A finite element algorithm for Exner s equation for numerical simulations of 2D morphological change in open-channels

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1 River, Coasal and Esuarine Morphodynamics: RCEM Tsinghua Universiy Press, Beijing A finie elemen algorihm for Exner s equaion for numerical simulaions of D morphological change in open-channels T.B. KIM Researcher, Environmenal Hydrodynamics Laboraory Deparmen of Civil and Environmenal Engineering, Yonsei Universiy 134 Shinchon-dong, Seodaemun-gu, Seoul, , Korea Y. CHOI MS Suden, Environmenal Hydrodynamics Laboraory Deparmen of Civil and Environmenal Engineering, Yonsei Universiy 134 Shinchon-dong, Seodaemun-gu, Seoul, , Korea S.-U. CHOI Professor, Environmenal Hydrodynamics Laboraory Deparmen of Civil and Environmenal Engineering, Yonsei Universiy 134 Shinchon-dong, Seodaemun-gu, Seoul, , Korea ABSTRACT: Recenly D numerical models have been proposed o simulae numerically he morphological change in open-channels. In general, he D numerical model for such purpose is comprised of hree pars, namely flow, sedimen ranspor, and morphology pars. In he presen sudy, for he flow analysis, he shallow waer equaions are solved using D characerisic dissipaive-galerkin mehod. In order o updae he morphological change, a similar finie elemen algorihm is proposed for he soluion of Exner s equaion. The proposed algorihm esimaes he morphological change based on sedimen loads a Gauss poins wihin he elemen. On he oher hand, he convenional mehod uses he value a a node, which may resul in non-unique values due o he disconinuiy of heir derivaives. The model is applied o wo problems: bed aggradaion due o excessive sedimen supply a he upsream and propagaion of a hump on he bed wihou sedimen supply a he upsream. Appropriae weighing of finie elemen scheme for he numerical soluion of Exner s equaion is also invesigaed. The proposed model is a decoupled model in a sense ha he bed elevaion does no change simulaneously wih he flow during each compuaional ime sep, and i is resriced o he case wih uniform sedimen, neglecing armoring or grain soring effecs. 1 INTRODUCTION Mos D numerical models for he simulaion of he bed elevaion change used he finie difference mehod. The finie volume mehod began o be used in 000s because of is excellen mass conservaion propery. I is well known ha he finie elemen mehod provides more flexibiliy in handling spaial domain han FDM or FVM. Neverheless, he finie elemen model has no been favored in he numerical simulaions of morphological change compared wih he finie difference mehods or finie volume mehods. Recenly, Vasquez e al. (008) presened he finie elemen model using riangular mesh for bed elevaion change in meandering rivers. However, he deailed finie elemen algorihm for bed elevaion change has no been proposed.

2 In his sudy, a finie elemen model for he flow and bed elevaion change is proposed. The shallow waer equaions and he Exner s equaion are solved by he finie elemen mehod. The shallow waer equaions are solved by D Characerisic Dissipaive-Galerkin (CDG) scheme, which belong o he family of Sreamline-Upwind / Perov-Galerkin (SU/PG) schemea. A new finie elemen algorihm for he Exner's equaion is also inroduced, and he new algorihm esimaes he equilibrium sedimen load no a a node bu wihin an elemen. In addiion, numerical experimens are carried ou o find appropriae weigh of Exner s equaion. For validaion, he developed model is applied o sraigh channel daa for bed aggradaion due o sedimen overloading (Soni e al., 1980) and he propagaion of a hump wihou sedimen supply a he upsream. The numerical model developed in he presen sudy is based upon he decoupled modeling approach assuming ha he ineracion beween flow and bed is ignorable during he compuaional ime sep. Also, he model is resriced o beds of uniform sedimen wihou armoring or grain soring effecs. NUMERICAL METHODS.1 Flow equaions For he flow analysis, he following D shallow waer equaions wih he effecive sress erms are adoped: U U U D x D y A B F 0 (1) x y x y where T U h p q () p p A gh 0 h h pq q p h h h (3) pq q p B h h h q q gh 0 h h (4) D x 0 p x p q y x (5)

3 D y 0 p q y x q y 0 z b gn F gh p p q x 7/3 h z b gn gh q p q y 7/3 h (6) (7) where h is flow deph, p and q are discharge per uni widh in x- and y-direcions, respecively, g is graviaional acceleraion, z b is bed elevaion measured from a cerain daum, n is Manning s roughness coefficien, and is urbulen viscosiy. Herein, he following parabolic eddy viscosiy model is used: Uh (8) 6 where U is he shear velociy, and is von Kármán consan (= 0.4). To solve he shallow waer equaions numerically, he finie elemen mehod is used. The weighed residual equaion of he shallow waer equaions akes he form such as U U U D D x y N A B F d 0 x y x y (9) where N denoes he weighing funcion. The various finie elemen schemes have been developed wih each unique forma. In his sudy, he Perov-Galerkin scheme is employed as following: Ni Ni Ni Ni x Wx y W y (10) x y where N i is basis or shape funcion for he i-h node, N i is weighing funcion for he i-h node, is weighing coefficien, W x and W y are weighing marices in he x- and y-direcions, respecively, and x and y are characerisic elemen lenghs in he x- and y- direcions, which are esimaed by means of Kaopodes (1984). In his sudy, following weighing marices suggesed by Ghanem (1995) are used. W x A A B, W y A B B (11) where A and B are advecion marix defined by Eqs. (3) and (4), respecively. For each elemen, Eq. (9) resuls nonlinear equaions, which are solved by using he Newon-Raphson mehod and unsymmerical fronal algorihm proposed by Hood (1976). 3

4 . Exner s equaion In order o updae he bed elevaion a each ime sep, he following Exner s equaion is solved: zb q q x 1 ' y p 0 x y (1) where p' is porosiy, and q x and q y are he x- and y-componens of oal sedimen load per uni widh, which are, respecively, expressed as q x q cos ; q q cos (13) y where is he direcion of sedimen ranspor and q is he oal sedimen load per uni widh. In he presen sudy, he following formula is used for he oal sedimen load: q b av (14) which was given in Soni e al. (1980). In Eq. (14), V is deph-averaged flow velociy, a and b are consans of values and 5.0, respecively, obained in Soni e al. s (1980) experimen. The weighed residual equaion of he Exner s equaion is given by zb 1 q q x y N d 0 1 p ' x y (15) By using he Green s Theorem in Eq. (15), he following equaion in he marix form can be obained as proposed by Kim and Choi (008): AΔzb D F (16) 1 p' ij e i j e A N N d (17) D q q d x y Ni Ni e ij e x y (18) e i e i x x y y (19) F N n q n q d where Γ e means he boundary of an elemen. The convenional mehod esimaes spaial variaion of mean velociy and bed opography a a node, which may resul in non-unique values due o he disconinuiy of heir derivaives. However, he proposed algorihm esimaes he change of such variables a Gauss poins wihin he elemen. In Eq. (15), N is he weighing funcion as in Eq. (9). Unlike he finie scheme for he flow equaion, a paricular scheme for he Exner s equaion has no been proposed nor discussed. Therefore, 4

5 Bubnov-Galerkin scheme in which he weighing funcion is he same as he basis funcion has been jus used. In his sudy, weighing funcion for he Exner s equaion is proposed and applied. A weighing funcion similar o he one used in PG scheme for he flow equaion is used. q N q N N N x y (0) x i y i i i q x q y If he weighing coefficien, ω, is zero, he weighing funcion by Eq. (0) resuls in BG scheme, if posiive, upwind weighing scheme, and if negaive, downwind weighing scheme, respecively. 3 APPLICATIONS 3.1 Bed aggradaion due o overloaded sedimen inpu Firs, he proposed model is applied o Soni e al. s (1980) experimen, in which hey carried ou bed aggradaion es due o overloaded sedimen supply. The experimen was conduced in a 0. m wide, 0.5 m deep, and 30 m long sraigh iling flume. Bed maerials and supplied sedimen paricles were uniform sands wih a median diameer of 0.3 mm. Afer he se up of a uniform flow condiion for he given discharge and slope, he sedimen supply rae was increased o a predeermined value by coninuously feeding excess sedimen a he upsream end of he flume. The bed and waer surface profiles were recorded a ime inervals varying from 10-0 minues. The measured profiles were averaged because of he presence of ripples and dunes on he bed. One of he cases in Soni e al. s (1980) experimens is seleced for he numerical simulaion. Iniial bed slope, waer discharge per widh, and waer deph were , 0.0 m /s, and 0.05 m, respecively. The amoun of sedimen supply a he upsream end of he flume is four imes larger han he equilibrium sedimen load rae. In his applicaion, boh BG and downwind schemes are used for he Exner s equaion. Figure 1 shows he resuls of he compued bed and waer surface elevaion profiles a various imes. Due o he overloaded sedimen supply a he upsream, bed elevaion increases and he range of bed elevaion change is exended oward downsream in ime. Simulaed resuls agree well wih experimenal daa by Soni e al. (1980), alhough waer surface profiles are slighly higher han measured daa. 3. Propagaion of hump on he bed In he presen secion, he propagaion of he hump on he bed is numerically simulaed. The flow condiion in Soni e al. s (1980) experimen is imposed wih a riangular hump as shown in Figure. I is expeced ha he hump on he bed propagaes in he downsream direcion wih diffusion and equilibrium sae reaches. Figure 3 shows bed elevaion profiles simulaed by BG scheme. I appears ha he hump on he bed spreads wih ime, reaching equilibrium aferwards. However, i is observed ha he disurbance of he bed elevaion propagaes no only in he downsream bu also upsream direcion. The propagaion owards upsream direcion seems o be caused by numerical oscillaions. Finally, he downwind or forward weighing scheme wih negaive weighing coefficien in Eq. (0) is applied. Figure 4 shows simulaed bed elevaion profiles a various imes. I can be seen ha he hump propagaes mainly in he downsream direcion and numerical oscillaions are noiceably and rapidly diminished wih ime. This is comparable wih he resul of BG scheme in Figure 3. 5

6 (a) a 15 min. (b) a 30 min. (c) a 40 min. Figure 1 Simulaed bed and waer surface profiles Figure Iniial bed profile wih a riangular hump 6

7 (a) a min. (b) a 5 min. (c) a 10 min. (d) a 0 min. Figure 3 Bed elevaion change simulaed by BG scheme (a) a min. (b) a 5 min. (c) a 10 min. (d) a 0 min. Figure 4 Bed elevaion change simulaed by downwind weighing scheme 4 CONCLUSIONS This sudy presens a finie elemen model for numerical simulaions of D morphological change of open-channels. The model compues he flow and morphological change by solving he shallow waer equaions and Exner s equaion, respecively. The D characerisic dissipaive-galerkin mehod is applied o solve he flow equaions, and a new algorihm is presened for he soluion of Exner s equaion. The new algorihm esimaes a unique value of he sedimen load wihin he elemen, which was no possible wih convenional mehods. The model was applied o a bed aggradaion problem due o sedimen over-load a he upsream, Soni e al. s experimen. I was found ha he model reproduce well bed elevaion change due o over-loaded sedimen supply a he upsream. Then, he model was applied o he propagaion of a hump on he bed. Boh BG and downwind 7

8 schemes simulaes well he hump propagaion. However, numerical oscillaions appear on he bed elevaion profile and propagae in he upsream direcion when he upwind scheme was used. 5 ACKNOWLEDGEMENTS This sudy was suppored by he 006 Core Consrucion Technology Developmen Projec (06KSHS-B01) hrough ECORIVER1 Research Cener in KICTEP of MLTM KOREA. REFERENCES Ghanem, A.H.M. 1995, Two-dimensional finie elemen modeling of flow in aquaic habias. Ph.D. Thesis, Universiy of Albera, Albera. Hood, P. 1976, Fronal soluion program for unsymmeric marices. Inernaional Journal of Numerical Mehods in Engineering, Vol. 10, pp Kaopodes, N.D. 1984, Two-dimensional surges and shocks in open channels. Journal of Hydraulic Engineering, ASCE, Vol. 110, No. 6, pp Kim, T.B. and Choi, S.-U. 008, Algorihm for D finie elemen modeling of bed elevaion change in a naural river. In Preceedings of he 8 h Inernaional Conference on Hydro-Science and Engineering, Nagoya Universiy, Nagoya, Japan, Sepember 9-1, 008. Soni, J.P., Garde, R.J., and Ranga Raju, K.G. 1980, Aggradaion in sreams due o overloading. Journal of he Hydraulics Division, ASCE, Vol. 106, No. HY1, pp Vasquez, J.A., Seffler, P.M., and Millar, R.G. 008, Modeling bed changes in meandering rivers using riangular Finie Elemens. Journal of Hydraulic Engineering, ASCE, Vol. 134, No. 9, pp