Numerical Solution of Overland Flow Model Using Finite Volume Method

Transcription

1 Sudies in Mahemaical Sciences ol. 4, No. 2, 212, pp DOI:1.396/j.sms ISSN [rin] ISSN [Online] Numerical Soluion of Overland Flow Model Using Finie olume Mehod Ummu Habibah [a],* [a] Mahemaics Deparmen, Universiy of Brawijaya, Malang, Indonesia. * Corresponding auhor. ummu_habibah@ub.ac.id Received 19 February, 212; acceped May, 212 Absrac Overland flow is one of Compuaional Fluid Dynamics (CFD) problem. In his paper we invesigae he waer level of overland flow ha is ofen occured afer rainfall on he land surface. Finie volume mehod is used o solve his problem. Quadraic upsream inerpolaion for convecive kineics (QUICK) sheme is used o have he discreiaion of he overland flow model because his sheme have been proved is numerical sabiliy. Numerical simulaion of he soluions is presened o describe he behaviour of his model. Key words Overland flow; Finie volume mehod; QUICK Ummu Habibah (212). Numerical Soluion of Overland Flow Model Using Finie olume Mehod. Sudies in Mahemaical Sciences, 4(2), Available from URL: hp:// DOI: hp://dx.doi.org/1.396/j.sms INTRODUCTION Rainfall is an aspec of he hydrologic cycle ha is imporan in he role of supplying waer in he world. Bu heavy rainfall wih long duraion can cause overland flow ha i poenially occur flood. Overland flow is waer on he he land surface ha flow afer rainfall. Overland flow ake place if he precipiaion level over he infilraion level o absorb waer. In order o know overland flow level, mahemaical model and is numerical soluion are needed o predic accuraely. Many numerical mehods were developed o solve he overland flow model. Mac Cormack and predicor corecor mehods was he mehod ha was used o have he numerical soluion of overland flow (Alhan e al., 25). Second-order Lax Wendroff and he hree-poin cenred finie difference schemes were used o ge he numerical soluion of overland flow (Goardi & enuelli, 2). Finie elemen mehod was used o have he numerical soluion of overland flow model (Jaber & Mohar, 23). Cubic-spline inerpolaion echnique (CSMOC scheme) was used o have he numerical soluion of overland flow model (Tsai & Yang, 25). In his paper, finie volume mehod is used o solve overland flow wih QUICK sheme because his mehod suiable for Compuaional Fluid Dynamics (CFD) problem. Furhermore, we simulae several condiion o show model performance. 32

2 Ummu Habibah/Sudies in Mahemaical Sciences, ol.4 No.2, NUMERICAL SOLUTION USING FINITE OLUME METHOD The physical model of overland flow can be seen in his figure. Figure 1 Overland Flow (Alhan e al., 25) Where: y = deph of waer q = he flow per uni widh Mahemaical model of overland flow is governed from physical laws include coninuiy and momenum equaions. This equaions is called governing equaion. This is based on Reynolds Transpor Theorem (Chow, dkk., 19). 2.1 Coninuiy Equaions Reynold Transpor Theorem is used o ge overland flow model is (Apsley, 27): d ( ) ( C A) Source (1) d adveksi n faces difusi Where: = volume of fluid r = mass of fluid f = consenraion C = convekiviy Γ = diffusiviy A = wide of surface Scalar ranspor of mass conservaion overland flow is d ( mass ) ne ouward mass flux d (2) A q ( i f ) x (3) 33

3 Ummu Habibah/Sudies in Mahemaical Sciences, ol.4 No.2, 212 Equaion (3) is coninuiy overland flow in conservaion form. Coninuiy equaions in non-conservaion form is y y y ( i f ) x x (4) Where is waer velociy, i is rainfall inensiy, x is disance, is ime and f is infilraion rae. 2.2 Momenum Equaions In a similar manner, momenum overland flow is derivaed from Reynold Transpor Theorem. 2 q ( ) q A y ) ga ga( S S f ) x x (5) Equaion (5) is momenum overland flow in conservaive form, and momenum overland flow in nonconservaion form is y g g( S S f ) ( i f ) x x y (6) Where g is acceleraion of graviy, is ime, S is fricion slope and f S is bed slope. 3. FINITE OLUME METHOD USING QUADRATIC USTREAM INTEROLATION FOR CONECTIE KINETICS (QUICK) SCHEME Coninuiy dan momenum equaions are solved simulanously. Numerical soluion of overland flow model using finie volume mehod is solved by inegraing he differenial equaion ha we have. The firs sep, we have o solve he governing equaions. If q = A hen q A =, so he equaion (3) become q q ( i f ) x Flux Source (7) Coninuiy equaion in (7) can be solved using QUICK scheme ha be ilusraed in figure (2) WW W w e E EE 1 2 x x Figure 2 Conrol Face of Conrol olume 34

4 Ummu Habibah/Sudies in Mahemaical Sciences, ol.4 No.2, 212 The firs sep inegrae equaion (7) over he conrol volume and ime inerval from o +, we have q q () dd d d ( i f ) d d x C C C From equaion () we have 1 ( ) ( ) e ( ) w ( ) q q qa qa d i f (9) In equaion (9), A is face area of he conrol volume, is is volume which equal o A x where x is he widh of he conrol volume. 1 1 Using QUICK sheme, qe = q + (3 qe 2 q qw ) and qw = qw + (3 q 2 qw qww ), equaion (9) may be wrien as (1) q q A q q q q A q q q q d i f ( ) ( ( (3 E 2 W ))) ( ( W (3 2 W WW ))) ( ) To evaluae he lef hand side of equaion (1) we make an assumpion he variaion of q, qe, qw, and q WW wih ime. We inegraed he flow per uni widh a ime or a ime + o calculae he ime inegral or combinaion of he flow per uni widh a ime or a ime +. We used weighed parameer q beween and 1 o approach he inegral of he flow per uni widh respec o ime as qd [ q (1 ) q ] (11) Using (11), equaion (1) we wrie as 1 1 ( q q ) A{ q (1 ) q [3 qe (1 ) qe 2 q (1 ) q 1 W (1 ) W ] W (1 ) W [3 (1 ) 2 W (1 ) W q q i f WW (1 ) WW ] ( ) (12) Equaion (12) dividing by A hroughouh, we have x ( q q ) ( q q E q q W q W q q W q WW ) q q q q q q q q i f x (1 )( E W W W WW ( ) (13) We can wrie equaion (13) as a ( q q ) ( q qe q qw qw q qw qww ) q q q q q q q q b (1 )( E W W W WW ) Where: (14) x a =, b = ( i f) x 35

5 Ummu Habibah/Sudies in Mahemaical Sciences, ol.4 No.2, 212 We can wrie equaion (14) as a (q q ) ( q qe qw qww ) (1 )( q qe qw qww ) b (15) Afer we have numerical soluion of coninuiy equaion, in a similar manner we do he discreion of momenum equaion in conservaif form. From equaion (5) by replacing A = q, we have q (q ) y ga ga( S S f ) y ga is source from momenum equaion, i be moved o righ hand side, hen we have x q (q ) y ga( S S f ) Flux (16) (17) Source define S = ga( S S f + y ), we have x q (q ) S x (1) Flux The equaion (1) is inegraed o and o he conrol volume, we have q d d C (q ) d d C Sd d (19) (Aq)e (Aq) w d S (2) C Equaion (19) is inegraed (q q ) Using QUICK sheme, equaion (2) can be wrie 1 (q q ) A { q (1 )q [3 qe (1 )qe 2 q (1 )q qw (1 )qw ] qw (1 )qw 1 [3 q (1 )q 2 qw (1 )qw qww (1 )qww ] S (21) Dividing by A, we have x 1 (q q ) { q (1 )q [3 qe (1 )qe 2 q (1 )q qw (1 )qw ] qw (1 )qw 1 [3 q (1 )q 2 qw (1 )qw qww (1 )qww ] S x We can wrie equaion (22) as 36 (22)

6 Ummu Habibah/Sudies in Mahemaical Sciences, ol.4 No.2, a ( q q ) ( q qe q qw qw q qw qww ) Sx q q q q q q q q (1 )( E W W W WW ) Or we can wrie as (23) a ( q q ) ( q qe qw qww ) Sx (1 )( q qe qw qww ) (24) Subsiue equaion (15) o (25), we have Sx b (25) We evaluae q = 1. This sheme is called fully implici. From equaion (24), we have a ( q q ) ( q qe qw qww ) b (26) Or we can wrie as ( a ) q qe qw qww b aq (27) Equaion (27) is numerical soluion of overland flow. To ge numerical soluion, domain is devided ino 5 nodes ha i describe number of node in conrol volume. The number of variabel equal o he equaions. The equaion change o marix equaion Mq = H, where M is coefisien of q, q is he flow per uni widh ha we wan o find, and H is value in righ hand side equaion (27). The marix form is a 3 3 q1 b aq 7 a 3 3 q2 b aq a q 3 b aq q a 4 b aq q 5 b aq 1 7 a 3 (2) 4. SIMULATION OF OERLAND FLOW MODEL Simulaion of overland flow model using synheic case can be seen in he example o demonsrae he heory ha is presened in he previous secion. Synheics Example Rainfall coninues wih he inensiy 3.2 cm/h over a 6 f. The slope of he land is.16. We wan o evaluae he flow per uni widh of overland flow in 5, 2, 3 and 9 minues. If we used x is 6 f and ime sep is 1 minues, The numerical soluion can be seen ino figure 3. 37

7 Ummu Habibah/Sudies in Mahemaical Sciences, ol.4 No.2, 212 Figure 3 Flow er Uni Widh of Overland Flow From he figure 3 we can see he flow per uni widh for each ime is increased, and a he end of he area we can see ha he flow per uni widh is in grea quaniies. I means ha waer flow o he lower land, and i can cause much waer accumulaion a he lower land. 5. CONCLUSION In his paper, finie volume mehod can be applied o ge he numerical soluion of overland flow model because his mehod suiable for CFD problem. Quadraic Upsream Inerpolaion for Convecive Kineics (QUICK) sheme is used o have discreiaion of overland flow model ha have been proved is sabliliy. And also, finie volume mehod is good mehod o solve CFD problem, specially for fluid problem because his model show he behavior of overland flow in he realiy problem. ACKNOWLEDGEMENT The auhors would like o say hank o Agus Suryano for discussion o provide valuable commens on he manuscrip. REFERENCES [[[ Alhan, C. M. K., Medina, M. A. dan Rao,. (25). On Numerical Modeling of Overland Flow. Applied Mahemaics and Compuaion, 166, [[[ Goardi, G. and enuelli, M. (2). An Accurae Time Inegraion Mehod for Simplified Overland Flow Models. Advances in Waer Resources, 31(1), [[[ Jaber, F., H. And Mohar, R., H. (23). Sabiliy and Accuracy of Two-Dimensional Kinemaic Wave Overland Flow Modeling. Advances in Waer Resources, 26(11), [[[ Tsai, T., L. And Yang, J., C. (25). Kinemaic Wave Modeling of Overland Flow Using Characerisics Mehod wih Cubic-Spline Inerpolaion. Advances in Waer Resources, 2(7), [[[ Apsley, D.D. (27). Quaniaive roperies of F.D. Schemes. Lecure Handou: CFD, Universiy of Mancheser, Mancheser. [[[ Chow,.T., Maidmen, D.R., Mays, L.W. (19). Applied Hydrology. New York: McGraw-Hill. [[[ Ferziger, J.H. dan eric, M. (22). Compuaional Mehods for Fluid Dynamics (Third, Rev. Ediion). New York: Springer. [[[ Henderson, F.M. (1966). Open Channel Flow. New York: The MacMillan Company. 3

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