The normal impact of a symmetric rigid body with an initially quiescent liquid half-space is considered using both Wagner theory and a model of viscous gas pre-impact cushioning. The predictions of these two theories are compared for a range of different body shapes. Both theories assume that the impactor has small deadrise angle. Novel solutions of the Wagner normal impact problem for a symmetric body with a power-law shape are presented, which generalize the well-known results for a parabola and a wedge. For gas cushioned pre-impacts, it is shown that a pocket of gas is entrained even for body shapes with a cusp at the body minimum. A scaling law is developed that relates the dimensions of the trapped gas pocket to the slope of the body. For pre-impact gas cushioning, surface tension is shown to smooth the liquid free-surface and delay the instant of touchdown for a smooth parabolic body, while for a wedge, increasing surface tension initially delays touchdown, before hastening touchdown as the importance of surface tension is increased further. For a flat-bottomed wedge, gas entrainment is again predicted in the gas-cushioning model although the location of initial touchdown, either on the transition between the wedge and the flat bottom or along the side of the wedge, now depends upon the parameters of the body shape.

1.
H.
Wagner
, “
Über stoß- und gleitvorgänge an der oberfläche von flüssigkeiten (Phenomena associated with impacts and sliding on liquid surfaces)
,”
Z. Angew. Math. Mech.
12
,
193
215
(
1932
).
2.
R.
Cointe
and
J.-L.
Armand
, “
Hydrodynamic impact analysis of a cylinder
,”
J. Offshore Mech. Arct. Eng.
109
,
237
243
(
1987
).
3.
A. A.
Korobkin
and
V. V.
Pukhnachov
, “
Initial stage of water impact
,”
Annu. Rev. Fluid Mech.
20
,
159
185
(
1988
).
4.
S. D.
Howison
,
J. R.
Ockendon
, and
S. K.
Wilson
, “
Incompressible water-entry problems at small deadrise angles
,”
J. Fluid Mech.
222
,
215
230
(
1991
).
5.
J. M.
Oliver
, “
Water entry and related problems
,” Ph.D. thesis,
University of Oxford
,
Oxford, United Kingdom
,
2002
.
6.
A. A.
Korobkin
and
A.
Iafrati
, “
Hydrodynamic loads during initial stage of floating body impact
,”
J. Fluids Struct.
21
,
413
427
(
2005
).
7.
R.
Purvis
and
F. T.
Smith
, “
Droplet impact on water layers: Post-impact analysis and computations
,”
Philos. Trans. R. Soc., A
363
,
1209
1221
(
2005
).
8.
S. D.
Howison
,
J. R.
Ockendon
,
J. M.
Oliver
,
R.
Purvis
, and
F. T.
Smith
, “
Droplet impact on a thin fluid layer
,”
J. Fluid Mech.
542
,
1
23
(
2005
).
9.
J.
Philippi
,
P.-Y.
Lagrée
, and
A.
Antkowiak
, “
Drop impact on a solid surface: Short-time self-similarity
,”
J. Fluid Mech.
795
,
96
135
(
2016
).
10.
S.-L.
Chuang
, “
Slamming of rigid wedge-shaped bodies with various deadrise angles
,” Technical Report 2268,
Department of the Navy, Structural Mechanics Laboratory
,
USA
,
1966
.
11.
K.
Hagiwara
and
T.
Yuhara
, “
Fundamental study of wave impact load on ship bow (1st report): Maximum impact pressures acting on a semi-cylindrical body like bow of a large ship
,”
J. Soc. Nav. Archit. Jpn.
1974
,
181
189
.
12.
F. J.
Huera-Huarte
,
D.
Jeon
, and
M.
Gharib
, “
Experimental investigation of water slamming loads on panels
,”
Ocean Eng.
38
,
1347
1355
(
2011
).
13.
D. H.
Peregrine
, “
Water wave impact on walls
,”
Annu. Rev. Fluid Mech.
35
,
23
43
(
2003
).
14.
O. M.
Faltinsen
and
A. N.
Timokha
,
Sloshing
(
Cambridge University Press
,
Cambridge
,
2009
).
15.
J. H. G.
Verhagen
, “
The impact of a flat plate on a water surface
,”
J. Ship Res.
11
,
211
223
(
1967
).
16.
S. K.
Wilson
, “
A mathematical model for the initial stages of fluid impact in the presence of a cushioning fluid layer
,”
J. Eng. Math.
25
,
265
285
(
1991
).
17.
F. T.
Smith
,
L.
Li
, and
G. X.
Wu
, “
Air cushioning with a lubrication/inviscid balance
,”
J. Fluid Mech.
482
,
291
318
(
2003
).
18.
R.
Purvis
and
F. T.
Smith
, “
Air-water interactions near droplet impact
,”
Eur. J. Appl. Math.
15
,
853
871
(
2004
).
19.
P. D.
Hicks
and
R.
Purvis
, “
Air cushioning and bubble entrapment in three-dimensional droplet impacts
,”
J. Fluid Mech.
649
,
135
163
(
2010
).
20.
P. D.
Hicks
and
R.
Purvis
, “
Liquid–solid impacts with compressible gas cushioning
,”
J. Fluid Mech.
735
,
120
149
(
2013
).
21.
S.
Mandre
,
M.
Mani
, and
M. P.
Brenner
, “
Precursors to splashing of liquid droplets on a solid surface
,”
Phys. Rev. Lett.
102
,
134502
(
2009
).
22.
P. D.
Hicks
and
R.
Purvis
, “
Gas-cushioned droplet impacts with a thin layer of porous media
,”
J. Eng. Math.
102
,
65
87
(
2017
).
23.
P. D.
Hicks
,
E. V.
Ermanyuk
,
N. V.
Gavrilov
, and
R.
Purvis
, “
Air trapping at impact of a rigid sphere onto a liquid
,”
J. Fluid Mech.
695
,
310
320
(
2012
).
24.
W.
Bouwhuis
,
M. H. W.
Hendrix
,
D.
van der Meer
, and
J. H.
Snoeijer
, “
Initial surface deformations during impact on a liquid pool
,”
J. Fluid Mech.
771
,
503
519
(
2015
).
25.
L.
Xu
,
W. W.
Zhang
, and
S. R.
Nagel
, “
Drop splashing on a dry smooth surface
,”
Phys. Rev. Lett.
94
,
184505
(
2005
).
26.
E. Q.
Li
,
K. R.
Langley
,
Y. S.
Tian
,
P. D.
Hicks
, and
S. T.
Thoroddsen
, “
Double contact during drop impact on a solid under reduced air pressure
,”
Phys. Rev. Lett.
119
,
214502
(
2017
).
27.
M. R.
Moore
,
J. R.
Ockendon
, and
J. M.
Oliver
, “
Air-cushioning in impact problems
,”
IMA J. Appl. Math.
78
,
818
838
(
2013
).
28.
M. R.
Moore
and
J. M.
Oliver
, “
On air cushioning in axisymmetric impacts
,”
IMA J. Appl. Math.
79
,
661
680
(
2014
).
29.
A. A.
Korobkin
, “
Inclined entry of a blunt profile into an ideal fluid
,”
Fluid Dyn.
23
,
443
447
(
1988
).
30.
A. A.
Korobkin
, “
The entry of an elliptical paraboloid into a liquid at variable velocity
,”
J. Appl. Math. Mech.
66
,
39
48
(
2002
).
31.
F. D.
Gakhov
,
Boundary Value Problems
(
Dover
,
Mineola, New York
,
1966
).
32.
L.
Duchemin
and
C.
Josserand
, “
Curvature singularity and film-skating during drop impact
,”
Phys. Fluids
23
,
091701
(
2011
).
33.
J. O.
Marston
,
I. U.
Vakarelski
, and
S. T.
Thoroddsen
, “
Bubble entrapment during sphere impact onto quiescent liquid surfaces
,”
J. Fluid Mech.
680
,
660
670
(
2011
).
34.
E. Q.
Li
and
S. T.
Thoroddsen
, “
Time-resolved imaging of a compressible air disc under a drop impacting on a solid surface
,”
J. Fluid Mech.
780
,
636
648
(
2015
).
35.
S.
Okada
and
Y.
Sumi
, “
On the water impact and elastic response of a flat plate at small impact angles
,”
J. Mar. Sci. Technol.
5
,
31
39
(
2000
).
36.
M.
Greenhow
, “
Wedge entry into initially calm water
,”
Appl. Ocean Res.
9
,
214
223
(
1987
).
37.
T.
Tveitnes
,
A. C.
Fairlie-Clarke
, and
K.
Varyani
, “
An experimental investigation into the constant velocity water entry of wedge-shaped sections
,”
Ocean Eng.
35
,
1463
1478
(
2008
).
38.
M.
Barjasteh
,
H.
Zeraatgar
, and
M. J.
Javaherian
, “
An experimental study on water entry of asymmetric wedges
,”
Appl. Ocean Res.
58
,
292
304
(
2016
).
39.
R.
Panciroli
,
A.
Shams
, and
M.
Porfiri
, “
Experiments on the water entry of curved wedges: High speed imaging and particle image velocimetry
,”
Ocean Eng.
94
,
213
222
(
2015
).
40.
P.
Yu
,
H.
Li
, and
M. C.
Ong
, “
Numerical study on the water entry of curved wedges
,”
Ships Offshore Struct.
13
,
885
898
(
2018
).
41.
A.
Driscoll
and
A.
Lloyd
, “
Slamming experiments—Description of facilities and details of impact pressure results
,” Technical Report R82002,
Admiralty Marine Technology Establishment, AMTE (Haslar)
,
Haslar, Gosport, Hants, UK
,
1982
.
42.
S. H.
Oh
,
S. H.
Kwon
, and
J. Y.
Chung
, “
A close look at air pocket evolution in flat impact
,” in
Proceedings of the 24th International Workshop on Water Waves and Floating Bodies
,
2009
.
43.
H. C.
Mayer
and
R.
Krechetnikov
, “
Flat plate impact on water
,”
J. Fluid Mech.
850
,
1066
1116
(
2018
).
44.
I. S.
Gradshteyn
and
I. M.
Ryzhik
, in
Table of Integrals, Series and Products
, edited by
A.
Jeffrey
(
Academic Press, Elsevier
,
Amsterdam
,
2000
).
45.
Handbook of Mathematical Functions
, 10th ed., edited by
M.
Abramowitz
and
I. A.
Stegun
(
Dover
,
New York
,
1972
).
46.
Wolfram Research
, Mathematica, Version 11.3,
Champaign, IL, USA
,
2018
.
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