Cavity dynamics in high-speed water entry
M. Lee,a) R. G. Longoria,b) and D. E. Wilson
Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas 78712-1063
~Received 18 September 1995; accepted 17 October 1996!
A method is presented for modeling the cavity formation and collapse induced by high-speed impact
and penetration of a rigid projectile into water. The approach proposes that high-speed water-entry
is characterized by a cavity that experiences a deep closure prior to closure at the surface. This
sequence in the physical events of the induced cavity dynamics is suggested by the most recent
high-speed water-entry experimental data, by results from numerical experiments using a
hydrocode, and by an understanding of the fundamental physics of the processes that govern surface
closure. The analytical model, which specifies the energy transfer for cavity production as
equivalent to the energy dissipated by velocity-dependent drag on the projectile, provides accurate
estimates for variables that are important in characterizing the cavity dynamics, and reveals useful
knowledge regarding magnitudes and trends. In particular, it is found that the time of deep closure
is essentially constant and independent of the impact velocity for a given projectile size, while the
location of deep closure has a weak dependence on impact velocity. Comparison of these analytical
results with experimental results from the literature and with results from numerical simulations
verifies the analytical solutions. © 1997 American Institute of Physics. @S1070-6631~97!02903-6#
The cavity formed during high-speed water entry by a
projectile can exhibit a wide range of dynamical response
characteristics, which depend primarily on the type of projectile and its impact velocity. Of particular interest are
physical events that occur at various stages of the entry that
not only influence the projectile motion but also the nature of
induced pressure waves in the late stages of a cavity collapse. By focusing attention on these characteristic measures,
namely the closure events, it is possible to evaluate how well
analytical models capture the relevant physics of these complex processes.
Cavity formation has been a topic of interest for some
time, especially in the area of hydroballistics, and related
water entry work can be found dating back to World War II.
Most of the work conducted before this period is for very
low speeds, whereas some hydroballistics studies, including
some conducted recently that were concerned with obtaining
design information for ordnance applications, examined impact velocities in the range of hundreds of meters per second.
Information on water entry beyond these subsonic speeds is
not available in the open literature, even though recent interest has turned to characterization of impact and penetration
phenomena resulting from supersonic projectiles. In particular, the work reported here is motivated by the U.S. Navy’s
interest in very high-speed ~i.e., greater than 1 km/s! solid
body entry into the ocean. Of particular interest are the dynamics of the cavity up to the time of closure, since this
event may lend insight to understanding the generation and
propagation of pressure waves in the late stages of water
entry by a projectile. Because of the intense computational
nature of this problem, and to gain insight into the depena!
Currently, Research Assistant Professor, Department of Mechanical Engineering, Naval Postgraduate School, Monterey, California.
b!
Corresponding author: Telephone: ~512! 471-0530; Fax: ~512! 471-8727;
Electronic mail: r.longoria@mail.utexas.edu
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Phys. Fluids 9 (3), March 1997
dence of certain parameters governing the water entry problem, an analytical model is developed in this paper that can
be used to approximate the cavity dynamics. This model can
be used to support long-range ~both in space and time! predictive studies for the applications discussed.
II. CAVITY FORMATION AND COLLAPSE
The dominant mechanism governing cavity formation
during water entry can be quantified by the kinetic energy
transferred from the projectile to the fluid. This energy transfer is driven by the dissipative processes associated with drag
on the projectile, and it can be used to describe the induced
cavity motion. As a solid body penetrates the free surface,
the cavity expands until the difference between pressure in
the surrounding fluid and that in the cavity balances the induced inertial effects and drives the fluid back to its undisturbed state. The ensuing cavity collapse can lead to a closure below the free surface ~a deep closure! or at the free
surface ~a surface closure!. The order in which these events
occurs influences the subsequent physical problem, and experimental studies have determined that the ambient conditions at the free surface, the impact velocity, and shape of the
projectile are critical parameters. Deep closure, or seal, occurs in very low-speed water entry problems, and was observed as early as 1918 by Mallock.1 This velocity regime,
which leads to a deep closure roughly midway between the
location of the projectile and the free surface, is classified by
Birkhoff and Zarantonello2 as very low-speed. They suggested that this velocity regime be classified by a range of
Froude number, 20,Fr,70, where Fr5V 2i /gD, g is gravitational acceleration, and D is the characteristic length ~e.g.,
the diameter of a sphere!.
These regimes are not well-defined, but for roughly
Fr.150, classified as low-speed,2 deep closure is preceded
by a surface closure. Under these conditions, the cavity
formed by the projectile continues to grow after a surface
1070-6631/97/9(3)/540/11/$10.00
© 1997 American Institute of Physics
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I. INTRODUCTION
Phys. Fluids, Vol. 9, No. 3, March 1997
and obliquely into water. He measured actual cavity underpressures ~at two different heights along the cavity! and
found that the estimation by 21r a V 2i was an order of magnitude too low. Abelson assumed that the cavity pressure
would drop by 1 atm when the impact velocity of the projectile was greater than 165 m/s based on extrapolation of
experimental data he obtained for impact velocities up to 76
m/s. More recent work by Wolfe and Gutierrez9 has also
examined cavity pressures for water entry with velocities in
the range of 150 to 300 m/s. The focus of these studies,
classified as high-speed, has been to provide information for
predicting the cavity dynamics for those cases leading to an
early surface closure.
There does not appear to have been any experimental or
theoretical studies on the cavity dynamics for very highspeed water entry; that is, beyond 400 m/s and above 1 km/s.
A study by Metzger10 includes analysis of one particular experiment for water entry with V i 5356 m/s, which is classified here as high-speed. This test involved an oblique water
entry at an angle of 70° from the surface, and is the highest
impact velocity reported for tests conducted by Albert May
in the late 1960s. Interestingly, this water entry led to a deep
closure rather than a surface closure. While the occurrence of
deep closure prior to a surface closure is typically associated
with very low-speed water entry, cavities for which deep
closure precedes surface closure at higher speeds were observed by Gilbarg and Anderson3 for cases in which the air
density was reduced. In these experiments, the suppression
of a surface closure was attributed to an effective reduction
in the Bernoulli effect. For high-speed water entry, the pressure drop can not be estimated using the incompressible Bernoulli equation. Those cases where impact velocity is greater
than V i 5345 m/s, and classified as high-speed, correspond
to a transition regime, and conditions may not induce an
accelerated surface closure. Indeed, this velocity corresponds
closely to Mach 1 for air flowing into the cavity from the
free surface, and the prevailing conditions would induce a
choked flow. Consequently, for very high-speed water entries, it will be assumed that this combination of effects will
result in the generation of a deep closure cavity before surface closure. First, there will be a relatively high-energy
deposition into the fluid that induces large velocities in the
cavity wall motion. Second, there will be a choked flow condition that will inhibit early surface closure, and instead promote early deep closure.
Figure 1 contains results that indicate how the time of
surface closure depends on impact velocity. The data points
in the low-speed range represent experimental measurements
made by Gilbarg and Anderson3 for water entry by a 1-in.
sphere. The experimental results are compared with predictions made using a model developed in this paper. For impact velocities less than 20 m/s, the cavity pressure is determined by the pressure drop due to the Bernoulli effect, which
was assumed to be 0.5r a V 2i •n, where n is 50. The value of
n was chosen based upon the results of Abelson8 who found
that the under-pressure 0.5r a V 2i was a poor approximation
and an order of magnitude smaller than measured values.
The use of n550 at low-speeds is obtained from Abelson’s
experimental results. For V i .
g 165 m/s ~high-speed!, the
Lee, Longoria, and Wilson
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closure has occurred. This type of water entry cavity has
been the subject of many experimental investigations. The
closure mechanisms were studied for low-speed impact by
Gilbarg and Anderson.3 They investigated the dependence of
air–water entry cavities on the atmospheric pressure at the
free surface for impact velocities between 15 ft/s and 100
ft/s, and for air pressures between 1/60 and 3 atmospheres.
They found that a main factor in controlling cavity formation
at low-speed entry was the nature of surface closure and that
Froude scaling was not useful in parameterizing the cavity
phenomena. Gilbarg and Anderson deduced from their experimental results that the product V i •T s was constant,
where T s is the time of surface closure following impact of
the surface. The value of T s depends on the parameters of the
problem and serves to signal the end and beginning of certain critical events in the life of this type of cavity. A general
trend indicated by existing experimental data in the literature, for example, is that increasing impact velocity lowers
the time of surface closure.
Modeling the closure mechanisms requires verification
of some basic assumptions about the cavity motion. In support of such endeavors, low-speed experiments were conducted by Birkhoff and Caywood,4 who used a photographic
technique ~novel at the time! to evaluate velocity fields in
water. They studied water entry by missiles and other cavity
motion problems and deduced that the motion of the cavity
wall is primarily radial or perpendicular to the cavity axis.
May5,6 investigated the vertical entry of steel spheres, examining the effect of atmospheric density and pressure ~above
the free surface!, the impact velocity, and the size and shape
of the missile ~projectile!. He found that at increasing depths
more energy appears to go into cavity production than is lost
by the missile. May’s observation was based on a model
using a constant drag coefficient, which might be valid for
low speeds, and he suggested the difference might be due to
the radial motion of the fluid from the cavity wall,6 as observed by Mallock1 and by Birkhoff and Caywood4 in their
photographic studies.
Birkhoff and Isaacs7 obtained a rough theoretical estimation of the time of surface closure. They found that
r a V i T s /D should be approximately constant, where r a is the
density of air at the free surface. The two major forces leading to the surface closure are an under-pressure caused by the
flow of air into the cavity behind the projectile, referred to as
the Bernoulli effect, and surface tension. For low-speed water entry, the local under-pressure at the splash neck is commonly estimated by the dynamic pressure term 21r a V 2i . This
under-pressure is a dominant factor in early surface closure.
After a surface closure, the cavity below the free surface
continues to expand due to inertial effects and from energy
transfer by the projectile at its instantaneous location. The
expansion leads to a drop in the cavity pressure ~usually
estimated by an isentropic process!. Eventually, deep closure
occurs as the cavity walls at a certain depth are driven to a
closed state by the pressure difference between the surrounding fluid and the cavity. Therefore, the time for deep closure
depends on the time at which the surface closure occurred.3,6
In related studies, Abelson8 focused on measurement of
the pressure in the cavity for projectiles entering vertically
where E p 50.5mV 2p , and z b is the projectile location. The
velocity decay coefficient, b (V p ), is defined as
b5b~ V p!5
~3!
Low velocity impact studies typically use a constant drag
coefficient. At higher velocities, the drag coefficient is highly
dependent on Mach number, M , especially in the transonic
regime. For spheres, the correlation of the drag coefficient11
shows that there are three ranges classified by velocity;11
roughly ‘‘subsonic’’ up to M 50.5, C d 50.384, ‘‘transonic’’
from 0.5 to 1.4, where C d 50.639610.5974(M 21)
2 0.1618(M 21) 2 20.7212(M 21) 3 , and ‘‘supersonic’’ beyond M 51.4, where C d 50.762410.2398(M 21 21/2.75)
20.475(M 21 21/2.75) 2 .
By numerically integrating Eq. ~1!, the velocitydependence of the drag forces can be considered, and the
velocity decay and kinetic energy loss of the projectile can
be evaluated as functions of the penetration depth, z. Using
these results, an equation for the cavity dynamics will be
determined in the next section.
B. Equation of cavity dynamics
pressure drop increases to 1 atm, and it is then held constant
at 1 atm for all cases based on the assumed choked condition. Results from experiments for high-speed water entry by
projectiles indicate the cavity pressure is close to the value
for water vapor pressure.9 Consequently, the results in Fig. 1
incorporate the assumption that the cavity pressure is zero
for high-speed cases. The trend shown in the transition region from the low-speed region up to 165 m/s and to the
high-speed region requires attention. The curve is drawn
through this region, however experimental results should be
conducted to confirm this predicted trend.
III. THEORY
A. Projectile dynamics
Consider a projectile with an initial velocity V i penetrating into a fluid along a straight trajectory in the 1 z direction.
For normal penetration of the free-surface of a half-plane, the
deceleration from the impact velocity can be described by
Newton’s second law,
m
d 2z
dV p
5
2 5m
dt
dt
1
( F z 5mg2 2 r w A 0 C d V 2p ,
~1!
where m is the projectile mass, z is the penetration axis, t is
time, V p is the penetration velocity of the projectile, r w is the
fluid density, A 0 is the projected area of the projectile,
C d 5C d (V p ) is a velocity-dependent drag coefficient, and g
is gravitational acceleration. Except for very low-speed water
entry, the effect of g will be negligible, and so it will be
ignored in the remaining analysis. The rate of change of
kinetic energy with respect to depth, dE p /dz b , can then be
expressed as
dE p
dV p
52mV p
5m b V 2p ,
dz b
dz b
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Phys. Fluids, Vol. 9, No. 3, March 1997
~2!
The combined effect of the projectile and the cavity on
the fluid motion is approximated by using distributed point
sources along the axis of penetration. These sources include
a time delay that accounts for the motion of the projectile
along its computed trajectory @via integration of Eq. ~1!#. The
source strength at any point along this trajectory is determined by employing energy conservation, where it is assumed that the kinetic energy loss of the projectile equals the
total energy ~kinetic plus potential! in a fluid section,2 as
illustrated in Fig. 2. Internal energy changes due to thermal
effects in the fluid are assumed negligible in this analysis.
This model is based on the work of Lundstrom,12 who drew
upon the basic cavity model of Birkhoff and Zarantonello,2
to model the pressure field generated by a subsonic tumbling
projectile in fuel tanks. The analysis presented here and in a
subsequent section extends this approach to describe the dynamics of the water entry cavity, with special attention given
to water entry at very high-speeds. Consequently, the methodology is useful, in principle, for estimating the cavity dynamics for any impact velocity and for any shape of the
projectile ~with suitable accounting for the drag coefficients!.
The radial and axial velocity components are found by
solving the linearized potential flow equation for distributed
point sources. The solutions obtained by Lee13 involve complex integral equations that must be solved numerically.
However, on the wall of the cavity the radial velocity has a
very simple form which is proportional to the source
strength, z , and is given by
u5
2
• z ~ j ,t ! ,
w
~4!
where w is the radial distance, z is the source strength, t is
time, and j is the penetration index, a variable that measures
the distance along the trajectory. The kinetic energy, E w , of
the fluid in the section within a finite radius, V, is then
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FIG. 1. The time of surface closure versus impact velocity. The data in the
low-speed range was measured by Gilbarg and Anderson3 for cavities
formed by a 1-in. sphere. In the low-speed range, the pressure drop due to
1
the Bernoulli effect was assumed to be 2r a V 2i •n, where n is 50. In highspeed range, the pressure drop was held constant at 1 atm. For all cases, the
predicted projectile motion assumes drag coefficients given by the Charters
and Thomas correlation.11
r wA 0C d~ V p !
.
2m
expression for the cavity pressure could be included. The
assumption made in the following analysis, however, considers the extreme case where P c is zero.
Each term in Eq. ~7! except for the source strength is
known. By introducing the variables A(z) and B(z) given by
@ A ~ z !# 2 5
F G
1 dE p
,
p P g dz b j
@ B ~ z !# 2 5
Pg
,
r wN
~8!
an expression for the source strength can be developed as
z 56 21 B ~ z ! A@ A ~ z !# 2 2 @ a ~ z !# 2 .
~9!
If the cavity wall velocity is denoted as ȧ, then the kinematic
boundary condition at the cavity wall is u w5a 5ȧ. Therefore,
the approximate velocity from potential theory @Eq. ~4!# and
the kinematic boundary condition together imply
FIG. 2. Cavity growth model. The location of the projectile is z b , and a is
the cavity radius at z5 j .
V
G
u 2 wdw dz.
a
Combining Eqs. ~9! and ~10!, an equation for the cavity
dynamics is determined as
~5!
Substituting Eq. ~4! into Eq. ~5! and integrating yields
dE w 54 pr w N z 2 dz,
~6!
where, N5ln(V/a), is a dimensionless geometric parameter,
and a is the cavity radius at the penetration depth. It can be
shown from the integral equations that this expression is a
reasonable approximation provided w is less than L/a, where
L is the length of the cavity. Beyond this region, the radial
velocity decays more rapidly than 1/w.
The ratio V/a is usually assigned a value in the range of
15–30, a range recommended by Birkhoff and Zarantonello.2
This is justified in that, for the special case where a projectile
is moving at a constant velocity, the cavity shape is determined for constant values of V/a in this range. For example,
Lundstrom12 used V/a515 to arrive at his results for subsonic projectiles.
Energy conservation requires that the kinetic energy loss
at z5 j equal the total energy stored in the fluid section of
height dz. In order to complete this expression, a potential
energy term must be included to account for energy stored in
the fluid due to the expansion of the cavity walls against the
hydrostatic pressure in the fluid section, P 0 (z). For the incremental volume defined by dz, the contribution by potential energy is given by dE z 5( P 0 (z)2 P c (z)) p a 2 dz. Here,
P c (z) is the cavity pressure at depth z, which will be assumed to have negligible variation in the 1z-direction as
suggested by the measurements of Abelson.8 Defining now,
P g [( P 0 2 P c ), we can express the conservation of energy
as,
F G
dE p
dz54 pr w N z 2 dz1 p P g a 2 dz.
dz b j
~7!
Combining Eq. ~7! and Eq. ~2! leads to a form similar to that
given by Birkhoff and Zarantonello.2 Note that if a variable
cavity pressure is to be considered, it is clear how such an
Phys. Fluids, Vol. 9, No. 3, March 1997
~10!
a ~ z ! ȧ ~ z ! 56B ~ z ! A@ A ~ z !# 2 2 @ a ~ z !# 2 .
~11!
Integration of Eq. ~11! with boundary condition a50, at the
time of projectile arrival, t b , yields
6 A@ A ~ z !# 2 2 @ a ~ z !# 2 5 @ A ~ z !# 2 @ B ~ z !#~ t2t b ! .
~12!
To account for the finite diameter of a projectile,
a5D/2 instead of a50, can be used for the integration of
Eq. ~11!, if necessary. Equation ~12! specifies the radial motion of the cavity as a function of time at any depth, z. In the
next section, this method is extended to study the cavity
dynamics in detail.
IV. EXTENSION OF THEORY AND RESULTS
A. Cavity radius
In this section, the previous analysis is used to describe
the cavity dynamics. From Eq. ~12!, an analytical expression
for the cavity radius can be derived as
a ~ z ! 5 A@ A ~ z !# 2 2 @ A ~ z ! 2B ~ z !~ t2t b !# 2 ,
for t.t b ,
~13!
which is valid at any depth, z. This equation describes the
cavity formation as a projectile reaches the depth, z. The
collapse of the cavity wall at any depth begins after it
reaches the maximum allowed radius. Note that a modified
form of Eq. ~13! can also be derived by changing the boundary condition from a50 to a5D/2 for the integration of Eq.
~11!.
Figure 3 shows the evolution of the cavity formed in
water at different times after an impact at the surface by a
sphere of density r p 52000 kg/m3 ~a value used throughout
the results reported in this paper! with an impact velocity of
V i 50.5 km/s. The analytical model estimates that deep closure will occur at a depth of 58 cm and between 38 ms and
40 ms following the surface impact. After this initial closure,
the cavity continues to close in the immediate vicinity both
above and below the location of deep closure. However, at
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F E
dE w 5 pr w
z 5 12 a ~ z ! ȧ ~ z ! .
maximum radius, w m , along the depth by setting ȧ to zero to
yield
w m 5A ~ z ! 5
A
1 dE p
5
p P g dz b
A
r w C d A 0 V 2p
}V p .
2p Pg
~14!
Note that the maximum cavity radius is proportional to the
local penetration velocity, assuming C d 'const and ignoring
the variation at P g with depth. Equation ~14! considers that
the total energy contained in the fluid when the maximum
cavity radius is reached is potential energy alone; this energy
subsequently acts to initiate the cavity collapse. Such a scenario represents a dynamical energy conversion process between the kinetic energy imparted by the projectile and potential energy stored in cavity formation.
B. Cavity wall velocity
The cavity wall velocity as a function of depth and time
is given by
FIG. 3. Evolution of the cavity formed in water after an impact as predicted
by the analytical model. Deep closure occurs at 58 cm below the free surface between 38 ms and 40 ms after impact. Results are for a sphere with
D52 cm, V i 50.5 km/s.
z50, as shown in the figure, the cavity continues to increase
in size due to the large amount of energy initially transferred
during impact.
The cavity reaches a maximum radius when the velocity
of the cavity wall becomes zero. From Eq. ~11!, we find the
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Phys. Fluids, Vol. 9, No. 3, March 1997
for t.t b ,
~15!
which results from differentiating Eq. ~13! with respect to
time. From this formulation, the predicted cavity wall velocity can be compared to measurements from an experimental
study as shown in Fig. 4~a!. This plot displays cavity velocity at a depth of 22 in. for vertical water entry by a 1-in.
sphere that has impacted the free surface at V i 573 ft/s. Although this is not a high-speed impact, it represents the only
data set found in the literature permitting this comparison.
The analytical prediction is contrasted with the experimental
results of May6 reproduced in Fig. 4~b!. In May’s experiments, the air pressure was reduced to 1/27 of an atmosphere
in order to prevent surface closure. Note that a negative cavity wall velocity corresponds to a collapse stage for the cavity. As a consequence of neglecting dissipative effects in the
analytical cavity formation model, Eq. ~15! has an inherent
symmetry in periods for cavity formation and collapse at any
depth. Note, however, that the experimental results show that
at a depth near deep closure the collapsing period is larger
than the formation period by about 25%. This asymmetry is
attributed to losses during the cavity formation where some
energy is converted into both pressure waves and dissipated
in the form of heat.
Figure 5 contains plots of the velocity of the cavity wall
for various depths as predicted by Eq. ~15!. There is a decrease in the period for cavity formation and collapse as
depth increases because cavities formed at relatively larger
depths receive less energy from the projectile than those
formed at shallower locations. In other words, the kinetic
energy transfer at deeper locations is significantly reduced.
It is appropriate, at this point, to recall the assumption
made in the analytical model that water is to be treated as an
incompressible fluid. This assumption is valid insofar as the
radial velocity of the cavity wall is small compared to the
speed of sound in water. Note that even though a large cavity
wall velocity may occur near the wall surface, the assumption of incompressibility is quite reasonable since this level
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B ~ z !@ A ~ z ! 2B ~ z !~ t2t b !#
da
5
,
dt A@ A ~ z !# 2 2 @ A ~ z ! 2B ~ z !~ t2t b !# 2
FIG. 5. Plots of the cavity wall velocity at depth ~a! 15 cm, ~b! 20 cm, and
~c! 25 cm as induced by a 2 cm diam sphere ~density, r p 52000 kg/m3 ) that
impacted the free surface at V i 50.5 km/s.
is only maintained for a relatively short period of the cavity
formation process. Note: Actually, the incompressible assumption is valid up to V i '700 m/s.
V. NUMERICAL SIMULATION
A. Description of hydrocode
Before presenting analytical results for deep closure, the
results presented thus far will be compared to those from
numerical simulations using an impact and penetration hydrocode. Hydrocodes are computer programs that numerically simulate nonlinear fluid–structure interaction, particularly physical problems that include shock generation and
propagation.14 Commercially available hydrocodes have become quite efficient, and in this study we used AUTODYN-2D
to study the complex physical phenomena that arises in highspeed water entry.
The conservation equations of mass, momentum, and energy, are coupled with specified constitutive laws and equaPhys. Fluids, Vol. 9, No. 3, March 1997
B. Verification of hydrocode
A difficulty faced in the study of very high-speed water
entry problems is that experimental results are scarce or not
available in the open literature. Indeed, a motivation for the
present study was a need to develop numerical and analytical
tools that would guide modern, large-scale experiments. It is
necessary to refer to studies conducted during the 1940s,
1950s, and 1960s, which reveal some critical characteristics
about the water entry problem. Specifically, useful information about the cavity formation and the shock stand-off distance can be inferred from the work of McMillen et al.16
McMillen et al.,16 conducted experiments in which the pressure fields generated by high-speed projectiles fired into water were measured. A shadowgraph technique was used to
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FIG. 4. Cavity wall velocity near deep closure at a depth of 22-in. for
vertical entry by a 1-in. diameter sphere obtained by ~a! present analytical
results, and ~b! experimental results of May.3 The impact velocity was
V i 573 ft/s, and in the experiments, the air pressure was 1/27 atmos. The
abscissa represents the time after the projectile has reached the specified
depth, while the ordinate represents the cavity wall velocity.
tion of state to arrive at a direct numerical solution of the
~fully nonlinear! problem. Commonly an Eulerian mesh is
generated to model the target, which is water in this case,
and a Lagrangian mesh is used for the projectile. This mesh
generation technique is useful for solid and fluid interaction
problems. The numerical mesh moves and distorts in the
Lagrangian scheme as the material flows, and so it has the
advantage of easily tracking the material interface. A disadvantage is that large deformation of the mesh requires rezoning and erosion algorithms to continue the calculation. With
an Euler grid, the material flows through a fixed mesh, so
there will be no numerical mesh deformation. The disadvantage of this scheme is the advection error associated with
multiple materials in a cell.
In this study a two-dimensional axisymmetric calculation was performed for normal water entry. The projectiles
were modeled by specifying a relatively high yield strength
so deformation would not occur. The water was modeled as
a fluid with no strength, with the equation of state ~EOS!
following a Mie–Gruneisen approximation. The parameters
of the EOS for water used in the numerical simulation are
summarized in Table I.15
TABLE I. Water EOS parameters.
r w 5999 kg/m3
c51480 m/s
s51.75
G50.28
C v 54.5 kJ/kg• K
U s 5c1s•u p , where
Initial density
Sound speed
Slope of U s 2u p
Grüneisen parameter
Specific heat
U s 5shock velocity
u p 5 particle velocity
FIG. 6. Comparison of pressure fields generated during very high-speed
water entry by a sphere. Data shown gauge the validity of results from
numerical simulation using AUTODYN in ~a! and ~c! against experimental
results in ~b! and ~d! as reported by McMillen et al.16 Cases ~a! and ~b! are
for subsonic impact by a sphere at V i 51.3 km/s, and cases ~c! and ~d! are
for supersonic impact at V i 51.8 km/s.
value less than that predicted by the analytical model. This
difference in the two results can be explained by identifying
that a portion of the total kinetic energy lost from the projectile produces a splash on the free surface in the hydrocode
C. Contrasting analytical and numerical results
Figure 8 summarizes results from a numerical simulation
of the same high-speed entry problem studied using the analytical model. There is an overall agreement with the analytical predictions shown in Fig. 3. In particular, the numerical
simulation predicts a deep closure, thus supporting the initial
assumption in the analytical model that a surface closure
would not precede the deep closure of the cavity. The results
in Fig. 8 do indicate, however, that the location of deep
closure is about 48 cm below the undisturbed free surface—a
TABLE II. Results of deep closure produced by a 2 cm sphere.
V i ~km/s!
T d ~ms!
z d ~cm!
V d ~m/s!
t d /T d
546
0.5
0.75
1.0
1.25
1.5
38.6
59.9
6.71
0.529
37.8
65.0
6.97
0.521
37.4
68.5
7.02
0.524
37.1
70.6
7.19
0.517
36.9
72.1
7.23
0.517
Phys. Fluids, Vol. 9, No. 3, March 1997
FIG. 7. A comparison of the velocity decay as found from a numerical
simulation using the hydrocode versus Eq. ~1! with a drag coefficient dependent on Mach number ~values from Charters and Thomas11!.
Lee, Longoria, and Wilson
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measure the pressure field surrounding and propagating from
a sphere as well as the instantaneous velocity of a sphere as
it penetrated ~normally! into a water target.
Numerical experiments were conducted to simulate the
physical experiments reported by McMillen et al. Figure 6
summarizes the results from these numerical simulations
along with the shadowgraphs measured by McMillen et al.16
Two useful bases for comparison of these results are the
shock stand-off ~SO! distance at a point in the time history of
the water entry and the instantaneous velocity (V). These
measures are contrasted in Fig. 6, and the numerical results
are found to compare very well with those measured and
reported by McMillen et al.
A study was also conducted to evaluate the applicability
of the drag coefficients from Charters and Thomas,11 which
were determined for spheres moving in air, to the water entry
problem. This evaluation compared the projectile motion
computed using Eq. ~1! ~with a velocity dependent drag coefficient from Charters’ and Thomas’ correlation! to the motion predicted by the hydrocode, where there decay is governed by a simulation of the governing equations taking into
account the material properties of the water and the impact
parameters. That is, the projectile dynamics predicted by the
hydrocode are dependent on fluid forces that are the integrated effect of the pressure field computed in a direct simulation of the Navier–Stokes equations. The results shown in
Fig. 7 show that, for high-speed water entry, the drag from a
simulation is comparable to that predicted using the Charters
and Thomas correlation.
T c ~ z ! 5t b ~ z ! 1
A~ z !
,
B~ z !
~16!
where t b (z) is the time it takes for the projectile to travel to
z, which is determined by integrating Eq. ~1!. The ratio
A(z)/B(z) is the cavity opening time ~the time for the cavity
to reach the maximum radius from zero!, where the values of
A(z) and B(z) are found from the numerical solution of Eq.
~1! along with Eq. ~2!.
One difficulty in determining and confirming an estimate
for the cavity collapse completion time is that there is not
much information available on the cavity collapse process,
especially for very high-speed water entry. As a first approximation, it is assumed that the cavity collapse period is equal
to the cavity formation period, even though this is not entirely realistic. Based on this assumption, however, an estimate for the cavity collapse completion time, T r , along the
depth of penetration is obtained from Eq. ~16!,
FIG. 8. Evolution of the cavity formed in water after an impact as predicted
by a hydrocode numerical simulation. Deep closure occurs at 48 cm below
the free surface between 38 ms and 40 ms after impact. Results are for a
sphere with D52 cm, V i 50.5 km/s.
simulation. This results in an increased hydrostatic pressure
in the region surrounding the cavity which would tend to
accelerate the deep closure event. The splash effect was not
included in the analytical model, so the effect of increasing
the surrounding pressure will not be evident in the predictions.
VI. DEEP CLOSURE ANALYSIS
The late stage of the cavity dynamics can involve a succession of closures along the cavity length, leading to a
‘‘pinching’’ process where many bubbles are generated.9
The closures produce closed cavities ~or bubbles! whose subsequent collapse can be a significant source of pressure
waves. In the class of high-speed water entry problems currently under discussion, the deep closure signals the end of
the cavity formation, and a description of this first deep closure is examined in this section. The time and location of
deep closure are determined by a direct numerical solution of
Eq. ~1!. In addition, scaling laws for these parameters are
developed using an approximation based on a constant b
value.
A. Time and location of deep closure
If it is assumed that the cavity begins to collapse at a
particular depth when the opening process of the cavity is
complete, the condition a5A in Eq. ~12! will yield an estimation for the cavity collapse initiation time at that depth,
T c (z), given by
Phys. Fluids, Vol. 9, No. 3, March 1997
A
.
B
~17!
Now, the time of deep closure, T d , is defined as the
minimum value for the cavity collapse completion time
along the entire depth of penetration. The solution of the
projectile dynamics using Eq. ~1!, along with Eq. ~17!, leads
to a numerical assessment of the time of deep closure. After
the first deep closure, the cavity is divided into two parts:
one is an open cavity from the location of deep closure to the
free surface, the other is a closed cavity that surrounds the
sphere, as shown in Fig. 3 or Fig. 8.
Figure 9 shows the time of deep closure for spheres impacting water at high speed as predicted by the method described. It is of interest to note that the time of deep closure
as estimated by Eq. ~17! is essentially independent of the
impact velocity. This invariance in the time of deep closure
was observed in the low-speed experiments of Gilbarg and
Anderson3 and May,6 under conditions where the pressure in
the cavity is about the same as that at the free surface
( P c ' P atm). By reducing the pressure above the free surface,
the deep closure event can be made to occur for low-speed
water entry before an early surface closure.
The numerical simulations predict that the time of deep
closure will slightly decrease with impact velocity. This effect can be attributed to the increase in hydrostatic pressure
from splash at the free surface, which would increase the
pressure in the fluid surrounding the cavity and induce an
early deep closure. It should be pointed out that 5 days of
computational time were typically required for each water
entry simulation, while the analytical predictions were computed in 30 s on an IBM RS/6000.
Plots of the velocity decay of a 2.0 cm sphere after water
entry at different impact velocities are given in Fig. 10, with
the results obtained from direct integration of Eq. ~1! and
taking into account the velocity-dependence in the drag coefficient. After about 5 ms following impact, each case
shows that the projectile velocity, V p , approaches the same
value regardless of the impact velocity. Therefore, the cavity
formation and collapse periods show a weak dependence on
the initial impact velocity.
Lee, Longoria, and Wilson
547
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T r 5t b ~ z ! 12•
FIG. 11. Ratio of traveling time to the location of deep closure (t d ) to the
time of deep closure (T d ) versus impact velocity (V i ). Results are for
spheres with diameters of ~a! 1.0 cm, ~b! 1.5 cm, and ~c! 2.0 cm.
The ratio of the traveling time to the location of deep
closure, t d , to the time of deep closure, T d , was computed
for spheres of different diameter using the analytical model.
Figure 11 indicates the invariance of this ratio with respect to
sphere diameter and impact velocity. For different impact
velocities, the predicted values for z d and V d ~the velocity at
deep closure! are summarized along with t d and T d in Fig. 6.
The time of deep closure, T d , is essentially independent of
the impact velocity, and the location of deep closure, z d ,
increases with impact velocity. The ratio of the traveling
time to the location of deep closure, t d , to the time of deep
closure, T d , is almost constant.
In Fig. 12, the location of deep closure is plotted against
the impact velocity of a sphere. This graph contrasts analytical predictions with results from hydrocode numerical simulations. The depth of deep closure, z d , is found to be a strong
function of sphere diameter and a weak function of impact
velocity. This trend was observed in the low-speed experiments by Gilbarg and Anderson3 and May.5 The values of
z d predicted by the hydrocode numerical simulations are less
than those predicted from the analytical model. Some of this
error can be attributed to the fact that the depth values were
estimated from the undisturbed free-surface in the hydrocode
results, thereby ignoring the splash. An alternative method
for estimating these values from these numerical results
would reduce this discrepancy.
FIG. 10. History of the projectile’s velocity decay. Most of the kinetic
energy for very high-speeds is expended within 5 ms as shown in the inlay.
Results are for impact by a sphere of 2.0 cm diam with impact velocity of
~a! 0.5 km/s, ~b! 1.0 km/s, and ~c! 1.5 km/s.
548
Phys. Fluids, Vol. 9, No. 3, March 1997
B. Scaling phenomena
Approximate scaling laws for high-speed water entry
problems can be obtained by considering a constant velocity
decay coefficient, b̄ 5 r w A 0 C d /2m p . Using this approximation, closed form expressions are derived for the time and
FIG. 12. Location of deep closure versus impact velocity. Results shown are
for a sphere of diameter ~a! 1.0 cm, and ~b! 2.0 cm.
Lee, Longoria, and Wilson
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FIG. 9. Time of deep closure versus impact velocity as predicted by the
analytical model and numerical simulation. Two cases are shown where the
sphere had a diameter of ~a! 1.0 cm, ~b! 2.0 cm.
location of deep closure. Using the results for cavity dynamics with constant b̄ , and then rewriting Eq. ~17! as a function
of penetration depth we find
T r5
e b̄ z 21
1
b̄ •V i
2
Pg
A
r wN
m b̄ •V i e 2 b̄ z .
p
~18!
The location where the deep closure occurs can be estimated
from Eq. ~18! by minimizing T r with respect to z such that
z d5
F A
1
ln
2 b̄
2
b̄
Pg
G
r wN
m b̄ •V 2i .
p
~19!
Now the time of deep closure is determined by substituting Eq. ~19! into Eq. ~18! to yield
a •V i 21
T d5
b̄ V i
1
F
G
a 2 / b̄ 1
1
5 2a2
,
a
Vi
b̄
~20!
FIG. 13. Ratio of the time of deep closure (T d ) to sphere diameter (D)
versus impact velocity (V i ). Results shown are for a sphere with diameter
~a! 1.0 cm, ~b! 1.5 cm, and ~c! 2.0 cm.
where
2 b̄
Pg
A
r wN
m b̄ .
p
VII. CONCLUSIONS
Through further reduction, Eq. ~20! becomes
T d 5C 0
r 1/2
w D
P 1/2
g
2
1 D
,
3 r̄ C d V i
~21!
where C 0 5(96N/27C d r̄ ) 1/4, r̄ 5 r w / r p , where r p is the projectile density, and P g 5 P 0 2 P c , where P 0 5 P atm
1 r w gz d . This equation can be rewritten in nondimensional
form as
SD
g
D
1/2
T d 5C 0
S
P 0 2 P atm
P 02 P c
DSD
1/2
D
zd
1/2
2
C1
,
Fr1/2
~22!
with the Froude number, Fr[V i / AgD, and C 1 [1/3r̄ C d .
For very low-speed impacts, P c ' P atm and we find
SD
g
D
1/2
T d 5C 0
SD
D
zd
1/2
2
C1
.
Fr1/2
~23!
An analytical model for the cavity dynamics has been
developed by using empirical velocity-dependent drag coefficients to describe the projectile dynamics. The energy dissipated in drag was used to approximate the energy expended
in cavity production. Although we have restricted our discussion to the case of high-speed impact by a sphere, the model
presented here can be extended for any body velocity or
shape. The applicability of these results to quantifying the
collapse pressure waves generated by high-speed water entry
is discussed in a companion paper currently under
preparation.17 Based on the results of the study reported in
this paper, the following conclusions can be drawn
~1! The time of deep closure is almost constant and independent of the impact velocity for a given sphere diameter, a
result reflected in scaling laws developed in this paper.
On the other hand, for very high-speed impacts P c '0,
Fr@1, and we obtain
SD
g
D
1/2
T d 5C 0
S
P 0 2 P atm
P0
DSD
1/2
D
zd
1/2
.
~24!
If the impact kinetic energy is relatively small such that the
closure event occurs near the surface ~as in the problems
studied here!, then
SD
g
D
or
1/2
T d 5C 0
S D
T d P atm
D rw
S DSD
r w gz d
P atm
1/2
D
zd
1/2
,
~25!
1/2
5C 0 5const.
The results given in Fig. 13 were determined from the analytical model ~with velocity-dependent b ) and indicate good
scaling of T d with D. A similar scaling of z d from Eq. ~19!
~not shown here! yields the results plotted in Fig. 14, indicating a weak dependence on impact velocity.
Phys. Fluids, Vol. 9, No. 3, March 1997
FIG. 14. Ratio of the location of deep closure (z d ) to sphere diameter
(D) versus impact velocity (V i ). Three cases are shown, where the diameter
of the sphere is ~a! 1.0 cm, ~b! 1.5 cm, ~c! 2.0 cm.
Lee, Longoria, and Wilson
549
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a 2[
~2! The location of deep closure is a weak function of impact velocity.
~3! Froude scaling is not valid for high-speed impact. On the
other hand, T d /D and z d /D are found to be nearly constant and independent of the impact velocity.
~4! The analytical predictions compare well with experimental measurements reported in the literature and with numerical simulations reported in this paper.
ACKNOWLEDGMENTS
The authors thank the Applied Research Laboratory at
the University of Texas at Austin for support throughout this
project, and the Institute for Advanced Technology for use of
computing facilities and software. In addition, we thank the
referees and the editors for their careful reading of the manuscript and for many helpful suggestions.
1
550
Phys. Fluids, Vol. 9, No. 3, March 1997
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Lee, Longoria, and Wilson
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6