Skip to main content

Unsteady forces on spheres during free-surface water entry

  • Tadd T. Truscott (a1), Brenden P. Epps (a2) and Alexandra H. Techet (a2)

We present a study of the forces during free-surface water entry of spheres of varying masses, diameters, and surface treatments. Previous studies have shown that the formation of a subsurface air cavity by a falling sphere is conditional upon impact speed and surface treatment. This study focuses on the forces experienced by the sphere in both cavity-forming and non-cavity-forming cases. Unsteady force estimates require accurate determination of the deceleration for both high and low mass ratios, especially as inertial and hydrodynamic effects approach equality. Using high-speed imaging, high-speed particle image velocimetry, and numerical simulation, we examine the nature of the forces in each case. The effect of mass ratio is shown, where a lighter sphere undergoes larger decelerations and more dramatic trajectory changes. In the non-cavity-forming cases, the forces are modulated by the growth and shedding of a strong, ring-like vortex structure. In the cavity-forming cases, little vorticity is shed by the sphere, and the forces are modulated by the unsteady pressure required for the opening and closing of the air cavity. A data-driven boundary-element-type method is developed to accurately describe the unsteady forces using cavity shape data from experiments.

Corresponding author
Email address for correspondence:
Hide All
1. Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
2. Aristoff, J. M. & Bush, J. W. M. 2009 Water entry of small hydrophobic spheres. J. Fluid Mech. 619, 4578.
3. Aristoff, J. M., Truscott, T. T., Techet, A. H. & Bush, J. W. M. 2010 The water entry of decelerating spheres. Phys. Fluids 22 (032102).
4. Asfar, K. & Moore, S. 1987 Rigid-body water impact at shallow angles of incidence. In Proceedings of the Sixth International Offshore Mechanics and Arctic Engineering Symposium, pp. 105–112. ASME, Virginia Polytechnic Inst. State Univ., Blacksburg, VA, USA.
5. Bergmann, R., van der Meer, D., Gekle, S., van der Bos, A. & Lohse, D. 2009 Controlled impact of a disk on a water surface: cavity dynamics. J. Fluid Mech. 633, 381409.
6. Birkhoff, G. & Isaacs, R. 1951 Transient cavities in air–water entry. Tech. Rep. 1490. Navord Rep.
7. de Boor, C. 1978 A Practical Guide to Splines. Springer.
8. Cleveland, W. S. 1979 Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74 (367), 829836.
9. Do-Quang, M. & Amberg, G. 2009 The splash of a solid sphere impacting on a liquid surface: numerical simulation of the influence of wetting. Phys. Fluids 21 (2), 022102.
10. Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics of transient cavities. J. Fluid Mech. 591, 119.
11. Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180183.
12. Eggers, J., Fontelos, M. A., Leppinen, D. & Snoeijer, J. H. 2007 Theory of collapsing axisymmetric cavity. Phys. Rev. Lett. 094502.
13. Epps, B. P. 2010 An impulse framework for hydrodynamic force analysis: fish propulsion, water entry of spheres, and marine propellers. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.
14. Epps, B. P. & Techet, A. H. 2007 Impulse generated during unsteady maneuvering of swimming fish. Exp. Fluids 43 (5), 691700.
15. Epps, B. P., Truscott, T. T. & Techet, A. H. 2010 Evaluating derivatives of experimental data using smoothing splines. In Proceedings of Mathematical Methods in Engineering International Symposium. MMEI, Lisbon Portugal.
16. Gaudet, S. 1998 Numerical simulation of circular disks entering the free surface of a fluid. Phys. Fluids 10 (10), 24892499.
17. Gekle, S., Gordillo, J. M., van der Meer, D. & Lohse, D. 2009 High-speed jet formation after solid object impact. Phys. Rev. Lett. 102, 034502.
18. Gekle, S., Peters, I. R., Gordillo, J. M., Meer, D. & Lohse, D. 2010 Supersonic air flow due to solid–liquid impact. Phys. Rev. Lett. 104, 024501.
19. Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121140.
20. Gilbarg, D. & Anderson, R. A. 1948 Influence of atmospheric pressure on the phenomena accompanying the entry of spheres into water. J. Appl. Phys. 19 (2), 127139.
21. Glasheen, J. W. & McMahon, T. A. 1996 Vertical water entry of disks at low Froude numbers. Phys. Fluids 8 (8), 20782083.
22. Goldman, D. I. & Umbanhowar, P. 2008 Scaling and dynamics of sphere and disk impact into granular media. Phys. Rev. E 77, 021308.
23. Gordillo, J. M. 2008 Axisymmetric bubble collapse in a quiescent liquid pool. I. Theory and numerical simulations. Phys. Fluids 20, 112103.
24. Grumstrup, T., Keller, J. B. & Belmonte, A. 2007 Cavity ripples observed during the impact of solid objects into liquids. Phys. Rev. Lett. 99, 114502.
25. Horowitz, M. & Williamson, C. H. K. 2008 Critical mass and a new periodic four-ring vortex wake mode for freely rising and falling spheres. Phys. Fluids 20, 101701.
26. von Kármán, T. 1929 The impact on seaplane floats during landing. Technical Notes 321. National Advisory Committee for Aeronautics, Aerodynamic Institute of the Technical High School, Aachen, Washington, D.C., USA.
27. Korobkin, A. A. & Pukhnachov, V. V. 1988 Initial stage of water impact. Annu. Rev. Fluid Mech. 20, 159185.
28. Lee, M., Longoria, R. G. & Wilson, D. E. 1997 Cavity dynamics in high-speed water entry. Phys. Fluids 9, 540550.
29. May, A. & Hoover, W. R. 1963 A study of the water-entry cavity. Unclassified NOLTR 63–264. United States Naval Ordinance Laboratory, White Oak, Maryland, USA.
30. Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Dover.
31. Moghisi, M. & Squire, P. T. 1981 An experimental investigation of the initial force of impact on a sphere striking a liquid surface. J. Fluid Mech. 108 (1), 133146.
32. Newman, J. N. 1977 Marine Hydrodynamics. MIT.
33. Raffel, M., Willert, C., Willert, C. E. & Kompenhans, S. 1998 Particle Image Velocimetry. Springer.
34. Saffman, P. 1995 Vortex Dynamics. Cabridge University Press.
35. Techet, A. H. & Truscott, T. T. 2011 Water entry of spinning hydrophobic and hydrophilic spheres. J. Fluids Struct. 27 (5–6), 716726.
36. Thoroddsen, S. T., Etoh, T. G., Takehara, K. & Takano, Y. 2004 Impact jetting by a solid sphere. J. Fluid Mech. 499, 139148.
37. Truscott, T. T. 2009 Cavity dynamics of water entry for spheres and ballistic projectiles. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA.
38. Truscott, T. T. & Techet, A. H. 2009a A spin on cavity formation during water entry of hydrophobic and hydrophilic spheres. Phys. Fluids 21, 121703.
39. Truscott, T. T. & Techet, A. H. 2009b Water entry of spinning spheres. J. Fluid Mech. 623, 135165.
40. Wagner, H. 1932 Phenomena associated with impacts and sliding on liquid surfaces. Z. Angew. Math. Mech. 12, 193235.
41. Worthington, A. M. 1908 A Study of Splashes. Printed by William Brendon and Son, Ltd; reprinted by Macmillian Co., New York, 1963 edn. Longmans Green and Co., Plymouth..
42. Yan, H., Liu, Y., Kominiarczuk, J. & Yue, D. P. 2009 Cavity dynamics in water entry at low Froude numbers. J. Fluid Mech. 641, 441461.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed