w wz 8«4y 2008 8 pp. 91 ~ 100 w w w e x Development of a Numerical Model of Shallow-Water Flow using Cut-cell System ½x Á Á Kim, Hyung-JunÁee, SeungOhÁCho, Yong-Sik Abstract Numerical implementation with a Cartesian cut-cell method is conducted in this study. A Cartesian cut-cell method is an easy and efficient mesh generation methodology for complex geometries. In this method, a background Cartesian grid is employed for most of computational domain and a cut-cell grid is applied for the peculiar grids where the flow characteristics are changed such as solid boundary to enhance the accuracy, applicability and efficiency. Accurate representation of complex geometries can be obtained by using the cut-cell method. The cut-cell grids are constructed with irregular meshes which have various shape and size. Therefore, the finite volume method is applied to numerical discretization on a irregular domain. The HC approximate Riemann solver, a Godunov-type finite volume method, is employed to discretize the advection terms in the governing equations. The weighted average flux method applied on the Cartesian cut cell grid for stabilization of the numerical results. To validate the numerical model using the Cartesian cut-cell grids, the model is applied to the rectangular tank problem of which the exact solutions exist. As a comparison of numerical results with the analytical solutions, the numerical scheme well represents flow characteristics such as free surface elevation and velocities in x-and y-directions in a rectangular tank with the Cartesian and cut-cell grids. KeyGwords : shallow-water equations, cut-cell method, finite volume method, HC approximate Riemann solver, TVD-WAF method Cartesian» w w x w rwš z w» w wš w. w» ³ w j» Cartesian txw e x y, z j» w p w ww w». w w w j» x, w» w w w e x w. HC Riemann w w yw, ew»w» w TVD-WAF» w. w w e x w» w w wƒ w ƒx w. w w e w» ³ w x- y- w y w y w. w :, w», w, HC Riemann w, TVD-WAF» 1. w e w, w x tx w e y j w e. w» mw k ³ew x txw» š wš, ¾ š z» w ƒ v w. rwš w ³ w j» w w» w ƒ ƒ w. w j» w w multi-block», t š t ywš k x w w š t e», ³ ƒ w e w x e». w»» w y ù, w» w v w w w ƒ w (Yang, 2000). w mw ywš z e wš e x w» wš w. * z Áw w w m œw (E-mail : john0705@hanyang.ac.kr) ** z Áy w œ w œw *** z Áw w œ w y œw ( ) 91
ü ew w ƒ y š. (2001) TVD» w 2 w w q e w, kz (2002) Roe» w w qƒ e w w. (2003) HC Riemann w TVD-WAF» w w w w w yw, ½ (2003) HC» TVD-WAF» yw Bellos (1992) q x e x xw. ½ y (2005) HC» MUSC-Hancock» w» w v w w e ³x y» w. w ü w w y w yw z, e x w. z e x ƒ v w. ³ w j» w š x tx y w w» w e wš, w» e x w e x w. w» p yw ƒ x ww w rw» (Causon, 2000; Causon, 2001; Qian, 2001). yw w w š t yw š x w (Qian, 2003). Zhou (2004) H(Harten, ax and van eer)» MUSC(Monotone Upstream-centered Schemes for Conservation aws)-hancock» e x w wš, 45 o 90 o q w CADAM(Concerted Action on Dam Break Modelling) v p(morris, 2000) x w. Gao (2006) w Roe» e x w w w Rayleigh-Taylor instability w j» w w, w q w w w w w. w w w xk j», w w xk e» w w yw e x w. yw» w Riemann w HC (Harten, ax, van eer and Contact wave)» w w, 1 e» w» w TVD-WAF(Total Variation Diminish-Weighted Average Flux)» w w»w. 2. w w» p ww, w txw z w». w w w w.» ³ w j» ƒx» txw. p yw, e ù p w ƒx w w txw, ƒx w» w ww w w ( 1 ). p w ww w w» w, w e w p w. x 1 š x x ƒ. š w tx, w w w w ƒ š ƒ. e e l» w ew 1. w 92 w wz «y
š, w ew. 2 r, wd ewš, dw š, d ƒ. w x w yw w w» w» š ƒ w w w w w. w, w š w z š rw w» ww. w w» w Ingram (2003) x wù š w ùkü z, ƒ e w w w w w w. P i = {( x 0, y 0 ),( x 1, y 1 ),,( x j, y j ),,( x n, y n )} (1) š w P i e ww» w (x i, y i ) (x j+1, y i+1 ) (x s, y s ) (x e, y e ) w wš, ƒx ƒ e w w. x ùkü š w P i t š t eƒ y w. w w sw w» w (2) w ó e (I s, J s ) (I e, I e ) w.», (x 0, y 0 ) x y t w. I s int x s x ------------- o y + 1 J x, s y = o s = ------------- + 1 y I e int x e x ------------- o y + 1, J e y = o x e = ------------- + 1 y (1) (2) w w» w w e w w w. w e 48ƒ ƒ w. w xkƒ 3ƒx l 5ƒx xk. 3. e x 3.1 x (3) x ùký. U E G ------ + ------ + ------- = S t x y x l U x y w l E G w S ƒƒ. h hu hv 0 U = hu, E = hu 2 + gh 2 /2, E = huv, S = ghs ox ghs fx hv huv hv 2 + gh 2 /2 ghs oy ghs fy (4) (4) h, u v ƒƒ x y w s³ ùkü. w sw S o S f ƒ ƒ w ùkü, Manning œ Chezy œ w w. ³ew w (3) yw» w (5) w w w. E G --- UdA+ ------dω + ------- dω = SdΩ t A Ω x Ω y Ω 3.2 Riemann w HC» 3 p š ww w, H» w w w w w». HC» Fraccarollo Toro(1995) w, Billet Toro(1997) HC» w k w. w w w w w w (6) ùký. (3) (5) U E ------ + ------ = 0 t x (6) w l š ƒ, q w (7) Riemann w. (6) 2. w Ũ = U for 0 S * U * U R U R for S R 0 for S 0 S * for S 0 S * R (7) w w e x 93
Riemann w w q (8). S l = min( u gh, u * gh * ) u + u R S * = u * = --------------- + gh 2 gh R S R = max( u R + gh R, u * gh * ) (7) (8) w w (9). E for 0 S E * = E + S ( U * U ) for S 0 S * E = E R * = E R + S R( U R * U R) for S * 0 S R E R for S R 0 2 y e» e w» w TVD w w. Toro(1999) WAF» w e x /2~ x /2 w. WAF E i + 1/2, j = x/2 x/2 E( Q i + 1/2, j( x, t/2) ) dx (8) (9) (10) HC» x/2~ x/2 3 p š w 4 w, (10) (11) ƒ ƒ s³w ùký. x š c 0 = 1, c 4 =1., (12) (11) wš w (13) w. k 1 E i + 1/2 = -- E 2 i + E i + 1 m ( ) 1 -- sign 2 ( c k )ψ i + 1/2 E i + 1/2 k = 1 4. x (13) e x mw» w w wƒ w ƒx w w w w., x» w t ew w y w., d w» t 30 o 45 o z g w wš e w w w e y w. e x w ƒ w 1 y w w ew w, l s w y š y w. e xk 5. d GA 5 (a) z w, GB GC z GA w w w. š xw» w k k E i 1/2 + WAF 4 = k = 1 ( k) β k E i + 1/2 (11) k ( k)» E i 1/2 j = E( Q (9) HC» s +, i + 1/2 ) ³ w β k (12). A k A k 1 β k = --------------------- = x 1 -- ( c 2 k c k 1 ) (12) (12) c k q S k w Courant ts k / 3. š ƒ 4. e w ƒ x 94 w wz «y
(3) (14) ƒ ƒ w. h T V T = h ij, u n, u t ( ) = V ij, ( u n, u t ) (14) 4.1 t w ƒx ynch Gray(1978) w w w ƒx ü w ew w. ƒ ƒx w ¼, k d š qšƒ ζ o (15) (17) w. ƒx ü 1/ 4» ü w, 2/4»» e s ƒ. 4/4»» sš w, k y ùkü. ζ o η() t = ----cos πx -----cos -----cos πy 2πt ------- + d 2 T ζ o gt ut () = -----------sin πx -----cos -----cos πy 2πt ------- 4 T ζ o gt vt () = -----------cos πx -----sin πy -----sin 2πt ------- 4 T T = 2/ gd (15) (16) (17) (18) ww 2 w, 2» e w. 5 d x y w y w w w w, 6 t =2T /4 t =4T /4 ƒx ü s t = T /4 t =3T /4 l s ùkü. d w e w w w, w e x j»ƒ w e mw y y wš y w. l s r, t =2T /4ƒ ù z» y š, ƒ» e š y w. 4.2 w w ƒx ƒx z k ƒ w z, w» w wš e w. z w» j»ƒ x = y =2.0m w,» ü w z ƒ w w w. e x w» w z ww (19) w (20)-(22) xw. θ ƒ z ƒ ùküš, x' y' z dw w œ t, x y z» œ t w. u' v' z t x' y' w ùkü, e w (23) x y w yw ew w. ¼ = 200.0 m d=5.0m, qš zo=1.0m» w y w. x = y =2.0m ³ w j» x' y' = cosθ sinθ sinθ cosθ x y (19) 5. d w w e ( z ) w w e x 95
6. l s ( z ) ζ o η() t ---- π xcosθ ysinθ = cos---------------------------------------cos ( + ) π -------------------------------------cos ( xcosθ ysinθ) 2πt ------- + d 2 T ζ o gt u () t ----------- π xcosθ ysinθ = sin---------------------------------------cos ( + ) π ------------------------------------- ( xcosθ ysinθ) cos 2πt ------- 4 T ζ o gt v () t = -----------cos---------------------------------------sin π( xcosθ + ysinθ) π -------------------------------------sin ( xcosθ ysinθ) 2πt ------- 4 T u' v' = cos( θ) sin( θ) sin( θ) cos( θ) u yv (20) (21) (22) (23) ³ w j» ƒx ww 30 o z e w ³ w w e w š w. 30 o z xk txw» w ww ùkù xkƒ ñ e š k ƒ. qš d =5.0m, ζ o =1.0m w e w, y w w w 7 ùkü, x w w e» w 8 t =2T /4 t=t /4 l s w. 9 10 y w 7. GB ( 100 100) 96 w wz «y
8. ³ w l s (30 o z ) 9. d w w e (30 o z ) y txw w w s l ƒ wš y w. 30 o z z txw» w 4 w w rw w. qš ƒ d =5.0m, ζ o =1.0m w. d e w w w 9 ùkü. w w w, w w e ƒ w w ewš y w. 10 t =2T /4 t =4T /4 ƒx ü s t = T /4 t =3T /4 l s ùkü. w xk w e, e ƒ w w ewš y w. l s r, x j» e ƒ w š š w w e x 97
y w., 200 m 200 m 45 o z ¼ ~ 141.42 m w» w w w. 45 o z txw» w w ³ 1/2 10. l s (30 o z ) ƒ ƒx xk. e qš 4.1 w d =5.0m, ζ o =1.0m w. d GC e w w w 11 ùkü. w w w, w w e ƒ 11. d w w e (45 o z ) 98 w wz «y
12. l s (45 o z ) w w ewš y w. 12 t =2T /4 t =4T /4 ƒx ü s t = T /4 t =3T /4 l s ùkü. l s r, ƒ x txwš, w l š tx š y w. 5. ƒx ³ š, x tx w w» w, š x x w w z e w w» w wš, w w e x w w e w. HC Riemann w w e w, TVD-WAF» w e x y w g. e x ƒ š w w š w ƒx ü w w. e x w» w t w ew w e ww, w x wš y w. t x 30 o 45 o w w w w r. ƒx ³ w w e w w w, w w e ƒ s l yw y w. w, ƒ l r, w w l w y x j š y w. w w y w w w y w š w e» w, w z š yw w q. š x ½ y, (2005) ³e x ƒ w š x w t. wm wz, wm wz, 25«, pp. 223-229. ½,, ½ w (2003) WAF» w w. w wz, w wz, 36«, pp. 777-785. kz,, (2002) w w. wm wz, wm wz, 22«, pp. 33-41., (2001) TVD e x :. w wz, w wz, 34«, pp. 187-195.,, (2003) TVD» w ew. w wz, w wz, 36«, pp. 597-608. Bellos, C., Soulis, J. V. and Sakkas, J. G. (1992) Experimental Investigation of Two-Dimensional Dam-Break Flows. J. of Hydraulic Research, Vol. 30, No. 1, pp. 47-63. Billet, S. J. and Toro, E. F. (1997) On the Accuracy and Stability of w w e x 99
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