilarity since locally near the wedge the fluid particle acceleration will be much greater than gravity, especially in the early stages of entry. As suggested by Mackie, 9 the effect of gravity vall be unimportant if t ' < v/g (since gravity introduces an extra variable gt/v which is neglected
Wedge entry into initially calm water," M. Greenhow m the self-similar assumption). The calculations here suggest that gravity may be ignored if t<<.v/2g, except perhaps in the jet region. Figure 2 shows the flow for a wedge of half-angle a of 30 ° entering the fluid at high speed. Starting from the mitml conditions with just the wedge vertex in calm water, the free surface quickly becomes self-similar and compares well with the results of Dobrovol'skaya.2a The particle at the jet t:p is treated as an ordinary free surface particle for the purposes for time-stepping the solution except that since in general the predicted position does not lie exactly on the wedge, the particle xs moved horizontally back onto the wedge surface at each time step from a posmon just outside the wedge envelope. This algorithm fulfills the condition, assumed by earher authors, that the particle imtially m contact with the wedge will always stay in contact, but results m flow with negatwe pressure on the upper
20
24-
V/ Figure 3. Development o f the flow when the ]et is allowed to leave the wedge surface. The point o f intersection o f the free and wedge surfaces is shown by e. Also shown on the arrowed curve are typical results from a computation with coarse spacing o f the free surface points, where although the let is poorly resolved, the rest o f the free surface is almost coincident with the more accurate calculation. c~= 30 °
ta¢
•
ca v~
K
I0
\
x "~
lo
lg
~o
3~;
Figure 4. Comparison o f the calculated pressure distribution (o) with that o f Dobrovol'skaya 2a shown as the solid line and terminating in o. The dashed lines are the pressure distributions for a wedge which is decelerating, a = 30 °. The wedge vertex is at dimensionless depth --1
,X
L
. xz(v"
F.
o., Lo
I
PRESSURE
COMPONENT
Figure 5. Development o f the pressure distribution .for a wedge which is decelerating. The numbers indicate the time step, o indicates the pressure distribution at time step 16 with constant wedge velocity and . . . . inidcates the pressure distribution for a decelerating wedge but with the jet retained on the wedge surface (note the small region o f negative pressure). The effect o f the initial conditions is dearly seen at time steps 4 and 8, a = 30 °
portion of the wetted wedge surface. Although the magnitude of the negative pressure is small compared with the pressures on the lower port:on of the wedge, the length of the wetted wedge surface where the pressure is negative is by no means small as can be seen in Fig. 5. Tins leads us to examine the physical conditions m the jet region more closely. Certainly the particle at the intersect:on point cannot move through the wedge surface. However, the pressure gradient within the very thin jet must be extremely small, and hence the fluid particles within the jet, and in particular the intersection point part:cle, will behave approxamately as free projectiles moving under graxaty alone. Clearly the numerically calculated pressure gradient within the jet is insufficient to keep it on the wedge surface, and doing so artlficxally, as in Fig. 2, may result m the negative pressure observed. In an effort to examine this effect further, some runs were made with :dentical numerical intervals in space and time, but with zero gravity. This resulted m the jet moving shghtly further up the wedge surface, and for the lower portion of the wetted wedge surface (below the undisturbed free surface level), the difference m the pressures was almost exactly hydrostat:c. Both of these effects were expected. In contrast, in the jet region, small negative pressures were again observed, with a magmtude only slightly smaller than for flows with gravity. This disagreement with the results of Dobrovol'skaya23 and Hughes ~ certainly indicates a weakness in the present scheme, probably assocxated with poor numerical resolution of the jet. With gravxty included, the negative pressure region may be a least partly physical but we can draw no firm conclusion from these numerical results. What happens if negative pressure actually occurs at the intersection point xs not clear. One possxbility lS that the free surface breaks up because of the pressure inversion across it as in Rayleigh-Taylor instability (see Taylora7). (The real situation would be further complicated by viscosity and surface tension which are neglected here .) This instability would eventually preclude further time stepping of the solution, and indeed calculations wath the intersection
Applied Ocean Research, 1987, Vol. 9, No. 4
217
Wedge entry into imlmlh' eahn water M (;reenh~m'
Figure 6a
High speed entry ol a wedge l~'ltll c~ = 30 °
cultles in resolving the jet nuinertcally muat bc lccognlsed, since its position and profile ale very' senslnve to the spacing between particles and the tlnre step ~,hosen in the plogIam. The same comment applies (to a lesser extent) to the position of the (new) Intersection point. Physically we might have expected this since near the intersection point the jet leaves the wedge surface tangentially upwards and hence the pressure gradient along the wedge surface here must be very small. On the other hand, the rest of the free surface and" pressure dlstrlbuhon on the wetted wedge surface is almost entirely Insensitive both to the numerical spacings and to the treatment of the jet This is the main point here since it permits practical and accurate numerical calculation for the majotlty of the flow for many time steps by avmdmg the abovementioned calculation breakdown when the jet is retained on the wedge If the (small) hydrostatic component is subtracted from the pressme along the wedge, then the agreement with the results of the zero-gravity calculations and of Dobrovol'skaya 2a is excellent except m the jet region, see Fig. 4 It IS usual to define a force coefficient CF as (9)
CF = Flpvat = F/pvgD
whine F is the actual fmce on the body a n d D is the depth of the vertex. The results here are influenced by the initial conditions for the early stages of entry but appear to conveige from above to CF = 0 97 for later stages. Dobrovol'skaya gives CF = 0 925 including a factor of 1/2 noted by Hughes, 24 a difference of about 5%.
Figure 6b High speed ento' ot a wedge with a = 30 °. Oblique I,lew
point kept on the wedge surface do break down near the let some time after this pressure inversion. However, Fig. 6b, taken together with the numerical work below suggests another possibility, namely that the jet fails away from the wedge surface introducing new free surface between the tip and the (new) Intersection point. Such a mechanism would reqmre the material tame derivative of pressure to be zero on a (moving) zero pressure contour and this has been examined by Longuet-tttgglns as as a model of the rear face of a breaking wave For wedge entry such a (semi-Lagrangmn) approach would need to satisfy the wedge surface condition and have intersecting free surfaces at the jet tip, which does at least seem possible. To explore this possibility numerically Figs. 2 and 3 show results obtained by treating the intersection point particle as an ordinary free surface particle without keeping it on the wedge. The new mtersectmn point is then taken to be on the wedge surface where the pressure vanishes. We notice first that the flow outside the jet region is almost ennrely unaffected, see Fig. 2, as is the pressure distribution on most of the wetted wedge surface. The form of the jet, wlueh ts calculated by following the fluid particles and employs no explimt smoothing, is extremely suggestive, especmlly when compared with the photographs from Greenhow and Lin 2 shown in Fig. 6. However, the diffi-
218
A p p h e d Ocean Research, 1987, Vol. 9, No 4
laa
-
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g
,,,,
,
,
,
f
as°3o •
,s"
Figure 7. Dimensionless pressures (o) f o r various wedge angles compared with the results o f Dobrovol'skaya =a and Hughes. ~ Small deadrise angle approximate theo,~es are also shown f o r ~ = 60 °. Borg 2' .. and Wagner a4(equation 2 hereto) . . . . The wedge vertex is at dimensionless depth --1
Wedge entry into initially calm water: M. Greenhow Also shown in Fig. 4 are two dotted graphs of the pressure distribution when the wedge does not have constant velocity but is free to accelerate under gravity, hydrodynamic and hydrostatic forces. This condition is typical for drop tests, and here the mass of the wedge is chosen to correspond with the Greenhow and I_an experiments. We notice that although at time step 24 the position of the wedge and free surface is almost indistinguishable from that shown in Fig. 3, and the velocity has decreased by only 6%, the acceleration of the wedge Is almost 3 g upwards. As expected from equation (2) this has an important effect on the pressures whose development is shown in Fig. 5, and hence on the value of CF (for time step 24, CF = 0.725). These results underline the importance of keeping the wedge acceleration small during drop tests, preferably by driving it through the free surface rather than by letting it fall through, and may account at least in part for the wide scatter in data for the force coefficients, especially for cylinder slamming where the forces wtll be even larger than for wedges slamming. We note that in the above calculations, the fluid mass was always conserved to within 5% of the mass of fired displaced by the wedge. It appears that Yim's 29 difficulty m conserving the fluid mass was primarily due to poor numerical resolution, which could not be improved satisfactorily because the jet was retained on the wedge surface and this quickly led to numerical breakdown with good resolution. Fortunately this problem is somewhat alleviated
Figure 8. Development o f self-similar flow Jbr high speed wedge entry with ~ = 9 ° from initial conditions shown in solid black, o denotes results o f Dobrovol'skaya ~ whtch show good agreement with the present calculation in the early stages o f entry. Note that the wedge has penetrated the free surface entirely in the later stages, leaving a cavity behind it
Ftgure 9. Complete penetratton o f the free surface by a wedge with a = 9 °. Note the slight closmg o f the cavity behind the wedge
by letting the jet separate from the wedge as described above. It should also be noted that for some of the calculations the mass error was only 1% to 2%, and this is much less than the mass of fluid above the level of vertical tangency of the free surface defining the jet region (about 5%). Therefore on this basis we conclude that even an approximate calculation of the jet IS preferable to smoothmg it out or simply removing it during the calculations. Since the numerical method follows the free surface particles (Lagrangian description) it is easy to see that Garabedian's a9 result for the constancy for arc length along the free surface between any two particles is accurately satisfied for the bulk of the free surface. In the jet region, however, the particles tend to spread out (see Fig. 3). The above discussion and comments apply to wedge half-angles of a = 9 ° and 15 °, the second of which was treated at length by Greenhow and Lin. 28 Figures 7 and 8 show the pressure distributions and free surface profile for a = 9 °. For both a = 9 ° and a = 15 °, the ejected jet actually moves inside the wedge envelope, but for ct = 15 °, Greenhow and Lin 28 replace the calculated jet by a jet which assumes that the jet particles do indeed move as projectiles. In view of the difficulty of resolving the jets more accurately and the fact that the exact jet form has virtually no significance for the rest of the flow, this seems to be a practical approach. However, for the complete penetration of the free surface by the wedge shown in the later stages of Fig. 8, the cavity formed behind the wedge is free to collapse reside the wedge envelope. This effect is also seen experimentally as in Fig. 9. We note that in the calculation for ot = 9 ° the jet velocities are low enough to be able to continue the calculations almost indefinitely (here to over 100 tune steps) and it is gratifying that the flmd mass is conserved to within 2% even at these late stages. Figure 7 also shows that the results of Dobrovol'skaya are m good agreement with the present calculation, whilst Fig. 10 compares the small ct analysis of Mackie 9 (equation (1) m
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Wedge entry into tnitlall3' cabn water: M. Greenhow
•- ~
~-,..;'~"
~
.__,
Figure 10. Present results f o r a wedge with a = 9 ° compared with the small a s y m p t o t w forms o f Mackie 9 . . . . and Borg 21 .
.
.
needed to give more accurate results. Finally we note that the tbrce coefficients Cf for c~ = 9 ° and 15 ° ale also ill good agreement with Dobrovol'skaya, being 0 034 and 0.l 3 respectively. In contrast to the excellent results above for wedge angles c~~< 30 °, we now discuss the case with c~ = 45 ° and compare the results with Hughes a4 As mlgh( be expected from Fig 11, very severe difficulties are encountered when is as large as 45 ° because of the fast m o w n g l e t Starting from the mltml conditions of Fig 12, the free surface development may be calculated quite realistically, but the jet causes the numerical procedure to break down before self-slmdar flow IS attained (I.e. the results are stdl Influenced by the mltml conditlons). This is thought to be the cause of the high pressures on the wedge surface (about double those of Hughes). Another shortcoming of the results is that the conservatmn of fluid mass is only approximately satisfied (to about 12%). By placing free surface particles so that the jet IS less well resolved, the procedure may be time stepped untd an approximately self-slmdar flow is attained. The pressures along the wedge shown m Fig. 7 are still too large and this is almost certainly a consequence of the poor mass conservation, see Fig. 12. More seriously (and despite considerable effort and computer time), no systematic method for improving the resolution of flows which become self-similar has been found for large c~ whach simultaneously gwes accurate free surface profiles and pressure dxstrlbutlons on the wedge surface This places some restrictions on the sensible use of the program )'or example conect results for a = 60 ° could not be obtained even approxamately. In wew of this it seems maportant to compare the existing theories )'or the pressure distributions, see Fig. 7. We see that the results of Borg 21 and Dobrovol'skaya 2a are m quite close agreement especmlly in the region of lugh pressure occurring close to where the free surface overturns For Wagner's ~4 theory (see equation (2)) the agreement IS not so good, although experiments of Chuang a° and Bisphnghoff and Doherty 4° support its use except for very small deadrlse angles (~< 10 °).
Figure 11. The high speed entry o f a wedge with a = 45 ° N o t e the long thin j e t close to the wedge surface
this paper), and Borg. 21 Neither result is particularly good for wedge angles as large as 9 ° , and we conclude that the practical utthty of these small angle theories is very limited indeed (see also Dobrovol'skaya's comments). The agreement of the pressure distribution on the wedge with a = 9 ° between Dobrovol'skaya's results and those of the present calculations is excellent except in the jet region as expected, and at the wedge vertex where better resolution would be
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Figure 12. Development o f the free surface f o r ~ = 45 ° f r o m the initial eonditions shown in solid black. --- -denotes the results o f Hughes (1972) while . . . . denotes results o f the numerical procedure with poor resolution needed to inhibit the l e t
Wedge entry into initially calm water: M. Greenhow
effects are important is shown in Fig. 13 where a displaced wedge oscillates about it equilibrium position generating waves. The wedge starts from rest at the initial position shown. The pressure is always positive on the wedge face and the solution may be continued almost indefimtely, the only limitation being reflections which would ulttmately return from the distant vertical boundaries. Clearly in the early stages shown here this has not yet happened and the calculations are accurate and realistxc. Of particular interest is the locus of the intersection point particle which moves in an almost elliptical path. For transient motion the mass of the wedge as well as its angle is an important parameter for determining the flow and, in particular, whether overturning occurs or not. Greenhow and Lin 28 give an example of a wedge with a = 15 ° but with four times the mass of the wedge in Fig. 13. This gives a wave which breaks some distance from the wedge, although the details of the overturning crest were not resolved• Figure 14 shows an example with a = 30 °
F •
"
"
~
•
Figure 13. Transtent motion o f a wedge wtth a 15 (a) The wedge initial position (solM black) is displaced f r o m its equilibrium position which has its vertex at x. The wedge moves downwards generating a wave. (b) The wedge moves up again. N o t e the locus o f the intersection point . . . . . From Greenhow and Lin 28
I
, /t[ A.Z'~, ~,..__ [ _ /'~,.Z__" Y ~ / " - . . ~ ¢ ~ ~-"~.~_.._ '/////// -
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....
r
,
4. SOME EXAMPLES OF NON-SELF-SIMILAR WEDGE MOTION This section presents results for the entry of a wedge for which the resulting flow is not self-similar, either because the wedge speed is slgmficantly variable, or because gravity is tmportant, or both. An interesting example where both
Figure 14. Transient motion and localised breaking f o r a wedge with a = 30 °
2o .1
1.......__._
0
20
}(
0
Figure 15. Transient motion o f a 60 ° wedge started f r o m rest below its equilibrium position with vertex position marked x. The numbers mdicate the time step and . . . . is the locus o f the intersection point
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Wedge ento' into mitzally cahn water M. Greenhow
", ,'\t -y Figure 16. Oblique entry o f a wedge at high constant speed. ~ = 7.5 ° and the wedge centreline is inclined 1 Z5 ° from the vertical The dotted lines and axes represent the non-dimensional pressure P/1/2pv 2 distrtbution along the wedge which is negative on the suction stde
where a wave is starting to form, but where the jet of fluid originally moving up the wedge side overturns and causes breaking (which stops the calculation). As expected this also happens with larger values of a, but ff the wedge is released from a position below its equilibrium position, this problem is postponed until the wedge begins to move downwards, see Fig. 15. Because of the large number of parameters involved m these flows where gravity is important (e.g. wedge angle and mass and lnmal conditions), exact criteria for the onset of breaking have not been established. It is noted, however, that the method is well suited to this type of problem, and other examples could be calculated without modification of the program. Finally an example of obhque entry at constant speed is shown m Fig. 16. This type of calculatmn is relevant to the problem of calculating the pressures on a partially submerged propellor where the blades enter (and exat) the water (see YimT). In contrast to the non-oblique entry cases, it is essential to remove the sharp point at the wedge vertex as shown in order to calculate realistic pressures. On the right hand, or suction side the pressures are negative which indicates that either cavitation or ventilation from the free surface will take place. More detailed study of this phenomenon should be possible using the present technique, but lies outside the scope of this paper.
5. CONCLUSIONS This paper has considered tune-dependent motions of wedges of various angles with both gravity and the nonlineanty of the boundary conditions on the wedge and free surfaces taken into account for the first time. The treatment of the intersection point due to Lin, Newman and Yue as has been extensively tested and shown to be satisfactory for a variety of transient motions capable of being treated by the versatile numerical approach. However, in cases where a jet is ejected, this intersection point treatment can result m negative pressures on the upper part of the wetted wedge surface. Since the jet is very thin, particles near the tip must behave very much as projectiles and may therefore be free to 'peel off' the wedge surface. The exact description of this phenomena is not yet clear but a possibility is the introduction of new free surface particles between the original mtersection point at the jet tip and the wedge surface - a somewhat unusual situation
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not encountered in, lot example, the trdnMent lnotloil cases where the intersection point is always the same fluid particle A modified numerical scheme suggests that tins occurs and achieves some success m simulating these flows. Howevm, it is shown that practical and accurate calculation of the rest of the fluid is possible without complete resolution ol the jet. The self-similarity assumed m earlier theories for high speed wedge entry has been validated very accurately for wedge half-angles ~ ~< 30 °, and approximately for c~ = 45 ° For larger wedge angles severe numerical problems were encountered.
ACKNOWLEDGEMENTS Some of the groundwork for this paper was done at the Dept. of Ocean Engineering M.I.T. where the photographs were taken with W.-M. Lan (see Greenhow and Lan~); the program was developed at the Dept of Mathematics, UmversIty of Manchester and the bulk of the computation was done at The Norwegmn Marine Technology Research Institute (MARINTEK) under an internal project 'Generation and Analysis of Waves and Currents m the Ocean Basin'. I wish to thank these restitutions as well as Prof. J. N. Newman, Dr W.-M. Lm and Dr T. Vlnje for thmr helpful suggesttons
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Wedge entry into initially calm water: M. Greenhow 16 17
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Fabula, A. G. Ellipse-fitting approximation of two-dimensional normal symmetric impact of rigid bodies on water, Proc. 5th Mid-Western Conf. Flutd Mech , pp 299-315, 1957 Faltmsen, O , Kjaerland, O., NCttveit, A and Vinje, T. Water impact loads and dynamic response of horizontal circular cylinders m offshore structures, OTC-paper 2741, Houston, Texas, 1977 Mlloh, T. Wave slam on a sphere penetratmg a free surface, J Eng. Maths 1981, 15 (3), 221 Shfffman, M. and Spencer, D.C The force of lmpact of a cone striking a water surface (vertical entry), Communs Pure Appl. Maths 1951, 4 , 3 7 9 Chuang, S.-L. Experiments on slamming of wedge-shaped bodies, J Shtp Res. 1967, 11 (3), 190 Borg, S. F. The maximum pressures and total forces on straightsided wedges with small deadrise, J Am. Soc Naval Engineers 1959, 71, August, 559 Gurevich, M. I. Theory of Jets in Ideal FluMs, Academic Press, pp. 435-446, 1965 Dobrovol'skaya, Z. On some problems of singularity flow of a fluid with a free surface, J. FluM Mech. 1969, 36 (4), 805 Hughes, O. F. Solution of the wedge entry problem by numerical conformal mapping, J FluidMech. 1972, 56 (1), 173 Garabedian, P. R. Obhque water entry of a wedge, Communs Pure Appl Math. 1953, V1, 157 Mackle, A. G. The water entry problem, Q. J Mech. Appl. Maths. 1968, XXII (1), 1 Johnstone, E. A. and Mackle, A. G. The use of Lagranglan coordinates in water entry and related problems, Proc. Comb. Phtl. Soc 1973, 74,529 Greenhow, M. and Lm, W.-M. Numerical simulation of nonhnear free surface flows generated by wedge entry and wavemaker motions, Proc. 4th Num. Conf Shtp Hydro, Washington DC, September, 1985
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Yim, B. Numerical solution of two-dtmensional wedge slamming with a nonlinear free-surface condition, Proc. 4th lnt Conf. Num. Ship Hydro., Washington, September, 1985 Nichols, B. D. and Hirt, C. W. Nonlinear hydrodynamic forces on floating bodies, Proc. 2nd Conf. Num. Shtp Hydro., Berkeley, pp. 382-394, 1977 Longuet-Hlggms, M. S. and Cokelet, E. D. The deformation of steep surface waves on water. I. A numerical method of computation, Proc. Roy. Soc. Lond A 1976, 350, 1 Greenhow, M., Vmje, T., Brevig, P. and Taylor, J. A theoretical and expertmental study of the capsize of Salter's Duck m extreme waves, J. FlutdMech. 1982, 118,259 Kravtchenko, J Remarques sur le calcul des amphtudes de la houle hneaire engendree par un batteau, Proc. 5th Conf. Coastal Eng, Grenoble, September, pp. 50-61, 1954 Peregrine, D. H. Flow due to a vertical plate moving in a channel, Unpubhshed notes l_an, W-M, Newman, J. N. and Yue, D. K. Nonlinear forced motions of floating bodies, Proc 15th Syrup Naval Hydro, Hamburg, Germany, Session 1, pp. 1-15 Greenhow, M. Survival of surface piercing wave energy devices m extreme waves, Proc. IUTAM Con/ Wave Energy, Lisbon, Portugal, July, 1985 Taylor, G. I. The instability of liquid surfaces when accelerated m a direction perpendicular to thetr planes. I, Proc. Roy. Soc. London A 1950, 201,192 Longuet-Higgms, M S. Parametric solutions for breaking waves, J FluM Mechantcs 1982, 121,403 Garabedlan, P R Asymptotic description o1 a free boundary at the point of separation, Proc Symp. Appl. Maths. 1965, 17, 111, ed R Finn, Am. Math. Soc. Pubhcation Blsphnghoff, R L. and Doherty, C S. Some studies of the Impact of vee wedges on a water surface, J Frankhn lnt 1952, 253,547
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