Aville online t www.sciencedirect.com Procedi Engineering 31 (01) 38 43 Interntionl Conference on Advnces in Computtionl Modeling nd Simultion Rpid filling of pipelines with the SPH prticle method Q. Hou *, L. X. Zhng, A. S. Tijsseling, A. C. H. Kruisrink c Deprtment of Mthemtics nd Computer Science, Eindhoven University of Technology, Eindhoven 5600 MB, The Netherlnds Deprtment of Engineering Mechnics, Kunming University of Science nd Technology, 650051 Kunming, P.R. Chin c School of Mechnicl, Mterils nd Mnufcturing Engineering, The University of Nottinghm, Nottinghm NG7 RD, UK Astrct The pper reports the development nd ppliction of SPH (smoothed prticle hydrodynmics) sed simultion of rpid filling of pipelines, for which the rigid-column model is commonly used. In this pper the wter-hmmer equtions with moving oundry re used to model the pipe filling process, nd mesh-less Lgrngin prticle pproch is employed to solve the governing equtions. To ssign oundry conditions with time-dependent (upstrem) nd constnt (downstrem) pressure, the SPH pressure oundry concept proposed recently in literture is used nd extended. Except for imposing oundry conditions, this concept lso ensures completeness of the kernels ssocited with prticles close to the oundries. As consequence, the oundry deficiency prolem encountered in conventionl SPH is remedied. The employed prticle method with the SPH pressure oundry concept ims to predict the trnsients occurring during rpid pipe filling. It is vlied ginst lortory tests, rigid-column solutions nd numericl results from literture. Results otined with the present pproch show etter greement with the test thn those from rigid-column theory nd the elstic model solved y the ox scheme. It is concluded tht SPH is promising tool for the simultion of rpid filling of pipelines with undulting elevtion profiles. 011 Pulished y Elsevier Ltd. Selection nd/or peer-review under responsiility of Kunming University of Science nd Technology Open ccess under CC BY-NC-ND license. Keywords: Rpid filling of pipelines; Undulting elevtion profile; SPH 1. Introduction Fluid trnsients in liquid-conveying pipelines involve lrge pressure vritions, which my cuse * Corresponding uthor. Tel.: +31-40-475546; fx: +31-40-44489. E-mil ddress: q.hou@tue.nl 1877-7058 011 Pulished y Elsevier Ltd. doi:10.1016/j.proeng.01.01.987 Open ccess under CC BY-NC-ND license.
Q. Hou et l. / Procedi Engineering 31 (01) 38 43 39 considerle dmge. Wter hmmer is proly the est known nd extensively studied phenomenon in this respect [1]. Rpid filling of n empty pipeline with undulting elevtion profile my occur under grvity nd y pumping. While the wter column is driven y high hed, ir is expelled y the dvncing wter column. If the generted ir flow is not seriously locked y vlves, the wter column grows with little dverse pressure nd my ttin high velocity. When the dvncing column impcts shrp end or prtilly closed vlve, severe wter-hmmer pressures occur [, 3, 4]. Also, wter column seprtion my occur t high elevtion points of the pipeline. It chnges the hydrulics significntly nd my cuse pressure surges more hrmful thn the initil wter hmmer when the seprted columns rejoin [5, 6]. Therefore, etter understnding of the rpid filling process is of high importnce. A relile model tht cn predict the mgnitude of the wter column velocities, the possile occurrence of column seprtion nd the induced overpressure in the system is highly desirle. For the 1D modelling of the rpid filling of pipelines, the rigid-column model [5] sed on set of ODEs is commonly used. It gives resonle results s long s the flow remins xilly uniform. When the wter column is distured somewhere in the system, pressure oscilltions long its length or even column seprtion my occur nd the rigid-column model will fil. The elstic model sed on set of PDEs for unstedy flow in conduits [1] is cple of deling with potentil fst trnsients in rpid pipe filling. However, the elstic model with moving oundry is difficult to solve using trditionl mesh-sed methods. A recent ttempt is the fully implicit ox or Preismnn finite-difference scheme, employed y Mlekpour nd Krney [7]. This method uses fixed sptil grid nd flexile temporl grid, where the Cournt numer is time dependent. The otined results gve cceptle greement with the lortory tests y Liou nd Hunt [5]. However, serious nd unsolved numericl convergence prolem occurred due to n uncontrollle lrge Cournt numer. In this pper, the SPH prticle method is employed to solve the full elstic model with moving oundry. The SPH computtions re compred with lortory mesurements, rigid-column theory nd numericl results of the ox scheme. Good greements re otined, especilly in the decelertion phse of the filling process, where the SPH results completely coincide with the lortory tests. The present Lgrngin prticle model, which tkes the moving oundry into ccount in nturl wy, is promising tool for slow, intermedite nd fst trnsients in the pipe filling process.. Governing equtions Consider pipeline equipped with vlve, with upstrem reservoir nd downstrem open to ir s sketched in Fig. 1. Two pipe segments with different slopes represent simple undulting elevtion profile. The vlve is locted t distnce L 0 from the inlet. After the vlve is opened, the wter will dvnce into the pipe. At the erly phse of the filling, the driving reservoir pressure domintes nd induces high ccelertion up to mximum velocity. With its length nd velocity incresing, inerti nd skin friction decelerte the wter column. Some time fter the wter column rrived t the end of the pipeline, stedy flow will develop. The following ssumptions re mde: The pipe segment tht hs een filled remins full nd well-defined front exists. This ssumption llows for one-dimensionl model to e used.
40 Q. Hou et l. / Procedi Engineering 31 (01) 38 43 Fig. 1. Definition sketch of filling of pipeline with undulting elevtion profile. The ir in the empty pipe cn flow out with negligile resistnce, nd consequently it hs no effect on the motion of the wter column. The Drcy-Weisch friction lw developed for stedy pipe flow cn e used. This is resonle ssumption for turulent pipe flows. The resistnce of the open vlve is negligile. The compressiility is tken into ccount through the wve speed, while the density remins constnt. The trnsient flow in pipe is governed y the following 1D continuity nd momentum equtions [1]: dp V c, (1) x dv 1 P V V g sin, () x D in Lgrngin form with dx V, (3) where P = pressure, V = velocity, = wter density, c = speed of sound, g = grvittionl ccelertion, = pipe inclintion ngle, = friction fctor, D = pipe dimeter, x nd t denote sptil coordinte nd time, respectively, nd d/ is the mteril derivtive. The initil conditions re V ( x,0) 0 nd P( x,0) g( H xsin ) R ( 0 x L0 ), (4) where L 0 is initil wter column length. The upstrem nd downstrem hed is tken respectively V V P(0, t) g( H K ) R nd P ( L( t), t) 0. (5) g g in which K = the entrnce loss coefficient nd L(t) = the wter column length. The velocity hed nd entrnce hed losses hve een included in the upstrem oundry condition. 3. SPH method In SPH the sptil derivtive of function f is pproximted y f f m dw ( x ) ( f f ) (6) x x dx with q 1.5q, 0 q 1, dw sign( x x ) 0.5 q, 1 q, (7) dx h 0, q,
Q. Hou et l. / Procedi Engineering 31 (01) 38 43 41 where the suscripts nd re the indices of the prticles, m nd re the mss nd density of prticle, W(x-x, h) is the kernel function with h the smoothing length, nd q=r /h with r = x -x the distnce etween the prticles. The kernel used herein is the cuic spline function; see [8] for detils out SPH. Replcing the sptil derivtives in Eqs. (1) nd () with the pproximtion (6), one otins the discrete SPH formultion given y the ODEs dp dw c m ( V V ), (8) dx dv 1 dw V V m ( P P ) g sin, (9) dx D The fct tht the density is nerly constnt hs een used in the derivtion of Eqs. (8) nd (9). To llevite possile oscilltions t shrp wve fronts, n rtificil viscosity term hs een dded to the momentum eqution. It hs the following form ch ( V V )( x x ) min ; 0. (10) r 0.01h The rte of chnge of prticle position is dx V. (11) 4. Boundry conditions To impose the oundry conditions given y Eq. (5), the novel SPH pressure oundry concept proposed y Kruisrink et l. [9] is employed nd extended. Assume tht t time t fluid prticle r (reservoir) is the one closest to the reservoir nd tht its velocity is V r (previous time step or initil vlue) (see Fig. ). To pply the upstrem x x is the initil fluid prticle spcing) is plced in the reservoir. Their velocity is V r nd their pressure is P g[ H (1 K) V /(g)]. in R r The numer of pressure inlet prticles, N pip, depends on the smoothing length h, s the kernel ssocited with prticle r needs to e fully supported. Since the rdius of the kernel is h, to meet the ove requirement n integer N pip > h/x must e tken. When pressure inlet prticle enters the pipe, it ecomes fluid prticle nd new pressure prticle is generted in the reservoir. The pressure oundry condition t the moving wter front cn e imposed in the sme wy. Suppose tht t time t prticle f (front) is locted t the wter front nd its velocity is V f (see Fig. ). A set of pressure prticles is plced downstrem of prticle f. The pressure of these prticles is zero, nd their velocity is set equl to V f. The numer of pressure outlet prticles, N pop, should e n integer lrger thn h/x too. With the defined pressure inlet nd outlet prticles, ll fluid prticles re fully nd properly supported. 5. Numericl results nd conclusion The SPH method is pplied to [5] experiments. One is referred to [5] for the detils of the test rig, which comprises 6.66 m long pipe of.9 mm inner dimeter. The clirted stedy friction fctor is 0.045 nd the entrnce-loss coefficient is 0.8. A relistic speed of sound c = 1000 m/s is used in SPH, nd there re out 650 prticles when the pipe is full. Figure 3 compres the predicted velocities ginst wter column length (mesured from inlet) with the mesurement of Liou nd Hunt [5], their rigidcolumn results nd the solution of Mlekpour nd Krney [7]. Among the results from the different models
4 Q. Hou et l. / Procedi Engineering 31 (01) 38 43 nd methods, the SPH solution grees the est with the mesurement, lthough the mximum velocity is not fully reched. The solution of the ox scheme [7] mtches the lte phse of the filling process well, ut under-predicts the velocity in the erly phse. The rigid-column results re presented in three different curves leled s 0, 10D nd 0D, where 10D nd 0D represent the length of virtul ) ) Fig.. Illustrtion of pressure prticles for () upstrem inlet condition nd () downstrem moving wter front. Velocity (m/s) 1.5 1 SPH 0 10D 0D Rigid-column model [5] Lortory tests (0 sets) [5] 0.5 Box scheme [7] 0 0 1 3 4 5 6 7 Column length (m) Fig. 3. Velocity vs. column length for lortory tests [5], rigid-column model [5], nd elstic model solved y the ox scheme [7] nd the present SPH method. pipe segment hed of the inlet [5]. In fct, etter solution cn e otined without dding ny virtul pipe if the velocity hed were included in the upstrem oundry condition used y Liou nd Hunt [5]. This hs lso een demonstrted in [7, 10, 11]. SPH seems to e vile method for simulting pipe filling processes. Although it hs een pplied herein to reltively slow filling process, wterhmmer due to possile column impct hs een tken into ccount in the formultion. Acknowledgements The first uthor is grteful to the Chin Scholrship Council (CSC) for finncilly supporting his PhD studies. The second uthor grtefully cknowledges the support from the Ntionl Nturl Science Founions of Chin (No. 50839003) nd Yunnn (No. 008GA07). References [1] Wylie EB, Streeter VL, Suo LS. Fluid Trnsients in Systems. Prentice-Hll, Englewood Cliffs, 1993. [] Guo Q, Song CSS. Surging in urn storm dringe systems. J. Hydrul. Engrg., 116, 1 (1990): 15337. [3] Zhou F, Hicks FE, Steffler PM. Trnsient flow in rpidly filling horizontl pipe contining trpped ir. J. Hydrul. Engrg., 18, 6 (00): 6534.
Q. Hou et l. / Procedi Engineering 31 (01) 38 43 43 [4] De Mrtino G, Fontn N, Giugni M. Trnsient flow cused y ir expulsion through n orifice. J. Hydrul. Engrg., 134, 9 (008): 139599. [5] Liou CP, Hunt WA. Filling of pipelines with undulting elevtion profiles. J. Hydrul. Engrg., 1, 10 (1996): 53439. [6] Bergnt A, Simpson AR, Tijsseling AS. Wter hmmer with column seprtion: A historicl review. J. Fluid. Struct., (006), 13571. [7] Mlekpour A, Krney BW. Rpidly filling nlysis of pipelines using n elstic model. 10th Int. Conf. on Pressure Surges, Edinurgh, UK, 008, 5395. [8] Liu GR, Liu MB. Smoothed Prticle Hydrodynmics: A Meshfree Prticle Method. World Scientific, Singpore, 003. [9] Kruisrink ACH, Perce F, Yue T, Cliffe A, Morvn H. SPH concepts for continuous wll nd pressure oundries. The 6th Int. SPHERIC Workshop, Hmurg, Germny, 011. [10] Mlekpour A, Krney BW. Rpid filling nlysis of pipelines with undulting profiles y the Method of Chrcteristics. ISRN Applied Mthemtics (011), Article ID 930460. [11] Rzk T, Krney BW. Filling of rnched pipelines with undulting elevtion profiles. 10th Int. Conf. on Pressure Surges, Edinurgh, UK, 008, 47387.