Modeling and Numerical Simulation of the Forces Acting on a Sphere During Early-Water Entry

7/30/2019 Modeling and Numerical Simulation of the Forces Acting on a Sphere During Early-Water Entry

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Modeling and numerical simulation of the forces acting on a sphere during early-water

entry

John Abrahama

John Gormanb

Franco Reseghettic

Ephraim Sparrowb

John Starkb

aUniversity of St. Thomas

School of Engineering

2115 summit Ave

St. Paul, MN 55105-1079, USA

[email protected]: 651-962-5766

fax: 651-962-6419

bUniversity of Minnesota

Department of Mechanical Engineering

111 Church St. SEMinneapolis, MN 55455-0111

cENEA, UTMAR-OSS,

Forte S. Teresa,

19032 Pozzuolo di Lerici, Italy

Abstract:

Mathematical modeling, absent simplifying assumptions and coupled with numerical simulation

has been implemented to determine the motions and forces experienced by a sphere penetrating awater surface from an air space above the surface. The model and simulation are validated by

comparisons with extensive experimental data and with trends from approximate analyses.

Although the present work adds to the understanding and quantification of the sphere as an entryobject, its major contribution is model development and validation to enable investigation of

water entry of objects of practical utility such as the expendable bathythermograph (XBT). The

XBT device is widely used in the determination of temperature distributions in large water

bodies such as oceans. The measured temperature distributions are, in turn, used to determinethe thermal energy content of oceans. During the course of the numerical simulations,

parametric variations were made of the sphere velocity, surface tension, flow regime (laminar orturbulent), and Reynolds number. The drag-coefficient results were found to be independent ofthese quantities. This outcome indicates that momentum transfer from the sphere to the adjacent

liquid is responsible for the drag force and that friction is

a secondary issue.


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1. IntroductionStudies of the impact of solid objects onto horizontal liquid surfaces have led to a detailed

understanding of the forces that act on the object in the very early stages of impact and on theshape and size of an air cavity formed adjacent to the object. This information is helpful for

predictions of the motion of objects that breach the surface as they pass into the liquid. There aremany applications for which such predictions are important, such as launching of torpedoes,motion of watercraft, and the deployment of oceanographic measuring devices. It is this last

application which has motivated the present study.

Each year, many thousands of oceanographic measurement devices are employed throughoutthe world. Those devices have a variety of shapes and sizes, and their launch conditions differ

significantly. For example, one of the most commonly used devices to measure ocean

temperatures for subsequent calculation of heat content is the expendable bathythermograph

(XBT). It can be launched from heights that vary between 2-30 meters above the surface of theocean. This large variation of launch heights has led to significant differences in the impact

velocity and the angle of entry.

As the XBT device travels through increasing ocean depths, it collects temperatureinformation which is conveyed to an on-ship computer by means of a trailing copper wire. The

wire unspools as the device falls, and canted fins engender a rotating motion that aids in the

unspooling process.For climate studies, it is important to accurately measure the ocean temperature distribution

throughout a specified depth. XBT timewise-temperature records, combined with the

instantaneous depth of the device, enable the determination of the thermal energy content of theocean which, in turn, facilitates closure of the Earths energy balance.

One shortcoming in the current use of the XBT device is that the depth is not measured

directly, but is inferred from a fall rate equation (derived under idealized experimental

conditions). Its coefficients were originally proposed by the manufacturers in the 1960's and

have been recalculated in experiments where the XBT depth-temperature profiles were comparedwith the results of more sophisticated and accurate devices. Those controlled experimental

conditions are not always encountered in the field, so that the global accuracy of XBTmeasurements in the historical archive is a hard problem to solve in the ocean heat content

calculations.

The present authors have created a methodology for predicting the rate at which XBT devicesfall through the water column (Stark et al., 2011; Abraham et al., 2011; 2012a; 2012b). That

model is able to deal with motion of fully submerged devices; however, it does not account for

the impact of water entry on the device motion.

As recognized in Abraham et al., (2012a), surface impact could affect the fall rate of thedevice through water and its inferred depth. This realization has motivated the present study. A

recent publication which focuses on the XBT device reinforces the importance of the entry forces(Xiao and Zhang, 2012). Those investigators included an entry calculation for their device andthen extended the solution to fully submerged motion. They concluded that the probe velocity at

entry had little impact on the subsequent rate of fall. On the other hand, that paper did not deal

with the motion of the object prior to impact with the water. Also, the verification of thoseresults with literature-based information on spheres only extended partway through the entry

process (normalized depth of 0.4 times the sphere radius), thereby avoiding issues related to the

trailing separation region. Furthermore, the calculations presented in that paper utilized a


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regularly shaped grid of elements and did not, therefore, use special boundary layer elements that

are typically employed to resolve flow in the boundary layer. Additionally, the type of XBT was

unspecified in the paper, it does not appear to be one of the standard XBT devices employed in

oceanographic temperature measurements. Consequently, while that paper was a significantstep forward, further advancements in this area are required.

The goal of the present work is to numerically explore the entry forces on a sphere that passesfrom air to water. The simulation will carried through to the situation in which the device is fullysubmerged in the water. Drag forces will be extracted at all instances and compared with

experiments from the literature. The sphere is chosen as the shape of interest here because it has

been more extensively studied than any other shape. Consequently, it can serve as an acceptedbaseline for the validation of the present physical model and simulation method. The successful

validation of the simulation model will justify its use for actual oceanographic devices.

The literature on the entry problem is rich and extends back more than a century. The first

significant study used novel photographic methods to illuminate the dynamics of the fluid flowin the cavity following sphere entry (Worthington and Cole, 1897). This work provided the seed

for a treatise (Worthington, 1908). A related issue was addressed by Von Karman (1929) in

connection with seaplane floats. Wagner (1932) investigated phenomena related to the impact ofobjects on liquid surfaces. Later, pioneering studies were able to extract drag forces on objects

passing into water (Watanabe, 1934; Gilbarg and Anderson, 1947; May and Woodhull, 1948;

May, 1951; 1952) and pressure distributions on the surfaces of objects during impact(Richardson, 1948).

Simultaneous with these early experiments, analytical methods and models were developed

that allowed predictions of entry forces, particularly in the very early stages of entry (Courant etal., 1945; Trilling, 1950; Shiffmann and Spencer 1945a; 1945b; 1947). During the following

decades, a number of analytical studies extended the available information to other shapes and to

oblique entry situations (McGehee et al., 1959; Nisewanter, 1961; Waugh and Stubstad, 1966;

Verhagen, 1967).

A revitalization of this research occurred in the 1970s with a focus on the dynamics of the aircavity formed upon the entry of the object into the fluid. Pressure measurements within the

cavity have been performed (Abelson, 1970). Also, new modeling strategies were devised todeal with non-spherical shapes (Hughes, 1972). Much of the research was sponsored by the

United States Government in order to understand the motion of naval weapons which undergo

liquid-surface entry (Baldwin, 1975a; 1975b; Baldwin and Steves, 1975; May, 1975; Koehlerand Kettleborough, 1977; Moghisi and Squire, 1981). Those studies used analytical and

experimental methods and investigated a variety of shapes including spheres, cylinders, wedges,

and Ogives.

A fourth generation of studies has emerged more recently that extend the availableinformation to high-speed entry problems (Shi and Takami, 2001; Gekle et al., 2009; Guo et al.,

2012). New analytical or numerical methods have been developed (Korobokin and Pukhnachov,1988; Park et al., 2003; Bergmann et al., 2009; Do-Quang and Amberg, 2010). Furthercontributions to cavity dynamics are reported (Duclaux et al., 2007; Grumstrip et al., 2007).

Finally, experiments continue to be performed to further investigate the issues that affect entry

forces (Truscott, et al., 2012)Despite this extensive history, it appears that the details of the distribution of drag on the

surface of a sphere passing into water have not been thoroughly investigated nor have flow

patterns in the neighborhood of the sphere been presented. The goal of this study is to develop a


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model and a concomitant numerical approach that can be used to definitively provide such

information. The devised methodology will utilize commercially available software and will

explore the impact of both laminar and turbulent flow conditions. The results will be compared

with the best available experimental data to validate the approach. If this methodology issupported by comparisons with experimental data, it can be used in conjunction with already

existing information for fully submerged objects in order to improve the identification of theinstantaneous depths of sensory devices.

2. Mathematical ModelThe mathematical model was formulated to take account of three-dimensional unsteady fluid

motions in air and water. The volume-of-fluid method (VOF) is used to separate the two fluid

regions. Zones completely filled by air are represented by VOFair= 1 (VOFwater= 0), while water

regions are defined by VOFwater= 1 (VOFair= 0). The VOF values of air and water add to 1throughout the entire solution domain. The air-water interface is identified by VOF = 0.5.

Coupling between the descending sphere and the respective fluids is due to fluid-solid

friction. The sphere is initially positioned in the air above the water-air interface. To initiate theaction, the sphere is given a vertical downward velocity. Gravity acts on both the sphere and on

the respective fluids; buoyancy effects are also accounted for.

In each fluid, the multi-dimensional, unsteady equations of fluid motion are solved. Thoseequations include conservation of mass and momentum. Separate numerical solutions were

performed for the respective regimes of laminar and turbulent flow. Subsequently, comparisons

will be made between the results for the respective flow regime models to elucidate whetherturbulent fluctuations in the fluid motion have a significant impact on the results.

A specific motivation for considering both laminar and turbulent flows is that in

practice, it is not known a priori what is the state of the fluid flow. In particular, under

controlled laboratory conditions, it is possible that the air and water are both slowly moving and

are laminar. On the other hand, for in-field deployments of devices, wind-driven airflow andwater currents may lead to turbulence. A more fundamental issue is that for flows over blunt

objects, the leading-edge boundary layer may be laminar, whereas turbulence may prevaildownstream.

For laminar flow, the governing equations for fluid flow are expressed by Eq. (1) and by

Eq. (2) without the quantity turb These equations respectively represent conservation of mass

and momentum. When turbulent flow is being analyzed, turb, the turbulent viscosity, must beincluded.

0

i

i

x

u(1)

3,2,1

jf

x

u

xx

p

x

uu

t

uj

i

j

turb

iji

j

i

j (2)

The symbol ui denotes the fluid velocity in one of the three coordinate directions, is the

density,p is the static pressure, andthe molecular viscosity. The symbol fj is a body-force


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term and allows inclusion of buoyancy forces. Note that the quantities andthat appear in Eq.(2) must be specific to air and water in the respective fluid regions.

For those situations where turbulent motions occur, additional equations are necessary to

quantify the turbulent viscosity turb. To this end, a number of phenomenological models have

been proposed, starting in 1972. The model chosen here, the Shear Stress Transport (SST) model

(Menter et al., 1994), has been shown to provide acceptable results for complex fluid flowsituations. Two supplementary quantities, andrespectively the turbulence kinetic energyand the specific rate of turbulence dissipation, were introduced by Menter from which turb

follows as

2,max SFaa

turb

(3)

The quantities andare found from two additional transport equations which are presented asEqs. (4) and (5).

i

turb

ii

i

xxP

x

u

t

1 (4)

and

iii

urbt

ii

i

xxF

xxSA

x

u

t

w

2

1

2

2

2 112 (5)

The term Prepresents the production of turbulence kinetic energy, while the terms arePrandtl-number-like coefficients for the respective transported variables. The quantity Sis the

magnitude of the shear strain rate, and the Fterms are blending functions which transition

between the model near the walland the model away from the wall. Equations (1)-(5)complete the definition of the SST methodology.

The solution domain encompassed the two fluids (air and water) and the sphere whichbegins its descent from a position slightly above the water surface. A schematic diagram of the

solution domain, Figure 1, also shows the initial position of the sphere. The dimensions were

selected to assure that the solutions were independent of the size of the domain. The mass of thesphere was 0.0335 kg, with a radius of 0.02 m. Values of the sphere initial velocity were varied

parametrically from 2 to 20 m/s, while the starting point of the center of the sphere was held

fixed at 0.03 m above the water surface. Properties for air and water are specified in Table 1.


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Fig. 1 The solution domain with annotations showing the physical size and boundary surfaces

Table 1 Properties of air and water

Air properties Water properties

Density (kg/m3) 1.19 997

Kinematic Viscosity (m2/s) 1.54e-5 8.93e-7

Dynamic Viscosity (N-s/m2) 1.83e-5 8.9e-4

Calculations were performed both with and without consideration of surface tension. For the

cases where surface tension was included, the value of the surface tension was 0.07 N/m.The motion of the sphere was determined by a force-momentum balance as embodied in

Newtons Second Law. The forces exerted on the sphere are due to fluid-solid interactions

(drag), gravity, and buoyancy. If the vertical component of Newtons second law is considered,there results

(6)


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Both the weight and the buoyancy force are strictly vertical, and the vertical component of the

drag force is readily evaluated. In Eq. (6), Vdenotes the time-dependent vertical velocity, andmis the mass of the sphere.

To complete the description of the problem, it is necessary to specify boundary conditions ateach of the surfaces which bound the solution domain. With reference to Fig. 1, the lowerboundary was designated as an opening which permits water to pass into and out of the domain.

There, entrainment (dragging of fluid by shear forces) was allowed. A similar condition was

imposed at the top of the domain. The nature of the opening boundary condition requires that thestatic pressure be specified at the boundary in question. In that regard, the pressure of the

ambient air was set equal to zero, and hydrostatic pressure was imposed at the upper and lower

boundaries when appropriate. At the vertical boundary of the solution domain, the static

pressure was specified, either zero (ambient air) or hydrostatic.Advantage was taken of symmetry so that the solutions were only carried out only within a

wedge that subtended 10 degrees. The two symmetry planes that bound the wedge prevent any

fluid flow across them and also require that the normal derivatives of the dependent variables bezero. The imposition of symmetry is supported by visual observations reported in numerous

experimental studies.

3. Solution ImplementationThe numerical solutions were performed by means of ANSYS CFX software, which is a

finite-volume code. That code provides access to several turbulence models. The one utilized

here is the Shear-Stress Transport model (SST) which has proven to be highly successful in

predicting complex fluid flows.

Two independent sets of numerical solutions were performed with the boundary conditions

imposed at different distances from the sphere. Comparisons between the two sets of solutionsdemonstrated total insensitivity to the different locations at which the boundary conditions were

imposed. The final configuration of the solution domain used for the numerical solutions is thatdisplayed in Fig. 1. The initial condition for the solutions was no motion in either fluid.

Numerical solutions were carried out until the sphere was fully immersed in the water. The

temporal duration of the solution was dictated by the magnitude of the initial velocity. Thetimewise step size was varied until the solution was independent of it .

The mesh which was used to discretize the solution domain was also a major focus issue,

particularly in the near vicinity of the sphere. Figure 2 has been prepared to quantify the fineness

of the mesh. The figure displays a sequence of magnifications to illustrate the mesh deploymentadjacent to the sphere surface. One zone which requires careful meshing is the sphere-fluid

interface. In addition to careful mesh refinement, special elements were employed at the spheresurface itself. These special elements, called boundary elements or inflation elements, arealigned with the solid surface to resolve the large gradients there. In order to accurately capture

the boundary layer flow, it was necessary to tailor the mesh to satisfy the refinement criterion y+

~ 1, wherey+ is a dimensionless distance measured from the fluid-wall interface. A moredetailed presentation of the impact of mesh and time step will be presented later.


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Fig. 2 Magnifications showing refined elements in the vicinity of the sphere-fluid interface.

4. Results and DiscussionThe key result which is sought from these simulations is the drag coefficient in the early

stages of entry. Specifically, focus is directed to the period between the instant of initial contact

of the sphere with the water surface to the moment when the sphere became fully submerged.The depth of penetration is commonly referenced to the size of the entry object (sphere radius),

so that the dimensionless penetration depth is defined as

(7)

The fully submerged condition corresponds to b > 2.

4.1. Drag Results

Values of the drag coefficient are obtained at any instant of time by nondimensionalization ofthe instantaneous vertical force Fexerted on the sphere by the fluid. Mathematically, the drag

coefficient is


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F

1

22

(8)

where Vis the instantaneous velocity of the sphere. The densityused in the denominator ofEq. (8) is that of water, and the symbolA represents the frontal area of the sphere. The thus-

determined drag coefficient is that required for use in dynamic models such as those that enableprediction of the depth of oceanographic devices (Stark et al., 2011; Abraham et al., 2011;2012a; 2012b).

A physical phenomenon that leads to very high drag coefficients at the beginning of the entry

is the acceleration of the water contiguous to the sphere. Figure 3 is a schematic diagram

prepared to show the water region which experiences that acceleration. This acceleration, whichrequires a transfer of momentum from the sphere to the fluid, results in a brief elevation of the

drag coefficient. The thus-accelerated water is typically referred to as a virtual mass.

Fig. 3 Illustration showing the liquid adjacent to the sphere which is accelerated.

With this discussion as background, entry-drag results are presented as shown in Fig. 4. This

set of results is an outcome of a study of spatial mesh- and time-step independence. Three sets

of results are shown, labeled respectively as,finer mesh, small time step; fine mesh, small timestep; andcoarse mesh, largetime step. The small and large time steps are 2e-5 and 2e-4

seconds, respectively. The coarse-mesh case encompassed 49,000 elements with the elements

nearest the sphere being1.5 mm thick. In contrast, the fine mesh case consisted of 664,000elements, with the elements nearest the sphere of 0.002 mm thickness. Despite these large

differences in the spatial mesh and the time step, the results are seen to be nearly the same overthe range of dimensionless depths set forth in the figure. A further refinement was carried outwith a mesh of 7,150,000 elements and a nearest wall element thickness of 0.001 mm. This

third-level of refinement was performed to verify the mesh-independence of the calculations.

Calculations were performed using a quad-core 3.1 GHz processer with 8GB RAM. The

coarse-mesh, large-time-step solutions required 12 clock hours for the calculations, whereas thefine-mesh, small time-step simulations required 210 hours for the laminar solution and 240 hours

for the turbulent calculation. The third level of refinement required 2500 hours (~105 days) for a


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full solution. Furthermore, the volume of data required to store all simulation variables at each

time step was approximately 4TB. As a consequence, the finer mesh, small step solutions were

only carried out for a portion of the entry cycle (up to b ~ 0.5). It can be seen that at least for this

portion of the entry problem, the results are quite similar, despite the more than 100-folddifference in mesh and solution time.

Fig. 4 Dependence of the predicted drag coefficient on mesh and time step. The small and large

time steps are 2e-5 and 2e-4 seconds, respectively. The coarse, fine, and finer meshes consisted

of 49,000, 664,000, and 7,150,000 elements, respectively.

The next set of results to be shown, which are presented in Fig. 5, compares the drag results

for the laminar and turbulent simulations. Also shown in the figure are experimental data

(Baldwin and Steves, 1975). These experiments were utilized because they cover the largestrange of dimensionless depths among those found in the literature. It should be noted that for the

simulation results displayed in this figure, the initial velocity of the sphere was 2 m/s (Reynolds

number ~ 80,000), and the specific gravity of the sphere was one. The experiments of Baldwinand Steves utilized a 3-inch-diameter (0.0762 m) and a 5-inch-diameter (0.127 m) sphere with

specific gravity values of 2.42 and 0.795, respectively. The initial velocities used in the Baldwin

and Steves study ranged from 15-23 ft/sec (4.6-7.0 m/s).The aforementioned differences in the operating conditions between the present simulations

and the Baldwin-Steves experiments is readily rationalized by the use ofdimensionless

parameters, as demonstated by the excellent agreement between the simulation and experimentalresults displayed in Fig. 5. The present results also agree very well with other experiments, such


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as those of Moghisi and Squire, 1981. Also significant is the fact that the spread among

experimental results from numerous investigations is greater than the differences between the

present results and those of Baldwin-Steves. The drag coefficient results of Figs. 4 and 5 (and

the results that follow thereafter) agree to within 5% with those predicted from a theory based onsimplifying assumptions due to Shiffmann and Spencer, (1945a; 1945b; 1947).

Another novel finding seen in Fig. 5 is the seeming insignificance of turbulence. Thisimportant outcome can be attributed to the dominance of momentum transfer between the sphereand the contiguous water as the source of drag, with friction being a secondary issue. The

simulations, which extend far beyond the experimental results, continue to show excellent

agreement between the laminar and turbulent solutions.When taken together, the excellent accord between the present simulations, literature-based

experiments, and theoretical predictions serve to validate the model and the simulation

methodology developed here.

Fig. 5 Drag coefficients in the entry zone, comparison of experimental results with laminar and

turbulent predictions.

In order to explore the possible dependence of the drag coefficient on velocity, simulationswere performed for initial velocities that varied from 2 20 m/s. This issue is addressed in Fig.

6, where results corresponding to initial velocities of 2 and 20 m/s are displayed. It is seen thatfrom the onset of impact to a dimensionless depth of one, the drag coefficients are nearlyidentical, indicating that over the investigated range of velocities, the drag coefficient is nearly

independent of velocity. This outcome is attributable to the fact that the momentum transfer

between the sphere and the contiguous water is proportional to V2. This conclusion is reinforced

both by theory (Shiffmann and Spencer, 1945a; 1945b; 1947) and by experiments (Moghisi and

Squire, 1981).


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The Reynolds numbers corresponding to the range of investigated velocities is approximately

80,000 800,000. It should be recognized that the displayed universal nature of the drag

coefficient is not expected to continue for deeper penetrations, particularly for cases which lead

to a trailing cavity. On the other hand, the recognition that the drag coefficients are universalover the time period when its values are highest is important for subsequent calculations of the

downward trajectory of the object. The results shown in Fig. 6 were implemented using thecoarse mesh and large time step model, as discussed in connection with Fig. 4.

Fig. 6 Comparison of drag coefficients for initial sphere velocities of 2 and 20 m/s.

Another relevant issue is whether surface tension has a significant effect on the drag

coefficient. In order to explore this issue, particularly during the duration of object entry whenthe drag coefficient is largest, two calculations have been performed. The results, presented in

Fig. 7, correspond to laminar flow with and without surface tension included. It is not surprisingthat surface tension has little impact on the results because the drag is primarily due to

acceleration of adjacent liquid (virtual mass).


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Fig. 7 Impact of surface tension on drag coefficient.

Of primary utility for the prediction of the trajectory of an object as it descends beneath thesurface of the water is a convenient representation of the drag coefficient during entry. Based on

the foregoingdetermined independence of the drag coefficient from the entry velocity and

surface tension, algebraic representations have been developed and are displayed in Fig. 8 Onefitted polynomial pertains to dimensionless depths 0 < b < 1, and another polynomial is

applicable for 1 < b < 2. The complex behavior of the drag coefficient made it impossible to

develop a single algebraic representation which would be suitable over the entire range ofdepths.


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Fig. 8 Functional relationships for drag coefficients in the entry region.

4.2. Flow Patterns

In addition to the presented quantitative information for the drag coefficient, the numerical

simulations also provide visualizations of the fluid patterns in both the air and the water regions.Such visualizations are presented in Figs. 9 (a) (e). Those figures respectively show the sphere

at different levels of penetration into the water region. The respective air and water regions areidentified by different shades of gray. The impact and initial penetration of the sphere into the

water are shown in the (a) and (b) parts of the figure. In particular, the latter shows the rise of

water along the periphery of the sphere and the subsequent formation of a cavity region at theupper surface of the sphere. The initiation of separation and cavity formation is seen in the (c)

(e) parts of the figure.


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(a) (b)

(c)

Figs. 9 (a) (e) Images of the water-air

regions during the entry process


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Another useful way to view the results, particularly with respect to impact forces, is to show

regions where water is moving in the upward or downward directions. The transference of

momentum to the adjacent water region, illustrated in Fig. 3, is responsible for a significant

portion of the impact forces. To provide the requisite information, Fig. 10 has been prepared.That figure shows four instances in time during the entry. For each image, two contours are

shown. Beneath each of the spheres, a large region of downward-flowing fluid is indicated andis connected with the legend at the right side of the image. To the side of the sphere, there is anupward flowing region of water which is linked to the legend at the left of the image. Both

legends are expressed in meters/second. Velocity magnitudes less than 0.01 m/s are not shown.

It is seen that both regions of flow grow during the entry process. It is also seen that thedownward fluid occupies a significantly larger space than does the upward flow. The time

values listed in the figure are referenced from the onset of sphere-water collision.

Figs. 10 (a) (d) Regions of upward and downward flow of water

5. Concluding RemarksThis investigation was motivated by the need to formulate and definitively validate a

physical model and its numerical implementation that can be used to accurately describe the entry

of a solid object into water from an air space above the air-water interface. Although the work

significantly advances the modeling of a sphere as the entry object, its greater significance is to

provide a validated model for the entry of other objects of practical utility such as oceanographictemperature measuring devices. The reason for the present focus on the sphere is that, among all

other object geometries, there is considerably more experimental data available for sphere entry.

Such data is a prerequisite for validation of numerical simulation models. The result of specialfocus here is the drag coefficient.

The drag coefficient presented here is the ratio of the drag force to the momentum flow rate.


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When plotted as a function of the dimensionless penetration depth, it was found that the drag

coefficient results are independent of the sphere velocity, surface tension, flow regime (either

laminar or turbulent), and Reynolds number. These remarkable findings can be attributed to the

fact that the drag force is due to momentum transfer between the moving sphere and the waterinto which sphere is intruding. Friction-based drag is negligible in the situation under study.

In a graphical presentation of the results, it was shown that in the very early stages of spherepenetration, very high values of the drag coefficients are encountered which lead to highretardation forces on the object. These high forces tend to slow the fall rate of objects that

impact the water surface. Correlation equation in algebraic form were developed to quantify the

relationship between the drag coefficient and the dimensionless depth of immersion of the sphereThe model and simulation methodology developed in this study can be extended to other

geometries such as expendable bathythermograph (XBT) devices which are used to measure

ocean temperature distributions. XBT devices are released from a wide range of heights above

the water surface with a corresponding extensive range of impact velocities on the surface. It isexpected that the model and methology set forth in this paper will facilitate the determination of

the impact and subsequent penetration of various objects of practical utility. The full penetration

results from the entry model will serve as the starting condition for a fully immersed model ofthe trajectory of the object.

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Modeling and Numerical Simulation of the Forces Acting on a Sphere During Early-Water Entry

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