Computers & Fluids 38 (2009) 183–193
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
An investigation of multiphase CFD modelling of a lateral sloshing tank Bernhard Godderidge a,*, Stephen Turnock a, Mingyi Tan a, Chris Earl b a b
Fluid-Structure Interactions Research Group, School of Engineering Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK BMT SeaTech, Grove House, 7 Ocean Way, Southampton SO14 3TJ, UK
a r t i c l e
i n f o
Article history: Received 29 April 2007 Received in revised form 26 September 2007 Accepted 19 November 2007 Available online 5 March 2008
a b s t r a c t A near-resonant, sway-induced sloshing flow in a rectangular tank is used to compare a homogeneous and inhomogeneous multiphase approach for fluid density and viscosity in a commercial CFD code. Dimensional analysis of the relative motion between the phases suggests the application of an inhomogeneous multiphase model whereas previous published work has used the computationally cheaper homogeneous (or average property) approach. The comparison between the computational and experimental results shows that the homogeneous model tends to underestimate the experimental peak pressures by up to 50%. The inhomogeneous multiphase model gives good agreement with the experimental pressure data. Examination of the relative velocity at the fluid interface confirms that the inhomogeneous model is the appropriate model to use for the simulation of a violent sloshing flow. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction When a tank is partially filled with a fluid and subjected to an external excitation force, sloshing occurs [1]. Ships with large ballast tanks and liquid bulk cargo carriers (e.g. oil tankers) are at risk of exposure to sloshing loads during their operational life [2]. The inclusion of structural members within the tanks dampens the sloshing liquid sufficiently in all but the most severe cases. This approach can not be used for liquefied natural gas (LNG) carriers and sloshing has thus evolved into a design constraint for this type of vessel [3,4]. Natural gas, consisting of typically 90% methane, is transported in liquefied form over long distances (>1600 km) as it is more economic than building a pipeline [5]. The liquefaction temperature of 163 °C requires a combination of suitable insulation and structural material to minimise heat transfer and withstand the applied loads. The accurate calculation of the sloshing loads is an essential element of the LNG tank design process [6]. The work of Abramson [7] underpins sloshing analysis and Ibrahim [8] gives a survey of sloshing modelling techniques. Three approaches are usually used to determine sloshing loads in naval architecture. (1) Experimentation is used by classification societies, among them Det Norske Veritas, Lloyd’s Register and the American Bureau of Shipping [6]. Correct scaling of the model sloshing loads is often difficult [7].
* Corresponding author. Tel.: +44 23 8059 6610; fax: +44 23 8059 3299. E-mail address:
[email protected] (B. Godderidge). 0045-7930/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2007.11.007
(2) Theoretical fluid dynamics models have been developed. A linear model for the aerospace industry was given by Graham and Rodriguez [9]. Faltinsen [10] developed a thirdorder theoretical sloshing model. The restriction imposed by the tank shape complexity has been overcome using boundary element methods. (3) A more general modelling technique is the solution of the Navier–Stokes equations using computational fluid dynamics (CFD). Some recent examples of CFD sloshing simulation include Aliabadi et al. [11], El Moctar [12], Hadzic et al. [13], Rhee [14] and Standing et al. [15]. Hadzic et al. [13], Rhee [14] and Standing et al. [15] used commercial CFD codes (COMET-now Star-CD, Fluent and CFX-5, respectively) to simulate sloshing in rectangular containers. While only a small (usually about five) number of oscillations are shown, good agreement between computational and experimental is observed. The fluid interaction models for the numerical simulation of sloshing can be implemented using the volume fraction of each fluid to determine the fluid mixture properties. This is a homogeneous multiphase model which is analogous to the volume of fluid (VOF) method developed by Hirt and Nichols [16]. A more general but computationally more expensive approach is an inhomogeneous multiphase model, where the solution of separate velocity fields for each fluid is matched at the fluid interfaces using mass and momentum transfer models [17]. Due to its comparatively low computational cost and good numerical stability, the homogeneous model has usually been preferred and was used in the sloshing studies in Refs. [11–15]. When studying problems with greater phase interaction or phase transfer, inhomogeneous multiphase models are more widely used
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[17,18], although Kunz et al. [19] studied sheet cavitation using a homogeneous model. The physics of a violent sloshing flow, including wave breaking, vapour entrapment and cushioning [20,21] may contradict the assumptions [22] inherent in the homogeneous model. The work outlined in this paper investigates the impact of the selection of multiphase model for a near-resonance sloshing flow in a rectangular 1.2 m by 0.6 m tank. Computational data obtained from the homogeneous and inhomogeneous multiphase models are compared to experimental pressure data from Hinatsu [23].
multiphase flow, conservation of mass for homogeneous multiphase flow is given as oðrqÞ o ðrqui Þ ¼ 0; ð6Þ þ ot oxi and the conservation of momentum is defined as o o op osij ðqui uj Þ ¼ þl þ bi ; ðqui Þ þ oxj ot oxj oxi
ð7Þ
with q¼
2 X
r l ql ;
ð8Þ
r l ll :
ð9Þ
l¼1
2. Computational modelling
and
2.1. Governing equations
l¼
2 X l¼1
An inhomogeneous viscous compressible multiphase flow with two phases a and b can be described by the conservation of mass for the compressible phase a o o ðrqui Þ ¼ m þ Cab ; ðrqÞ þ ot oxi
ð1Þ
where Cab is mass transfer between the phases and m mass sources, q density, r volume fraction and ui velocity of phase a. The corresponding equation for conservation of momentum for phase a is given as o o op oðrsij Þ ðrqui uj Þ ¼ r þ þ M C þ M a þ bi ; ðrqui Þ þ ot oxj oxi oxj
ð2Þ
where bi are body forces, Ma forces on the interface caused by the presence of phase b, l the dynamic viscosity, the term MC ð¼ Cab ubi Cba ui Þ interphase momentum transfer caused by mass transfer and the stress tensor sij is expressed as oui ouj : ð3Þ sij ¼ l þ oxj oxi The interface momentum transfer term Ma needs to be considered in greater detail as it links the fluid velocity fields. This term may be modelled by a linear combination of known forces acting on the fluid interface, such that Ma ¼ MD þ MV þ MB þ ML þ MW ;
ð4Þ
where MD is drag force, MV virtual mass force, MB Basset force, ML lift force due to fluid rotation and MW wall lubrication force [17,24]. Due to its complicated nature, the Basset force is generally ignored in practical multiphase analysis [17]. The virtual mass force is used to model the interaction of small, subgrid-scale particles with the surrounding fluid. This is ignored in the present analysis. The lift force is generated by fluid rotation around particles. The correct modelling of wall lubrication force requires a fine grid [24], making its inclusion in transient simulations impractical. The interphase drag force MD is expressed using the drag coefficient CD ¼
D 1=2qjU a U b j2 A
;
ð5Þ
where A is interfacial area, D drag, q density and j Ua Ubj velocity between the phases a and b. For the current Newtonian flow regime, a drag coefficient of 0.45 is used [17]. Eqs. (1) and (2) are computationally expensive as the number of conservation equations to be solved doubles with an additional fluid. A simplification is given with homogeneous multiphase flow. In this case it is assumed that the relative motion between the phases can be neglected [22]. Thus, the interface momentum transfer in Eq. (4) becomes large, but the velocity field is identical for both phases and only one set of conservation of momentum equations needs to be solved. Applying this simplification to the governing equations for inhomogeneous
2.2. CFD model 2.2.1. Numerical implementation The model given by Eqs. (1)–(7) is implemented in a commercial CFD code.1 The governing equations (1), (2) for the inhomogeneous or Eqs. (6), (7) for the homogeneous multiphase model are discretised using the finite volume method [25]. Fig. 3 shows the control volume A–B–C–D–E–F–G–H obtained from hexahedral element discretisation of a computational domain. The locations marked by crosses are the centres of the faces of the control volume. Conservation of mass for phase a (Eq. (6)) with no mass sources is discretised using Gauss’s divergence theorem and an implicit second-order backward Euler scheme [24,25] as X V 3 1 ðrqa ui ni Þnk Sk ¼ 0: ð10Þ ðqa rÞn2 ðqa rÞn1 þ ðqa rÞn 2 þ dt 2 2 k where dt is time step, superscript n the time step currently computed, n np the npth time step before n, ni the unit vector orthogonal to the kth area of a control volume Sk and V is the volume of a control volume. The phases a and b must fill the available volume so that ra þ rb ¼ 1:
ð11Þ
Eq. (11) can be combined with Eq. (6) to give a volume continuity equation instead of writing an additional mass conservation equation for phase b [25] " # X 2 X 1 V 3 n 1 n2 n n1 þ ð12Þ ðrl ql ui ni Þk Sk ¼ 0: q 2ql þ ql ql ot 2 l 2 k l¼1 The equations of momentum conservation, Eq. (2) for inhomogeneous and Eq. (7) for homogeneous multiphase flows, are discretised in a similar manner. For simplicity, only the discretisation of Eq. (7) is shown, but an analogous result can be written for Eq. (2). As in the equations discretised previously, an implicit second-order backward Euler scheme is used for the time derivative. The discretised momentum equation for phase a is written as X X oðqui Þ X ðqui uj nj Þnk Sk ¼ pnk lij nj Sk þ qn g i V þ ðsnij ljl nl ÞSk ; þ ot k k k ð13Þ where oðqui Þ V 3 1 ¼ ðqui Þn 2ðqui Þn1 þ ðqui Þn2 ; ot dt 2 2
ð14Þ
1 The simulations were carried out using ANSYS CFX-10.0 running on a 64 bit, 2.2 GHz processor with 2 GB of RAM as part of the University of Southampton’s Iridis 2 computational facility.
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lij is the unit vector in the xi direction (and j is a dummy subscript) and p the pressure. Density q, viscosity l and the stress tensor sij are defined by Eqs. (8), (9) and (3), respectively. The equations governing the k e turbulence model can be found in [26]. The flow quantities are computed at the node points, but the discretisation of the governing equations requires the quantities at the integration points of the control volume associated with each node, as shown in Fig. 3. The present model uses a high-resolution advection scheme which varies between first- and second-order depending on the spatial gradient. For a scalar quantity U the advection scheme is written in the form UIP ¼ UUP þ brU R;
ð15Þ
where UIP is the value at the centre of an integration surface, UUP the value at the upwind vertex and R the vector from the upwind vertex to the integration point (see Fig. 3). The model is a secondorder, upwind-biased scheme for b = 1, but reverts to first-order when b = 0 [25]. The computation of b follows an approach similar to that given by Barth and Jesperson [27], which aims to maintain b as close to unity as possible while ensuring that the computed U at the integration points are bounded [25]. Since the present model uses a colocated grid for pressure and velocity, an interpolation scheme based on that proposed by Rhie and Chow [28] is used [25]. Gradients are computed at integration points using tri-linear shape functions defined in [24]. The advection scheme can be modified to improve the resolution of a free surface. By allowing b > 1, while enforcing the boundedness of quantities at the integration points, the advection scheme compresses the ‘thickness’ of the free surface [25]. The compressiveness of the advection scheme arises from the fact that it is anti-diffusive when b exceeds unity [25]. The advection scheme does not rely on reduced the time step size for its compressiveness, which is an important consideration for extended transient simulations. The ability of this scheme to maintain the sharpness of a free surface over extended time scales has been shown by Zwart [25]. The discretised equations yield a 6 6 coupled system at each node for a two phase system. The resulting system is solved with a coupled solver [25]. Each step of the solution is iterated on the full grid as the algebraic multigrid solution strategy has been found to result in computational instabilities. The solution for the two phase system is obtained by solving the equations for volume continuity (12) and conservation of momentum (13), solving the mass conservation equation (6) for water, solving Eq. (11) for air and, solving the turbulence kinetic energy and dissipation equations [25]. The air–water system is not strictly coupled in this procedure2 which introduces the need for additional iterations to obtain a solution [24]. For every time step in the transient simulation, the iterative solution of each sub-system in the above list is obtained using the incomplete lower–upper (ILU) factorisation technique [24]. Depending on the flow field and time step size, between five to ten iterations are required for a converged solution. Thirty sloshing oscillations were solved with approximately 15,000 time steps. 2.2.2. Multiphase model In considering computational efficiency alone, the homogeneous multiphase model will be the most effective but the interaction between the phases is ignored. The homogeneous multiphase
model is used in most sloshing simulations [11,13–15]. When the water impacts a tank wall, a small air pocket usually remains. This behaviour is observed in experimental studies of sloshing [21] as well as the present computational investigation. The properties of this bubble and surrounding fluid can be used to determine a suitable multiphase model. Brennen [22] provides guidance using a size parameter X and a mass parameter Y in conjunction with the particle Reynolds number. They are defined as R mp ; ð16Þ X ¼ 1 qc v l mp 2mp 1þ Y ¼ 1 ; ð17Þ qc v qc v and the particle Reynolds number RN;a ¼
The follow-on code CFX-11 can solve the fully coupled system (i.e. combine steps 1–3 in a single system).
ð18Þ
The terms and numerical data for a typical air bubble in water observed during sloshing impact, given in Table 1, are used. Brennen [22] finds that if either the condition X Y2 or X Y/ (UR/mc) is violated, the inhomogeneous multiphase model (Eqs. (1) and (2)) should be used. Using data both from the CFD results described within and experimental findings by Lugni et al. [21] given in Table 1 it is found that X Y2. However, the second condition X Y/(UR/mc) is not satisfied. This suggests that the use of an inhomogeneous multiphase model is required for the analysis of this violent sloshing problem. 2.2.3. Computational models The computational grid used, shown in Fig. 1, contains 9605 hexahedral elements, with the first node offset 0.3 mm from the wall with a boundary layer mesh 20 nodes thick in the adjacent 35 mm from the tank wall. The element diagonal varies between 10 mm near the wall and 25 mm at the tank centre. Grid independence of the results is confirmed in Table 4 where the reference grid contains 13,761 hexahedral elements with 28,000 nodes. As the computational domain is fully enclosed, a reference pressure was set by defining one node at the centre (0.6 m, 0.6 m) of the top wall (see Fig. 2) as a one cell wide outlet with pstatic = 1 atm. This was found to be a more stable approach than using the explicit specification of the pressure at the corresponding node. The computational parameters were selected based on the sensitivity study by Godderidge et al. [29]. It was found that the second-order time marching scheme is most appropriate, as mass and momentum are conserved over a large, O(104), number of time steps. The magnitude of each time step was controlled dynamically using the root mean square of the local cell Courant number CN computed over the entire velocity field. A maximum threshold value of CN, RMS = 0.1 was identified in the validation study by
Table 1 Quantities used in Eqs. (16) and (17) for multiphase analysis Quantity l mp mc qc R U v X Y2 Y/(UR/mc) a
2
jU b U a jR : ma
Length scale (m) Particle mass (kg) Kinematic viscosity (Ns m1)a Density (kg m3)b Particle radius (m) Characteristic velocity (m s1) Particle volume (m3)
CFD data
Lugni et al. [21]
1.2 1.23 106 1.00 106 1000 0.01 4 4.19 106 1.66 102 9.93 101 1.25 105
1.0 2.66 107 1.00 106 1000 0.006 5 9.05 107 5.99 103 9.93 101 3.32 105
This applies to water, the suspending fluid. A particle is defined as a finite piece of the dispersed phase, e.g. an air bubble in water. b
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Fig. 1. Computational grid.
Table 2 CFD model description and parameters
Fig. 2. The sloshing problem (all dimensions in m).
Fig. 3. Location of the integration points and nodes.
Godderidge et al. [29] and the convergence criteria was applied in line with the recommendations in [24]. The computational parameters used in the simulations are summarised in Table 2. 2.3. Mass conservation The most severe sloshing occurs when the natural frequency of the sloshing fluid and the excitation frequency coincide. The
Parameter
Setting
Water Air Sloshing motion Turbulence model Wall boundary Spatial discretisation Temporal discretisation Timestep control Convergence control
Incompressible fluid Ideal gas Body force approach ke No slip Gradient-dependent first or second-order Second-order backward Euler RMS Courant number = 0.1 RMS residual 2 105
natural sloshing frequency of a potential flow in an open rectangular tank excited by pure sway may be written as pg h tanh p ; ð19Þ x2n ¼ L L where g is the gravitational constant, h the filling level, L the tank length and xn the first natural frequency. As the filling level h is directly related to the amount of fluid in the tank, it is evident that any numerical loss or gain of fluid mass will influence the computational result. Not only will there be a change of static pressure, but the proximity of the excitation frequency to the resonance frequency is changed as well, influencing the dynamic pressure contribution. Thus, the simulation of violent sloshing using a viscous fluid model requires the conservation of fluid mass over a large number of iterations and time steps. Post-impact fluid fragmentation, wave overturning, breaking and the attachment of a thin layer of fluid on tank walls are particularly challenging for the mass conservation of finite volume codes. The effects can be mitigated by improving the mesh resolution and/or using a higher-order time marching scheme. However, both result in increased computational cost per time step. A second, more popular solution is to formulate the governing equations to force global conservation of mass. This gives rise to Eq. (12). However, mass conservation is enforced only for each individual time step and error in fluid mass can start to build, especially when simulating a large number of time steps. If the errors are related and monotonic (cumulative error), the exponential increase of the total error can be related to the residual e according to
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Fig. 4. Sloshing impact, free surface contours shown: (a) [t = 5.85T]; (b) [t = 5.99T]; (c) [t = 6.01T]; and (d) [t = 6.08T].
Fig. 5. Water centre of gravity displacement.
Table 3 CFD simulation summary
Table 4 Results of the grid independence study (from Godderidge et al. [29])
Parameter
Homogeneous
Inhomogeneous
Simulation length Simulation CPU time CPU time per simulated second Forced mass correction
45.0 s 2.897 105 s 1.42 h/s No
52.5 s 1.520 106 s 3.54 h/s No
Nodes 53.1 s 1.251 106 s 3.27 h/s Yes
5600 12,000
Difference relative to the finest grid (%) P4
P6
P9
4.83 1.44
4.29 1.45
0.72 0.44
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Fig. 6. Water mass fraction.
Fig. 7. Power spectra of pressure data at P4 and P6: (a) [P4] and (b) [P6].
mðtÞ ¼ ð1 þ eÞn ; m0
ð20Þ
where m0 is the fluid mass at t = 0 and n the time step. Assuming that the errors at each time step are independent of each other
and have a Gaussian distribution (independent error), the envelope of the mass error evolution is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðtÞ 1 ð1 þ eÞ2ðnþ1Þ ¼e : ð21Þ m0 1 ð1 þ eÞ2
B. Godderidge et al. / Computers & Fluids 38 (2009) 183–193
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Fig. 8. Pressure history at P4 between oscillation 20–30: (a) homogeneous multiphase flow and (b) inhomogeneous multiphase flow (mass correction).
The error propagation can be eliminated by adjusting the mass conservation equations to compare the fluid mass at time step n to the initial fluid mass. The implementation in the governing equations is complicated by the addition of inflow and outflow boundary conditions. A more pragmatic solution is the addition of liquid to the computational domain, called forced mass conservation, to counteract numerical diffusion. Faltinsen and Rognebakke [30] report the implementation of this type of scheme by shifting the position of the free surface according to the fluid mass gained or lost. An interface sharpening function can be employed to maintain a clearly defined free surface and enforce global mass conservation [31]. Both schemes directly affect the free surface position as part of the correction procedure and require adjustment of the source code. However, altering the source code is not always possible or desirable and a more readily implementable solution has to be found. Fluid impacts during sloshing can result in large fluid velocities at the free surface [21,32]. The effect of altering the fluid mass in the free surface volume elements to enforce global mass conservation on conservation of momentum and hence fluid velocity has not been examined. Therefore, it may be preferable to add lost fluid in less violent regions of the flow field, usually found near the tank bottom. In the current study, if mass is lost it is addressed by the introduction of a mass source evenly distributed across the bottom wall. The inflow rate depends on the explicitly specified initial water mass. A typical mean inflow rate 0.2% of the initial water mass per second (5 106 m0 per time step) of simulated time was observed. At each timestep the inflow rate is recomputed to
compensate the water mass imbalance with a constant inflow rate _ ¼ Dm=Dt over one second. This scheme was required only for the m inhomogeneous multiphase model.
3. Sloshing problem The sloshing problem modelled in this study is illustrated in Fig. 2. The tank dimensions, locations of the pressure monitor points and axis system orientation are shown. The results obtained from the computational model are compared to experimental steady state sloshing pressures given by Hinatsu [23]. The pressure results are recorded once the sloshing flow reached a steady state. At monitor points P4 and P6 the pressure history is periodic and suitable for comparison to the computational results. However, the experimental pressure history at P9 varies by more than 100% of the mean peak and for three of seven oscillations no impact is observed. Thus, the pressure history at P9 appears sensitive to the flow evolution. The comparatively low sampling rate of 1 kHz, compared to 17 kHz in similar studies by Rognebakke and Faltinsen [33] may add further uncertainty to the experimental results. Therefore the experimental and computational results are not compared at P9. The tank motion is given as 2p t ; ð22Þ x ¼ A sin T where A = 0.015 m is the displacement amplitude, T = 1.404 s the sloshing period and t the elapsed time. In the current case, the tank motion is in the x-direction only, as indicated in Fig. 2.
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Fig. 9. Pressure history at P6 between oscillation 20–30: (a) homogeneous multiphase flow and (b) inhomogeneous multiphase flow (mass correction).
4. Results Fig. 4 shows the water motion during a sloshing impact. Immediately before impact a jet of fluid forms just below the top wall in the right corner of Fig. 4b. As the water impacts the top wall, a large local pressure peak is observed at the top right corner of the tank in Fig. 4c. After impact the resulting water column breaks as indicated in Fig. 4d. For resonant sloshing a more violent flow field is expected. A free surface smearing index is proposed to assess the extent of interface smearing during the simulation. The minimum distance of contour c2 (corresponding to rwater = 0.50) to contours c1 (rwater = 0.05) and c3 (rwater = 0.95) is calculated along contour c2 and compared to the mesh element diagonal dmesh at the corresponding locations on contour 2. An ideal value of this index is unity, as it indicates that the free surface is captured within one element. The free surface smearing index at element i is given as DFSi ¼
ci1 ci2 þ ci2 ci3 i
dmesh
:
ð23Þ
The mesh diagonal has been chosen as length scale since two different computational models are compared on the same grid. Other applications, such as grid independence studies, would employ a different length scale such as the reference grid element diagonals or tank height. An average value of DFS is computed at each time interval of 0.01 s (not each time step) and subsequently over the duration
of the simulation. A straight line (ax + b) fit to the obtained DFS gave b = 1.57 for the inhomogeneous and b = 2.27 for the homogeneous multiphase model. For both models, a = O(103), indicating that the free surface sharpness does not change during the simulation. A value of 2.14 was obtained for b for the reference grid in the grid independence study. While the inhomogeneous multiphase model resolves the free surface more sharply than the homogeneous multiphase model the mean difference between the models is 0.7dmesh. This corresponds to a static pressure of approximately 50 Pa. The homogeneous multiphase flow model conserved the fluid mass for the duration of the simulation, with the water mass fracremaining constant at unity. The water centre of gravity tion mðtÞ m0 motion, given in Fig. 5, shows that the flow field becomes periodic after about 20 oscillations. The initial transient stage lasts approximately ten oscillations and a secondary transient stage is observed between oscillations 10 and 20. As the experimental pressure data is for the steady state, pressures are compared in the region between oscillations 20 and 30. The computationally more expensive alternative is the inhomogeneous multiphase model with associated costs of the models used given in Table 3. Fig. 6 shows the water mass m(t) in the sloshing container. During the first 2–3 oscillations, fluid sloshing is calm compared to the subsequent flow field and the fluid mass fraction remains constant at unity. Later the flow becomes more violent and after 30 oscillations (15,000 time steps), nearly 10% of the total water mass has been lost. The rate of decrease of fluid mass follows the behaviour of a cumulative error (Eq. (20)). The
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191
Fig. 10. Velocity field difference: (a) tank free surface position; (b) local free surface position; (c) difference in velocity vector direction; and (d) difference in velocity vector magnitude.
gradient of the mass loss during the oscillations 20–30 corresponds to a residual of 2 105, as specified in Table 2. The mass loss does not behave as an independent error (Eq. (21)). Since the code always computes the volume fraction of air as a function of the water volume fraction, a small error in the iterative
solution is permitted to propagate and build with an increasing number of time steps. The effect of the error can be reduced with a smaller permissible residual. A more effective solution may be to alternate the solution of the mass conservation equation for air and water (and compute the volume fraction of the other
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phase), thus encouraging the error to behave in a manner similar to the independent error in Fig. 6. From the above results it is evident that the main weakness of the inhomogeneous multiphase model is the poor conservation of mass over extended timescales. This is addressed using the forced mass conservation procedure outlined in Section 2.3. The implementation of forced mass conservation resulted in the water mass fraction remaining at unity throughout the simulation. Oscillations 20–30 are again a suitable basis for comparing computational and experimental results. Only the results from the homogeneous and inhomogeneous mass corrected models will be considered in greater detail.
DU ¼
jU air j jU water j ; 0:5ðjU air j þ jU water jÞ
ð25Þ
where the velocity vector U = (u, v). Before fluid impact, shown in Fig. 10d, the relative velocity is greatest near the left wall and DU is small. During impact the peak relative velocity has increased to equal the mean flow velocity. Near the wall, DU reaches a peak value of 0.8. Once the deflected jet has formed, the relative velocity is greater than the mean flow velocity, and 10% of the jet velocity. The difference in velocity magnitude is greatest at the front of the post-impact jet. As is the case with the directional difference, a large DU is observed at the base of the post-impact jet.
4.1. Pressure spectrum Fig. 7 compares the power spectra of the experimental and computational homogeneous and inhomogeneous steady state pressures. At P4, shown in Fig. 7a, good agreement between the inhomogeneous CFD results and the experimental data is observed near the excitation period. The peak value is well predicted, but experimental and computational high-frequency noise is observed as well. However, the homogeneous model underpredicts the peak by approximately 50%. Fig. 7b shows the analogous plot for P6. This time, the experimental peaks are at the excitation period T and at approximately 0.55T. Again the inhomogeneous mass corrected model predicts both peaks, but the trough between the peaks is not matched. The homogeneous model underestimates the experimental power spectrum by an order of magnitude. 4.2. Pressure history Fig. 8 depicts the pressure time history at P4 and confirms the observations from the spectral analysis. The homogeneous flow model underestimates the experimental data3 by up to 25%. The inhomogeneous model data in Fig. 8b matches the experimental pressure history, albeit the pressure peak value in the time series is underestimated by about 6%. The observations at P6 in Fig. 9 are similar, with the homogeneous result underpredicting the experimental pressure in excess of 50%. The inhomogeneous model produces a better match, but the double-peak shape of the experimental data is not replicated. The static pressure component is more significant at P4 than at P6, where the predictions of the homogeneous model are poor. This indicates that the dynamic properties of the sloshing flow are not simulated correctly by the homogeneous multiphase model. A principal assumption of the homogeneous model, outlined in Section 2.1, is that the relative motion between the phases can be ignored. The difference in air and water velocity fields at the fluid interface, given as rair = 0.5 = rwater, is examined in greater detail. The free surface position near the impact is shown in greater detail in Fig. 10a and the free surface displacement in the entire tank is given in Fig. 10b. The difference in velocity vector direction, shown in Fig. 10c, is defined as U air U water : ð24Þ Dh ¼ cos1 jU air jjU water j The directional difference is greatest immediately prior to the water impacting the top wall. Once the post-impact jet is formed, the velocity vectors are well aligned near the front of the jet, but differ by up to 30° near the base of the jet. Fig. 10d shows the difference in the magnitude of the air and water velocity 3 The experimental data depicted was obtained by conducting a Fourier decomposition of the raw data from Hinatsu [23].
5. Conclusions It has been shown that for the CFD analysis of violent sloshing flows an inhomogeneous multiphase model is the most appropriate to use. Dimensional analysis outlined by Brennen [22] was used to identify the inhomogeneous model as the most suitable. The relative velocity between the phases, which is ignored in the homogeneous multiphase model, has been found to be significant compared to the global velocity field. Neglecting the motion between the phases results in an incorrect velocity field and causes the sloshing pressure peaks to be underestimated by in excess of 50%. The steady state pressure histories and the power spectrum from the inhomogeneous multiphase solution compare well to the experimental data. While the pressure history from the homogeneous model appears to agree with experimental data during the initial transient phase, the comparisons of the experimental pressure history with the computational steady state pressure history illustrates the shortcomings of the homogeneous multiphase model. However, the inhomogeneous multiphase model is 2.3 times more computationally expensive than the homogeneous multiphase model. The maintenance of the correct fluid mass and resolution of the free surface over extended time scales is a challenge for the present volume of fluid code. A free surface smearing index is proposed and is used to show that the free surface sharpness remains constant throughout the simulations. The observed pressure difference is not attributed to smearing of the free surface. The mass imbalance encountered when using the inhomogeneous model was overcome by the introduction of a mass source at the bottom tank wall. The advantage of this approach is that its implementation does not require modification of the source code and there is no mass error build-up. A better solution to the problem may be either the use of a fully coupled solution algorithm or the alternating computation of mass conservation of each fluid. Knowledge of peak pressure is an essential component of LNG tank design. When modelling violent sloshing, the inhomogeneous multiphase model gives the most faithful representation of the flow physics. Due to the higher computational cost of the inhomogeneous multiphase model, the homogeneous model is useful for an initial CFD analysis of sloshing. The application of the inhomogeneous model to simulate a sloshing flow should be preceded by dimensional analysis of the flow physics obtained from CFD or experimentation to determine the suitability of the multiphase model. Acknowledgements This work was carried out under the auspices of the Engineering Doctorate programme at the University of Southampton, with support from the Engineering and Physical Sciences Research Council (UK) and BMT SeaTech Ltd. The authors also acknowledge the
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