Drag Coefficients of Steel Spheres Entering Water Vertically

May, Albert; Woodhull, Jean C.

doi:10.1063/1.1715027

Cited by 103 publications

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“…May & Woodhull (1948) conclude that the cavity shape is not dependent on the nose shape of the projectile for a given drag force. Shi, Itoh & Takami (2000) image bullets shot vertically into a tank at 342 m s −1 , qualitatively considering the bullets' supercavitating behaviour; they find that for blunt leading edge projectiles, the after-body shape can significantly affect the splash formation.…”

Section: Water-entry Problemmentioning

confidence: 92%

“…Their work, like much of the existing theoretical work done to determine the force at impact, only considers impact up to a maximum penetration depth of half a sphere diameter. May & Woodhull (1948, 1950 note that the drag coefficient declines gradually towards a value between 0.25 and 0.3 when cavity is formed; the precise shape of the curve appears to depends on the specific gravity of the impacting object.…”

Section: Water-entry Problemmentioning

confidence: 99%

“…Today, digital high-speed cameras are widely used for imaging water-entry hydrodynamics. Extensive experimental investigations of water entry by spheres and projectiles are presented in Bell (1924), May & Woodhull (1948, 1950, Richardson (1948), May (1951), May & Hoover (1963) and Abelson (1970). …”

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confidence: 99%

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Water entry of spinning spheres

2009

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The complex hydrodynamics of water entry by a spinning sphere are investigated experimentally for low Froude numbers. Standard billiard balls are shot down at the free surface with controlled spin around one horizontal axis. High-speed digital video sequences reveal unique hydrodynamic phenomena which vary with spin rate and impact velocity. As anticipated, the spinning motion induces a lift force on the sphere and thus causes significant curvature in the trajectory of the object along its descent, similar to a curveball pitch in baseball. However, the splash and cavity dynamics are highly altered for the spinning case compared to impact of a sphere without spin. As spin rate increases, the splash curtain and cavity form and collapse asymmetrically with a persistent wedge of fluid emerging across the centre of the cavity. The wedge is formed as the sphere drags fluid along the surface, due to the no-slip condition; the wedge crosses the cavity in the same time it takes the sphere to rotate one-half a revolution. The spin rate relaxation time plateaus to a constant for tangential velocities above half the translational velocity of the sphere. Non-dimensional time to pinch off scales with Froude number as does the depth of pinch-off; however, a clear mass ratio dependence is noted in the depth to pinch off data. A force model is used to evaluate the lift and drag forces on the sphere after impact; resulting forces follow similar trends to those found for spinning spheres in oncoming flow, but are altered as a result of the subsurface air cavity. Images of the cavity and splash evolution, as well as force data, are presented for a range of spin rates and impact speeds; the influence of sphere density and diameter are also considered. IntroductionThe water-entry problem, by itself, is directly relevant to many different applications: from ballistics (May 1975) and ship slamming (Faltinsen & Zhao 1997) to skipping stones (Rosellini et al. 2005) and Basilisk lizards (Glasheen & McMahon 1996). One of the geometrically most simple objects that can be studied is the sphere. This canonical shape impacting on the free surface does not, however, yield simple hydrodynamic results, and the results are even more complex when spin is introduced (Truscott & Techet 2006). An experimental study of the impact of a sphere, spinning transverse to its velocity, on a water surface is presented herein, offering a first look into how spin can affect water-entry behaviour. Figure 1 shows a comparison of the non-spinning (a) and spinning (b) impact of a standard billiard ball on the free surface. The air cavity and splash formed by the spinning sphere vary distinctly from the axisymmetric cavity formed with no spin. The subsurface air cavity bends along the trajectory of the spinning sphere, and the splash † Email address for correspondence: ahtechet@MIT.EDU curtain grows vertically and collapses asymmetrically. For the spinning water-entry problem, valuable insight into the physics can be drawn from both water entry and spinning sphere re...

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Section: Water-entry Problemmentioning

confidence: 92%

Section: Water-entry Problemmentioning

confidence: 99%

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Water entry of spinning spheres

2009

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“…Such experimental results were very helpful in approving analytical results and relations. May and Woodhull [4][5][6] published three important papers to find the drag coefficient and the scaling relationship in water entry of sphere projectiles by considering the Reynolds and Froude numbers. They used an experimental method to study the air cavity formed during projectile impact on water using highspeed photography.…”

Section: Experimental Studiesmentioning

confidence: 99%

Numerical Investigation of Water Entry Problem of Pounders with Different Geometric Shapes and Drop Heights for Dynamic Compaction of the Seabed

et al. 2018

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The water entry problem of three-dimensional pounders with different geometric shapes of cube, cylinder, sphere, pyramid, and cone was numerically simulated by the commercial software Abaqus, and the effects of pounder shape and drop height from the free surface of water on deepwater displacement and velocity as well as pinch-off time and depth were investigated. An explicit dynamic analysis method was employed to model fluid-structure interactions using a Coupled Eulerian-Lagrangian (CEL) formulation. The simulation results are verified by showing the computed shape of the air cavity, displacement of sphere, pinch-off time, and depth which all agreed with the experimental results. The results reveal that the drag force of water has the highest and lowest effect on cubical and conical pounders, respectively. Increasing the pounder drop height up to the critical height leads to increased pounder velocity while impacting the model bed and more than the critical drop height has a reverse effect on pounder impact velocity. Pinch-off time is a very weak function of pounder impact velocity; but pinch-off depth increases linearly with increased impact velocity.

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“…Problem of impacts and ricochets of solid bodies against water surface have been received a considerable amount of attention [1,2,3,4,5,6]. In the early stage, the problem was of importance in naval engineering concerning the impacts of canon balls on sea-surface [7].…”

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confidence: 99%

Theoretical and Numerical Approach to “Magic Angle” of Stone Skipping

Nagahiro

Hayakawa

2005

Phys. Rev. Lett.

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We investigate the condition for the bounce of circular disks which obliquely impacts on fluid surface. An experiment [ Clanet, C., Hersen, F. and Bocquet, L., Nature 427, 29 (2004) ] revealed that there exists a "magic angle" of 20• between a disk's face and water surface in which condition the required speed for bounce is minimized. We perform three-dimensional simulation of the diskwater impact by means of the Smoothed Particle Hydrodynamics (SPH). Futhermore, we analyze the impact with a model of ordinal differential equation (ODE). Our simulation is in good agreement with the experiment. The analysis with the ODE model gives us a theoretical insight for the "magic angle" of stone skipping.PACS numbers: 45.50. Tn, 47.11.+j, 47.90.+a Problem of impacts and ricochets of solid bodies against water surface have been received a considerable amount of attention [1,2,3,4,5,6]. In the early stage, the problem was of importance in naval engineering concerning the impacts of canon balls on sea-surface [7]. Investigations then revealed that there exists a maximum angle of incidence θ max for impacts of spheres, above which the rebound does not occur [8]. Besides, it was empirically found that the θ max relates to specific gravity of sphere σ as θ max = 18/ √ σ. This relation was theoretically explained using a simple model of an ordinal differential equation (ODE) [8,9]. In military engineering today, the problem of water impacts may be not as important as that of a century ago, however, recently it attracts renewed interest under the studies of locomotion of basilisk lizards [10] and stone-skip [11].This study is motivated by experimental study of stone-skip -bounce of a stone against water surfaceby C. Clanet et. al. [12]. They investigated impacts of a circular disk (stone) on water surface and found that an angle about φ = 20• between the disk's face and water surface would be the "magic angle" which minimizes required velocity for bounce. In this paper, we study theoretically and numerically the oblique impact of disks and water surface. Our simulation successfully agrees with the experiment. Moreover, we apply an ODE model [17] to the disk-water impact and obtain an analytical form of the required velocity v min and maximum angle θ max as a function of initial disk conditions.To perform a numerical simulation of the disk-water impact, we solve the Navier-Stokes equation using the technique of Smoothed Particle Hydrodynamics (SPH) [13,14]. Fig. 1 is the snapshots of our simulation. The SPH method is based on Lagrangian description of fluid and has an advantage to treat free surface motion. Several representation of the viscous term have been proposed for this method. In this work, we adopt an artificial viscous term [15] which is simple for computation * Electronic address: nagahiro@cmpt.phys.tohoku.ac.jp and sufficiently examined with Couette flow [16]. In our simulation, we neglect surface tension and put the velocity of sound of the fluid, at least, 25 times larger than the incident velocity of the disk.In ...

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Drag Coefficients of Steel Spheres Entering Water Vertically

Cited by 103 publications

References 2 publications

Water entry of spinning spheres

Water entry of spinning spheres

Numerical Investigation of Water Entry Problem of Pounders with Different Geometric Shapes and Drop Heights for Dynamic Compaction of the Seabed

Theoretical and Numerical Approach to “Magic Angle” of Stone Skipping

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