Critical Free Surface Flows in a Sloshing Tank

  • Scolan, Y.M (ENSTA Bretagne)
  • 투고 : 2018.09.30
  • 심사 : 2018.11.27
  • 발행 : 2018.12.31


There are many issues in fluid structure interactions when dealing with the free surface flows in a sloshing tank. For example the problem of how yielding a highly nonlinear wave with a simple forced motion over a short duration is of concern here. Nonlinear waves are generated in a rectangular tank which is forced horizontally; its motion consists of a single cycle of oscillation. One of the objectives is to end up with a shape of the free surface yielding a wide range of critical flows by tuning few parameters. The configuration that is studied here concerns a plunging breaker accompanied with a critical jet where great kinematics are simulated. The numerical simulations are performed with a twodimensional code which solves the fully nonlinear free surface boundary conditions in Potential Theory.


E1GPC1_2018_v4n4_163_f0001.png 이미지

Fig. 1. Left: dambreaking case, successive free surface profiles (red lines). Initial free surface deformation (green line) of Gaussian type: y = h + ae−r(x−L)2with L = 4m (length of the tank), h = 0.2m, a = 0.9m, r = 2m−2. Right: profile at the last stage of the overturning crest, terminology and notations used for the description of the plunging breaker.

E1GPC1_2018_v4n4_163_f0002.png 이미지

Fig. 2. Variation of the velocity modulus ||$\vec{\nabla}\phi$|| (left) lagrangian acceleration $\frac{d\vec{\nabla}\phi}{dt}$ (right) in terms of time and arclength along the free surface with origin at the left wall. Superimposed blue curve: location of the crest, superimposed black curve: location of the maximum acceleration, superimposed green curve: location of the maximum velocity. See figure (1) for computational

E1GPC1_2018_v4n4_163_f0003.png 이미지

Fig. 3. Time variation of the fluid energy components, potential energy: green line, kinetic energy: red line, error on the energy conservation (with factor 104: blue line. See figure (1) for computational data.

E1GPC1_2018_v4n4_163_f0004.png 이미지

Fig. 4. Horizontal motion of the rectangular tank. Half sinus with period T matched to third order polynomials at the beginning and the end. Amplitude of the motion : A.

E1GPC1_2018_v4n4_163_f0005.png 이미지

Fig. 5. Forced horizontal motion of the tank as illustrated in figure (4). Successive free surface profiles (red lines). Amplitude of the forced motion A = 0.273m, period T =1.975s.

E1GPC1_2018_v4n4_163_f0006.png 이미지

Fig. 6. Forced horizontal motion of the tank as illustrated in figure (4). Successive free surface profiles (red lines). Amplitude of the forced motion A = 0.2331m, period T = 1.725s. Length of the tank L = 1.08m, mean water level h = 0.22833m. Green dots: maximum of velocity modulus on each free surface profile.

E1GPC1_2018_v4n4_163_f0007.png 이미지

Fig.7. Variation of the velocity (left) and acceleration (right) at the free surface in terms of time and arclength. Superimposed cuves, blue: tip of the crest, black: maximum acceleration, green: maximum velocity.

E1GPC1_2018_v4n4_163_f0008.png 이미지

Fig. 8. Left: time variation of the maxima of velocity (green) and acceleration (red) at the free surface. Right: time variation of the maxima of acceleration at the free surface in terms of tsing − t with tsing = 1.77175s with full logscales. Superimposed curve : f (t) = 0.7/(tsing −t)1.24. See figure(6) for computational data.

E1GPC1_2018_v4n4_163_f0009.png 이미지

Fig. 9. Time variation of the energy components: potential, kinetic, total. The instant t =1.41s corresponds to the maximum of the potential energy. See figure (6) for computational data.

E1GPC1_2018_v4n4_163_f0010.png 이미지

Fig. 10. Successive free surface profiles (red lines) obtained from a restart at t0 = 1.63s. Green dots : maximum of velocity modulus on each free surface profile.

E1GPC1_2018_v4n4_163_f0011.png 이미지

Fig. 11.Variation of the velocity (left) and acceleration (right) at the free surface in terms of time and arclength. Superimposed curves, blue : crest, black : maximum acceleration, green : maximum velocity.

E1GPC1_2018_v4n4_163_f0012.png 이미지

Fig. 12. Pressure components, top left : -$\phi$,t, top right : $\frac{1}{2}$\vec{\nabla}2$\phi$, bottom : $\frac{p}{\rho}$ = -$\phi$,t - $\frac{1}{2}$\vec{\nabla}2 $\phi$ - gy, at time instant t = 0.1555s. Units: m2/s2.

E1GPC1_2018_v4n4_163_f0013.png 이미지

Fig. 13. Left : free surface profile at time instant t = 0.1555s. Right: variation of the normal and tangential components of the velocity with the arclength. Marks located at points of either horizontal tangent or vertical tangent. Correspondance of the color code in both figures.

E1GPC1_2018_v4n4_163_f0014.png 이미지

Figure 14: Time variations of the maximum velocity (left) and maximum acceleration (right) at the free surface for all restarts instants to ∈ [1.41s : 1.70s].


  1. Bogaert, H., Leonard, S., Brosset, L., Kaminski, M.L., 2010 Sloshing and scaling: results from the SLOSHEL project, Proc. ISOPE, Beijing, China.
  2. Brosset, L., Mravak, Z., Kaminski, M. Collins, S. and Finnigan, T., 2009, Overview of SLOSHEL project, Proc. ISOPE, Osaka, Japan.
  3. Brosset, L., Lafeber, W., Bogaert, H., Marhem, M., Carden, P. and Maguire, J., 2010 Mark III panel subjected to a flip-through wave impact: results from the SLOSHEL project. Proc. ISOPE, Maui, Hawaii, USA.
  4. Constantin A., 2015, The time evolution of the maximal horizontal surface fluid velocity for an irrotational wave approaching breaking. Journal of Fluid Mechanics, 768, 468-475.
  5. Cooker, M.J., Peregrine, D.H., 1990. A model of breaking wave impact pressures. In: Proceedings of 22nd ASCE Conference Coastal Engineering, Holland, pp. 1473-1486.
  6. Dommermuth D. G., Yue D. K. P., Lin W. M., Rapp R. J., Chan E. S. and Melville W. K., 1988, Deep-water plunging breakers: a comparison between potential theory and experiments. Journal of Fluid Mechanics, 189, 423-442.
  7. Greenhow M., 1983, Free-surface flows related to breaking waves. Journal of Fluid Mechanics 134, 259-275.
  8. John F., 1953, Two-Dimensional Potential Flows with a Free Boundary, Communications on pure and applied mathematics, Vol. VI, 497-503.
  9. Karimi M.R., L. Brosset, J.-M. Ghidaglia, M.L. Kaminski, 2016, Effect of ullage gas on sloshing, Part II: Local effects of gas-liquid density ratio. European Journal of Mechanics B/Fluids, 57, 82-100.
  10. Longuet-Higgins M. S. & Cokelet E. D. 1976, The deformation of steep surface waves on water. I. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1-26.
  11. Longuet-Higgins M. S., 1980, A Technique for Time-Dependent Free-Surface Flows Proceedings of the Royal Society of London. Series A, 371, 441-451.
  12. Longuet-Higgins M. S., 1980, On the forming of sharp corners at a free surface, Proc. R. Soc. Lond. A, 371, 453-478.
  13. Longuet-Higgins M. S., 1981, On the overturning of gravity waves, Proc. R. Soc. Lond. A, 376, 377-400.
  14. Longuet-Higgins M. S., 1982, Parametric solutions for breaking waves, Journal of Fluid Mechanics 121, 403-424.
  15. Longuet-Higgins, M.S., and Oguz, H.N. 1997, Critical jets in surface waves and collapsing cavities. Phil. Trans. R. Soc. Lond., A 355, 625-639.
  16. Longuet-Higgins M. S. , 2001, Vertical jets from standing waves; the bazooka effect, A.C. King and Y.D. Shikhmurzaev (eds.), IUTAM Symposium on Free Surface Flows, pp 195-203, Kluwer Academic Publishers.
  17. Miller, R. L., 1957, Role of vortices in surf zone prediction: sedimentation and wave forces. In Beach and Nearshore Sedimentation (ed. R. A. Davis & R. I. Ethington), pp. 92-114. SOCE.con. Paleontologists and Mineralogists Spec. Publ. 24.
  18. New, A.L., McIver P,. & Peregrine D. H. 1985, Computations of overturning waves. Journal of Fluid Mechanics 150, 233-251
  19. Peregrine, D.H., Cokelet, E.D. and McIver, P., 1980, The fluid mechanics of waves approaching breaking. Proc. 17th Coastal Engng.Conf. ASCE, Sydney, Vol. 1, 512-528.
  20. Scolan Y.-M., O. Kimmoun, H. Branger, F. Remy, 2007, Nonlinear free surface motions close to a vertical wall. 22nd International Workshop on Water Waves and Floating Bodies, Plitvice, Croatia, 15 April - 18 April 2007.
  21. Scolan, Y.-M., 2010, Some aspects of the flip-through phenomenon: A numerical study based on the desingularized technique J. Fluid Struc., 26, Issue 6, 918-953.
  22. Scolan Y.-M., M. R. Karimi, F. Dias, J.-M. Ghidaglia, J. Costes, 2014, Highly nonlinear wave in tank with small density ratio. 29th International Workshop on Water Waves and Floating Bodies, Japan, April 2014.
  23. Scolan Y.-M., 2016, The kinematics in a plunging breaker revisited. 32nd International Workshop on Water Waves and Floating Bodies, Dalian, China, April 2016, pp 173-176.
  24. Scolan Y.-M. & L. Brosset, 2017, Numerical Simulation of Highly Nonlinear Sloshing in a Tank Due to Forced Motion, International Journal of Offshore and Polar Engineering (ISSN 1053- 5381) Vol. 27, No. 1, March 2017, pp. 11-17;
  25. Sulem, C. Sulem P.-L. and Frisch, H., 1983 Tracing complex singularities with spectral methods Journal of Computational Physics, vol. 50, April 1983, p. 138-161.
  26. Vinje, T. and Brevig, P. 1981, Numerical simulation of breaking waves. Adv. Water Resources, 4, 77-82.
  27. Yasuda T., Mutsuda H., Mizutani N. 1997, Kinematics of overturning solitary waves and their relations to breaker types Coastal Engineering , 29, Issues 3-4, 317-346.