Journal of Hydraulc Structures J. Hydraul. Struct., 2016; 2(2):22-34 DOI: 10.22055/hs.2016.12849 Numercal nvestgaton of free surface flood wave and soltary wave usng ncompressble SPH method Mohammad Zounemat-Kerman 1 Habbeh Sheybanfard 2 Abstract Smulaton of free surface flow and sudden wave profle are recognzed as the most challengng problem n computatonal hydraulcs. Several Euleran/Lagrangan approaches and models can be mplemented for smulatng such phenomena n whch the smoothed partcle hydrodynamcs method (SPH) s categorzed as a proper canddate. The ncompressble SPH (ISPH) method hres a precse ncompressble hydrodynamc formulaton to calculate the pressure of flud, and the numercal soluton s obtaned by usng a two-step sem-mplct scheme. Ths study presents an ISPH method to smulate three free surface problems; (1) a problem of sudden dam-break flood wave on a dry bed wth and obstacle n the downstream, (2) a test case of the gradual collapse of the water column on a wet bed and (3) a case of soltary wave propagaton problem. The model has been confrmed based on the results of experments for the dam-break problems (n whch was set up by the authors) as well as the collapse of the water column test case and analytcal calculatons for the soltary wave smulaton. The computatonal results wth a mean relatve error less than 10%/4% for the wave heght/wave front poston, demonstrated that the appled ISPH flow model s an approprate modelng tool n free surface hydrodynamc applcatons. Keywords: Free-surface flows, Dam-break, Soltary wave, Incompressble smoothed partcle hydrodynamcs Receved: 04 October 2016; Accepted: 17 November 2016 1. Introducton Free surface flows and currents are accompaned wth deformaton and fragmentaton of the free surface and sold boundares. These fragmentatons and Deformatons n dealng wth massve flows of dam-break are exacerbated. There are two maor approaches of Euleran and Lagrangan technques for modelng of free surface flows. However, most of numercal modelng and studes of flud flow are manly based on the Euleran mesh based formulatons such as fnte dfference, fnte-volume and fnte-element methods. Although the ablty of these approaches for a wde range of numercal problems have been proven, but these models mght be lmted for complcated phenomena wth large fragmentatons and deformatons due to ther mesh adaptablty and connectvty restrctons [1]. In recent years, Lagrangan numercal models have been further developed and ther usage s becomng ncreasngly attractve. 1 Assocate professor, Water Engneerng Department, Shahd Bahonar Unversty of Kerman, Kerman, Iran, Emal: zounemat@uk.ac.r (Correspondng author) 2 M.Sc of water structures, Water Engneerng Department, Shahd Bahonar Unversty of Kerman, Kerman, Iran, 0913939928, Emal: habbeh.sheyban@gmal.com
Numercal nvestgaton of free surface flood wave. 23 In a Lagrangan method, advecton term s calculated by drect movement of partcles. Therefore, there s no numercal dffuson [12]. The SPH numercal models, whch enable the smulaton of complcated and complex free surface flows, are consdered as a purely Lagrangan approach [11]. The SPH was developed by Lucy, Gngold and Monaghan (1977) n astrophyscs to perusal the mpact of stars and galaxes and boldes on planets [16]. Recently, ncompressble SPH (ISPH) and weakly compressble SPH (WCSPH) methods (due to ther flexblty and power as numercal methods) have been appled to a whole of flud flow problems (such as the study of gravty currents [4], wave propagaton [2], wave nteracton on the coastal and offshore structures [8]; [7]) and dam-break flow [15], [6], [7], [17] and [13]). In ths respect, Monaghan (1999) used weak compressble SPH Approach for modelng soltary wave along the offshore [12]. Colagross and Landrn (2003) mplemented SPH method to treat two-dmensonal nterfacal flows wth dfferent fluds separated by sharp nterfaces [2]. Atae-Ashtan and Shobeyr (2008) presented an ncompressble-smoothed partcle hydrodynamcs (ISPH) formulaton to smulate mpulsve waves generated by landsldes [4]. The results proved the effcency and applcablty of the ISPH approach for smulaton of these knds of complex free surface problems. Shao (2010) nvestgated wave nterplay wth porous meda by ISPH procedure [21]. Zhu (2009) used ISPH technque for nvestgatng wave nteracton by porous meda. Lnd et al (2012) ntroduced a generalzed algorthm for valdatons and stablty of propagatng waves as well as mpulsve flow [18]. Džebo et al (2014) descrbed two dfferent ways of defnng the terran roughness n SPH smulatons performed for smulatng dam-break phenomenon at the storage reservor of a hydropower plant n Slovena [7]. Lnd et al (2015) presented a new ncompressble-compressble (water-ar) smoothed partcle hydrodynamcs method to model wave slam [19]. Altomare et al (2015) appled the valdaton of an SPH-based technque for wave loadng and strke on coastal structures [6]. Altomare et al (2015) used the SPH-based technque for examnaton of the nteracton for wave and coastal structures. Such as vertcal structures and storm return walls [6]. Jonnsson et al (2015) modeled dam-break phenomenon on a wet bed wth SPH method. In ths paper, the ISPH method has been used for modelng free surface flow. As presented by [20], n the ISPH numercal method, the governng equatons of contnuty and momentum are solved n a twostep proecton method. In the ISPH, the pressure term s mplctly resulted by solvng the Posson equaton n the second proecton step. As a result, pressure fluctuaton would be less n the ISPH than n comparson wth the WCSPH method (where slght compressblty s appled va the equaton of state) [20]. In ths respect, three test cases of 1) dam-break flow over dry bed and encounterng an obstacle, 2) gradual collapsng of the water column over a wet bed, and 3) a soltary wave were modelled n ths research usng ISPH model. The accuracy of the ISPH model to smulate flood waves and soltary wave flows s assessed wth contrastng the calculated results wth those measured by expermental observatons and analytcal calculaton. The governng equaton of moton, ISPH formulaton, solvng algorthm and results of the advanced model are shown n the followng sectons. 2. Governng equatons of flow and dscretzaton Governng equatons of wave flow are ust mass conservaton equatons and momentum conservaton as follows: 1 D. u 0 Dt Du 1 2 p u fb (2) Dt where s densty, u s velocty vector, P s pressure, and fb s an external force. Dscretzaton of pressure terms and vscosty n Naver-Stokes (NS) equatons by smoothed partcle hydrodynamcs method s lke below: (1)
24 M. Zounemat-Kerman, H. Sheybanfard 1 p p ( p ) ( ) ( m w ) m. w 2 2 (3) 1 p p ( p ) m ( ). w 2 2 (4) 4 m ( ). 2 r w ( u) ( u u ) 2 2 2 ( ) ( r ) Where p =p -p, r =r -r, s vscosty coeffcent and (whch s usually equal to 0.1h) s a small number ntroduced for keepng the denomnator no-zero durng the computaton. 3. ISPH formulaton The ISPH technque s formed by ntegral nterpolaton for approxmaton equatons. The extent of functon at r pont s explaned below (6): f ( r0) f ( r) ( r0 r) dr Usng ths ntegral n numercal analyss s practcally mpossble. Therefore, Drac Delta functon exchanges wth smoothed functon and amount of quantfy for partcle s calculated usng the followng ntegral (7): f ( r0) f ( r) ( r0 r) dr Ths equaton s dscretzed as below (8): m f ( r ) f ( r ) w ( r r, h) (8) Where s support doman, m and are mass and densty of partcle, m s volume element assocated wth partcle, w s smoothng functon, r s poston vector, h s smoothng length whch resulted the wdth of kernel. In ths paper, h=1.2dr was used, where dr s ntal partcle spacng. Smoothng functon (9) used n ths artcle (cubc splne) was recommended by [15]. 10 3 2 3 (1 3 q q ), f 0 q 1 7 2 2 4 h 10 w (2 q) 3, f 0 q 2 (9) 28 h 2 0, otherwse q s a parameter equal to r /h. Crcle radus of central partcle (kh) wll take 2h values. For fndng partcles n the nfluence doman of a central partcle, f the number of total partcles s n, then n 2 computaton s needed that wll cause ncreased calculaton load. Therefore, lnked lst method s used that means searchng for 9 squares around the central partcle, aganst usng total research of all computatonal domans [21]. In ths method, the computatonal doman s separated nto squares wth sze 2h and each partcle belongs to a square (Fgure 1). It s clearly vsble that the neghbor partcles of the central partcle are only n 9 squares on every sde of the partcle nterest. (5) (6) (7)
Numercal nvestgaton of free surface flood wave. 25 Fgure 1 Dvdng of the Computatonal doman nto squares wth 2h szes for fndng partcles n the nfluence doman of the central partcle. 4. Solvng algorthm In ths study, the ISPH method wth a two-step sem-mplct algorthm has been utlzed to solve the governng equatons. In the frst step, accordng to the dmensons of the problem and ntal dstance between partcles, fxed partcles and mass of all partcles s calculated usng (10) [12]. m 0 (10) w( r r ), h ) Then, mass of all partcles can be determned accordng to (11). m w( r r, h) (11) In ths step, Naver-Stokes equatons by gnorng pressure terms (n the absence of ncompressble condton) are solved to calculate velocty of partcles. Afterwards, velocty u and poston partcle r are calculated (12-13) [21]. u ut u (12) r rt u t (13) Δu denotes the ncrement of partcle velocty at the predcton step; Δt s the tme ncrement; u t and r t are the partcle velocty and condton at tme t; and u and r are ntermedate partcle velocty and condton, respectvely. In the next step (correcton step), the pressure force, whch has been gnored n the last step (predcton step), s used to revse the partcle velocty as the followng: 1 u p t 1 t (14) u t 1 u u (15) where Δu stands for the ncrement of partcle velocty at the correcton step; ρ s ntermedate partcle densty whch was calculated after the predcton step; and p t+1 and u t+1 are the partcle pressure and partcle velocty at tme t+1. By gettng average, the fnal poston of the partcle s centered n tme as below: u 1 u t r t 1 r t t (16) 2 In the above relaton, r t and r t+1 denote postons of the partcle at tme t and t+1. The Posson equaton s used to calculate the pressure as below:
26 M. Zounemat-Kerman, H. Sheybanfard 1.( ) 0 p t 1 2 (17) 0 t Where ρ 0 s ntal constant densty at each of the partcles. The advantage of the ISPH method s that large pressure fluctuatons can be decreased because of usng a strct hydrodynamc formulaton. The pressure n (17) can be solved by ether any avalable drect or teratve solvers. However, the teratve solvers have shown to be more robust n case of great numbers of partcles. 5. Boundary Condtons (sold walls and free surface) 5.1. Sold boundares For modellng sold wall, the sold mpermeable boundares are acted by unmovng wall partcles. They not only prevent the entrance of nner flud partcles nto sold wall but also balance the pressure of them. Because of the pressure balance of flud partcles, Posson's equaton wll be solved for these partcles. It should be mentoned that two rows of dummy partcles out of these walls are placed for gnorng wall partcles as free surface partcles. The veloctes of wall partcles are arranged zero to ndcate the unmovable boundary condton. 5.2. Free surface In the ISPH model, the partcle densty s used to determne the free surface and after that, a zero pressure s gven to the free surface partcles. Snce there s no flud partcle exstng n the outsde area of the free surface, thereafter the densty on the surface partcle should drop sgnfcantly [17]. For recognzng free surface partcles, f the calculated partcle densty satsfes the followng condton, the partcle s recognzed as a surface partcle [5]. (18) 0 In (18), s free surface parameter and 0.8< <0.99. In ths paper, =0.97 was used. 6. Modellng of free surface flows wth ISPH model Three free surface problems have been consdered to show the model's capabltes and to nvestgate ts accuracy. Two dam-break sudden and gradual flood wave problems and the soltary wave n open-channel on a stff bed are used to llustrate the model's capacty to calculate the freesurface poston and the dstort of the water surface. The numercal results have been contrasted to expermental measurements and analytcal soluton. 6.1. Modellng of sudden dam-break flood wave Flow aganst an solated obstacle Ths experment was done by the frst author n the Cvl Engneerng Laboratory of the Islamc Azad Unversty (Sran Branch, Iran). The flume dmensons are ndcated n Fgure 2. Because of the mmedate Removablty of the gate from the water column, t could be consdered as a sudden dam-break. In addton to recent computatons, water densty was consdered 1000 kg /m3 and water dynamcs vscosty was about 0.001 N.s/m3. For performng ths smulaton, 3229 partcles wth ntal between-partcle dstance of 0.006 m were used. Comparson of expermental results and the results of numercal modellng the dam-break flow n contrast to a rgd obstacle (as shown n Fgure 3) shows the poston of partcles and free-surface shape, and pressure dstrbuton after a 0.1, 0.3, 0.5 and 0.7 s smulaton. Fgure 4 shows the pressure dstrbuton resulted from the numercal model for smulaton of 0.3 and 0.7 s. It may be seen that the predcted free-surface shape has an acceptable adaptaton of the expermental observatons. For better understandng and more accurate analyss of results, normalzed dagram of tme n comparson to poston proceedng of wave front of numercal model wth the dstance of 0.006 m between partcles and expermental model s gven n Fgure 5.
Numercal nvestgaton of free surface flood wave. 27 Fgure 2 Intal geometry of sudden collapse of free water column (dam-break) n contrast to a rgd obstacle (dam-break) wth ntal heght of 25 cm. (a) (b) Fgure 3 Comparson of (a) numercal results and (b) expermental observatons; the observed dgtzed freesurface poston s denoted by the black crcle n (a).
28 M. Zounemat-Kerman, H. Sheybanfard Fgure 4 calculated Pressure contours by the ISPH model n 0.3 s and 0.7 s for the dam-break flow n contrast to a rgd obstacle test case. Fgure 5 Comparson of numercal and expermental normalzed graph of the locaton of the wave front for the dam-break wave flow aganst a rgd obstacle test case. TABLE 1. Evaluaton of the ISPH numercal model va relatve error statstc for the dam-break flood wave aganst an solated obstacle test case. Tme(s) Relatve error of Relatve error of wave heght % wave poston % 0.1 7.9 11.9 0.3 5.73-2.18 0.5 13.01-1.59 0.7 10.18 2.5 Mean % 9.15 3.29 Results were analyzed consderng the poston of proceedng wave front and flow heght aganst tme. For evaluatng the numercal method, mean relatve error was used (Table 1). Table 1 shows that mean relatve error n the computaton of the poston of the wave front was 3.29%. Furthermore, mean relatve error n computng flow heght was less than 9.15%. Comparson of numercal and expermental results shows good accuracy n flow smulaton of dam-break test case wth a downstream obstacle. 6.2. Modellng gradually collapse of Water column (dam break) on a wet bed In ths test case, the expermental observatons were taken from [10]. The expermental setup s mentoned n Fgure 6. In the ntal tme, water column wth a depth of 0.15 m s blocked by a gate. The ntal downstream water depth s set as 0.018 m. The flood wave propagates as the gate opens completely from above wth a constant speed of 1.5 m/s. As for the numercal smulaton, n total, 5930 partcles wth an ntal dstance of 0.004 m were used.
Numercal nvestgaton of free surface flood wave. 29 Fgure 6 Intal geometry of gradually collapse of free water column (dam-break) wth ntal heght of 0.15 m and gate velocty of 1.5 m/s. Fgure 7 compares the numercally calculated water surface wth the dgtzed expermental water surface n four dfferent tme steps. As may be seen from Fgure 7, there s some dscrepancy between the numercal consequences and the expermental observaton n the ntal stage of the free-surface change. However, t should be noted that the mushroom-shaped water surface observed n the numercal results has also been reported by [14], [3] and [9]. In general, the calculated free watersurface features are n acceptable agreement wth the expermental observatons. Fgure 7 Comparson of (above) numercal results and (below) expermental observatons (reprnted from [20]); the observed dgtzed free-surface poston s denoted by the black crcle n (a). Fgure 8 shows the poston of partcles, free-surface shape, and pressure dstrbuton after a 0.281 and 0.531 s smulaton. Normalzed dagram of tme n comparson to poston proceedng of wave front of numercal model wth the dstance of 0.004 m between partcles and expermental model s gven n Fgure 9. Fgure 8 Pressure contours calculated by the ISPH model n 0.281 s and 0.531 s after the frst tme for gradually collapse of the water column test case. Summary of the numercal model evaluaton usng relatve error statstc crteron s shown n Table 2. Table 2 shows that mean relatve error n the computaton of the poston of the wave front was 1.73%. Furthermore, mean relatve error n computng flow heght was less than 8.3%. Comparng the numercal results and expermental observatons, confrms the ablty of the developed numercal model n flow smulaton of gradual flood wave over wet bed.
normalzed poston of wave front (x/l) 30 M. Zounemat-Kerman, H. Sheybanfard 3.00 2.50 2.00 1.50 1.00 0.50 0.00 expermental ISPH model 0 2 4 6 t(2g/l)^0.5 normalzed tme Fgure 9 Comparson of numercal and expermental normalzed dagram of the poston of the wave front for gradually collapse of water column case. TABLE 2 Evaluaton of the ISPH numercal model va relatve error statstc of the poston of the wave front for gradually collapse of water column test case. Tme (s) Relatve error of Relatve error of Wave heght % Wave poston % 0.156 7.4 2.12 0.281 8.2 1.6 0.406 7.8 1.23 0.531 10 2.0 Mean % 8.3 1.73 6.3. Modelng of soltary wave In ths part, the accuracy of ISPH flow model s assessed by analogzng the numercal and theoretcal wave heghts of a soltary wave propagatng on an solated bed. Consderng the followng relaton (whch s derved from the Boussnesq equaton), the ntal wave stuaton was arranged by dentfyng the partcles near the computatonal entrance regardng an analytcal soltary wave profle and velocty derved from the Boussnesq equaton [21]. 2 3a ( x, t) asec h [ ( x ct)] 3 (19) 4d where η= water surface elevaton, a=wave ampltude, d=water depth and c g( d a) s the soltary wave celerty. The horzontal velocty under the wave profle s calculated by (20) u g d (20) In ths study, a soltary wave wth the wave ampltude a=0.06 m s consdered. The water depth s 0.2 m and the ntal partcle spacng s 0.01 m. The frst stuaton was arranged by dentfyng the partcles near the computatonal entrance wth the wave profle of Eq. (19), zero vertcal velocty and horzontal velocty of Eq. (20). As for the numercal computaton, the 5200 partcles were placed on a regular grd wth square cells, n a way that those that were above the profle were elmnated. Ths condton leads to a slghtly rough ntal profle such that the flud rapd adusts to the soltary wave profle. Fgure 10 ndcates the comparsons of the modeled wave profles and the analytcal answers at varous tmes.
Numercal nvestgaton of free surface flood wave. 31 Fgure 10 Comparsons of modeled (the blue ponts) and analytcal (the black ponts) soltary wave profles at 0.1, 0.4, 0.7, 1 and 1.2 s. Results of smulatons are ndcated acceptable agreement between the numercal and the analytcal calculatons. However, accordng to the results at t=1 s (where the wave deals the rght wall) a reducton n the accuracy of the numercal model can be observed. Ths problem s due to an error whch s usually occurs n the numercal soluton of Posson's equaton. Fgure 11 Calculated pressure contours by the ISPH numercal model n 0.4 s and 1.2 s. In Fgure 11, the pressure contours at the tme of 0.4 s and 1.2 s are shown. As may be observed the pressure contours are almost regular. At the tme of 0.4 s, the pressure at the peak of wave approxmately equals to the hydrostatc pressure measured at ths pont (=2900 pa). Fgures 12 and 13 respectvely ndcate the normalzed dagram of tme n comparson to poston of soltary wave and
32 M. Zounemat-Kerman, H. Sheybanfard the normalzed dagram of tme n comparson to soltary wave heght. Fgure 12 Normalzed dagram of tme compared to condton of the soltary wave front and ts comparson wth the analytcal results for the soltary wave test case. Fgure 13 Normalzed dagram of tme compared to heght of the soltary wave and ts comparson wth the analytcal results. The comparson for the wave heght between the calculated results and observaton n terms of relatve error are presented n Table 3. TABLE 3 Evaluaton of the ISPH numercal model va relatve error statstc of the poston of the wave front for the soltary wave test case. Tme(s) Relatve error of Relatve error of wave wave heght % front poston % 1 1.6-1.05 0.4 3.3-2.04 0.7-5 -0.69 1-11 -2.45 1.2-6.6-1.9 Mean % 5.5 1.62 7. Concluson The purpose of present study was to provde nsght nto mesh-free ISPH smulaton of free surface hydraulcs problems and challenge the capabltes of ths method n the feld of computatonal flud dynamcs. The ISPH model was appled for flow smulaton of three test cases of dam-break flood wave, gradual collapse of water column and soltary wave. Governng equatons on wave flows were solved usng ISPH dscrete method wth two-step fractonal algorthm. The ntal dstance of 0.006 m for the dam-break test, 0.004 m for gradual water collapse and 0.01 m for soltary wave case were consdered
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