Computational investigation of the external excitation frequency effect on liquid sloshing phenomenon

Transcription

1 Computatonal nvestgaton of the external exctaton frequency effect on lqud sloshng phenomenon Abdallah Bouabd *, Zed Drss, Laboratory of Electro-Mechanc Systems (LASEM) Natonal School of Engneers of Sfax (ENIS) B.P. 1173, km 3.5 Soukra, 3038 Sfax, TUNISIA Abstract: - In ths work, the Computatonal Flud Dynamc (CFD) method was used to smulate the lqud sloshng phenomenon n a rectangular tank. The tank was subjected to an external snusodal exctaton. Partcularly, we are nterested on the study of the external exctaton frequency effect. The hydrodynamc parameters descrbng the flow lke the velocty feld, the average velocty, the statc and the dynamc pressure were presented. All numercal results were conducted usng the commercal CFD code "FLUENT". The volume of flud (VOF) method based on the fnte volume method was used to smulate ncompressble vscous two phase flow n a tank partally flled wth lqud. The results show that the lqud sloshng sgnfcantly depends on the frequency value. For a weak value of frequency, the lqud moves slowly. Wth the ncrease of the frequency, the lqud sloshng becomes more volent. A good agreement has been shown by comparng our numercal results wth the expermental one. Key-Words: - External exctaton, frequency, turbulent flow, fnte volume method, VOF 1 Introducton When a contaner partally flled wth lqud s subjected to an external exctaton, the phenomena of sloshng are present. These phenomena are nterest of many engneerng sectons such as petroleum cylndrcal tanks, road tankers, aerospace vehcles and lquefed natural gas carrers. Thus, the control of sloshng phenomena n dfferent types of contaner s very mportant. In fact, the sloshng can causes structure problems. Ths phenomenon of lqud sloshng n the partally flled contaner s a result of dfferent external exctatons. Therefore, t has been studed for many exctaton types. For example, Akyldz et al. [1] developed an expermental setup to study the phenomena of sloshng n a rectangular tank subjected to an external snusodal exctaton. They studed sloshng for dfferent values of fll depth and the baffle effect. Moln et al. [] nvestgated expermentally and numercally the phenomenon of sloshng n a tank excted wth horzontal and rollng movement. Yan et al. [3] dscussed expermentally the effect of lateral and longtudnal exctatons on the sloshng of the lqud n a partally flled tank. Jn et al. [4] studed expermentally the effect of horzontal perforated plates wth dfferent values of frequency and ampltude of an external snusodal exctaton. Ther study revealed that the ampltude and the frequency of the external exctaton affect the behavor of the sloshng. Also, they showed that the use of the perforated horzontal plate can sgnfcantly reduce the sloshng n the tank. Godarz et al. [5] nvestgated numercally the moton of the lqud n a partally flled contaner subjected to harmonc and sesmc exctatons. Hou et al. [9] analyzed numercally the lqud sloshng phenomenon n a rectangular tank subjected to external exctatons. The volume of flud (VOF) and the dynamc mesh technque are used to predct the deformaton of the free surface and to mpose the external exctatons respectvely. They compared the effect a sngle exctaton and the effect of multple coupled exctatons. They concluded that the sloshng s more mportant for the case of multple coupled exctatons and the case of the resonant frequency. Mezan et al. [11] studed the problem of sloshng n rectangular tank subjected to an external snusodal exctaton n the vertcal drecton. The sloshng phenomenon s also studed n a tank subjected to a lateral snusodal movement by many researchers [6-8]. Akyıldız et al. [10] analyzed the sloshng phenomena n a tank subjected to rotatonal flow for dfferent values of rollng frequences. Nowadays, The CFD (computatonal flud dynamcs) s used n several studes felds [1-15]. Especally, Yu el al. [16] used the CFD technque to study the effect of the geometry varatons on the nozzle flow. They concluded that the flow s consderably affected by the geometry confguraton E-ISSN: 4-347X 1 Volume 11, 016

2 n the nozzle. Andersson at al. [17] nvestgated the free surface evoluton n a reservor durng spllng usng the CFD technque. Mnasr et al. [18] studed the free surface deformaton caused by movng cylnders. They used the movng mesh technque to mpose the cylnders moton and the VOF method to predct the evoluton of the free surface. In the other hand, the CFD technque s used on the study of the sloshng phenomena by many researchers. For example, Srram et al. [19] nvestgated the lqud sloshng phenomenon n a tank subjected to horzontal and vertcal exctaton usng the CFD method. Godderdge et al. [0] analyzed the moton of the lqud n a rgd tank subjected to lateral movement usng the CFD technque. Bouabd et al. [1] nvestgated the moton of the lqud n a rectangular tank wth the commercal software Fluent.They studed effect of the vertcal baffle heght. Ther results showed that the sloshng phenomenon decrease wth the ncrease of the baffle heght. Accordng to the lterature revew, t s clear that the study of external exctaton parameters s very nterestng. In ths paper, we are nterested on the study of the lqud sloshng phenomenon n a rectangular tank subjected to an external snusodal exctaton usng the CFD technque. The VoF method s used to predct the free surface evoluton. The external exctaton frequency effect on the sloshng volence s studed and analyzed. The numercal results lke the velocty feld, the statc pressure and dynamc pressure are presented and dscussed. Geometrcal arrangement The consdered geometry s a rectangular tank as shown n fgure 1. Ths geometrcal arrangement s smlar to Pangrahy [] applcaton. In fact, the geometry s defned by the length L=0.6 m and the heght equal to H=0.6 m. The tank s partally flled wth lqud wth the heght h=0.1 m. 3 Numercal model 3.1 Governng equatons The phenomenon of lqud sloshng n a rgd tank for an ncompressble, mmscble flud s governed by the contnuty and the Naver-Stockes equatons wrtten as follows: ( u ) 0 (1) t x p ( u ) ( u u j ) t x x j u u j ( ) F x x x j j () Where u represent the velocty components, ρ represents the densty, p represents the pressure, µ represents the vscosty and F s the external body force. X F ρg j ρ (3) t The turbulence knetc energy k and ts dsspaton rate ε are gven by the followng equatons: μ k ρ k ρ k u μ t x x σ x t j k j G G ρε Y S (4) k b M k μ ε ρ ε ρ εu μ t x x σ x t j ε j ε ε C G C G C ρ S k k 1ε k 3ε b ε ε (5) G k and G b are the generaton of turbulence knetc energy (kg.m -1.s -3 ) respectvely due to the mean velocty gradents and buoyancy. Y M s the contrbuton of the fluctuatng dlataton n compressble turbulence to the overall dsspaton rate. The turbulent vscosty μ t (Pa.s) s gven by: k μ ρc (6) ε t μ The default values of the constants C 1ε, C ε, C μ, σ k and σ ε are presented n table 1. Fg. 1 Geometrcal arrangement Table 1. Model constants C 1ε C ε C μ σ k σ ε E-ISSN: 4-347X Volume 11, 016

3 The external exctaton s defned as follow: X Asn( ωt ) (7) Where A and ω represents the ampltude and the frequency, respectvely. The ampltude s fxed as A=0.1m for all smulaton cases n ths study. Accordng to Akyldz et al. [3], the natural frequency s gven by the followng equaton: ng nh n tanh( ) (8) L L Where L represents the length of the tank, h represents the lqud depth and n represents the mode number. For the lqud sloshng phenomenon, the resonance frequency s dfferent to the natural frequency of the flud gven by equaton (8) for n=1 [3]. The natural frequency n ths applcaton s equal to f p =0.79 Hz. Four dfferent values of frequency were studed (table ). Table. Frequency value Frequency Value (Hz) Rato f /f p f f f f The free surface s defned as the zero level set of a level-set functon Ф gven by the followng equaton: Ф x, t 0 Ф x, t 0 (9) Ф x, t 0 The free surface evoluton s gven by the transport equaton defned as follow: Ф Ф u 0 (10) t x 3. Numercal method The commercal CFD code "Fluent" has been used to present the local flow characterstcs n the tank. The VoF method (Volume of Flud) s used to dentfy the two fluds; water and ar. For the external exctaton, a User Defned Functon (UDF) s developed on language C++ and nterpreted n Fluent. The Naver-Stokes equatons and the standard k-ε turbulence model equatons were solved usng a fnte volume dscretzaton method [4-5]. 4 Numercal results and dscussons In ths secton, we presented the numercal results descrbed the turbulent flow: the free surface evoluton, the velocty felds, the average velocty, and the dstrbuton of the statc and the dynamc pressure. The numercal results are presented for four dfferent values equals to f 1 =0.5 Hz, f =0.5 Hz, f 3 =0.79 Hz and f 4 =0.9 Hz. For each frequency, the hydrodynamcs parameters are gven over tme: at t=t/4, t=t/, t=3t/4 and t=t. 4.1 Free surface evoluton The free surface elevaton for the dfferent values of frequency s presented at t=t/4, t=t/, t=3t/4 and t=t as shown n fgures, 3, 4 and 5 respectvely. Globally, we note that the lqud moves n the same sense of the external exctaton. In fact, at t=t/4 and t=t/ the lqud moves from the rght to the left for the dfferent values of the frequency. At t=3t/4 and t=t, the lqud changes the movement sense and moves from the left to the rght. In the other hand, such results ndcate that the sloshng behavour strongly depends on the frequency value. Indeed, the mnmum moton of lqud s observed for the mnmum value of frequency equal to f 1 =0.5 Hz, whereas for the maxmum sloshng has been observed for the frequency value equal to f 4 =0.9 Hz. For f 1 =0.5 Hz, the mnmum free surface deformaton has been observed for the dfferent nstances. The sloshng ncreases for f =0.5 Hz. For the natural frequency f 3 =0.79 Hz, the movement of the lqud s more mportant than for f 1 =0.5 Hz and f =0.5 Hz and lower than for f 4 =0.9 Hz. Where u represents the local velocty. E-ISSN: 4-347X 3 Volume 11, 016

4 4. Velocty felds Fg. Free surface evoluton at t=t/4. Fg. 3 Free surface evoluton at t=t/ Fg. 4 Free surface evoluton at t=3t/4. Fg. 5 Free surface evoluton at t=t. The velocty feld for the dfferent values of frequency s shown n fgures 6, 7, 8 and 9 for the dfferent nstances consdered n ths study. Globally, we note that the lqud moton creates a recrculaton zones n the tank. In fact, for the f1 frequency a small recrculaton zone appeared at t=t/4. In ths case, the maxmum value can be observed on the free surface close to the left wall. However, the velocty s near zero n the rest of the tank whch means that lqud moton s very weak. At t=t/, the recrculaton zone appeared on the free surface close to the rght wall. The maxmum value of the velocty appeared n the recrculaton zone. Also, we note that the velocty s dfferent to zero closes the left wall on the free surface. At t=3t/4, the recrculaton zones are more mportant. At t= T, many recrculaton zones appear on the free surface close to the left and the rght wall. In the other hand, we note that the hghest value s observed at t=t, whereas the lowest value s observed at t=t/. For the second value of frequency f =0.5 Hz, at t=t/4 the velocty value s more mportant n the bottom of the tank. Also, we note that a large recrculaton s observed on the free surface. At t=t/, the hghest value has been observed on the free surface close to the wall. Two recrculaton zones on the free surface close to the left wall appear at t=3t/4. At t=t, a recrculaton zone appears close to the left and the rght wall on the free surface of the lqud. For the natural frequency f 3 =0.79 Hz, a large recrculaton zone appears on the free surface n the center of the tank, whereas a small recrculaton zone appears close to the left wall at T/4. At t=t/, the lqud moton s more mportant. Three recrculaton zones appear. Indeed, the frst one s located n the center of the tank, the second appear close the left wall and the thrd appear n the rght corner n the bottom of the tank. The hghest value s observed n the bottom of the tank. At t=3t/4, one zone of recrculaton s observed on the free surface. At t=t, two recrculaton zones appear. The frst s located n the rght half of the tank where the maxmum value s noted. For f 4 =0.9 Hz, the behavor of the velocty s smlar to the case of f 3 =0.79 Hz at t=t/4. At t=t/, the hghest value s observed on the free surface close to the left wall. Three recrculaton zones have been observed. At t=3t/4, a large recrculaton zone s located on the free surface. At t=t, two small recrculaton zones have been observed. From such results, we conclude that the velocty feld depends strongly on the external exctaton frequency. Also, such results ndcate that the velocty feld vares over tme. In E-ISSN: 4-347X 4 Volume 11, 016

5 fact, for each value of frequency the velocty evaluate over tme. Fg. 6 Velocty feld at t=t/4. Fg. 9 Velocty feld at t=t Magntude velocty The dstrbuton of the Magntude velocty caused by the moton of the lqud n the tank for dfferent values of frequency s shown n fgures 10, 11, 1 and 13. Such results ndcate that the dstrbuton of the Magntude velocty depends on the external exctaton frequency. In fact, at t=t/4 the value of the average velocty ncreases wth ncrease of the frequency. At ths nstant, the maxmum value has been noted for f 4 =0.9 Hz, whereas the mnmum value can be observed for f 1 =0.5 Hz. At t=t/ and t=t, the lqud return to change the sense due to the change of the external exctaton sense. Therefore, the maxmum value can be observed for f =0.5 Hz at t=t/ and for f 1 =0.5 Hz at t=t. At t=3t/4, the maxmum value s obtaned for f 4 =0.9 Hz. These results revealed that the locaton of the hghest value of the mean velocty depends on the movement sense. Fg. 7 Velocty feld at t=t/. (c) f 3 =79 (d) f 4 =0.9 Fg. 10 Magntude velocty at t=t/4. Fg. 8 Velocty feld at t=3t/4. E-ISSN: 4-347X 5 Volume 11, 016

6 Fg. 11 Magntude velocty at t=t/. Fg. 14 Dstrbuton of statc pressure at t=t/4. Fg. 1 Magntude velocty at t=3t/4. Fg. 15 Dstrbuton of statc pressure at t=t/. Fg. 13 Magntude velocty at t=t. 4.4 Statc pressure Fgures 14, 15, 16 and 17 show the dstrbuton of the statc pressure caused by lqud sloshng at dfferent probes. From these results, we note that the locaton of the compresson zones depends on the external exctaton frequency value. Snce the drecton of movement of the flud s the same durng a half perod, the locaton of the compresson zones s the same one of the prevous nstant n a half perod. In fact, at t=t/4 and t=t/ the compresson zones are located n the bottom of tank n the left corner, whereas t s located n the rght corner at t=3t/4 and t=t. In the other hand, we note that the value of the statc pressure ncreases wth the ncrease of the movement frequency at t=3t/4 and t=t. Also, we note that the hghest value of the statc pressure s maxmal for f 3 =0.79 Hz At t=t/ and t=3t/4. Fg. 16 Dstrbuton of statc pressure at t=3t/4. E-ISSN: 4-347X 6 Volume 11, 016

7 Fg. 19 Dstrbuton of Dynamcs pressure at t=t/. Fg. 17 Dstrbuton of statc pressure at t=t. 4.5 Dynamc pressure The dstrbuton of the dynamc pressure n the tank for the consdered values of frequency s presented n fgures 18, 19, 0 and 1. From these results, t s clear that the dynamc pressure dstrbuton depends sgnfcantly on the external exctaton frequency. In fact, the maxmum value of the dynamc pressure s observed for the maxmum value of the frequency for t=t/4, t=t/ and t=3t/4. However, the mnmum value s obtaned for the mnmum value of frequency. At t=t, the maxmum value has been noted for f 3 =0.79 Hz. Also, the locaton of the maxmum value n the tank vared wth moton of the lqud n the tank. In addton, we note that the maxmum value of the dynamc pressure appears n the recrculaton zones. Fg. 0 Dstrbuton of dynamc pressure at t=3t/4. Fg. 18 Dstrbuton of Dynamcs pressure at. t=t/4. Fg. 1 Dstrbuton of dynamc pressure at t=t. 5 Comparson wth expermental results Fgure shows the varaton of the statc pressure for the dfferent values consdered n ths study. Partcularly, we compare our numercal results wth the expermental results of Pangrahy et al. [] for the case of the frequency equal to f 1 =0.5 Hz. The comparson gves a good agreement whch confrm the valdty of our numercal model. Such results ndcate that the varaton of the pressure s characterzed by a snusodal behavor for the dfferent values of frequency. In the other hand, we note that the perod of the pressure vared wth the varaton of the frequency. In fact, t ncreases wth the decrease of the frequency value. The maxmum value of the pressure s observed for f 4 =0.9 Hz and the mnmum value s obtaned for f 1 =0.5 Hz. For the natural frequency equal to f 3 =0.79 Hz and the frequency f 4 =0.9 Hz, t has been noted that the value of the statc pressure are very close. E-ISSN: 4-347X 7 Volume 11, 016

8 p(pa) Concluson f 1 f 3 Expermental f f t(s) Fg. 4 Statc pressure profle In ths paper, we studed the lqud sloshng phenomena n a rectangular tank subjected to an external exctaton. The VOF method s used to predct the free surface evoluton over tme. Partcularly, the effect of the external exctaton frequency s nvestgated. In fact, four dfferent values of frequency are examned. The results show that the sloshng phenomena depend strongly on the frequency value. Indeed, all numercal parameters are affected by the frequency value. The free surface evoluton becomes more mportant wth the ncrease of the external exctaton frequency. The lqud moves slowly for a weak value of frequency, whereas the lqud sloshng becomes more volent wth the ncrease of the frequency. For the velocty, we conclude that the recrculaton zones become more mportant wth the ncrease of the frequency. Also, the velocty value ncrease wth the ncrease of the external exctaton frequency. The same concluson can be drawn for the statc and the dynamc pressure. The results are affected strongly by the value of movement frequency. In fact, the locaton of the compresson zones on the tank depends on the frequency values. From these results, we conclude that the sloshng phenomena n the rgd tank depend on the external exctaton parameters, especally on the frequency for an external snusodal exctaton. An nterestng perspectve s to develop an expermental setup to study the lqud sloshng n other representatve cases. Further work n ths context s n progress. References: [1] Akyldz, H., Unal, N. E., Expermental nvestgaton of pressure dstrbuton on a rectangular tank due to the lqud sloshng. Ocean Engneerng, 3, 005, [] Moln, B., Remy, F., Expermental and numercal study of the sloshng moton n a rectangular tank wth a perforated screen. Journal of Fluds and Structures, 43, 013, [3] Jn, H., Lu, Y., L, J. H., Expermental study on sloshng n a tank wth an nner horzontal perforated plate, Ocean Engneerng, 8, 014, [4] Yan, G., Rakheja, S., Sddqu, K., Expermental study of lqud slosh dynamcs n a partally flled tank, Journal of Fluds Engneerng, 7, 009, [5] Goudarz, M. A., Sabbagh-Yazd, S. R., Investgaton of nonlnear sloshng effects n sesmcally excted tanks, Sol Dynamcs and Earthquake Engneerng, 43, 01, [6] Koh, C. G., Luo, M., Gao, M., Ba, W. Modellng of lqud sloshng wth constraned floatng baffle, Computers and Structures, 1, 013, [7] J, Y. M., Sh, Y. S., Park, J. S., Hyun, J. M. Experments on non-resonant sloshng n a rectangular tank wth large ampltude lateral oscllaton, Ocean Engneerng, 50, 01, 10 [8] Jung, J. H., Yoon, H. S., Lee, C.Y., Shn, S. C., Effect of the vertcal baffle heght on the lqud sloshng n a three-dmensonal rectangular tank, Ocean Engneerng, 44, 01, [9] Hou, L., L, F., A Numercal Study of Lqud Sloshng n a Two-dmensonal Tank under External Exctatons, Journal of Marne Scence and Applcaton, 11, 01, [10] Akyıldız, H., Unal, N. E., Aksoy, H., An expermental nvestgaton of the effects of the rng baffles on lqud sloshng n a rgd cylndrcal tank, Ocean Engneerng, 59, 013, [11] Mezan, B., Ourrad, O., Capllary effect on the sloshng of a flud n a rectangular tank submtted to snusodal vertcal dynamcal exctaton, Journal of hydrodynamcs 6: 014, [1] Navd, N. N., Mahmood, F. G., Numercal smulaton of fllng process of natural gas onboard vehcle cylnder. Journal of the Brazlan Socety of Mechancal Scences and Engneerng, 01, DOI /s [13] Dlson, J. S. Walter L. W., Rolf Bertrand Schroeter, Cleton Rodrgues Texera, Influence of the cuttng edge mcro-geometry of PCBN tools on the flank wear n orthogonal E-ISSN: 4-347X 8 Volume 11, 016

9 quenched and tempered turnng M steel, Journal of the Brazlan Socety of Mechancal Scences and Engneerng, 013, DOI /s [14] Alexandre, M. W., Reyolando, M. B., Jose M. B., The frst frequency of cantlevered bars wth geometrc effect: a mathematcal and expermental evaluaton, Journal of the Brazlan Socety of Mechancal Scences and Engneerng, 013, 35, [15] Elmer, M. G., Leandro, G. S., Vncus, M., Danlo, C. R., Marcello, A. M. Verfcaton and accuracy comparson of commercal CFD codes usng hydrodynamc nstablty, Journal of the Brazlan Socety of Mechancal Scences and Engneerng, 013, 36, [16] Yu, Y., Shademan, M., Barron, R. M., Balachandar, R., CFD study of effects of geometres varatons on flow n a nozzle, Engneerng applcatons of computatonal flud mechancs, 6, 01, [17] Andersson, A. G., Andersson, P., Landström, T. S. CFD modelng and valdaton of free surface flow durng spllng of reservor n down scale model, Engneerng applcatons of computatonal flud mechancs, 7, 013, [18] Mnasr, C., Hafsa, Z., Omr, M., Maalel, K., A movng grd model for smulaton of free surface behavor nduced by horzontal cylnders ext and entry, Engneerng Applcatons of Computatonal Flud Mechancs 4, 010, [19] Srram, V., Sannasraj, S. A., Sundar, V., Numercal smulaton of D sloshng waves due to horzontal and vertcal random exctaton, Appled Ocean Research, 8, 006, [0] Godderdge, B., Turnock, S., Tan, M., Earl, C., An nvestgaton of multphase CFD modelng of a lateral sloshng tank, Computers & Fluds, 38, 009, [1] Bouabd, A., Drss, Z., Abd, M. S., Vertcal Baffles Heght Effect on Lqud Sloshng n an Acceleratng Rectangular Tank, Internatonal Journal of Mechancs and Applcatons, 3, 013, [] Pangrahy, P. K., Saha, P. K., Maty, U. K., Expermental studes on sloshng behavor due to horzontal movement of lquds n baffled tanks, Ocean Engneerng, 36, 009, 13- [3] Akyldz, H., Unal, N. E., Sloshng n a three-dmensonal rectangular tank: numercal smulaton and expermental valdaton, Ocean Engneerng, 33, 006, [4] Drss, Z., Bouzgarrou, G., Chtourou, W., Kchaou, H., Abd, M. S., Computatonal studes of the ptched blade turbnes desgn effect on the strred tank flow characterstcs, European Journal of Mechancs B/Fluds, 9, 010, [5] Drss, Z., Mlayeh, O., Drss, D., Maaloul M, Abd, M. S., Numercal smulaton and expermental valdaton of the turbulent flow around a small ncurved Savonus wnd rotor, Energy, 74, 014, E-ISSN: 4-347X 9 Volume 11, 016