Cooker, M. J.

Korobkin, A. A.

Zekri, H. J.

English

University of East Anglia digital repository

Liquid sloshing and impact in a closed container withhigh fillingH.J. Zekri1, A.A. Korobkin1, M.J. Cooker11School of Mathematics, University of East Anglia, Norwich, UKE-mail: h.zekri@uea.ac.uk1 IntroductionViolent liquid sloshing is of concern to cargo tank designers due to the problems of safety in extremeloadings. It is an interesting topic with still some disputable problems. For example Abramson, Bass,Faltinsen and Olsen [1] investigated the sloshing and resultant loads in liquid natural gas carriers fordifferent tank geometries and liquid fill depths. Ibrahim [3] provided a comprehensive study withexamples of free tank motions. Cooker [2] analysed with experiments a horizontal rectangular wavetank which swings at the lower end of a pendulum. Ten, Malenica and Korobkin [5], presented a semi-analytical approach for fluid-structure interactions inside tanks in different impact situations withhigh and low fillings. Our work is about the sloshing of standing waves inside a closed highly filledcontainer. Unsteady two-dimensional (2-D), irrotational flow is treated. The liquid-roof interactionis discussed with and without the effect of gravity and a comparison is made. The short-time modelof the liquid-roof impact is governed by a Mixed Boundary Value Problem (MBVP), which is solvednumerically and using the asymptotic methods.2 Mathematical formulationIn figure 1, a stationary highly filled rectangular tank, containing an inviscid incompressible liquid isshown. The flow is 2-D and irrotational and surface tension force is neglected. The container has heightH and length 2L and lies in the region y˜ ≥ 0. Here y˜ = 0 is the bottom and y˜ = H is the roof of thecontainer, y˜ = H−h is the still water level, and x˜ = ±L are the rigid impermeable walls. The lengths,time, velocity potential and pressure are scaled by H,√H/g, h√Hg and ρgh, respectively, where gis the gravitational acceleration and ρ is the constant density. The small parameter  = h/H  1 isresponsible for linearisation. In non-dimensional variables (without tilde), the initial shape of the freesurface is given by the equation y = f(x), f(x) = 1 −  + 2λ∑∞n=1 an cos(knx), an and kn are knownconstants and λ = LH .x˜y˜x˜ =Lx˜ =-L oy˜ = Hy˜ = 0h = 0.05H Free surface’s level stateRigid roofFigure 1: Container description with free surface at its level state.Considering the container to be without the rigid roof, in non-dimensional linearised initial bound-ary problem at the leading order the velocity potential φl(x, y, t) and the surface elevation ηl(x, t) readφl(x, y, t) =∞∑n=1a¯n cosh(kny) cos(knx) sin(wnt), (1)ηl(x, t) =2λ∞∑n=1an cos(knx) cos(wnt), an = −a¯nωn cosh(kn), (2)where the angular frequencies ωn are related to the wave numbers kn = npi/λ by the dispersionrelation ωn =√kn tanh(kn), for n ∈ N . In the next section we introduce the rigid roof to the tank,we decompose the velocity potential φ(x, y, t) and the surface elevation η(x, t) into two parts. Thishas been done by adding unknown functions: a so called correction velocity potential φc(x, y, t) and acorrection surface elevation ηc(x, t) resulting in φ = φc + φl and η = ηl + ηc respectively.3 Semi-analytical solutionAt the leading order, we derive the linearised non-dimensional MBVP, with gravity and the rigid roofincluded. During the early stage of the impact, with some asymptotic analysis, we have found thatgravity has a small influence. We define δ  1 to stretch the time t = t∗+ δtˆ, t = t∗ instant of impact,consequently the other variables, x = δ1/2xˆ, y = δ1/2yˆ, φc(x, y, t) = δ1/2φˆc(xˆ, yˆ, tˆ), xc0(t) = δ1/2xˆc0(tˆ)and ηl(x, t) = δηˆl(xˆ, tˆ). By combining the dynamic and kinematic boundary conditions we arrive at(without hat)∂2φc∂t2+ δ32∂φc∂y= 0 |x| > xc(t), y = 0, (3)where x = xc(t) is where the free surface meets the roof. The corresponding limiting problem as δ → 0is depicted in Figure 2. This problem does not account for gravity.φc = 0φc = 0y = 0∂φc∂y = −∂2ηl∂t2∂2φc∂x2+ ∂2φc∂y2= 0φc −→ 0 as x2 + y2 −→∞• •x = −xc0(t) x = xc0(t)Figure 2: The MBVP at the leading order.Working with stretched variables, we can approximately replace the wall and the bottom conditionswith the far-field condition given in Figure 2. With the condition introduced by Wagner [6] at thecontact points and the displacement potential Φ(x, y, t) =∫ t0 φc(x, y, τ)dτ introduced by Korobkin [4],we are able to calculate semi-analytically the leading order position of the contact point, x = xc0(t),during impact. Consequently the hydrodynamic pressure distribution can be found (see Figure 3).4 Numerical solutionA collocation method was used for the symmetric flow considered in the original coordinates xand y. The unknowns, velocity potential φ(x, y, t), surface elevation η(x, t) and the pressure on theroof p(x, 0, t) are respectively presented by Fourier series. The domain is discretized into N regularlyspaced nodes (N = 350 in our calculations).The combined kinematic and dynamic boundary conditions (with gravity included) lead to a sys-tem of the form~η = ~ηl −A−1B ~P −A−1M−1∑m=1Gtm~p tm , (4)where the initial pressure ~p t1 , found in the previous section, with respect to this algorithm is given.The tri-diagonal matrix A is associated with unknown and known surface elevation vectors, ~η and~ηl respectively. The matrix G is calculated at every time step while the matrix B is independent oftime and depends only on the time step length. The pressure ~P and surface elevation ~η are to bedetermined at the instant t = tM . However, the fact that on the free surface, xc0(tM ) < x < 1, wehave p(x, tM ) = 0 and on the impact region, 0 < x < xc0(tM ), we have η(x, tM ) = 1, with somerearrangements makes the system (4) solvable on its own. The free surface and the wetted region aredistinguished and updated at each time step by calculating the position of x = xc0(tM ) as part ofthe solution. The pressure vector ~p tm is known from the previous time steps t = tm, for 1 ≤ m ≤M−1.Continuing with the stretched variables introduced from Section 3, we study the gravity influenceon the length of the wetted region during the impact stage, that is xc0(t) + δ · xc1(t). The correctiondue to gravity xc1(t) in Figure 4b, is found to be almost completely insignificant at the early stage ofimpact. As time goes on the effect of gravity is that it decreases the length of the wetted region andeven at the very late period of this stage, its effect is found to be small. Figure 4a shows the wettedlength with and without gravity. More results of the numerical simulation will be presented at theworkshop.5 AcknowledgementsH. Zekri gratefully acknowledges the financial support received from Kurdistan Regional Government-Iraq for his study.References[1] H. N. Abramson, R. Bass, O. Faltinsen, and H. Olsen. Liquid slosh in LNG carriers. In Symposiumon Naval Hydrodynamics, 10th, Proceedings, Pap and Discuss, Cambridge, Mass, June 24-28,1974., number Proceeding, 1976.[2] M. J. Cooker. Water waves in a suspended container. Wave Motion, 20(4):385–395, 1994.[3] R. A. Ibrahim. Liquid sloshing dynamics: theory and applications. Cambridge University Press,2005.[4] A. Korobkin. Formulation of penetration problem as a variational inequality. Din. Sploshnoi Sredy,58:73–79, 1982.[5] I. Ten, Sˇ. Malenica, and A. Korobkin. Semi-analytical models of hydroelastic sloshing impact intanks of liquefied natural gas vessels. Philosophical Transactions of the Royal Society A: Mathe-matical, Physical and Engineering Sciences, 369(1947):2920–2941, 2011.[6] H. Wagner. Phenomena associated with impacts and sliding on liquid surfaces. Z. Angew. Math.Mech, 12(4):193–215, 1932.Figure 3: Semi-analytical pressure distribution without gravity at: (a) t = 1.0790; (b) t = 1.5760.Figure 4: Contact point position with correction due to gravity, here t = 0 is the impact time: (a) The leading ordercontact point, blue line,and the contact point with correction due to gravity, red line; (b) details of correction xc1(t) dueto gravity,for the contact point position.

Liquid sloshing and impact in a closed container with high filling

Abstract

Similar works

Full text

Available Versions

University of East Anglia digital repository