Comparing Sloshing Phenomena in a Rectangular Container with and without a Porous Medium Using Explicit Nonlinear 2-D BEM-FDM

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1 Transacton B: Mechancal Engneerng Vol. 7, No., pp. 93{0 c Sharf Unversty of Technology, Aprl 00 Comparng Sloshng Phenomena n a Rectangular Contaner wth and wthout a Porous Medum Usng Explct Nonlnear -D BEM-FDM Abstract. M. Abbaspour ; and M. Ghods Hassanabad The sloshng phenomena n a partally lled tank can aect ts stablty. Modcatons of tank nstablty due to the movement of the tank carrer, are key desgn ponts for the stablty of a carrer. Even though the sloshng phenomenon has already been nvestgated usng the BEM-FDM technque, the research n ths paper covers ths phenomenon n a porous meda, whch s new n -D coordnates. For ths purpose, a Laplace equaton has been used for potental ow, and knematc and dynamc boundary condtons have been appled to the free surface. Also, a formulaton has been developed for a free surface n porous meda. BEM has been used for solvng the governng equaton and FDM dscretzaton has been used for knematc and dynamc free surface boundary condtons and for tme marchng. Theoretcal results have been vered wth expermental data collected n ths study. The results show an acceptable agreement between theory and experment, and the rapd dampng property n the sloshng phenomena by usng porous materal n the water, as expected. Also, these results llustrate that the derved formula n ths research are applcable and true. Keywords: Boundary Element Method (BEM); Fnte Derence Method (FDM); Porous meda; Potental ow; Lqud free surface. INTRODUCTION Flud ow models nvolvng a deformng doman, n general, and free-surface ow models such as sloshng, n partcular, have a smlar challenge n ther numercal smulaton. These knds of problem, especally wth an nterface-trackng approach, need movng mesh or regeneraton of the doman mesh at each tme step. Some dcultes, such as mesh regeneraton and usng more complex meshes, ncrease the soluton tme. The best approach for decreasng soluton tme s to use the BEM method to solve free-surface ows especally n potental ows. Sloshng s one of the free-surface ows nvestgated for many applcatons. In the storage tanks of LNG tankers, sloshng produced by sea waves can aect the stablty of the shp. The same phenomenon may. School of Mechancal Engneerng, Sharf Unversty of Technology, Tehran, P.O. Box , Iran. *. Correspondng author. E-mal: m-abbaspourjamejam.net Receved 5 Aprl 009; receved n revsed form 5 July 009; accepted October 009 happen when a traler carryng any type of lqud passes over a road curve or a ramp. Hence, nvestgatng the sloshng phenomena n these stuatons s a desgn key pont for ensurng safety. Many researchers have nvestgated sloshng phenomena by derent theoretcal and expermental methods. Faraday [] began the rst nvestgaton of ud sloshng n 83. Then, n 883, Ryle [] assessed sloshng phenomena on a vbratng bed. In the 0th century, many researchers studed sloshng phenomena n horzontal, vertcal and rotatonal moton (such as sway, heave and roll), -D and 3-D moton, and many other problems. In recent years, a wde range of artcles have focused on sloshng, some of whch have nvestgated the stablty of contaners due to sloshng. Lqud sloshng n shp tanks has been analyzed by Carou and Casella []. Nasar et al. [3] set up an expermental study on lqud sloshng dynamcs n a tank carryng barge. Wu [4] has nvestgated the resonance of sloshng n a tank. Lu and Ln [5] solved, numercally, lqud sloshng n tanks n three dmensons. The nte element method has been appled to solve the sway

2 94 M. Abbaspour and M. Ghods Hassanabad lnear and nonlnear moton of a -D contaner by Vrlla et al. [6]. Hamano et al. [7] have used the boundary element method to model large ampltude standng waves n -D vessels. The boundary element method and the nte derence method have been used to model water wave moton wth lamnar boundary layers n a rectangular contaner by Jamal [8]. In recent years, a smple porous materal (named Exess [9]) has been ntroduced as an ant-exploson product n storage tanks. In addton, these materals can mprove the stablty behavor of ther carrers. The advantage of ths materal s that they only occupy.4 to.7 percent of the contaner space. In ths study, a new approach has been presented to solve free surface potental ow under non-lnear boundary condtons n a rectangular contaner, wth and wthout porous materal. Moreover, a BEM- FDM model has been developed for sloshng n porous meda. METHODS Mathematcal Formulaton There s a wde range of moton that leads to sloshng phenomena. As a practcal and mportant case, the maneuverng of a lqud tank carrer followed by a straght movement has been nvestgated. As a result of free surface experences, the sloshng starts n a parabolc shape as shown n Fgure. Based on statc relatons n ud mechancs, the ntal condton of a ud free surface would have a parabolc shape, whch can be expressed as: z =! g (R + x) R! g ;! = U R ; () n whch R, U and g are radus of curvature, carrer velocty and gravtatonal acceleraton, respectvely; x Fgure. Schematc representaton of crcular moton of rectangular contaner. and z are horzontal and vertcal coordnates (Fgure ). Followng the two dmensonal free surface sloshng non-lnear theory, the governng equaton and boundary condtons are expressed as below [0]. By assumng rrotatonal ow, the governng equaton s a Laplace equaton n the form of: r = 0; () n whch s the potental functon. The knematcs boundary condton on the free surface s: t = z x x ; (3) n whch s wave ampltude. Ths relaton s converted to []: t = cos n ; (4) n whch n s normal to the free surface and s the angle the free surface makes wth the horzontal lne. Applyng Bernoull's equaton on a free surface provdes the dynamc free surface boundary condton: t = B(t) g ( ) : (5) Usng Equatons 4 and 5, the dynamc free surface boundary condton s converted to []: t x = B g s n s n tan # ; (6) n whch s s measured along the free surface and (=t) x ndcates the rate of change at a constant horzontal poston, but followng the free surface vertcally. The boundary condton on sold walls s: n = 0: (7) It s worth notng that the mentoned formulas are sutable for a contaner wth only one type of ud. If a lqud contaner s lled by a porous materal, such as Exess, the related equatons wll der. Assumng rrotatonal ow, the governng equaton n the porous meda s the same Laplace equaton. Also, the same s true for the knematc boundary condton of a free surface and walls,.e. Equatons 4 and 7.

3 Applcaton of -D BEM-FDM for Sloshng n Porous Meda 95 The dynamc boundary condton on a free surface, however, ders from the above equatons (Equatons 5 and 6). Neld et al. have presented the extenson of Darcy's law, accordng to the Woodng paper, that can be appled for dynamc boundary condtons. The extenson of Darcy's law s []: V t + (V:r)V = rp f ~v g; (8) f k n whch P, V, ~v, f, k, and g are pressure, ntrnsc average velocty, velocty vector, ud densty, permeablty, vscosty and gravtatonal acceleraton, respectvely. When the Duput-Forchhemer relatonshp s used, the above equaton becomes []: ~v t + (~v:r)~v = rp f ~v g; (9) f k n whch s porosty. Ths equaton was obtaned by analogy wth the Naver-Stokes equaton. The rrotatonal ow theory leads to the relatonshp [3]: (~v:r)~v = r(~v:~v); (0) where: ~v = r; () n whch and ~v are potental functon and velocty vector, respectvely. By substtutng Relatons 0 and n 9, the equaton s smpled to: r t + (r) + P + f and hence: + gz = 0; k f () t + (r) + P + + gz = B(t): (3) f k f The above equaton s smlar to Equaton 5 n general form. Equaton 5 s Bernoull's equaton whch has been obtaned by a speced method [3]. In ths study, a smlar method has been used for dervng Equaton 3 n a porous medum, whch can be used for free surface dynamc boundary condtons. Usng the same method for convertng the dynamc free surface boundary condton to a form of Equaton 6, the followng equaton s obtaned: = B g t k s x n # s n tan : (4) Therefore, all formulas are equal for ordnary sloshng and sloshng n porous meda, except under free surface dynamc boundary condtons. In other words, only Equaton 5 ders from Equaton 3. Boundary Element Method In order to solve the Laplace equaton wth all boundary condtons, the Green functon s used and the governng equaton can be transformed to the boundary ntegral equaton shown as [4]: Z (p)= where: (p; q) (q) n q (q) (p; q) ds q ; n q (5) = ln r; (6) n whch p s the source pont and s the fundamental soluton n the boundary element method. In the boundary element method, after establshng the ntegral equaton, t should be converted to an algebrac system of equatons for obtanng potental functons and ther dervatves on the nodes of the boundary. The followng equaton s concluded from the ntegral Equaton 5 [4]: [H] nn [] n = [G] nn [ n ] n ; (7) n whch H and G are coecent matrces; and n are potental functons of boundary nodes and ther normal dervatves to the boundary, respectvely; and n s node numbers on the boundary. There are n unknown values n the set of Equaton 7. These unknown values should be decreased to n unknown values. There are some Neumann boundary condtons (Equaton 7) on the boundary of ud n a rectangular contaner, whch determne the values of n on the walls. Also, there are Robn boundary condtons (Equatons 6 and 4) on the boundary of ud n a rectangular contaner, whch determne the relatons between and n on the free surface. These Robn boundary condtons should be dscretzed for each element because the speccaton of each node should be placed n Equaton 7. The nte derence dscretzng method s used for ths purpose. Fnte Derence Method Drchlet and Neumann quanttes of boundares were obtaned by dscretzng the ntegral equaton n the boundary element method. Tme marchng and the shape of the free surface at each tme step requres the dscretzng and solvng of the knematc and dynamc boundary condtons of the free surface. The nte derence method was selected for dscretzng the free

4 96 M. Abbaspour and M. Ghods Hassanabad surface boundary condtons. The shape of the free surface was moded by splne nterpolaton at each tme step. The knematc boundary condton on a free surface (Equaton 3) of a contaner, wth and wthout porous meda, s dscretzed as below []: + = + t a cos + n + + a cos n # ; (8) n whch a s between 0 and. As seen n Equaton 8, both teratons and + are at the rght hand sde of the equaton. If 0 s replaced n a, then the rght hand sde of Equaton 8 s only ncluded by varables n teraton and the formulaton wll be explct. In the same manner f a =, then the formulaton wll be mplct, and f 0 < a <, then the formulaton wll be composed of explct and mplct. Equaton 8 s used for determnng the new poston of the free surface nodes n tme marchng and for applyng t nto Equatons 9 and 0. The dynamc boundary condton on the free surface (Equaton 6) of a contaner wthout porous meda s dscretzed, as below []: + + t c s bag cos + t + tan + # n g t b( a) g t cos c t s s tan n c t s n n + = # + ; (9) n n whch b and c are between 0 and. For the same reasons mentoned above, f a = b = 0, then the formulaton of Equaton 9 wll be explct. The dynamc boundary condton on the free surface (Equaton 4) of a contaner wth porous meda s dscretzed after some lengthy and new operatons, as below: + d t + + t K f = c s bag cos + t + # tan + + n ( d) k f b( a) g t cos ( c) t s n s g t tan c t n n s # + : (0) n Equatons 9 and 0 follow the same condtons for mplct or explct states. Equatons 9 and 0 are used for determnng the potental functon and ts dervatve of free surface nodes n tme marchng. Also, the dervatve of the potental functon relatve to s (tangental lne) s obtaned by nte derence approxmaton n the prevous tme step (teraton ). Some Numercal Consderatons Three constant parameters (a, b, c) were ntroduced n Equatons 8 to 0. a, b and c are between zero and. If these three parameters are equal to, the soluton wll be fully mplct, because all terms n Equatons 8 to 0 are n teraton +. However, f they are equal to zero, the soluton wll be fully explct, because all terms n Equatons 8 to 0 wll be n teraton. Derent quanttes of these parameters determne the weght of derent mplct and explct terms n the equatons. In ths study, the fully explct formulatons (a = b = c = 0) were chosen due to ther easness. Also, Table shows the other speccatons of the sloshng model. Pressure on the sde walls for calculatng nondmensonal forces can be obtaned by Bernoull's equaton (Equaton 5) for a contaner lled wth lqud only and derved equaton (Equaton 3) for a contaner wth porous materal. In ths study, the non-dmensonal forces on the sde walls have been computed n terms of non-dmensonal tme.

5 Applcaton of -D BEM-FDM for Sloshng n Porous Meda 97 Table. The man parameters used n the numercal method. Parameters Quantty Descrptons U (m/s) Carrer velocty L (m) 0. L s the length of rectangular contaner H (m) 0. The heght of rectangular contaner R (m) Radus of curvature t (sec) 0.0 Tme step No. F. elements 40 Number of free surface elements No. T. elements 0 Number of total elements on the boundares 0.4 The porosty of doman whch s near to the low dense wre crmps doman K (m ) 8 0 The permeablty of doman whch s near to the low dense wre crmps doman Expermental Method In the present study, an expermental set up was appled for verfyng the numercal data obtaned n ths study. As shown n Fgure, a parabolc plane was used to shape the water free surface of the contaner as a parabola. Ths parabola matches the numercal dmensons of ths research and Equaton. Also, low dense wre crmps have been used as the porous materal. For ntalzng the experment, the parabolc plane was put on the water free surface. Then, ths plane was taken away quckly n order to observe the tank surface behavor. The sloshng phenomenon occurred due to gravty, surface tenson etc. wth gravty as the domnant factor. The edges of the parabolc plane were sealed to prevent water penetraton. Also, n ths smple setup, unwanted sde eects such as sloshng due to ntense ar ow under the plane were prevented by makng some holes on the upper part of the parabolc plane. The surface tenson, however, s unavodable. It s the man reason for derences between numercal and expermental results, but t can be detected from gravtatonal waves by observaton. RESULTS Fgures 3 and 4 show some real pctures of free surface moton, compared to numercal results. Cyan curves show the numercal data of ths research. As shown n Fgure 3a, n the 0. sec after the ntal condton, the free surface of the water n the experment concdes wth the numercal sample. In Fgure 3b, after sec, the water surface reaches near ts equlbrum surface. In ths case, some waves are seen on the water surface, but the man lne of the free surface concdes wth theoretcal results. In Fgure 3c, the free water surface passes from ts equlbrum lne, and hence the other end of the free surface goes up. As shown n ths gure, gnorng the waves wth small wave lengths, the free surface of the expermental data concdes properly wth numercal results. Also, accordng to Fgures 3d to 3k, the numercal moton of the free surface curve concdes acceptably wth expermental wave shapes. Accordng to Fgure 4a, n the case of usng porous materal n the contaner, the cyan curve extracted from the numercal results under ntal condtons concdes wth the free surface of the water. In Fgure. Image of expermental devce.

6 98 M. Abbaspour and M. Ghods Hassanabad Fgure 3. Images of water sloshng from 0. s to 0 s (comparng numercal and expermental data). Fgure 4b, after 0. sec, the computed curve moves wth the same pattern of the expermental free surface moton. Also, n other parts of Fgure 4, there s a good match between expermental and numercal results. As shown n Fgure 4, the resulted wave s completely damped after a complete perod, whch s.08 sec. Fgure 5 shows the non-dmensonal forces on the sde walls, wth respect to non-dmensonal tme, whch s obtaned by the numercal calculaton of the present research. The rapd dampng property of porous materals s llustrated n Fgure 5 very well, as expected. In ths gure, L s 0 cm, s about

7 Applcaton of -D BEM-FDM for Sloshng n Porous Meda 99 Fgure 4. Images of water sloshng from 0 sec to.6 sec n porous meda (comparng numercal and expermental data). 000 kg/m 3, g s 9.8 m/s and after around.4 sec the force s damped. Fgure 6 shows the rght end of the free surface dsplacement, expermentally and numercally, n the contaner lled just by water. Ths gure shows an acceptable agreement between experment and computaton. Fgure 7 shows the rght end of the free surface dsplacement, expermentally and numercally, n the contaner lled wth porous materal. Ths gure shows a very good concdence between experment and computaton. In ths gure, the ampltude damps n around.35 sec. DISCUSSION AND CONCLUSION Kukner and Baykal have nvestgated sloshng phenomena due to trangular ntal free surface. They showed that utlzng trangular corrugated bottoms may help to regulate the ow n tanks [5]. Corrugated bottoms were not used n the present research; nstead, porous materal was placed n the contaner n order to ncrease the stablty of the tank. In addton, a parabolc shape was used for the ntal free surface. The research of Kukner and Baykal [5], as well as the present research, tres to ntroduce a method to augment the stablty of tanks. It seems

8 00 M. Abbaspour and M. Ghods Hassanabad Fgure 5. Comparson of non-dmensonal wall forces n terms of non-dmensonal tme n a rectangular contaner wth porous materal and only lled by water. that the method of the present study s more useful, because of the more rapd dampng of sloshng forces. One of the most mportant problems n shp LNG tanks s sloshng. Graczyk and Moan studed ths subject and showed that the level of free surface aects the sloshng forces on the tank walls [6]. In ths research, a parabolc shape was used for the ntal free surface, whch can be smlar to the result of shp maneuverng. Accordng to the results of ths research, t s proposed that, to decrease ths sloshng eect, t s proper to use porous materals. In the present research, some derences were observed between expermental and numercal results. The experment ndcated that when the parabolc plane s taken away, gravty and surface tenson create two types of wave. The rst wave s due to gravty, wth a wave length of 4L (contaner length s L); n ths case, the ampltude of free surface moton n the mddle of the contaner s about zero. The second wave s due to surface tenson; wth a wave length of about L (contaner length s L). Ignorng the eects of the second wave leads to an acceptable smlarty between the theoretcal and expermental results. In concluson, usng porous meda s an excellent dea for rapd dampng of the sloshng phenomenon. Expermental results ndcated that the wave damps n about one second n a porous contaner, whle takng about twenty seconds wthout a porous medum. A smlar trend occurs for wave forces exerted on the contaner walls. REFERENCES Fgure 6. Free surface rght end dsplacement for comparng numercal and expermental results n a contaner lled just wth water. Fgure 7. Free surface rght end dsplacement for comparng numercal and expermental results n the contaner wth porous materal.. Ibrahm, A., Lqud Sloshng Dynamcs: Theory and Applcatons, Cambrdge Unversty Press, pp (005).. Carou, A. and Casella, G. \Lqud sloshng n shp tanks: a comparatve study of numercal smulaton, Marne Structures,, pp (999). 3. Nasar, T. et al. \Expermental study of lqud sloshng dynamcs n a barge carryng a tank, Flud Dynamcs Research, 40, pp (008). 4. Wu, G.X. \Second-order resonance of sloshng n a tank, Ocean Engneerng, 34, pp (007). 5. Lu, D. and Ln, P. \A numercal study of threedmensonal lqud sloshng n tanks, Journal of Computatonal Physcs, 7, pp (008). 6. Vrlla, J.C. et al. \Lnear and nonlnear -D nte element analyss of sloshng modes and pressures n rectangular tanks subject to horzontal harmonc motons, Journal of Sound and Vbraton, 3, pp (008). 7. Hamano, K. et al. \Boundary element smulaton of large ampltude standng waves n vessels, Engneerng Analyss wth Boundary Elements, 7, pp (003).

9 Applcaton of -D BEM-FDM for Sloshng n Porous Meda 0 8. Jamal, M. \BEM modelng of surface water wave moton wth lamnar boundary layers, Engneerng Analyss wth Boundary Elements, 30, pp. 4- (006). 9. \Ant exploson systems, 4man.htm. 0. Dean, R.G. and Dalrymple, R.A., Water Wave Mechancs for Engneers and Scentsts, 3rd Edton, World Scentc, pp (99).. Lu, P.L. and Lggett, J.A. \Applcaton of boundary element methods to problems of water waves, n Developments n Boundary Element Methods-, P.K. Banerjee and R.P. Shaw, Eds., Appled Scence Publshers LTD, pp (98).. Neld, D.A. and Bejan, A., Convecton n Porous Meda, 3rd Edton, Sprnger, Scence+Busness Meda Inc, pp. 8-5 (006). 3. Whte, F.M., Flud Mechancs, 6th Ed., McGraw-Hll Scence Engneerng (006). 4. Katsekadels, J.T., Boundary Elements: Theory and Applcatons, Frst Ed., Elsever Scence Pub. Co. (00). 5. Kukner, A. and Baykal, M.A. \Wave sloshng n corrugated bottom tanks wth two-dmensonal ow, Ocean Engneerng, 6, pp (999). 6. Graczyk, M. and Moan, T. \A probablstc assessment of desgn sloshng pressure tme hstores n LNG tanks, Ocean Engneerng, 35, pp (008). BIOGRAPHIES Madjd Abbaspour receved hs BS degree n mechancal engneerng from Sharf Unversty of Technology (SUT) n 973, hs MS n thermal energy from the Massachussets Insttute of Technology (MIT) n 975, and hs PhD degree n cvl and envronmental engneerng from Cornell Unversty n 980, wth a mnor n ocean engneerng. Snce then he has served n SUT as a faculty member. He has publshed more than 3 books n hs related eld and more than 50 papers n respected journals and nternatonal conference proceedngs. In 995, he was honored to receve an award for the best publshed book n the eld of engneerng, enttled \Envronmental Engneerng. Prof. Abbaspour s the chef edtor of the ISI ranked \Internatonal Journal of Envronmental Scence and Technology (IJEST). He has won many academc awards and also the natonal medal of mert for outstandng research actvtes (997). He has two regstered nventons n the eld of ocean and marne engneerng (009). He also won the 0th and th Kharazm nternatonal awards, respectvely, n 997 and 999, n the eld of research and nnovaton. Madjd Ghods Hasanabad receved a BS n mechancal engneerng (sold mechancs) from Tehran Unversty n 000 and a MS n mechancal engneerng (ud mechancs) from Sharf Unversty of Technology, Tehran, n 003. He has been workng snce 004 on hs doctoral project n `free surface ows', and `boundary element method' at Sharf Unversty of Technology. He has publshed more than 5 papers n journals and conference proceedngs. He has one regstered nventon n the eld of refrgeraton (008) and n 00 he won the Iranan mechancal engneers socety award for the best BS thess n sold mechancs.