EXTENDED ABSTRACT

Submitted By G BHARAT V R REDDY 14CE65R22

The air-water sloshing problem: Fundamental analysis and parametric studies on excitation and fill levels K.P.Thiagarajan, D.Rakshit, N.RepalleThe UWA Oceans Institute and School of Mechanical Engineering, The University of Western Australia, Perth, WA 6009,AustraliaABSTRACTThe problem of liquid sloshing in partially filled transport containers has gained recent attention. The impact pressure on container walls due to sloshing in partially filled containers depend on several parameters such as fill level, amplitude of oscillation of tank. Stability of containers In this paper linear potential theory for sloshing in 2 dimensional rectangular tank is studied.Computational models were developed using the finite volume approximation for fluid domain and VOF technique for the liquid air interface.INTRODUCTIONAs the demand for LNG has increased so has the demand for larger vessels to carry LNG. Ship owners and transporters are aiming to transport LNG in these carriers at partially full level. Due to the motion of the sea, the free surface of the partially filled fluid moves and imparts impact pressures on the inner surfaces of the containers. These forces can sometimes lead to the structural failure of the container. Various parameters affecting the impact pressures are studied.OBJECTIVEThe paper studies the fundamental sloshing phenomenon occurring in a 2 dimensional rectangular container. A linear inviscid potential theory is developed and response amplitude operators are derived.Computational fluid dynamics methods are then applied for sloshing in rectangular tank where the two phases are air and water. The term response amplitude operator(RAO) is introduced as the motion amplitude per wave amplitude.THEORETICAL DEVELOPMENTThe formulation is of the small amplitude wave theoryThe governing equation2 =0Boundary conditions

The solution to the above problem is n2= g (n/a)tanh(nh/a)where n is the mode number.The generalized free surface equation and the velocity potential may be written as

To bring the excitation into picture Euler Lagrange equation is applied

The final equation turns out to be qn + n2qn = (2(1-cosn)/n)tanh(nh/a)X Response Amplitude operator RAOx = =|( 2/ 2- n2)cos()|VALIDATION OF FORMULATIONWarnitchai and Pinakaew conducted experiments were conducted on sloshing in a rectangular tank of length 400mm and width 200mm placed on a shaker table. A wire mesh was placed half way in the tank to increase damping of the motion. Results were noted at 30% fill condition when the tank was excited at various frequencies while maintaining a constant amplitude of 2mm. At this amplitude motion is considered linear.The results obtained from the theory matched with the experimental values. The agreement in the off resonant regions is reasonable and well predicted by the linear theory. In the resonant region experimental and numerical data are sparse.

COMPUTATIONAL MODELThe computational solution of the fluid dynamics problem requires the solution of conservation of mass and conservation of momentum equations.In the present situation modelling is done by assuming the flow to be turbulent. To reduce the complexity of simulations by time averaging, turbulence effects are incorporated in terms of mean quantities of flow. With the introduction of time averaging procedure the instantaneous velocity becomes ut=U + u where U is the mean and u is the fluctuating component.We obtain Continuity equation Momentum Equation

The additional represents the turbulent stresses.

The Reynolds stresses are calculated with the knowledge of turbulent kinetic energy k and dissipation rate . The interface between liquid and gas is tracked by volume of fluid (VOF) method. It relies on the fact that two or more phases are not interpenetrating and for each new phase a new variable i.e the volume fraction of the corresponding phase is introduced. Based on the local volume fraction of the qth fluid the appropriate properties and variables are assigned to each cell within the domain. A single momentum equation is solved throughout the domain and the resulting velocity field is shared among the phases. In the case of turbulent quantities a single set of transport equations is solved and the turbulence variables are shared by the phases throughout the field. COMPUTATIONAL PROCEDUREA structured moving grid-mesh is used to simulate 2D sloshing flow in rectangular tank for various fill levels ranging 10% to 95%. In order to model the free surface boundaries, a dynamic mesh algorithm is utilized. In this method, the mesh is updated for each set of successive iteration from the given time varying boundary conditions. The boundary sway motions considered in the numerical simulations are created using a piston crankshaft arrangement. The relation between the crankshaft speed and the piston stroke with the sway period and amplitude is developed using a user defined sub routine.The variables monitored are pressure and free surface elevation. Pressures are measured along the right wall and on the roof, at points spaced 45mm apart. At each of these points, the pressure is obtained as the vertex average of all the cells within 2.5mm strip. The peak pressure obtained over the cycles was averaged to get an indicative maximum pressure at that location. For the wave height two iso surfaces were created, coincident with the instantaneous free surface and the bottom. Mesh convergence studies are also done.RESULTS AND DISCUSSIONSThree main parameters are analysed. These are Excitation frequency, Excitation amplitude Liquid fill level.INFLUENCE OF EXCITATION FREQUENCY At lower fill level, the agreement predicted by the linear theory is in close agreement with the simulations except at near resonant conditions. The average of pressure peaks tends to be lower than the theoretical values. The data around resonance is highly non-linear and the simulated results. A close examination of the free surface elevation at second natural frequency shows that free surface motion is non-linear. INFLUENCE OF EXCITATION AMPLITUDEIn the resonant and near resonant regions the excitation amplitude and damping are expected to play an important role. It is also important in differentiating linear and non-linear regimes. It is seen that for amplitudes upto 2.5 % tank length there is general agreement between theory and simulations. Beyond this there is deviations in theory and simulated values as non-linear behaviour creeps in. Increasing the amplitude of sway increases the sloshing height of water in the tank. For the higher amplitudes, the water can be seen to hit the roof at one instant and leave the bottom of the tank dry at another instant in time.INFLUENCE OF TANK FILL LEVELAt higher fill levels, the impact with the roof can cause pressure changes, which significantly deviate from linear predictions. Irrespective of the frequency and fill beyond 50% the maximum pressure tends to occur at a fixed point on the roof. It is felt that this phenomenon could be caused by the topology of the overturning free surface, which may have deterministic features. CONCLUSIONSValuable insight was provided by the numerical and theoretical calculations but undisputable results havent been found.It was interesting to note that 20% and 80% fill levels had somewhat higher pressures than other conditions.At fills beyond 50% , the maximum pressure occurred at affixed point on the roof.