Critical Free Surface Flows in a Sloshing Tank

Abstract

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.

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.

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

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.

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.

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.

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.

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.

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 t_sing − t with t_sing = 1.77175s with full logscales. Superimposed curve : f (t) = 0.7/(t_sing −t)^1.24. See figure(6) for computational data.

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.

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.

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.

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.

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.

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