1-Dimensional Advection-Diffusion Finite Difference Model Due to a Flow under Propagating Solitary Wave

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1 014 4th Internatonal Conference on Future nvronment and nergy IPCB vol.61 (014) (014) IACSIT Press, Sngapore I: /IPCB V mensonal Advecton-ffuson Fnte fference Model ue to a Flow under Propagatng Soltary Wave Muhammad A. Fauz 1 and Fauz A. Zaky 1 epartment of ceanography, Faculty of arth Scence and Technology, Bandung Insttute of Technology, Indonesa epartment of ceagneerng, Faculty of Cvl and nvronmental ngneerng, Bandung Insttute of Technology, Indonesa Abstract. Advecton dffuson phenomena has been commonly observed n coastal areas. ur am s to nvestgate the soltary wave effect to advecton dffuson of a substance n a near shore shallow water wth an open channel. Consderng that the model s an open channel, one dmensonal approach s appled. The model soluton s obtaned by combnng Korteweg e Vres (KdV) equaton wth advecton dffuson equaton. The KdV equaton s numercally solved usng sem-mplct Crank-Ncolson scheme whle the advecton dffuson equaton s numercally solved by usng an upwnd Forward Tme Central Space (FTCS) scheme. The results show that there s a rapd dsperson of the substance when the soltary wave comes nto contact wth the source Keywords: advecton-dffuson, soltary wave, fnte dfference method. 1. Introducton Transport and dsperson phenomena whch are descrbed by advecton-dffuson equaton have been common problems and observed n a wde range of ndustral and engneerng applcatons [1]. Somehow, t s mportant to nvestgate the advecton-dffuson process of a contamnant especally near shore. Many contamnants from ndustral actvty end up at the coastal zone. Ths advecton-dffuson phenomena also plays an mportant role n keepng the balance of the coastal envronment such as n heat exchange and salnty transport phenomena. P N Flow recto Land and Coastal Area P N C A Sourc Land and Coastal Area C A Fg. 1: Sketch of the open channel In ths paper, our am s to nvestgate advecton-dffuson process of a substance near shore due to a flow under a propagatng soltary wave. We consder a smple case of advecton-dffuson phenomena n 1 dmensonal doman such as n an open channel where the soltary wave enters from the open ocean. 5

2 The model s obtaned by solvng 1 dmensonal Korteweg e-vres (KdV) and 1 dmensonal advecton-dffuson equaton wth the help of fnte dfference method. In ths study, the soltary wave s used because t s been proven by Munk [] that a soltary wave theory gves a better approxmaton for near shore wave than lnear wave theory. ne dmensonal KdV equaton for an even bottom read as [] C 0 (1) C 1 Ch where C gh, and wth h s the water depth, g s the gravtatonal acceleraton h 6 and s the surface elevaton. ne dmensonal advecton-dffuson equaton can be expressed as F F F u A () where F descrbes the concentraton of a substance, u u, t represents velocty of a flud flow as a functon of space and tme and A s the dffusvty constant. quaton (1) and () wll be solved by usng fnte dfference to produce 1 dmensonal advectondffuson numercal model.. Numercal Method.1. Soltary Wave model From KdV quaton The KdV equaton (1) wll be solved by usng Crank-Ncolson scheme. By followng [4], the fnal form of equaton (1) after the dscretzaton s read as b b1 a0 a1 a d () where the left sde coeffcents are n 6 a0 C n 4 a1 1 C 1 n 4 b1 1 C 1 a b and the rght sde coeffcent s descrbed as C n n 4 6 d 4 quaton () s then solved by Gaussan lmnaton method consderng that the model doman s small enough so t doesn t need much computatonal tme. Intal condton for (1) s a,0 asec h k x0, where k 4h wth x 0 s the ntal peak poston and a s the wave ampltude where the wave propagates from left to rght. For the rght boundary condton, an extrapolaton boundary s appled to make the wave travels wthout reflecton at the boundary... Soltary Wave Test For the soltary wave test, consder a wave flume wth 90 m length and wth a constant depth 1 m. We choose wave ampltude 0. m, m and s wth wave peak poston ( x 0 ) at m from left boundary wth smulaton tme 5 s. The comparson betweeumercal and analytcal s provded n Fg. (a) where both of them are almost perfectly concded along ts propagaton. 6

3 (a) (b) Fg. : Comparson betweeumercal model (sold lne) and analytcal soluton (dashed lne) for varous t (a) and relatve ampltude as the wave travels (b) It can be seen from Fg. (b) that there s almost no ncrease or decrease n the ampltude as the wave propagates. Wth a 0. 0 m where a s the numercal ampltude whch s fluctuatng along the propagaton. The sgnfcant error occurs only at t = 5 s, see Fg. (a) when the wave reaches the rght boundary. But overall, the numercal model s n good agreement wth the analytcal soluton and the rght sde extrapolaton boundary gves a good result wthout any reflecton... Coupled KdV and Advecton-ffuson Model An upwnd dscretzaton for the advecton term accordng to [5] and FTCS dscretzaton for the dffusve term n equaton () s where u u n F F 0.5 F F F F F n F F 0.5 and (4) A wth u s obtaned from the soltary wave model by usng the lnear wave theory relaton whch s wrtten as u, t, t g / h. Fg. : Plot of concentraton aganst tme at pont x = 9.85 for explct upwnd, mplct upwnd, FTCS, and CTCS method We also compare the method above wth another method such as mplct upwnd, FTCS, and CTCS for a pure advecton process wth unform velocty 0.1 m/s whch starts at t = 0 (Fg. ). The result shows that the mplct and explct upwnd gves a better approxmaton than the FTCS and CTCS method. It s seen that both FTCS and CTCS methods resultng a negatve concentraton whch s mpossble to happen. A tme 7

4 lag s also exsts n FTCS and CTCS scheme at early smulaton tme where the concentraton s stll mg/l although the flow s already exst (the flow s already exst at t = 0 as mentoned before). Although the mplct upwnd method s really stable [5], we choose the explct upwnd method rather than mplct upwnd method because the explct method produces a shorter computatonal tme. As long as the model parameter satsfy the stablty crteron, the result wll be almost perfectly concded wth the mplct method (see Fg. ). The coupled model uses the same wave set up as the frst soltary wave test wth same space and tme resoluton ( x, t ) but wth length of channel 75 m and smulaton tme 1 s. There are two scenaros smulated by ths coupled model. The frst scenaro consders only the advecton process and the second scenaro consders both of advecton and dffuson process wth A 10 m / s (we take a small dffuson constant to keep the advecton process vsble). Both scenaro use an ntal condton F 0 mg/l at x wth nsulated boundary condtons at both sdes. We assume that there s no dsperson to the open ocean at the boundary.. Results For the tme seres plot, we show only the source pont tme seres at x = 9.85 m. From the frst scenaro, t can be seen from Fg. 4 (a) that a rapd decrease happens at a tme nterval between 5 s 8 s. It s confrmed by the tme seres plot (see Fg. 4 (b)) that the rapd decrease begns at around t = 5 s to t = 8 s when the flow reaches the source pont. Ths rapd decrease at the source pont s also ndcatng that a rapd dsperson occurs (because the concentraton s conserved). It can also be seen that before the flow reaches the source pont (before t = 5 s), there s no decrease n concentraton at ths pont. After t = 8 s, the concentraton doesn t decrease anymore although the flow stll exsts at the source pont (Fg. 4 b). It happens because the concentraton has already dspersed rapdly to the rght sde at nterval 5 s to 8 s. It makes the concentraton at the source pont s almost 0 mg/l at t = 8 s. It can be seen from Fg. 4 (a) that the peak poston has moved nto a pont between 0 m 0.5 m wth the concentraton at the source poston s almost 0 mg/l at t = 8 s (black sold lne). Fg. 4: Plot of concentraton (a) aganst length of channel for varous t (a) and plot of concentraton (b) (upper) and velocty (lower) aganst tme at pont x = 9.85 (b) for the 1 st scenaro For the second scenaro, t can be seen from Fg. 5 (a) that before t = 7 s the dffuson s more domnant than the advecton. Although f we see carefully at t = 7 s, a small advecton process has already begun. It s ndcated by the wdenng of the sold red lne curve to the rght at ts lower regon. It s also confrmed by Fg. 5 (b) where the flow begns to reach the source pont at around t = 6 s. It means the advecton process s startng to occur at t > 6 s. But at those nterval, the flow velocty s stll small enough so the dffuson phenomena s stll domnatng. After t > 6.5 s, the advecton process domnates the dsperson whch almost has the same trend as the frst scenaro, see Fg. 5 (b) and Fg. 4 (b). And we can also see that after t = 8 s the concentraton decrement s approachng zero although the flow stll exsts. It s caused by the smlar reason as the frst scenaro that at t > 8 s, the concentraton at x = 9.85 m s almost 0 mg/l whch s ndcated by sold black lne n Fg. 5 (a). 8

5 We have to nvolve the water partcle movement to explan ths rapd dsperson. Unlke the lnear shallow water theory, the water partcle movement under a propagatng soltary wave doesn t make a horzontal oscllatng translatonal movement, see [6]. It makes a moton that followng the drecton of the wave propagaton. So t makes the contamnated water partcles always move followng the drecton of the wave propagaton and dssemnate the partcle of the substance to the rght contnuously. 4. Conclusons ne mensonal advecton-dffuson model s used to smulate how fast a substance s dspersed nto ts envronment n an open channel caused by a flow under soltary wave. Wth a small dffuson coeffcent, t can be seen from the model that the advecton holds a bg role n dspersng the substance rapdly. The water partcle movements under a propagatng soltary wave s a key n explanng the rapd dsperson of a substance. 5. Acknowledgements We would lke to thank to L. H. Wryanto for a very valuable dscussons about KdV model and also to Mutara R. Putr for gvng oceanographc modellng lectures. 6. References (a) Fg. 5: Plot of concentraton aganst length of channel for varous t (a) and plot of concentraton (upper) and velocty (lower) aganst tme at pont x = 9.85 (b) for the nd scenaro [1] Sngh, KM, Tanakan M. n exponental varable transformaton based boundary element formulaton for advecton-dffuson problems, ng. Anal Bound lem, 000; 4:5-5 [] Munk, W. H. A Soltary Wave Theory and Its Applcaton for Surf Problems. Annals of the New York Academy of Scences, 1949, 51, cean Surface Waves. [] ngemans, M.W. Water Wave Propagaton over Uneven Bottoms. World Scentfc, Publshng Co. Pte. Ltd. [4] Wryanto, L.H., Warsoma johan. Metoda Beda Hngga pada Persamaan KdV Gelombang Interface, Jurnal Matematka, Vol. 9, [5] Kowalk, Z., T.S. Murty. Numercal Modelng of cean ynamcs, World Scentfc Publshng Co. Pte. Ltd [6] Borluk, H, Henrk Kalsch. Partcle dynamcs n the KdV approxmaton, Wave Moton 49 (01) (b) 9