simulation of three dimensional liquid-sloshing models

Kasetsart J. (Nat. Sci.) 46 : 978 - 995 (2012)

Department of Mechanical Engineering, Faculty of Engineering, Mahidol University, Nakhon Pathom 73170, Thailand.* Corresponding author, e-mail: [email protected]

Received date : 23/07/12 Accepted date : 12/10/12

Simulation of Three Dimensional Liquid-Sloshing Models usingC++ Open Source Code CFD Software

Ekachai Chaichanasiri and Chakrit Suvanjumrat*

ABSTRACT

The open source computational fl uid dynamic (CFD) codes—open fi eld operation and manipulation (OpenFOAM)—were developed to simulate the three dimensional sloshing of liquid inside moving rectangular tanks. Equations governing the two-phases of fl uid were used in these simulations and the Volume of Fluid method was used to capture the free surface of the liquid sloshing. The fi nite volume method was used to solve these problems. The solutions of the simulations were validated using experiment results based on the free surface height and the pressure developed on the wall of the tanks. It was found that the average error of the surface height and the pressure obtained from the simulations were 2.33% and 6.65%, respectively. Keywords: computational fl uid dynamics (CFD), fi nite volume, Volume of Fluid (VOF) method,

sloshing, open fi eld operation and manipulation (OpenFOAM)

INTRODUCTION

Sloshing is the phenomenon that occurs when a partially filled liquid moves inside a container. It can cause many problems in transportation especially in tanker trucks on highways, liquid tank cars on railroads and liquid cargos in ocean-going vessels. The effect of the severe sloshing is pressure developing on the internal wall of the container which for many liquids results in a loss of maneuvering stability during transportation. Whenever this pressure is greater than the endurance limit of the container wall, the container will be damaged. Therefore, studying liquid sloshing is important to the design of appropriate container shapes to reduce the effect of pressure. Many researchers have used computational fl uid dynamic (CFD) software which is based on

fl uid dynamics theory to analyze the fl uid fl ow and sloshing behavior (Lee et al., 2007; Bernhard et al., 2009; Rebouillat and Liksonov, 2010; Jung et al., 2012). Unfortunately, the CFD software is mostly commercial software whose license fees are too expensive and this prohibits further study. Development of CFD code using open source software (OSS) is an alternative which is economical for studying liquid sloshing. The Open Field Operation and Manipulation (OpenFOAM) software is OSS and is based on the CFD method. It is written in C++ code under the GNU general public license which is a good reason for using it to study liquid sloshing (OpenFOAM, 2009). The current study involves the development and implemention of the C++ code in the OpenFOAM software for analysis of liquid sloshing inside moving rectangular tanks. The developed codes can be applied to design the appropriate containers

Kasetsart J. (Nat. Sci.) 46(6) 979

which reduce the intense sloshing pressure when tanker trucks are overturned during an accident. Moreover, this research will motivate CFD programmers to use the open source CFD software for application development. This will save the cost of buying commercial software.

MATERIALS AND METHODS

Mathematical model A two-phase fl ow can be classifi ed as a gas-solid, gas-liquid and liquid-solid combination. The sloshing of liquid inside moving tanks is considered and computed based on the gas-liquid interface. The basic equation of fl uid motion for each continuous phase of the sloshing surface can be obtained by fi rst identifying the appropriate fundamental principles from conservation of mass and momentum (Yeoh and Tu, 2010). The basic mass conservative equation for the liquid sloshing inside the moving tanks is given by Equation 1:

�� + ∇ ∙ (�� ) = 0 (1)

where �� ≡ [�� , �� , � ], t is time and u, v and w are the velocity in the x, y and z direction, respectively. The conservation of momentum of the kth phase can be written in the compact form as Equation 2:

�� + �� ∙ ∇Uik = −∇�� + ∇� �� +�,�� + �,�� (2)

where ∇Pk is the pressure gradient and ∇�� is the shear stress. The shear stress may be taken to be proportional to the time rate of strain for the Newtonian fluid. The shear stress component for kth phase according to Newton’s law of the viscosity is written as Equation 3:

�� = !� �� + ��⎛⎝

⎛⎝

(3)

where xi is the moving frame coordinates and μk is the dynamic viscosity for the kth phase. The moving frame is a rectangular shape as shown in Figure 1 and considered to be a rigid body. A single particle is regarded and assigned the mass equal to the total mass of the rigid body (Fox and McDonald, 1998). The center of mass in the moving frame with respect to the oxyz frame is located by the position vector relative to a fi xed frame, OXYZ. The geometry shown in Figure 1 expresses the relation equation of the kth phase particle as given by Equation 4:"� = #� + �� (4)

where Ri is a vector between the initial point of the moving frame and the fi xed frame. A particle of the kth phase is located relative to the moving frame and the fi xed frame denoted by the position vectors ri and Xi, respectively. The velocity of a particle is relative to an observer in the OXYZ frame and is written as Equation 5:

$� = �#�� + {�� + (%� × �� ) } (5)

where �� is the translational velocity of the kth phase particle and %� is the rotational velocity of the kth phase particle with respect to the moving frame. The acceleration of a particle relative to an observer in the OXYZ frame is written as Equation 6:

�� = �#�� + �� + 2%� × �� + %� × (%� × �� )

+ �%�� × �� (6)

where �� is the translational acceleration of the kth phase particle. The external force term in Equation 2 for a moving frame relative to the fi xed frame is given by Equation 7:�,�� = &� �� (7)

where mk is the total mass of the kth phase fl uid inside the moving frame.

Kasetsart J. (Nat. Sci.) 46(6)980

The interface separating the two phases of fluid is defined as the transitional region between the fl uids. The phase indicator function αk(x,y,z,t) is defi ned to distinguish the phases that are present within the fl uid fl ow by Equation 8:

'� (�, �, (, �) = )1, � (�,�,() � � ��ℎ -ℎ�� &� (�)0, ��ℎ�� (8)

The particular values of αk are associated with each fl uid and propagated using a volume of fl uid (VOF) function (Gopala and Berend, 2008; Wang et al., 2004) as shown in Equation 9:

�'�� + �� ∇'� = 0 (9)

where �� is the interface velocity. The free surface boundary is assigned for the free surface. The conservation of mass at the interface can be derived by multiplying Equation 1 by αk which is written as Equation 10 and hence Equation 11:

'� �� + '� ∇ ∙ (�� ) = 0 (10)

�� '�� = �� '�� + '� �� (11)

Equation 11 can be rewritten as Equation 12:

�� '�� + '� ∇ ∙ (�� ) = �� '�� (12)

Substituting Equation 9 into Equation 12 yields Equations 13–15:

�� '�� + '� ∇ ∙ (�� ) = −�� ∇'� (13)

�� '�� + ∇ ∙ ('� �� ) − �� ∇'� = −�� ∇'� (14)

�� '�� + ∇ ∙ ('� �� ) = �� (�� − �� )∇'� (15)

The ∇αk is zero everywhere except at the interface for the conservation of mass for the two-fl uid model. The conservation of momentum at the interface can be obtained by multiplying Equation 2 by the phase indicator function as shown in Equation 16:

'� �� + '� �� ∙ ∇Uik= −'� ∇�� + '� ∇� �� + '� ��,�� +'� ��,�� (16)

The local acceleration term in Equation 16 can be employed in Equation 15 for the time derivative term and rewritten according to Equation 17:

�� '� �� = �� '� �� + �� ∙ ∇('� �� ) −�� (�� − �� )∇'� (17)

The advection term can be written as Equation 18:'� �� ∙ ∇�� = ∇ ∙ ('� �� ⊗ �� ) − �� ∙ ∇('� �� ) (18)

and hence as Equation 19:

�� ⊗ �� = / �� ⃑ ⃑ ⃑ ⃑⃑⃑�� 4 (�� + �� + � �) (19)

Figure 1 System of the moving frame. A particle is located relative to the moving frame (oxyz) and the fixed frame denoted by the position vectors ri and Xi, respectively. Ri is a vector between the initial point of the moving frame and the fi xed frame. The velocity of the particle is relative to an observer in the OXYZ frame.

y

x z

Y X Z

Xi

Ri

ri

o

O

Particle


Substituting Equations 17–19 into Equation 16 yields Equation 20:�� '� �� + ∇ ∙ ('� �� ⊗ �� )= −∇('� �� ) + ∇('� � �� ) + '� ��,�� +'� ��,�� +�� ∇'� + (�� − �� )∇'� + �� (�� − �� )∇'�

− � �� ∇'� (20)

where �� is the interfacial pressure which can occur at any point along the interface.The interfacial momentum source (Ωk) can be rewritten as Equation 21:

Ωk = �� (�� − �� )∇'� + �� ∇'� +(�� − �� )∇'� − � �� ∇'� ≃ Fik (21)

where Fik is the total force on the interface.

Finite volume discretization Figure 2 represents a typical space which is separated by hexagonal cells. A computational point P is in the centroid of the cell of interest which shares faces with the neighboring cells. Si is the vector normal to the face and pointing outward from the cell. The vector �78 connects

the computational point P with the neighbor point N. The purpose of the discretized equation is to transform one or more of the governing equations into a corresponding system of algebraic equations. The approximated solution will be solved in space and time domains. This process is facilitated by using the governing equations which are written in terms of the general differential equation or the general transport equation (Versteeg and Malalasekera, 2007) as shown by Equation 22:�� ∅��:&� ;��

+ ∇< 4∙ �� ∅�>�� :��& = ∇ ∙ Γ� ∇∅�;�� :��& +4

Sk@�� :��&⏞ (22)

where øk is the dependent variable of the kth phase, ρk is the density of the kth phase, �� is the vector of velocity of the the kth phase, Γk is the diffusion coeffi cient of the kth phase and Sk is the source term of the kth phase. The corresponding values of øk, Γk and Sk for the mass, momentum and VOF equation are summarized in Table 1. The general differential equation is formulated by integrating over the interested control volume (∀P) and time as shown by Equation 23:

B CB �� ∅�� ∀� + B ∇ ∙ (�� ∅� )� ∀�H �� =�+∆��B JB ∇ ∙ (�� Γk∇∅� )�∀�K ��+∆�� + B LB Sk�∀�M ��+∆�

�

(23) In this study, the Euler implicit time differencing scheme is used exclusively and the assumed linear variation of øk within a time step is given by Equation 24:

B �� ∅�� ∀� = (�� ∅�� )�+∆t − (�� ∅�� )�∆� ∀� (24)

where ∅�� is the general variable of the kth phase

Figure 2 Typical space domain of fi nite volume discretization. A computational point P is in the centroid of the cell of interest which shares faces with the neighboring cells. Si is the vector normal to the face and pointing outward from the cell. The vector �78 connects the computational point P with the neighbor point N.

Y X

Z

P

N

Si


in the interested cell and Δt is the incremental time step. The discretization of the convection term is performed by integrating over a cell and transforming the volume integral to a surface integral using Gauss’s theorem as shown in Equation 25:

B ∇ ∙ (�� ∅� )� ∀�= B(�� )∅�� @ = N(�� )∅��

��=1 @ (25)

where ∅�� is the general variable of the kth phase on the faces of the interested cell and is the surface area of the interested cell. The general variable of the kth phase can be evaluated in the variety of ways. The Upwind differencing scheme is used in this study (Rogers and Kwak, 1988). This method determines ∅�� on the face from the direction of fl ow according to Equations 26 and 27:

∅�� = O∅�� ≥ 0∅7� �� < 0 (26)

when � = (�� ) @ (27)where F is the mass fl ux through the face of the interested cell. The diffusion term is integrated over a cell and linearized as follows by Equations 28 and 29:

B ∇ ∙ (�� Γ� ∇∅� )� ∀�= B(�� Γ� ∇∅�� ) �@ = N(�� Γ� )∇∅��

��=1 @ (28)

and

∇∅� @ = ∅7� − ∅��|�7R | |@| (29)

where |�7R | is the length of vector between the centre of the interested cell and the centre of a neighboring cell. The terms in the general transport equation which cannot be treated as convection or diffusion terms are included as the source terms. The source terms may be a function of the dependent variable which is linearized and takes the form of Equation 30:

@� = @�� + @�� ∅�� (30)

where @�� is explicit source term; @�� is implicit source term.Integration of Equation 30 over a cell gives Equation 31:

B @� �∀� = @�� ∀� + @�� ∅�� ∀� (31)

Using Equations 24, 25, 28 and 31 and by assuming that the cells do not change in time, then Equation 23 can be rewritten as Equation 32:

(�� ∅�� ) �+∆t − (�� ∅�� ) �∆� ∀� + N (�� )∅��

�=1 @ =

N (�� Γ� )∇∅��

�=1 @ + @�� ∀�+@�� ∅�� ∀� (32)

The discretization of the generic transport equation for ∅�� can be written in terms of any cells. The coeffi cients of the interested cell and its

Table 1 Generic form of the general differential equation. Conservative Equation øk Γk Sk

Mass 1 0 0 Momentum �� μk −∇�� +��,�� +��,�� VOF αk 0 0øk = Dependent variable of the kth phase; �� = Vector of velocity of the the kth phase; Γk = Diffusion coeffi cient of the kth phase; and Sk = Source term of the kth phase; VOF = Volume of fl uid; αk = Phase indicator function; μk = Viscosity of the kth phase.


neighbors are grouped and this results in Equations 33 and 34:

�� (∅�� ) �+∆� = N �7� (∅7� ) �+∆�7�=1 + @�� − ∇�� (33)

and

�� = N �7�7

�=1 − @�� (34)

where ∅7� is the general variable of the kth phase in the neighboring cells, �� is the central coeffi cient of the kth phase and �7� is the neighboring coeffi cient of the kth phase. This system of linear algebraic equations can be expressed in a matrix form as Equation 35:

[T][∅� ] = [@] (35)

where [A] is a sparse matrix with coeffi cient �� on the diagonal �7� , [∅� ] is the column vector of the dependent variable and is the source vector. A problem associated with the velocity and pressure becoming decoupled can be solved by interpolating the discretized momentum equation onto the cell faces and applying the continuity constraint to the cell face velocities (OpenFOAM, 2009). The discretized momentum equation is obtained by rearranging Equation 33 to give Equations 36 and 37:

(�� ) ��+∆� = V(�� ) �� − 1�� ∇�� (36)

and

V(�� ) � = N �7� (�� ) 7�+∆�7�=1 + @�� (37)

This formulation of the discretized momentum equation is used to predict the face value of the velocity by isolating the contribution of the pressure when interpolating it to the control volume face. The contribution of the pressure gradient at the face is then added explicitly to (�� ) ��+∆� by calculating it directly from the

pressure value at the nodes sharing the face. For simplicity the superscript t + Δt can be dropped and the face value of the velocity is defi ned as Equation 38:

(�� ) � = CV(�� )�� H� − C∇�� H� (38)

where the face values other than the pressure gradient are calculated by using linear interpolation as shown by Equations 39 and 40:

CV(�� )�� H� = W� CV(�� )�� H� + (1 − W�) CV(�� )�� H7 (39)

C 1�� H� = W� C 1�� H� + (1 − W�) C 1�� H7 (40)

The discretized continuity equation has the form of Equation 41:

B ∇ ∙ �� ∀�= N ��

�=1 @ = 0 (41)

Substitution of (�� ) � from Equation 38 into Equation 41 gives Equation 42:

N CV(�� )�� H��

�=1 @ = N C∇�� H��

�=1 @ (42)

The terms containing the pressure gradient over the face are calculated similarly to the diffusion term of the momentum equation. The pressure in Equation 42 is now defi ned in terms of a cell and its nearest neighbor and it is possible to reformulate the pressure equation in terms of these values as follows in Equation 43:

�� = N �7� ��7

�=1 + @�� (43)

Furthermore, the velocity at the face given by Equation 38 is also used for the calculation of the conservative volumetric fl uxes at the face as shown by Equation 44:

�� = ( @�� ) � (44)


The no-slip condition is treated at the wall boundary which has a velocity at the wall of zero or is equal to that of the wall itself for the laminar fl ow. The wall shear stress is included in the momentum equation as a source term and calculated from the straightforward assumption that the velocity varies linearly from the wall to the nearest cell node as shown by Equation 45:

� �� = ! ��|� �� |

(45)

where |� �� | is the magnitude of the normal distance of the nearest cell node from the wall and μ is the dynamic viscosity. The velocity and pressure have a strongly linear coupling which has been a research topic for several years. The PISO (Pressure Implicit with Splitting of Operator) algorithm is a pressure-velocity algorithm specially developed for the non-iterative computation of unsteady compressible flows and is used in the present study (Issa, 1986). The PISO algorithm can be described as follows: (a) Momentum Prediction: Equation 36 is solved fi rstly with a guessed fi eld Pk* normally the pressure fi eld of the previous time step. The solution of the momentum equation gives a new velocity field ��∗ which does not satisfy the continuity equation. (b) Pressure Solution: These predicted ��∗ are used to assemble V(��∗) in Equation 37 which is needed for the pressure Equation 43. The solution of the pressure equation gives rise to a new pressure fi eld Pk**. (c) Explicit Velocity Correction: Equation 44 gives a new set of conservative volumetric fl uxes consistent with the new pressure fi eld. The new pressure fi eld is used in Equation 36 to do an explicit correction on velocity. The new velocity ��∗∗ is now consistent with the new pressure fi eld. The velocity in a control volume is given by Equation 36 which means that the velocity not only depends on the pressure gradient but also on

V(�� ), which includes the contribution from the neighboring cells. The PISO algorithm uses ��∗∗ to calculate V(��∗∗). These give rise to Pk*** which in turn is used to calculate ��∗∗∗. This iteration over the last two steps continues until a pre-defi ned tolerance is met.

Development and Implementation The sloshing C++ code of the two-fl uids inside the rectangular tank is performed by creating applications that fall into two categories—utilities and solvers. The utilities involve the data manipulation for creating the space domain of the 3-dimensional rectangular tanks. Two types of tanks are studied—a tank without a baffl e and a tank with a vertical baffl e. The dimensions of the tank without a baffl e are: width = 300 mm, height = 240 mm, depth = 240 mm. The dimensions of the baffl e are: width = 300 mm, height = 120 mm, thickness = 5 mm. The vertical baffl e is installed at the center and mounted on the bottom of the tank. The space domain of the tank with a baffl e is achieved by excluding the baffl e volume in the tank without a baffl e space domain. The space domain of a tank without and with the baffl e is divided using uniform hexagonal cells with Δx = Δy = Δz = 0.05 mm. The numbers of cells are 38,800 and 38,400 for the tank without and with the baffl e, respectively. The space domains of this study are shown in Figure 3. The fi lling ratios of water were 50% and 75%. The transport properties of the fl uid that fl ows inside and across the cells of the tanks are shown in Table 2. The cell movement in this study used the motion function which was developed to the whole cells using Equation 46:

: = T Ysin J2Z:� + [K − sin [\ (46)

where T is the translation of the tank, A is the distance or amplitude of the translation of 1 cycle, TP is the translation period and β is the translation phase.


The translations of the tank movement are received by fi tting Equation 46 to the experiment which has three elevations. In total, 12 cases will be studied and validated by the experiment and are described in Table 3. The solver category is contained in the PISO algorithm which was described in previous section.

Validation setting The experiment was set up to validate the CFD models. The rectangular acrylic tanks were placed on a moving plate which was fi xed to the linear guide ways. The movement of the plate was controlled by voltage supplied to the servo motor which was fi xed to the linear guide ways through the ball screw. The tanks were moved laterally following the positions in the graph of Figure 4 by the supplied voltages which have three levels. The experiments were carried out with the tanks set up under the conditions according to the CFD study cases. The sequences of the water sloshing inside the tanks were recorded by a Logitech webcam

camera. The Kyowa pressure transducer model PGM-10KE was located at a height of 60 mm from the bottom of tanks to record the pressure on the tank wall. The apparatus of the experiment is shown in Figure 5.

RESULTS AND DISCUSSION

Figure 6 shows the sequences of the 50% fi ll ratio water sloshing inside the experimental moving tanks. The heights of the surface level on the tank wall indicate the severity of the sloshing. In the tank without a baffl e, when the moving speed was increased from level 1 to level 3 the sloshing surface had a severe fl ow as shown in Figures 6A–6C. The severity of the surface fl ow was reduced by the baffl e and is shown by the comparison of the surface height on the side wall of tanks between Figures 6A–6C (baffl e) and Figures 6D–6F (no baffl e), respectively. When the water fi ll ratio was increased to 75% the severity of the surface flow also

Figure 3 Description of the 3-dimensional space domain of (A) rectangular tank without a baffl e and (B) rectangular tank with a baffl e.

Table 2 Properties of water and air. Fluid type μ (Pa s) ρ (kg m–3) Water 8.9 × 10–4 1,000 Air 1.48 × 10–5 1μ = Viscosity of fl uid; ρ = density of fl uid.

A B

240 mm

240 mm 300 mm

120 mm

147.5 mm 300 mm

240 mm

240 mm5 mm


Figure 5 Apparatus used for the liquid sloshing experiment. The +T arrow shows the positive translation.Figure 4 Controlled displacements for the tank

motions of the tank translations of T1, T2 and T3, respectively.

increased according to the tank speeds as shown in Figures 7A–7C. The vertical baffl e still reduced the severity of the sloshing (Figures 7D–7F) but not as effectively as at the 50% fi ll ratio. The three speeds of the CFD tanks were controlled according to the input position in Figure 4, then Equation 46 was fi tted with the constants which are described in Table 4. In order to investigate the liquid sloshing

inside the tanks, the simulations were runs for 13 sec. The sequences of the CFD tank without and with the baffl e having a 50% and 75% fi ll ratio are shown in Figures 8 and 9. It was found that the surface sloshing of the CFD corresponded with the experiment. The heights of the surface on the left wall of the experiment were detected using SloshDetector software (Suvanjumrat and Puttapitukporn, 2010). The comparisons between

Table 3 Detail of the computational fl uid dynamic (CFD) cases studied. Filling ratio (%) Tank translation 50 75 T1 T2 T3

1 ☼ ☼ 2 ☼ ☼ 3 ☼ ☼

4 ☼ ☼ 5 ☼ ☼ 6 ☼ ☼

7 ☼ ☼ ☼ 8 ☼ ☼ ☼ 9 ☼ ☼ ☼

10 ☼ ☼ ☼ 11 ☼ ☼ ☼ 12 ☼ ☼ ☼

The symbol ☼ is the characteristic used to indicate that set in the CFD models.

CFD case Baffl e


cases without and with the baffl e in various tank translations are shown in Figures 11A–11C and Figures 11D–11F, respectively. The pressure variation with time is the harmonic form; then the root mean square (RMS) method is used to fi nd the comparison of the pressure between the CFD and experiment. The RMS of the pressure signal which was measured on the left wall of the 3-dimensional tank with and without a baffl e cases throughout the 13 sec period of three experiment are shown in Table 6.

Table 5 Average error from the experimental results of the calculated computational fl uid dynamic (CFD) surface height variation with time on the left wall of tanks from three tank translations (T1, T2, T3).

Filling ratio Average error of CFD compared with experiment (%) (%) T1 T2 T3 Tank without a baffl e 50 1.25 4.01 6.27 75 1.65 1.03 4.35 Tank with a baffl e 50 1.16 1.70 1.81 75 1.00 1.38 2.40

Table 4 Constant values of Equation 46 for tank translations T1, T2 and T3. Tank translation A (mm) TP (sec) β (radius) T1 100 4.00 π/2 T2 100 2.67 π/2 T3 100 2.00 π/2A = Distance or amplitude of the translation of 1 cycle; TP = the translation period; β = the translation phase.

the simulation and experimental surface height on the left wall are shown by the graphs in Figure 10. The surface height variation with time on the left wall of the CFD models had an average error compared with the experimental surface height as shown in Table 5. The pressure results calculated from CFD were extracted from the cell which had the same position as the pressure transducer. Figure 10 shows the comparison of pressure variation with time between the simulation results and the experimental data. The comparisons for the

CFD model

Table 6 Comparison of pressure between the calculated computational fl uid dynamic (CFD) and experiment results.

Root mean square of pressure (kPa) CFD Experiment 1 0.54 0.50 0.04 8.69 2 0.52 0.50 0.02 3.71 3 0.39 0.59 0.20 34.30 4 1.13 1.12 0.01 0.27 5 1.11 1.10 0.01 0.87 6 0.99 1.10 0.10 9.65 7 0.54 0.50 0.04 8.01 8 0.53 0.51 0.02 3.20 9 0.51 0.52 0.01 0.36 10 1.08 1.10 0.02 1.52 11 1.07 1.15 0.08 6.95 12 1.02 1.05 0.03 2.27

Study case Differential value % error


Figure 6 Sequential images of the water 50% fi ll ratio sloshing inside the tank without a baffl e (A–C) and with a baffl e (D–F) moving with the translations of T1, T2 and T3, respectively. The ⇒ and ⇐ symbols show the movement direction of tanks.


Figure 7 Sequential images of the water 75% fi ll ratio sloshing inside the tank without a baffl e (A–C) and with a baffl e (D–F) moving with the translations of T1, T2 and T3, respectively. The ⇒ and ⇐ symbols show the movement direction of tanks.


Figure 8 CFD results of the water 50% fi ll ratio sloshing inside the tank without a baffl e (A–C) and with a baffl e (D–F) moving with the translations of T1, T2 and T3, respectively. The ⇒ and ⇐ symbols show the movement direction of tanks.


Figure 9 CFD results of the water 75% fi ll ratio sloshing inside the tank without a baffl e (A–C) and with a baffl e (D–F) moving with the translations of T1, T2 and T3, respectively. The ⇒ and ⇐ symbols show the movement direction of tanks.


Figure 10 Comparison of the surface height of the water sloshing on the left wall of tank from three tank translations (T1, T2, T3) from the experiment (Exp) and the computational fl uid dynamic (CFD) calculations (A) without a baffl e at a 50% water fi ll ratio, (B) without a baffl e at a 75% water fi ll ratio, (C) with a baffl e at a 50% water fi ll ratio, and (D) with a baffl e at a 75% water fi ll ratio.

The CFD and experimental study cases in which there was a 50% water fi ll ratio and the baffl e showed an interesting phenomenon. The liquid sloshing inside the tank exhibited prominent behavior in that the heights of the free wave surface decreased and become linear because the shallow fl ow of water across the tip of a baffl e during tank moved from the left to right. The pressure signals also decreased and became linear when the height of the baffl e and the level of water

were equally which can be observed both in the CFD and experimental results. The sequences of the water fl ow across the tip of the baffl e are shown using the VOF equation in Figure 12. The water has α = 1 and the air has α = 0, respectively.

CONCLUSION

In this study, the fi nite volume method (FVM) for the multi-phase fl ow was developed.

Time (sec)

Time (sec)

Time (sec)

Time (sec)

Surf

ace

heig

ht (m

m)

Surf

ace

heig

ht (m

m)

Surf

ace

heig

ht (m

m)

Surf

ace

heig

ht (m

m)


Figure 11 Comparison of experimental results with and without baffl es and the calculated computational fl uid dynamic (CFD) results of the pressure exerted on the left wall of tank by three tank translations (T1, T2, T3) by sloshing with (A) 50% water ratio and T1, (B) 50% water ratio and T2, (C) 50% water ratio and T3, (D) 75% water ratio and T1, (E) 75% water ratio and T2, (F) 75% water ratio and T3

Time (sec)

Time (sec)

Time (sec)

Time (sec)

Time (sec)

Time (sec)

Pres

sure

(kPa

)Pr

essu

re (k

Pa)

Pres

sure

(kPa

)

Pres

sure

(kPa

)Pr

essu

re (k

Pa)

Pres

sure

(kPa

)


The Volume of Fluid (VOF) method was used to defi ne surface of water. The 3-dimensional liquid-sloshing models were implemented and validated by experiments. The height of the surface of the sloshing on the left wall of tank was compared to the fi nite volume models. The surface height variation with time of the CFD models had an overall average error of 2.33% compared to the experimental surface height. The impact pressure depends on the water height and the excitation of motion of the tank. The developed CFD code successfully estimated the pressure on the left wall of the tank compared with the measured data. Since the pressure variation with time has a sinusoidal form, the root mean square method can be used to determine the magnitude of pressure in the time period of 0 ≤ t ≤ 13 sec. The CFD pressure had an overall average error of 6.65% compared to the recorded data of the pressure transducer. The slosh severity depends on various

factors like the depth of the water, the translations of tank and the tank geometry. The CFD calculations and the experiment using the study cases which had a 50% water fi ll ratio and a baffl e showed an interesting phenomenon. The pressure decreased and became linear when the height of the baffl e was set equal to the level of the water. The study cases in this research validated and verifi ed the use of the codes developed in the OpenFOAM software for liquid sloshing analysis, which can be used for the design of novel baffl es which can reduce the intense sloshing surface when tanker trucks turn and thus reduce the possibility of accidents.

ACKNOWLEDGEMENTS

The authors wish to thank Asst. Prof. Tumrong Puttapitukporn for the use of the experimental devices at the Mechanical Engineering Department, Kasetsart University.

Figure 12 Sequence of the 50% fi ll water fl ow across the baffl e tip inside the tank moving by the translations of T3. The α symbol shows the constant of the fl uid phase. The ⇒ and ⇐ symbols show the movement direction of tanks.


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simulation of three dimensional liquid-sloshing models

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