- Email: [email protected]
Numerical studies on sloshing in rectangular tanks using a tree-based adaptive solver and experimental validation
Ocean Engineering 82 (2014) 20–31
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Numerical studies on sloshing in rectangular tanks using a tree-based adaptive solver and experimental validation Hai-tao Li, Jing Li n, Zhi Zong, Zhen Chen School of Naval Architecture, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China
ar t ic l e i nf o
a b s t r a c t
Article history: Received 18 October 2012 Accepted 19 February 2014 Available online 13 March 2014
As liquid cargo transport developing, hull structural stability and strength are highlighted in various engineering areas due to tanks partially filled with fluid. Sloshing may be accounted as unpredictable force imposed on the whole faces of the tank walls. Water movement in rectangular tanks, under rolling and horizontal excitations, is investigated by numerical and experimental approaches. A parallel code, which directly discretizes the incompressible Navier–Stokes equations, coupled with VOF and tree-based adaptive algorithm, is employed to simulate the behavior of 2D fluid motion. A series of experiments are carried out to measure the pressures on the tank walls and under the water. A good agreement is shown in this paper and the values obtained by computations are validated. Through this, some studies are done on the basis of different excitation frequencies and filled levels. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Sloshing VOF Adaptive algorithm Numerical simulation
1. Introduction Sloshing is considered as a motion of water in partially filled tanks under an external excitation, which usually represents a violent phenomenon with high amplitudes of water surface and strong localized impacts on the tank walls when the excitation frequency is close to the natural frequency of this system. This phenomenon is particularly evident under relatively low water depth. The critical depth to length ratio is 0.3368 (Faltinsen and Timokha, 2009). Sloshing is also regarded as water wave movement of the free surface which is normally sorted into different types according to their shapes: standing wave, traveling wave, hydraulic jump, breaking wave and the combination of themselves. When the resonance occurs, it is difficult to separate them from each other, because of the strong nonlinear and arbitrary phenomenon experienced by the surface. Moreover, the instantaneous loads can be significantly high which may cause structural damage in turn and even engender sufficient torque to impede the stability of the vehicle. To that end this paper considers that under relatively low water depth how the surface shape changes according to different external frequencies and what extent are the loads predicted by numerical means against experimental method. To assess sloshing loads and the free-surface elevation, theoretical approaches based on linear and nonlinear potential flow theory and laboratory experimentation based on scaled models
n
Corresponding author. Tel.: þ 86 411 84708451; fax: þ 86 411 84708451x8036. E-mail address: [email protected] (J. Li).
http://dx.doi.org/10.1016/j.oceaneng.2014.02.011 0029-8018 & 2014 Elsevier Ltd. All rights reserved.
have been employed by many researchers. Although the sloshing technology developed for space applications is not directly applied to ship cargo tanks, Abramson (1966) studied the sloshing problem systematically, in terms of the space industries, by potential flow theory with different shapes of fuel tanks for launch vehicles. A series of analytical solutions were derived based on the linear method, and corresponding experiments were conducted. Faltinsen (1974), Faltinsen (1978) and Faltinsen (2000) applied potential flow theory of an incompressible liquid and developed nonlinear multimodal method for sloshing, and some classic formulae were obtained, which have been widely used today (Waterhouse, 1994), but only restricted to cases in simple shape tanks with the free surface elevation of a single-valued function. Akyildiz and Ünal (2005) and Akyildiz and Ünal (2006) designed an experiment which was set up to observe the behavior of liquid sloshing in rectangular container subjected to external excitation with a relatively deep water depth. Rafiee et al. (2011) studied experimentally sloshing phenomena in a thin rectangular tank (with width-tolength ratio of 0.0769) under a sway excitation, and only shallow water depth was considered. Since the advent of computers, people started to solve fluid mechanics problem by using computational techniques. Numerical simulations have become important and widely adopted technique to deal with strong nonlinear sloshing. Wu et al. (1998) analyzed the sloshing waves using finite element method (FEM). Shallow water theory is an important mathematical model when hydraulic jump occurs, which was employed by Nakayama and Washizu (1981) to present a nonlinear analysis of inviscid and incompressible fluid motion in a tank subjected to forced oscillation. At the same time,
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
both FEM and boundary element method (BEM) were combined in. Besides the aforementioned techniques, approaches based on Navier–Stokes equations have been applied to study liquid sloshing. Reynolds averaged Navier–Stokes (RANS) equations were compared with shallow water theory by Armenio and La Rocca (1996). Moreover, Liu and Lin (2009) studied sloshing using volume of fluid (VOF) method coupled with turbulent modeling. Xue and Lin (2011) similarly analyzed the surface elevation and velocity field of liquid sloshing by spatially averaged Navier–Stokes (SANS) equations coupled with virtual boundary force (VBF) method. The smoothed particle hydrodynamics (SPH) method featured mesh-free has been also applied to the study of sloshing flow. Rafiee et al. (2011) and Chen et al. (2013) both developed improved SPH techniques to simulate the sloshing flow by using more accurate integration and more applicable treatment of the boundary conditions. However, to our knowledge, adaptive algorithm used in this field is rarely seen. Liang and Borthwick (2009) validated a Godunov-type shallow flow solver on adaptive quadtree grids by simulating linear sloshing motions in a vessel with a parabolic bed. A parallel program Gerris Flow Solver with VOF method based on adaptive quadtree grids was created by Popinet (2003), Popinet (2009) and Popinet (2010). A condensed introduction of this opensource code is summarized in Section 3. The motivation of this paper is to demonstrate that Gerris can indeed simulate the violent fluid motion of sloshing and predict the accurate pressure. Therefore, both numerical simulation and laboratory experiment were carried out to study sloshing problems. We focus on giving comparisons of loads between the computations and laboratory experiments under the filling ratios falling into the vicinity of the critical depth mentioned in the first paragraph. For the purpose of that, we designed a series of tests of water sloshing in a rectangular tank. Because of the characteristic of the tank with a square bottom, the 3D effects cannot be ignored during our analysis. Thus, the free surface wave shape is another important aspect of this study.
2. Mathematical model The fluid motion of sloshing can be described by the incompressible continuity equation and Navier–Stokes equations: ∂u ρ ð1Þ þ u ∇u ¼ ρF ∇p þ ∇ 2μD ∂t ∇u¼0
ð2Þ
where, u, p, ρ are velocity, pressure, density, respectively. D the deformation tensor defined as Dij ¼ ð∂i uj þ ∂j ui Þ=2 . ρF the external forces in 2D can be derived and written in the moving coordinate system (Celebi and Akyildiz, 2002), as shown in Fig. 1, as follows: 2 F 1 ¼ g sin θ þ x2 θ€ þ x1 θ_ þ2θ_ u2 A€
ð3Þ
2 F 2 ¼ g cos θ x1 θ€ þ x2 θ_ 2θ_ u1
ð4Þ
where A and θ are the translational and rotational amplitude of the non-inertial frame with regard to the considered time, respectively, A ¼ A0 sin ðωs t þ εs Þ
ð5Þ
θ ¼ θ0 sin ðωp t þ εp Þ
ð6Þ
where, A0 ; θ0 ; ωs ; ωp ; εs ; εp are the translational and rotational amplitudes, frequencies and phase differences, respectively.
21
Fig. 1. Definition sketch.
3. Numerical computation The numerical method used in Gerris has been described in detail in Popinet (2003), where a finite volume method (FVM) is applied to solve the governing equations. VOF approach is employed to capture the free-surface and to reconstruct it, and the spatial discretization of the computational domain relies on the quadtree (in 2D) adaptive method based on square grids hierarchically. Second order up-wind schemes is used to address the convective term (Bell et al., 1989). Projection and multi-grid method are used to solve the pressure Poisson equation, respectively. In the following we will summarize the main characteristics of the technique. 3.1. Spatial discretization Square finite volumes, organized hierarchically as a quadtree, are used to discretize the domain spatially. Fig. 2 indicates an example of spatial discretization. Since the quadtree structure is applied, the adaptive algorithm is rather simple to process. It is permitted to have different variables for the refinement criterion, for example, the norm of the local vorticity vector. A cell can be refined whenever hJ∇ U J 4τ max J U J
ð7Þ
where, the size of a grid is represented by the length h, and τ is the threshold value, which meets 0 o τ o1. Then, a simple criterion based on vorticity is built here. Compared with structural uniform grids, the adaptive algorithm can reduce time and memory space needed by computation significantly. The coarsening step is processed every time the high resolution is not necessary, and the refinement step is processed whenever the high gradient of velocity or the strong contortion rate of the free-surface occurs, since it is imperative to enhance the computational precision to guarantee the efficiency and accuracy. The attribute of square grids is inherited by the quadtree based algorithm. This square shape of finite volume cells allows values at any points to be interpolated and allows the free-surface reconstructing conveniently, and higher accuracy is guaranteed by this as well. Fig. 2 shows the development of the adaptive grids in the domain of one computational result (water depth¼ 30 cm, frequency ¼ 1:05ω0 ) in this paper. From this it can be seen that
22
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 2. Grids for computation. (a) t ¼ 0.05 s and (b) t¼ 3.95 s.
the grids intersected with the immersed solid boundary and water surface are all finest, which can ensure the accurate description of the tank wall and the interface. Far from the walls and the freesurface, larger cells are present and the spatial resolution decreases accordingly. The vorticity and the gradient of volume fraction are both used as the adaptive criteria which meet a derived condition of Eq. (7). The mesh is adapted at each time step to follow evolving of them. 3.2. Temporal discretization A classical fractional-step projection method (Brown et al., 2001) is used to discretize the time domain, thus the governing equations can be written as
ρn þ 1=2
hu u i h i n n þ un þ 1=2 ∇un þ 1=2 ¼ ∇ μn þ 1=2 ðDn þ Dn þ 1=2 Þ þ ρF Δt
un þ 1 ¼ un
Δt ∇p ρn þ 1=2 n þ 1=2
∇ un þ 1 ¼ 0
ð8Þ ð9Þ
ð10Þ
where, un is a provisional velocity. Projection method used to solve Poison equation is considered. Specific interpolation schemes are required at different levels of the finite volume cells. More details are discussed in Popinet (2003). 3.3. VOF method A classic VOF scheme (Gueyffier et al., 1998) is applied to ensure the procedure work well. The free-surface can be captured and reconstructed in the incompressible quadtree adaptive grids with different resolutions (Popinet, 2009). This effective and efficient method is first proposed by Debar (1974) and then consummated by Hirt and Nichols (1981). The basic rule is: define the tracer T and 1 T as the water and air volume fraction in the computational domain, respectively. For every independent cell, the sum of the two ingredients is 1, and the time dependence of T is governed by the equation: ∂T þ∇ ðTuÞ ¼ 0 ∂t
ð11Þ
Fig. 3. Wave profile for rolling frequency ¼ 0:9ω0 and water depth¼ 30 cm. (a) t¼ 3.6 s and (b) t ¼ 3.7 s.
Therefore, the density and the kinetic viscosity involved in Eq. (1) can be written as
ρðTÞ ¼ ρ1 T þ ρ2 ð1 TÞ
ð12Þ
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 4. Experiment setup. (a) Testing rig, (b) Gearbox, (c) Motor, (d) Inverter, (e) Collector, (f) Sensors and (g) One of the holes.
23
24
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
μðTÞ ¼ μ1 T þ μ2 ð1 TÞ
ð13Þ
3.4. Numerics and turbulence modelling
Fig. 5. Test tank dimensions. Table 1 Model conditions. Rolling frequency (ω0 )
Water depth (cm)
1 2 3 4 5 6 7 8
0.35 0.35 0.65 0.65 0.80 0.80 1.10 1.10
30 45 30 45 30 45 30 45
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
numerical data experimental data Pressure(KPa)
Pressure(KPa)
Model label
The configuration of numerical computation organized in a parameter file follows Popinet (2012). The undisturbed freesurface is the starting position before rolling. For the present experiment, only one box is refined as the computational domain, and its size sets to be 1.4 to fit the height of the tank. The detailed dimension of the tank will be described in Section 4. The maximum level of resolution is 7. No-slip boundary conditions apply on the tank walls using GfsSolid where a full C function is applied to implicitly define the surface. Given the large Weber number (over 94.7), the effect of the free-surface tension is neglected. To realize the variation in acceleration of Eqs. (3) and (4), a source term is associated to the variable of velocity. Considering the high Reynolds number for some cases during sloshing tests (R 106 ), a smaller scale than the mesh size the energy transfer at is expected (approximately the order of 1=R). The physical viscosity is contained within our numerical model, and in practice numerical schemes always have some numerical viscosity due to higher order errors associated with the discrete representation of the solution. This numerical dissipation can describe turbulent subgrid energy transfer instead of more complex Large Eddy Simulations semi-empirical modes (Popinet et al., 2004). The complex profile of the free-surface, for example, wave breaking and hydraulic jump (see Fig. 3), can be freely simulated since the VOF method can easily tackle the surface deformation and merging. However, when the excitation frequency is near the resonance computation may not
0
2
4
6
8
10
12
14
16
18
20
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
numerical data experimental data
0
2
4
6
8
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
14
16
18
20
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
Pressure(KPa)
Pressure(KPa) 2
4
6
8
10
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
18
20
t(s)
numerical data experimental data
0
12
numerical data experimental data
t(s)
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
10 t(s)
Pressure(KPa)
Pressure(KPa)
t(s)
12
14
16
18
20
t(s) Fig. 6. History of pressure values for rolling frequency ¼ 0:35ω0 and water depth¼30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1. (b) Sensor 2 and (c) Sensor 3.
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
14
16
t(s) Fig. 7. History of pressure values for rolling frequency ¼ 0:35ω0 and water depth¼ 45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
be carried out after several periods due to the very strong nonlinear phenomena. Thus, as a first study, this paper will focus on the pressure prediction just as described in Section 1, while a more comprehensive study will be certainly conducted in the future.
4. Experimental investigations Sloshing motion is investigated by the experimental approach based on changing excitation frequencies. Different frequencies significantly affect the phenomenon of fluid in a partially filled tank. Specifically, the resonance occurs when the external frequency is close to the natural frequency of the fluid. The formula (in 2D) is given by gπ d ω20 ¼ tanh π ð14Þ l l
1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
installed. Fig. 4(a) illustrates the location of the instruments. The mechanism of this system is using the inverter (see Fig. 4(d)) to control the rotor speed. Through the transformation of the gearbox, power is passed to the rotation axis to roll the platform on request. Four pressure sensors (see Fig. 4(f)) were used to monitor the local pressure distribution on the tank wall and in the fluid itself. Before installation of the sensors, three holes were drilled on the tank carefully to guarantee the water-tightness (see Fig. 4(g)). Each sensor was linked to the data collector (see Fig. 4(e)) by which the data was passed into the computer and was converted from voltage to pressure. Together with other instruments in our laboratory it will provide a stringent test of the numerical method. All the pressure gauges were sampled 100 times every second with the range of 10 KPa. The minimum and maximum accuracy are 0.01%FS and 0.05%FS respectively. A model plexiglass tank was deliberately installed on the platform to serve the experiment. In practice, only the data from three sensors were analyzed except the one in the center. Dimensions of the tank and locations of the sensors are shown in Fig. 5. In our study, water was pumped into the tank. The excitation motion was rotational and the reference point was on the center of the bottom. The upright position was always the initial position for each test. The aspect we concentrate on is using the experiment outcomes to validate our numerical computation rather than focusing
numerical data experimental data
Pressure(KPa)
Pressure(KPa)
where ω0 is the first order natural frequency, l is the length of the tank, and d is the depth of water. To investigate the pressure distribution under different situations and validate the numerical method, we took a wide range of external frequencies into account. Two different fill levels, 0.3 m and 0.45 m, and the rolling amplitude, 50, were involved in our study. The steel structural platform and the driving device associated with a motor (see Fig. 4(c)) and a gearbox (see Fig. 4(b)) were
0
2
4
6
8
10
12
25
1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
14
numerical data experimental data
0
2
4
6
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
14
0
Pressure(KPa)
Pressure(KPa) 2
4
6
8
12
14
2
4
6
8
10
12
14
t(s)
numerical data experimental data
0
10
numerical data experimental data
t(s)
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
8 t(s)
Pressure(KPa)
Pressure(KPa)
t(s)
10
12
14
t(s) Fig. 8. History of pressure values for rolling frequency ¼ 0:65ω0 and water depth¼ 30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
14
t(s) Fig. 9. History of pressure values for rolling frequency ¼ 0:65ω0 and water depth¼ 45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
on the experimentation itself. Therefore we deliberately conducted each case of sloshing tests once and no uncertainty analysis was carried out. Moreover, ignoring statistic analysis of experimental results of same conditions is a common choice for some relevant studies when these two study means are employed to investigate sloshing problems (Panigrahy et al., 2009; Khezzar et al., 2009). While a huge discrepancy may be seen (two solutions under similar excitation frequency: one was to vary it from lower to higher frequency, and another in the opposite direction, Faltinsen, 2000) we only started the motor after letting the water rest and keep the speed constant quickly in order to avoid this ‘Jump’. Certainly, repeating tests may bring in better comparisons and to benefit uncertainty assessment procedure, however, it may be significant only for some particular uses, for example, to find the first pressure peak mounting of the gauge (Souto-Iglesias and Pérez-ArribasBotia-Vera, 2011).
5. Results and discussion
Pressure(KPa)
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
numerical data experimental data
0
2
4
6
8
10
12
Pressure(KPa)
numerical data experimental data
0
2
4
6
8
10
12
Pressure(KPa)
t(s)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
0
2
4
6
8
10
12
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
numerical data experimental data
0
2
4
6
8
10
12
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
numerical data experimental data
0
2
4
6
8
10
12
t(s)
t(s)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
numerical data experimental data
t(s)
Pressure(KPa)
Pressure(KPa)
In this paper, a series of tests are designed to investigate the sloshing motion under different excitation frequencies and fill levels. Sometimes violent nonlinearity is expected, and some corresponding computational experiments are done by which a
1.8 1.5 1.2 0.9 0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 -1.5
1.8 1.5 1.2 0.9 0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 -1.5
t(s)
Pressure(KPa)
26
numerical data experimental data
0
2
4
6
8
10
12
t(s) Fig. 10. History of pressure values for rolling frequency ¼ 0:80ω0 and water depth¼ 30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
Fig. 11. History of pressure values for rolling frequency ¼ 0:80ω0 and water depth¼45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
good agreement is obtained. The model test runs are according to Table 1. Figs. 6–13 show the changing pressure values after subtracting the initial static pressure. All the curves are built by Gnuplot with ‘csplines’ smooth technique. We find that more smooth curves are obtained as the frequency is lower from Figs. 6 to 13, since the 3D effect turns to be more significant at high frequencies tests. Moreover, the violent nonlinear situation can be an obstruction to the stability of the numerical algorithm, thus some salient numerical dispersion is represented in Figs. 12 and 13. The initial acceleration of the electric motor can spend some time during which the computational process has experienced several periods at the higher frequency. Given this, we should do some translation of axes, and that is why in some higher frequency cases, experimental data does not start from the origin point and the two kinds of data cannot agree well at some initial periods, as shown in Figs. 8–13. So, the computational method can offer us a more reliable approach for a specific problem than the experimental method does, when some variables do not influence any results of ideal cases which reveal the nature. Figs. 6–13 show that the peak points Sensor 1 and 2 measured have little difference in the phase, and a higher local pressure is predicted in the vicinity of the wall than, sometimes as twice as,
2.8 2.4 2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 -1.6 -2
numerical data experimental data Pressure(KPa)
Pressure(KPa)
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
0
1
2
3
4
5
2.8 2.4 2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 -1.6 -2
27
numerical data experimental data
0
1
2
numerical data experimental data Pressure(KPa)
Pressure(KPa)
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 0
1
2
3
4
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2
5
0
Pressure(KPa)
Pressure(KPa) 1
2
5
1
2
3
4
5
t(s)
numerical data experimental data
0
4
numerical data experimental data
t(s)
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2
3 t(s)
t(s)
3
4
5
1
2
3
4
5
t(s) Fig. 13. History of pressure values for rolling frequency ¼ 1:10ω0 and water depth¼45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
3
fill level = 30cm fill level = 45cm
2.5
Pressure(KPa)
that in the internal water. At the same time, values measured by Sensor 3 experience an opposite trend. Fig. 14 indicates that the peak pressure with small water depth is higher than those with higher water depth, and it is much significant during the resonance. Some characteristics can be found, for instance, a steady increase between 0.1 and 0.75 times natural frequency, a sequential sharp boost before a plateau comes at 0:9ω0 , a rapid fall after that, and the resonance point is a little bit lower than ω0, say, almost 0:95ω0 , which are similarly portrayed in the former theoretical analysis (Faltinsen, 1976). Not only the pressure, the free-surface shapes between numerical and experimental results are also well matched, one of which is shown in Fig. 15 by comparison. Some phenomena are investigated by experiments. Specifically, 3D effect is an interesting factor, which can thwart the 2D numerical validity because the square tank was employed which may lead to significant representation of 3D surface patterns (Wang, 1989), unless the 3D simulation is used. Figs. 16 and 17 indicate that 3D effect comes out later as the fill level rises, and it appears earlier as the excitation frequency increases. In our tests, 0:75ω0 is a critical frequency for the first appearance of 3D effect after several measurable periods. Although it seems to be much significant at higher excitation frequencies, it is partial because the droplets falling onto water surface lead to changes in fluid motion due to resonant effect before the motor reaching the constant
numerical data experimental data
0
t(s) Fig. 12. History of pressure values for rolling frequency ¼ 1:10ω0 and water depth¼ 30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2
2 1.5 1 0.5 0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency Ratio
1
1.1 1.2 1.3 1.4
Fig. 14. Max numerical pressure values measured by Sensor 1 of the first three periods for different oscillation frequencies with two kinds of water depth.
speed. It is hard to reach a steady state during our laboratory experiments though the excitation direction is parallel to the front tank wall. General speaking, this rolling motion of the tank may trigger a 2D flow and reach the stable state of wave evolution as
28
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 15. Water surface shape comparison between experiments and computations for the case of 30 cm water depth and 0:8ω0 excitation frequency at 3.7 s computational time.
Fig. 17. Three dimensional sloshing patterns for rolling frequency ¼ 0:75ω0 and water depth¼ 45 cm. (a) t ¼22 s and (b) t¼ 22.2 s.
wave is observed at frequency 0:5ω0 in Fig. 18. An interesting part in Fig. 19 is the surface wavelet seems easier to occur for the higher fill level under a relatively low frequency, even though it is not a major concern for sloshing. Surge and wave breaking play an important role in sloshing problems accompanying strong nonlinearity. It can be seen that small difference in water depth leads a huge discrepancy at occurrence of surge, wave breaking and hydraulic jump. For example, wave breaking occurs at the center of the horizontal surface for two opposite forward traveling waves and at the position where a surge pulses close to the lateral wall at frequencies of 0:75ω0 and 0:8ω0 (see Fig. 20), respectively, with 30 cm water depth, yet there is no wave breaking in the corresponding cases with 45 cm water depth (see Fig. 21).
Fig. 16. Three dimensional sloshing patterns for rolling frequency ¼ 0:75ω0 and water depth¼ 30 cm. (a) t¼ 15 s and (b) t¼ 15.12 s.
6. Conclusions
our simulation shows. However, the instability due to the square bottom of the tank may affect the liquid motion. Consequently, swirling and diagonal waves dominate. This is the major point where 3D effects differ from 2D sloshing motions. Therefore, 2D numerical simulations take their shortcomings compared with 3D experiments when a long duration is available. Water wave is another factor we take into account. In cases with lower excitation frequencies, we hardly find traveling wave, while the standing wave dominates. In our experiments, traveling
The tree-based VOF approach was employed to simulate water motion in the rolling tank. The adaptive algorithm is efficient to deal with the high resolution needed for surface capturing and reconstruction. Good agreement was obtained by comparison between numerical and experimental tests regarding the local forces by changing the excitation frequency and water depth. Discussions related to the numerical results were carried out. We also investigated the phenomena observed during a series of experiments, concentrating on the water surface patterns: three dimensional effect and water waves.
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
29
Fig. 18. Traveling wave for rolling frequency ¼ 0:5ω0 and water depth¼ 45 cm. (a) t¼ 93 s, (b) t¼ 93.16 s and (c) t ¼93.24 s.
Fig. 19. Surface wavelet patterns for rolling frequency ¼ 0:35ω0 . (a) Water depth¼ 30 cm and (b) Water depth¼45 cm.
Fig. 20. Surge and wave breaking for rolling frequency ¼ 0:8ω0 and water depth¼ 30 cm. (a) t¼ 25 s, (b) t¼ 25.04 s, (c) t ¼25.08 s, (d) t ¼25.12 s, (e) t ¼25.16 s, (f) t¼ 25.2 s, (g) t¼ 25.24 s, (h) t¼ 25.28 s and (i) t ¼25.32 s
30
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 21. Surface patterns for rolling frequency ¼ 0:8ω0 and water depth¼45 cm. (a) t¼ 29 s, (b) t ¼29.04 s, (c) t¼ 29.08 s, (d) t ¼29.12 s, (e) t¼ 29.16 s, (f) t¼ 29.2 s, (g) t¼ 29.24 s, (h) t ¼ 29.28 s and (i) t ¼29.32 s.
Thus, we can study the sloshing problem with relatively low water depth and strong non-linearity by this tree-based VOF approach. Actually this approach offers us a reciprocal way to enhance the studies on sloshing and also the numerical method itself through solving hydrodynamic problems. Experimental method is also an effective means to observe significant phenomena during studies.
Acknowledgments The present work is supported by the National Natural Science Foundation of China (51221961, 51279030) the National Key Basic Research Special Foundation of China (2010CB832704, 2013 CB036101) and the Scientific Project for High-Tech Ships: Key Technology Research on Semi-planing Fore-placed-outrigger Trimaran.
References Abramson, H.N., 1966. The Dynamic Behavior of Liquids in Moving Containers. Technical Report. NASA. SP-106. Akyildiz, H., Ünal, E., 2005. Experimental investigation of pressure distribution on a rectangular tank due to the liquid sloshing. Ocean Eng. 32, 1503–1516. Akyildiz, H., Ünal, E., 2006. Sloshing in a three-dimensional rectangular tank: numerical simulation and experimental validation. Ocean Eng. 33, 2135–2149. Armenio, V., La Rocca, M., 1996. On the analysis of sloshing of water in rectangular containers: numerical study and experimental validation. Ocean Eng. 23, 705–739. Bell, J.B., Colella, P., Glaz, H.M., 1989. A second-order projection method for the incompressible Navier Stokes equations. J. Comput. Phys. 85, 257–283. Brown, D.L., Cortez, R., Minion, M.L., 2001. Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168, 464–499. Celebi, M.S., Akyildiz, H., 2002. Nonlinear modeling of liquid sloshing in a moving rectangular tank. Ocean Eng. 29, 1527–1553.
Chen, Z., Zong, Z., Li, H., Li, J., 2013. An investigation into the pressure on solid walls in 2d sloshing using SPH method. Ocean Eng. 59, 129–141. Debar, R., 1974. Fundamentals of the KRAKEN Code. Technical Report. LLNL. UCIR760. Faltinsen, O.M., 1974. A nonlinear theory of sloshing in rectangular tanks. J. Ship Res. 18, 224–241. Faltinsen, O.M., 1976. Applicability and limitations of theoretical methods for description of slosh phenomena. In: Papers: Seminar on Liquid Sloshing, Det Norske Veritas, Oslo, p. Paper No. 4. Faltinsen, O.M., 1978. A numerical nonlinear method of sloshing in tank with twodimensional flow. J. Ship Res. 22, 193–202. Faltinsen, O.M., 2000. Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J. Fluid Mech. 407, 201–234. Faltinsen, O.M., Timokha, A.N., 2009. Sloshing (Hydrodynamics), 1st edition Cambridge University Press, New York Gueyffier, D., Nadim, A., Li, J., Scardovelli, R., Zaleski, S., 1998. Volume of fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152, 423–456. Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225. Khezzar, L., Seibi, A.C., Goharzadeh, A., 2009. Water sloshing in rectangular tanks can experimental investigation & numerical simulation. Int. J. Eng. 3, 174–184. Liang, Q., Borthwick, A.G., 2009. Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography. Comput. Fluids 38, 221–234. Liu, D., Lin, P., 2009. Three-dimensional liquid sloshing in a tank with baffles. Ocean Eng. 36, 202–212. Nakayama, T., Washizu, K., 1981. The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems. Int. J. Numer. Methods Fluids 17, 1631–1646. Panigrahy, P.K., Saha, U.K., Maity, D., 2009. Experimental studies on sloshing behavior due to horizontal movement of liquids in baffled tanks. Ocean Eng. 36, 213–222. Popinet, S., 2003. Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries. J. Comput. Phys. 190, 572–600. Popinet, S., 2009. An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866. Popinet, S., 2010. The gerris flow solver. 〈http://gfs.sf.net〉. Popinet, S., 2012. Syntax reference. 〈http://gfs.sourceforge.net/wiki/index.php/ Object_hierarchy〉. Popinet, S., Smith, M., Stevens, C., 2004. Experimental and numerical study of the turbulence characteristics of air flow around a research vessel. J. Atmos. Ocean. Technol. 21, 1575–1589.
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Rafiee, A., Pistani, F., Thiagarajan, K., 2011. Study of liquid sloshing: numerical and experimental approach. Comput. Mech. 47, 65–75. Souto-Iglesias, A., Pérez-ArribasBotia-Vera, F., 2011. A set of canonical problems in sloshing: Part 0: experimental setup and data processing. Ocean Eng. 38, 1823–1830. Wang, Y., 1989. Numerical Simulation of Sloshing in Arbitrary Containers (Ph.D. thesis), Dalian University of Technology.
31
Waterhouse, D.D., 1994. Resonant sloshing near a critical depth. J. Fluid Mech. 281, 313–318. Wu, G.X., Ma, Q.W., Taylor, R.E., 1998. Numerical simulation of sloshing waves in a 3d tank based on a finite element method. Appl. Ocean Res. 20, 337–355. Xue, M., Lin, P., 2011. Numerical study of ring baffle effects on reducing violent liquid sloshing. Comput. Fluids 52, 116–129.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Numerical studies on sloshing in rectangular tanks using a tree-based adaptive solver and experimental validation Hai-tao Li, Jing Li n, Zhi Zong, Zhen Chen School of Naval Architecture, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China
ar t ic l e i nf o
a b s t r a c t
Article history: Received 18 October 2012 Accepted 19 February 2014 Available online 13 March 2014
As liquid cargo transport developing, hull structural stability and strength are highlighted in various engineering areas due to tanks partially filled with fluid. Sloshing may be accounted as unpredictable force imposed on the whole faces of the tank walls. Water movement in rectangular tanks, under rolling and horizontal excitations, is investigated by numerical and experimental approaches. A parallel code, which directly discretizes the incompressible Navier–Stokes equations, coupled with VOF and tree-based adaptive algorithm, is employed to simulate the behavior of 2D fluid motion. A series of experiments are carried out to measure the pressures on the tank walls and under the water. A good agreement is shown in this paper and the values obtained by computations are validated. Through this, some studies are done on the basis of different excitation frequencies and filled levels. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Sloshing VOF Adaptive algorithm Numerical simulation
1. Introduction Sloshing is considered as a motion of water in partially filled tanks under an external excitation, which usually represents a violent phenomenon with high amplitudes of water surface and strong localized impacts on the tank walls when the excitation frequency is close to the natural frequency of this system. This phenomenon is particularly evident under relatively low water depth. The critical depth to length ratio is 0.3368 (Faltinsen and Timokha, 2009). Sloshing is also regarded as water wave movement of the free surface which is normally sorted into different types according to their shapes: standing wave, traveling wave, hydraulic jump, breaking wave and the combination of themselves. When the resonance occurs, it is difficult to separate them from each other, because of the strong nonlinear and arbitrary phenomenon experienced by the surface. Moreover, the instantaneous loads can be significantly high which may cause structural damage in turn and even engender sufficient torque to impede the stability of the vehicle. To that end this paper considers that under relatively low water depth how the surface shape changes according to different external frequencies and what extent are the loads predicted by numerical means against experimental method. To assess sloshing loads and the free-surface elevation, theoretical approaches based on linear and nonlinear potential flow theory and laboratory experimentation based on scaled models
n
Corresponding author. Tel.: þ 86 411 84708451; fax: þ 86 411 84708451x8036. E-mail address: [email protected] (J. Li).
http://dx.doi.org/10.1016/j.oceaneng.2014.02.011 0029-8018 & 2014 Elsevier Ltd. All rights reserved.
have been employed by many researchers. Although the sloshing technology developed for space applications is not directly applied to ship cargo tanks, Abramson (1966) studied the sloshing problem systematically, in terms of the space industries, by potential flow theory with different shapes of fuel tanks for launch vehicles. A series of analytical solutions were derived based on the linear method, and corresponding experiments were conducted. Faltinsen (1974), Faltinsen (1978) and Faltinsen (2000) applied potential flow theory of an incompressible liquid and developed nonlinear multimodal method for sloshing, and some classic formulae were obtained, which have been widely used today (Waterhouse, 1994), but only restricted to cases in simple shape tanks with the free surface elevation of a single-valued function. Akyildiz and Ünal (2005) and Akyildiz and Ünal (2006) designed an experiment which was set up to observe the behavior of liquid sloshing in rectangular container subjected to external excitation with a relatively deep water depth. Rafiee et al. (2011) studied experimentally sloshing phenomena in a thin rectangular tank (with width-tolength ratio of 0.0769) under a sway excitation, and only shallow water depth was considered. Since the advent of computers, people started to solve fluid mechanics problem by using computational techniques. Numerical simulations have become important and widely adopted technique to deal with strong nonlinear sloshing. Wu et al. (1998) analyzed the sloshing waves using finite element method (FEM). Shallow water theory is an important mathematical model when hydraulic jump occurs, which was employed by Nakayama and Washizu (1981) to present a nonlinear analysis of inviscid and incompressible fluid motion in a tank subjected to forced oscillation. At the same time,
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
both FEM and boundary element method (BEM) were combined in. Besides the aforementioned techniques, approaches based on Navier–Stokes equations have been applied to study liquid sloshing. Reynolds averaged Navier–Stokes (RANS) equations were compared with shallow water theory by Armenio and La Rocca (1996). Moreover, Liu and Lin (2009) studied sloshing using volume of fluid (VOF) method coupled with turbulent modeling. Xue and Lin (2011) similarly analyzed the surface elevation and velocity field of liquid sloshing by spatially averaged Navier–Stokes (SANS) equations coupled with virtual boundary force (VBF) method. The smoothed particle hydrodynamics (SPH) method featured mesh-free has been also applied to the study of sloshing flow. Rafiee et al. (2011) and Chen et al. (2013) both developed improved SPH techniques to simulate the sloshing flow by using more accurate integration and more applicable treatment of the boundary conditions. However, to our knowledge, adaptive algorithm used in this field is rarely seen. Liang and Borthwick (2009) validated a Godunov-type shallow flow solver on adaptive quadtree grids by simulating linear sloshing motions in a vessel with a parabolic bed. A parallel program Gerris Flow Solver with VOF method based on adaptive quadtree grids was created by Popinet (2003), Popinet (2009) and Popinet (2010). A condensed introduction of this opensource code is summarized in Section 3. The motivation of this paper is to demonstrate that Gerris can indeed simulate the violent fluid motion of sloshing and predict the accurate pressure. Therefore, both numerical simulation and laboratory experiment were carried out to study sloshing problems. We focus on giving comparisons of loads between the computations and laboratory experiments under the filling ratios falling into the vicinity of the critical depth mentioned in the first paragraph. For the purpose of that, we designed a series of tests of water sloshing in a rectangular tank. Because of the characteristic of the tank with a square bottom, the 3D effects cannot be ignored during our analysis. Thus, the free surface wave shape is another important aspect of this study.
2. Mathematical model The fluid motion of sloshing can be described by the incompressible continuity equation and Navier–Stokes equations: ∂u ρ ð1Þ þ u ∇u ¼ ρF ∇p þ ∇ 2μD ∂t ∇u¼0
ð2Þ
where, u, p, ρ are velocity, pressure, density, respectively. D the deformation tensor defined as Dij ¼ ð∂i uj þ ∂j ui Þ=2 . ρF the external forces in 2D can be derived and written in the moving coordinate system (Celebi and Akyildiz, 2002), as shown in Fig. 1, as follows: 2 F 1 ¼ g sin θ þ x2 θ€ þ x1 θ_ þ2θ_ u2 A€
ð3Þ
2 F 2 ¼ g cos θ x1 θ€ þ x2 θ_ 2θ_ u1
ð4Þ
where A and θ are the translational and rotational amplitude of the non-inertial frame with regard to the considered time, respectively, A ¼ A0 sin ðωs t þ εs Þ
ð5Þ
θ ¼ θ0 sin ðωp t þ εp Þ
ð6Þ
where, A0 ; θ0 ; ωs ; ωp ; εs ; εp are the translational and rotational amplitudes, frequencies and phase differences, respectively.
21
Fig. 1. Definition sketch.
3. Numerical computation The numerical method used in Gerris has been described in detail in Popinet (2003), where a finite volume method (FVM) is applied to solve the governing equations. VOF approach is employed to capture the free-surface and to reconstruct it, and the spatial discretization of the computational domain relies on the quadtree (in 2D) adaptive method based on square grids hierarchically. Second order up-wind schemes is used to address the convective term (Bell et al., 1989). Projection and multi-grid method are used to solve the pressure Poisson equation, respectively. In the following we will summarize the main characteristics of the technique. 3.1. Spatial discretization Square finite volumes, organized hierarchically as a quadtree, are used to discretize the domain spatially. Fig. 2 indicates an example of spatial discretization. Since the quadtree structure is applied, the adaptive algorithm is rather simple to process. It is permitted to have different variables for the refinement criterion, for example, the norm of the local vorticity vector. A cell can be refined whenever hJ∇ U J 4τ max J U J
ð7Þ
where, the size of a grid is represented by the length h, and τ is the threshold value, which meets 0 o τ o1. Then, a simple criterion based on vorticity is built here. Compared with structural uniform grids, the adaptive algorithm can reduce time and memory space needed by computation significantly. The coarsening step is processed every time the high resolution is not necessary, and the refinement step is processed whenever the high gradient of velocity or the strong contortion rate of the free-surface occurs, since it is imperative to enhance the computational precision to guarantee the efficiency and accuracy. The attribute of square grids is inherited by the quadtree based algorithm. This square shape of finite volume cells allows values at any points to be interpolated and allows the free-surface reconstructing conveniently, and higher accuracy is guaranteed by this as well. Fig. 2 shows the development of the adaptive grids in the domain of one computational result (water depth¼ 30 cm, frequency ¼ 1:05ω0 ) in this paper. From this it can be seen that
22
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 2. Grids for computation. (a) t ¼ 0.05 s and (b) t¼ 3.95 s.
the grids intersected with the immersed solid boundary and water surface are all finest, which can ensure the accurate description of the tank wall and the interface. Far from the walls and the freesurface, larger cells are present and the spatial resolution decreases accordingly. The vorticity and the gradient of volume fraction are both used as the adaptive criteria which meet a derived condition of Eq. (7). The mesh is adapted at each time step to follow evolving of them. 3.2. Temporal discretization A classical fractional-step projection method (Brown et al., 2001) is used to discretize the time domain, thus the governing equations can be written as
ρn þ 1=2
hu u i h i n n þ un þ 1=2 ∇un þ 1=2 ¼ ∇ μn þ 1=2 ðDn þ Dn þ 1=2 Þ þ ρF Δt
un þ 1 ¼ un
Δt ∇p ρn þ 1=2 n þ 1=2
∇ un þ 1 ¼ 0
ð8Þ ð9Þ
ð10Þ
where, un is a provisional velocity. Projection method used to solve Poison equation is considered. Specific interpolation schemes are required at different levels of the finite volume cells. More details are discussed in Popinet (2003). 3.3. VOF method A classic VOF scheme (Gueyffier et al., 1998) is applied to ensure the procedure work well. The free-surface can be captured and reconstructed in the incompressible quadtree adaptive grids with different resolutions (Popinet, 2009). This effective and efficient method is first proposed by Debar (1974) and then consummated by Hirt and Nichols (1981). The basic rule is: define the tracer T and 1 T as the water and air volume fraction in the computational domain, respectively. For every independent cell, the sum of the two ingredients is 1, and the time dependence of T is governed by the equation: ∂T þ∇ ðTuÞ ¼ 0 ∂t
ð11Þ
Fig. 3. Wave profile for rolling frequency ¼ 0:9ω0 and water depth¼ 30 cm. (a) t¼ 3.6 s and (b) t ¼ 3.7 s.
Therefore, the density and the kinetic viscosity involved in Eq. (1) can be written as
ρðTÞ ¼ ρ1 T þ ρ2 ð1 TÞ
ð12Þ
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 4. Experiment setup. (a) Testing rig, (b) Gearbox, (c) Motor, (d) Inverter, (e) Collector, (f) Sensors and (g) One of the holes.
23
24
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
μðTÞ ¼ μ1 T þ μ2 ð1 TÞ
ð13Þ
3.4. Numerics and turbulence modelling
Fig. 5. Test tank dimensions. Table 1 Model conditions. Rolling frequency (ω0 )
Water depth (cm)
1 2 3 4 5 6 7 8
0.35 0.35 0.65 0.65 0.80 0.80 1.10 1.10
30 45 30 45 30 45 30 45
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
numerical data experimental data Pressure(KPa)
Pressure(KPa)
Model label
The configuration of numerical computation organized in a parameter file follows Popinet (2012). The undisturbed freesurface is the starting position before rolling. For the present experiment, only one box is refined as the computational domain, and its size sets to be 1.4 to fit the height of the tank. The detailed dimension of the tank will be described in Section 4. The maximum level of resolution is 7. No-slip boundary conditions apply on the tank walls using GfsSolid where a full C function is applied to implicitly define the surface. Given the large Weber number (over 94.7), the effect of the free-surface tension is neglected. To realize the variation in acceleration of Eqs. (3) and (4), a source term is associated to the variable of velocity. Considering the high Reynolds number for some cases during sloshing tests (R 106 ), a smaller scale than the mesh size the energy transfer at is expected (approximately the order of 1=R). The physical viscosity is contained within our numerical model, and in practice numerical schemes always have some numerical viscosity due to higher order errors associated with the discrete representation of the solution. This numerical dissipation can describe turbulent subgrid energy transfer instead of more complex Large Eddy Simulations semi-empirical modes (Popinet et al., 2004). The complex profile of the free-surface, for example, wave breaking and hydraulic jump (see Fig. 3), can be freely simulated since the VOF method can easily tackle the surface deformation and merging. However, when the excitation frequency is near the resonance computation may not
0
2
4
6
8
10
12
14
16
18
20
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
numerical data experimental data
0
2
4
6
8
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
14
16
18
20
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
Pressure(KPa)
Pressure(KPa) 2
4
6
8
10
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
18
20
t(s)
numerical data experimental data
0
12
numerical data experimental data
t(s)
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
10 t(s)
Pressure(KPa)
Pressure(KPa)
t(s)
12
14
16
18
20
t(s) Fig. 6. History of pressure values for rolling frequency ¼ 0:35ω0 and water depth¼30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1. (b) Sensor 2 and (c) Sensor 3.
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
14
16
t(s) Fig. 7. History of pressure values for rolling frequency ¼ 0:35ω0 and water depth¼ 45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
be carried out after several periods due to the very strong nonlinear phenomena. Thus, as a first study, this paper will focus on the pressure prediction just as described in Section 1, while a more comprehensive study will be certainly conducted in the future.
4. Experimental investigations Sloshing motion is investigated by the experimental approach based on changing excitation frequencies. Different frequencies significantly affect the phenomenon of fluid in a partially filled tank. Specifically, the resonance occurs when the external frequency is close to the natural frequency of the fluid. The formula (in 2D) is given by gπ d ω20 ¼ tanh π ð14Þ l l
1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
installed. Fig. 4(a) illustrates the location of the instruments. The mechanism of this system is using the inverter (see Fig. 4(d)) to control the rotor speed. Through the transformation of the gearbox, power is passed to the rotation axis to roll the platform on request. Four pressure sensors (see Fig. 4(f)) were used to monitor the local pressure distribution on the tank wall and in the fluid itself. Before installation of the sensors, three holes were drilled on the tank carefully to guarantee the water-tightness (see Fig. 4(g)). Each sensor was linked to the data collector (see Fig. 4(e)) by which the data was passed into the computer and was converted from voltage to pressure. Together with other instruments in our laboratory it will provide a stringent test of the numerical method. All the pressure gauges were sampled 100 times every second with the range of 10 KPa. The minimum and maximum accuracy are 0.01%FS and 0.05%FS respectively. A model plexiglass tank was deliberately installed on the platform to serve the experiment. In practice, only the data from three sensors were analyzed except the one in the center. Dimensions of the tank and locations of the sensors are shown in Fig. 5. In our study, water was pumped into the tank. The excitation motion was rotational and the reference point was on the center of the bottom. The upright position was always the initial position for each test. The aspect we concentrate on is using the experiment outcomes to validate our numerical computation rather than focusing
numerical data experimental data
Pressure(KPa)
Pressure(KPa)
where ω0 is the first order natural frequency, l is the length of the tank, and d is the depth of water. To investigate the pressure distribution under different situations and validate the numerical method, we took a wide range of external frequencies into account. Two different fill levels, 0.3 m and 0.45 m, and the rolling amplitude, 50, were involved in our study. The steel structural platform and the driving device associated with a motor (see Fig. 4(c)) and a gearbox (see Fig. 4(b)) were
0
2
4
6
8
10
12
25
1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
14
numerical data experimental data
0
2
4
6
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
14
0
Pressure(KPa)
Pressure(KPa) 2
4
6
8
12
14
2
4
6
8
10
12
14
t(s)
numerical data experimental data
0
10
numerical data experimental data
t(s)
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
8 t(s)
Pressure(KPa)
Pressure(KPa)
t(s)
10
12
14
t(s) Fig. 8. History of pressure values for rolling frequency ¼ 0:65ω0 and water depth¼ 30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
numerical data experimental data
0
2
4
6
8
10
12
14
t(s) Fig. 9. History of pressure values for rolling frequency ¼ 0:65ω0 and water depth¼ 45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
on the experimentation itself. Therefore we deliberately conducted each case of sloshing tests once and no uncertainty analysis was carried out. Moreover, ignoring statistic analysis of experimental results of same conditions is a common choice for some relevant studies when these two study means are employed to investigate sloshing problems (Panigrahy et al., 2009; Khezzar et al., 2009). While a huge discrepancy may be seen (two solutions under similar excitation frequency: one was to vary it from lower to higher frequency, and another in the opposite direction, Faltinsen, 2000) we only started the motor after letting the water rest and keep the speed constant quickly in order to avoid this ‘Jump’. Certainly, repeating tests may bring in better comparisons and to benefit uncertainty assessment procedure, however, it may be significant only for some particular uses, for example, to find the first pressure peak mounting of the gauge (Souto-Iglesias and Pérez-ArribasBotia-Vera, 2011).
5. Results and discussion
Pressure(KPa)
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
numerical data experimental data
0
2
4
6
8
10
12
Pressure(KPa)
numerical data experimental data
0
2
4
6
8
10
12
Pressure(KPa)
t(s)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
0
2
4
6
8
10
12
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
numerical data experimental data
0
2
4
6
8
10
12
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
numerical data experimental data
0
2
4
6
8
10
12
t(s)
t(s)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
numerical data experimental data
t(s)
Pressure(KPa)
Pressure(KPa)
In this paper, a series of tests are designed to investigate the sloshing motion under different excitation frequencies and fill levels. Sometimes violent nonlinearity is expected, and some corresponding computational experiments are done by which a
1.8 1.5 1.2 0.9 0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 -1.5
1.8 1.5 1.2 0.9 0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 -1.5
t(s)
Pressure(KPa)
26
numerical data experimental data
0
2
4
6
8
10
12
t(s) Fig. 10. History of pressure values for rolling frequency ¼ 0:80ω0 and water depth¼ 30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
Fig. 11. History of pressure values for rolling frequency ¼ 0:80ω0 and water depth¼45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
good agreement is obtained. The model test runs are according to Table 1. Figs. 6–13 show the changing pressure values after subtracting the initial static pressure. All the curves are built by Gnuplot with ‘csplines’ smooth technique. We find that more smooth curves are obtained as the frequency is lower from Figs. 6 to 13, since the 3D effect turns to be more significant at high frequencies tests. Moreover, the violent nonlinear situation can be an obstruction to the stability of the numerical algorithm, thus some salient numerical dispersion is represented in Figs. 12 and 13. The initial acceleration of the electric motor can spend some time during which the computational process has experienced several periods at the higher frequency. Given this, we should do some translation of axes, and that is why in some higher frequency cases, experimental data does not start from the origin point and the two kinds of data cannot agree well at some initial periods, as shown in Figs. 8–13. So, the computational method can offer us a more reliable approach for a specific problem than the experimental method does, when some variables do not influence any results of ideal cases which reveal the nature. Figs. 6–13 show that the peak points Sensor 1 and 2 measured have little difference in the phase, and a higher local pressure is predicted in the vicinity of the wall than, sometimes as twice as,
2.8 2.4 2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 -1.6 -2
numerical data experimental data Pressure(KPa)
Pressure(KPa)
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
0
1
2
3
4
5
2.8 2.4 2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 -1.6 -2
27
numerical data experimental data
0
1
2
numerical data experimental data Pressure(KPa)
Pressure(KPa)
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 0
1
2
3
4
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2
5
0
Pressure(KPa)
Pressure(KPa) 1
2
5
1
2
3
4
5
t(s)
numerical data experimental data
0
4
numerical data experimental data
t(s)
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2
3 t(s)
t(s)
3
4
5
1
2
3
4
5
t(s) Fig. 13. History of pressure values for rolling frequency ¼ 1:10ω0 and water depth¼45 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
3
fill level = 30cm fill level = 45cm
2.5
Pressure(KPa)
that in the internal water. At the same time, values measured by Sensor 3 experience an opposite trend. Fig. 14 indicates that the peak pressure with small water depth is higher than those with higher water depth, and it is much significant during the resonance. Some characteristics can be found, for instance, a steady increase between 0.1 and 0.75 times natural frequency, a sequential sharp boost before a plateau comes at 0:9ω0 , a rapid fall after that, and the resonance point is a little bit lower than ω0, say, almost 0:95ω0 , which are similarly portrayed in the former theoretical analysis (Faltinsen, 1976). Not only the pressure, the free-surface shapes between numerical and experimental results are also well matched, one of which is shown in Fig. 15 by comparison. Some phenomena are investigated by experiments. Specifically, 3D effect is an interesting factor, which can thwart the 2D numerical validity because the square tank was employed which may lead to significant representation of 3D surface patterns (Wang, 1989), unless the 3D simulation is used. Figs. 16 and 17 indicate that 3D effect comes out later as the fill level rises, and it appears earlier as the excitation frequency increases. In our tests, 0:75ω0 is a critical frequency for the first appearance of 3D effect after several measurable periods. Although it seems to be much significant at higher excitation frequencies, it is partial because the droplets falling onto water surface lead to changes in fluid motion due to resonant effect before the motor reaching the constant
numerical data experimental data
0
t(s) Fig. 12. History of pressure values for rolling frequency ¼ 1:10ω0 and water depth¼ 30 cm at sensor positions: 1, 2 and 3, respectively. (a) Sensor 1, (b) Sensor 2 and (c) Sensor 3.
2 1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2
2 1.5 1 0.5 0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency Ratio
1
1.1 1.2 1.3 1.4
Fig. 14. Max numerical pressure values measured by Sensor 1 of the first three periods for different oscillation frequencies with two kinds of water depth.
speed. It is hard to reach a steady state during our laboratory experiments though the excitation direction is parallel to the front tank wall. General speaking, this rolling motion of the tank may trigger a 2D flow and reach the stable state of wave evolution as
28
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 15. Water surface shape comparison between experiments and computations for the case of 30 cm water depth and 0:8ω0 excitation frequency at 3.7 s computational time.
Fig. 17. Three dimensional sloshing patterns for rolling frequency ¼ 0:75ω0 and water depth¼ 45 cm. (a) t ¼22 s and (b) t¼ 22.2 s.
wave is observed at frequency 0:5ω0 in Fig. 18. An interesting part in Fig. 19 is the surface wavelet seems easier to occur for the higher fill level under a relatively low frequency, even though it is not a major concern for sloshing. Surge and wave breaking play an important role in sloshing problems accompanying strong nonlinearity. It can be seen that small difference in water depth leads a huge discrepancy at occurrence of surge, wave breaking and hydraulic jump. For example, wave breaking occurs at the center of the horizontal surface for two opposite forward traveling waves and at the position where a surge pulses close to the lateral wall at frequencies of 0:75ω0 and 0:8ω0 (see Fig. 20), respectively, with 30 cm water depth, yet there is no wave breaking in the corresponding cases with 45 cm water depth (see Fig. 21).
Fig. 16. Three dimensional sloshing patterns for rolling frequency ¼ 0:75ω0 and water depth¼ 30 cm. (a) t¼ 15 s and (b) t¼ 15.12 s.
6. Conclusions
our simulation shows. However, the instability due to the square bottom of the tank may affect the liquid motion. Consequently, swirling and diagonal waves dominate. This is the major point where 3D effects differ from 2D sloshing motions. Therefore, 2D numerical simulations take their shortcomings compared with 3D experiments when a long duration is available. Water wave is another factor we take into account. In cases with lower excitation frequencies, we hardly find traveling wave, while the standing wave dominates. In our experiments, traveling
The tree-based VOF approach was employed to simulate water motion in the rolling tank. The adaptive algorithm is efficient to deal with the high resolution needed for surface capturing and reconstruction. Good agreement was obtained by comparison between numerical and experimental tests regarding the local forces by changing the excitation frequency and water depth. Discussions related to the numerical results were carried out. We also investigated the phenomena observed during a series of experiments, concentrating on the water surface patterns: three dimensional effect and water waves.
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
29
Fig. 18. Traveling wave for rolling frequency ¼ 0:5ω0 and water depth¼ 45 cm. (a) t¼ 93 s, (b) t¼ 93.16 s and (c) t ¼93.24 s.
Fig. 19. Surface wavelet patterns for rolling frequency ¼ 0:35ω0 . (a) Water depth¼ 30 cm and (b) Water depth¼45 cm.
Fig. 20. Surge and wave breaking for rolling frequency ¼ 0:8ω0 and water depth¼ 30 cm. (a) t¼ 25 s, (b) t¼ 25.04 s, (c) t ¼25.08 s, (d) t ¼25.12 s, (e) t ¼25.16 s, (f) t¼ 25.2 s, (g) t¼ 25.24 s, (h) t¼ 25.28 s and (i) t ¼25.32 s
30
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Fig. 21. Surface patterns for rolling frequency ¼ 0:8ω0 and water depth¼45 cm. (a) t¼ 29 s, (b) t ¼29.04 s, (c) t¼ 29.08 s, (d) t ¼29.12 s, (e) t¼ 29.16 s, (f) t¼ 29.2 s, (g) t¼ 29.24 s, (h) t ¼ 29.28 s and (i) t ¼29.32 s.
Thus, we can study the sloshing problem with relatively low water depth and strong non-linearity by this tree-based VOF approach. Actually this approach offers us a reciprocal way to enhance the studies on sloshing and also the numerical method itself through solving hydrodynamic problems. Experimental method is also an effective means to observe significant phenomena during studies.
Acknowledgments The present work is supported by the National Natural Science Foundation of China (51221961, 51279030) the National Key Basic Research Special Foundation of China (2010CB832704, 2013 CB036101) and the Scientific Project for High-Tech Ships: Key Technology Research on Semi-planing Fore-placed-outrigger Trimaran.
References Abramson, H.N., 1966. The Dynamic Behavior of Liquids in Moving Containers. Technical Report. NASA. SP-106. Akyildiz, H., Ünal, E., 2005. Experimental investigation of pressure distribution on a rectangular tank due to the liquid sloshing. Ocean Eng. 32, 1503–1516. Akyildiz, H., Ünal, E., 2006. Sloshing in a three-dimensional rectangular tank: numerical simulation and experimental validation. Ocean Eng. 33, 2135–2149. Armenio, V., La Rocca, M., 1996. On the analysis of sloshing of water in rectangular containers: numerical study and experimental validation. Ocean Eng. 23, 705–739. Bell, J.B., Colella, P., Glaz, H.M., 1989. A second-order projection method for the incompressible Navier Stokes equations. J. Comput. Phys. 85, 257–283. Brown, D.L., Cortez, R., Minion, M.L., 2001. Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168, 464–499. Celebi, M.S., Akyildiz, H., 2002. Nonlinear modeling of liquid sloshing in a moving rectangular tank. Ocean Eng. 29, 1527–1553.
Chen, Z., Zong, Z., Li, H., Li, J., 2013. An investigation into the pressure on solid walls in 2d sloshing using SPH method. Ocean Eng. 59, 129–141. Debar, R., 1974. Fundamentals of the KRAKEN Code. Technical Report. LLNL. UCIR760. Faltinsen, O.M., 1974. A nonlinear theory of sloshing in rectangular tanks. J. Ship Res. 18, 224–241. Faltinsen, O.M., 1976. Applicability and limitations of theoretical methods for description of slosh phenomena. In: Papers: Seminar on Liquid Sloshing, Det Norske Veritas, Oslo, p. Paper No. 4. Faltinsen, O.M., 1978. A numerical nonlinear method of sloshing in tank with twodimensional flow. J. Ship Res. 22, 193–202. Faltinsen, O.M., 2000. Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J. Fluid Mech. 407, 201–234. Faltinsen, O.M., Timokha, A.N., 2009. Sloshing (Hydrodynamics), 1st edition Cambridge University Press, New York Gueyffier, D., Nadim, A., Li, J., Scardovelli, R., Zaleski, S., 1998. Volume of fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152, 423–456. Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225. Khezzar, L., Seibi, A.C., Goharzadeh, A., 2009. Water sloshing in rectangular tanks can experimental investigation & numerical simulation. Int. J. Eng. 3, 174–184. Liang, Q., Borthwick, A.G., 2009. Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography. Comput. Fluids 38, 221–234. Liu, D., Lin, P., 2009. Three-dimensional liquid sloshing in a tank with baffles. Ocean Eng. 36, 202–212. Nakayama, T., Washizu, K., 1981. The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems. Int. J. Numer. Methods Fluids 17, 1631–1646. Panigrahy, P.K., Saha, U.K., Maity, D., 2009. Experimental studies on sloshing behavior due to horizontal movement of liquids in baffled tanks. Ocean Eng. 36, 213–222. Popinet, S., 2003. Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries. J. Comput. Phys. 190, 572–600. Popinet, S., 2009. An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866. Popinet, S., 2010. The gerris flow solver. 〈http://gfs.sf.net〉. Popinet, S., 2012. Syntax reference. 〈http://gfs.sourceforge.net/wiki/index.php/ Object_hierarchy〉. Popinet, S., Smith, M., Stevens, C., 2004. Experimental and numerical study of the turbulence characteristics of air flow around a research vessel. J. Atmos. Ocean. Technol. 21, 1575–1589.
H.-t. Li et al. / Ocean Engineering 82 (2014) 20–31
Rafiee, A., Pistani, F., Thiagarajan, K., 2011. Study of liquid sloshing: numerical and experimental approach. Comput. Mech. 47, 65–75. Souto-Iglesias, A., Pérez-ArribasBotia-Vera, F., 2011. A set of canonical problems in sloshing: Part 0: experimental setup and data processing. Ocean Eng. 38, 1823–1830. Wang, Y., 1989. Numerical Simulation of Sloshing in Arbitrary Containers (Ph.D. thesis), Dalian University of Technology.
31
Waterhouse, D.D., 1994. Resonant sloshing near a critical depth. J. Fluid Mech. 281, 313–318. Wu, G.X., Ma, Q.W., Taylor, R.E., 1998. Numerical simulation of sloshing waves in a 3d tank based on a finite element method. Appl. Ocean Res. 20, 337–355. Xue, M., Lin, P., 2011. Numerical study of ring baffle effects on reducing violent liquid sloshing. Comput. Fluids 52, 116–129.