URBAN FLOODS MODELLING AND TWO-DIMENSIONAL SHALLOW WATER MODEL WITH POROSITY

  _{SIPIL ’ MESIN ’ARSITEKTUR ’ELEKTRO} e k

URBA N FLO O DS MO DELLING A ND TWO - DIMENSIO NA L SHA LLO W WA TER MO DEL

WITH PO RO SITY

• * Rud i He rma n

  A b stra k

Po la g e na ng a n m e m iliki d a m p a k ya ng sa ng a t b e sa r p a d a wila ya h p e rko ta a n d i d a e ra h

d a ta ra n sung a i. Da la m m e m p e rhitung ka n p e rub a ha n d a n p e ng ura ng a n d e b it a kib a t a d a nya

b a ng una n d a n ko nstruksi ya ng la in p a d a wila ya h d a ta ra n sung a i, p e ng g una a n m o d e l a lira n

d a ng ka l d e ng a n c e la h m e rup a ka n sua tu ha l ya ng d iuta ra ka n. Untuk ha l ini m o d e l sum b e r a ir

ya ng khusus untuk m e ng e ksp re sika n ke hila ng a n e ne rg i d a la m wila ya h p e rko ta a n d ip e rluka n.

Fo rm ula ya ng d iusulka n a d a la h m e m b a nd ing ka n d a ta uji c o b a ya ng d ip e ro le h d a ri sa lura n

la b o ra to rium p a d a ska la ya ng d ise sua ika n d e ng a n wila ya h p e rko ta a n. Pa d a ting ka t a kura si

ya ng sa m a , m e to d e ini m e m p e rliha tka n p e nuruna n ya ng sig nifika n d a la m p e rhitung a nnya

d ib a nd ing ka n d e ng a n m e ng g una ka n g rid ya ng le b ih ke c il p a d a sim ula si kla sik.

  Kata kunc i: d a ta ra n sung a i, mo d e l fisik, mo d e l d ua -d ime nsi Abstrac t

Flo o d p la ins with urb a n a re a s ha ve sig nific a nt e ffe c ts o n inund a tio n p a tte rns. A sha llo w wa te r

mo d e l is p re se nte d, with p o ro sity to a c c o unt fo r the re d uc tio n in sto ra g e a nd in the e xc ha ng e

se c tio ns d ue to the p re se nc e o f b uild ing s a nd o the r struc ture s in the flo o d p la ins. A sp e c ific so urc e

te rm re p re se nting he a d lo sse s sing ula ritie s in the urb a n a re a s is ne e d e d . The p ro p o se d fo rmula tio n

is c o mp a re d to e xp e rime nta l d a ta o b ta ine d fro m c ha nne l e xp e rime nts o n a sc a le o f a n

urb a nize d a re a . The me tho d is se e n to re sult in sig nific a nt re d uc tio n o f c o mp uta tio na l e ffo rt

c o mp a re d to c la ssic a l simula tio ns using re fine d g rid s, with a simila r d e g re e o f a c c ura c y.

  Ke y wo rds: inund a tio n, p hysic a l mo d e l, two -d ime nsio na l mo d e l

  a nd c o m p uta tio na l tim e . Ho we ve r,

1. Intro d uc tio n

  c o nsid e ring la rg e urb a n a re a s c a n le a d Urb a n a re a s a re o fte n vulne ra b le to a d ra m a tic inc re a se o f b o th b e c a use o f the c o njunc tio n o f a hig h c o m p uta tio na l c e ll num b e r a nd tim e . c o nc e ntra tio n o f inha b ita nt a nd

  The use o f la rg e -sc a le m o d e ls in e c o no m ic a c to rs a nd a ha za rd o us suc h situa tio ns c o nstitute s a n a lte rna tive , c o nte xt (im p e rvio us a re a s, p ro xim ity o f b e c a use the re q uire d d e ta ile d m e shing rive rs e tc ). A p re ve ntio n a p p ro a c h le a d s o f the urb a n a re a (to re p re se nt to the d e ve lo p m e nt o f flo o d m a p p ing , c ro ssro a d s a nd sq ua re s) is b yp a sse d with va rio us a p p lic a tio ns in the fie ld o f thro ug h a p o ro us zo ne re p re se nta tio n. urb a n e ng ine e ring : kno wle d g e o f

  The p o ro sity c o nc e p t wa s first p re se nte d e xp o sure to risk, re g ula tio n o f urb a n b y He rvo ue t e t a l. [1] a nd De fina e t a l p la nning , e la b o ra tio n o f c risis [2]. The intro d uc tio n o f p o ro sity in the m a na g e m e nt sc e na rio s fo r e xa m p le . sha llo w wa te r e q ua tio n m o d ifie s the

  Am o ng the e xisting num e ric a l m o d e ls, e xp re ssio ns fo r the fluxe s a nd so urc e two -d im e nsio na l sha llo w wa te r m o d e ls te rm s. Fo llo wing G uino t a nd So a re s- se e m to b e the b e st c o m p ro m ise

  Fra za o [3], a sp e c ific so urc e te rm b e twe e n flo w d e sc rip tio n, d a ta ne e d s

• Sta f Pe ng a ja r Jurusa n Te knik Sip il Fa kulta s Te knik Unive rsita s Ta d ula ko , Pa lu

  In wha t fo llo ws, φ is a ssum e d to d e p e nd o n the sp a c e c o o rd ina te s o nly. The to p o g ra p hic a l so urc e te rm s c a n b e writte n a s : …….(3) whe re z b is the b o tto m e le va tio n. The first te rm o n the rig ht-ha nd sid e e q . (3) a c c o unts fo r va ria tio ns in the b o tto m le ve l. The re sulting fo rc e o n the wa te r b o d y is e xe rte d o nly a fra c tio n

  Jurna l SMARTe k, Vo l. 4, No . 4, No p e m b e r 2006: 253 - 259

  re p re se nting he a d lo sse s d ue to sing ula ritie s in the urb a n a re a s (c ro ssro a d s in p a rtic ula r) is ne e d e d . This sp e c ific so urc e te rm is d e rive d fro m g e ne ra l Bo rd a he a d lo ss fo rm ula a nd ta ke s the fo rm o f a te nso r.

  De p th a nd surfa c e ve lo c ity m e a sure m e nts ha ve b e e n c o nd uc te d o ve r a p hysic a l m o d e l o f a n urb a n a re a . The se e xp e rim e nta l d a ta a re c o m p a re d to va rio us num e ric a l sim ula tio ns, inc lud ing c la ssic a l b i-d im e nsio na l sim ula tio n a nd d iffe re nt la rg e -sc a le b i- d im e nsio na l sim ula tio ns. Se c tio n 2 is d e vo te d to a sho rt p re se nta tio n o f the g o ve rning e q ua tio ns o f la rg e -sc a le m o d e ls. Se c tio n 3 c o nsists o f the d e sc rip tio n o f the p hysic a l m o d e l. Se c tio n 4 tre a ts the c o m p a riso n b e twe e n e xp e rim e nta l m e a sure m e nts a nd num e ric a l re sults. Se c tio n 5 g ive s so m e c o nc lud ing a nd p ro sp e c tive re m a rks.

2. Lite ra tur Re vie w

  2.1 G o ve rning Eq ua tio ns o f the Num e ric a l Mo d e l

  The sha llo w wa te r e q ua tio ns with p o ro sity c a n b e writte n in c o nse rva tio n fo rm a s ;

  ( ) ( ) ( ) S G φ y F

• ∂ ∂

  φ x U φ t

  ….(1) …………………..……………….(2) Whe re g is the g ra vita tio na l a c c e le ra tio n, h is the w a te r d e p th, u a nd v a re the x- a nd y- ve lo c itie s re sp e c tive ly,

  to ta l se c tio n o f the c o ntro l vo lum e . The se c o nd te rm o n the rig ht-ha nd sid e a c c o unt fo r the lo ng itud ina l va ria tio ns in the p o ro sity. The e ne rg y lo sse s te rm is a ssum e d to re sult fro m (i) the b o tto m a nd wa ll she a r stre ss, a c c o unte d fo r b y Stric kle r’ s la w a nd (ii) the e ne rg y lo sse s trig g e re d b y the flo w re g im e va ria tio ns a nd the m ultip le wa ve re fle c tio ns d ue to o b sta c le s o r c ro ssro a d s, a c c o unte d fo r b y c la ssic a l he a d lo ss fo rm ula tio n. It c a n b e writte n a s; ……………………………...(4) Whe re K is the Stric kle r c o e ffic ie nt a ssum e d to b e iso tro p ic in the fo llo wing , s xy , s xy , s yx a nd s yy a re c o e ffic ie nts a c c o unting fo r the lo c a l he a d lo sse s d ue to urb a n sing ula ritie s. The se fo ur c o e ffic ie nts invo lve d in a te nso r a re ne c e ssa ry to re p re se nt he a d lo sse s in a stre e t ne two rk no t a lig ne d o n the syste m c o o rd ina te s, Eq . (4) re d uc e s to ; ………………………………….(5)

  φ o f the

• ∂ ∂

  S o ,y a re the to p o g ra p hic a l so urc e te rm s a rising fro m the b o tto m slo p e s a nd p o ro sity va ria tio ns in the x- a nd y- d ire c tio ns re sp e c tive ly, S fx a nd S fy a re the so urc e te rm s a rising fro m e ne rg y lo sse s in the x- a nd y- d ire c tio ns, re sp e c tive ly [3].

  φ is the p o ro sity, S o ,x a nd

  In wha t fo llo ws, Eq . (1) is d isc re tise d using the finite -vo lum e

  = ∂ ∂

• A c la ssic a l 2D sim ula tio n, whe re b uild ing s a re re p re se nte d a s im p e rvio us b o und a rie s, a nd
• A p o ro sity 2D sim ula tio n, whe re

  φ

  The fre e surfa c e p ro file s fo r the thre e sim ula tio ns we re p lo tte d a lo ng the lo ng itud ina l a xis y = 0, with the e xp e rim e nta l d a ta (Fig ure -2). The d a she d line re p re se nts the lim its o f the sq ua re b o rd e ring the 22 b uild ing s, a nd the g ra y-fille d re c ta ng le s d e lim it the b uild ing s p o sitio ns o n the sp e c ifie d lo ng itud ina l a xis. The C r p ro file a p p e a rs to b e ve ry c lo se to re p re se nt the influe nc e o f the urb a n a re a . Two a lte rna tive so lutio ns a re c o nc e iva b le , e ithe r to a rtific ia lly re d uc e the Stric kle r

  4.1 Influe nc e o f the m e shing d e nsity o f the urb a n a re a Two p o ro sity sim ula tio ns ha ve b e e n c o m p a re d , o ne with a re fine d m e shing o f the urb a n a re a , a nd the o the r with a c o a rse o ne . No lo c a l he a d lo sse s a re invo lve d in this p a ra g ra p h. In wha t fo llo ws, the d iffe re nt sim ula tio ns will b e re c a lle d b y two le tte rs a c ro nym , the first le tte r ind ic a te s the sim ula tio n typ e (Re fine d o r C o a rse ). The c ha nne l wa s d ivid e d into thre e zo ne s, the c ha ra c te ristic m e sh le ng ths o f whic h c a n b e fo und in ta b le -1. With the PR sim ula tio n, the urb a n a re a is m e she d e xa c tly a s in the C R sim ula tio n, b ut the insid e a re a o f the b uild ing s is m e she d to o .

  o f the urb a n a re a wa s c a lc ula te d a s the ra tio o f the to ta l p la ne a re a o f the b uild ing s to the a re a o f the sq ua re d e lim iting the 22 b uild ing s. This le a d s to a va lue o f 0.45.

  φ

  Va rio us va lue s o f he a d lo sse s c o e ffic ie nts s x a nd s y we re te ste d fo r re p re se nting the influe nc e o f the stre e t ne two rk. The p o ro sity

  1. The b uild ing a re no t e xp lic itly re p re se nte d in this sim ula tio n.

  is stric tly le ss tha n

  e q ua l to 1 e ve rywhe re e xc e p t in the urb a n a re a w he re

  Urb a n Flo o d s Mo d e lling a nd Two -Dime nsio na l Sha llo w Wa te r Mo d e l with Po ro sity (Rud i He rma n)

  φ is

  φ is unifo rm ly e q ua l to 1.

  Two typ e s o f num e ric a l sim ula tio ns we re und e rta ke n:

  4. C o m p a riso n Be twe e n Exp e rim e nta l Me a sure m e nt a nd Num e ric a l Re sults

3. De sc rip tio n o f the Physic a l Mo d e l

  Fig ure -1. Exp e rim e nta l se t-up : c ha nne l d im e nsio ns (m ) a nd urb a n d istric t la yo ut.

  1/ 3 / s.

  The e xp e rim e nta l se t-up is lo c a te d in the la b o ra to ry o f the C ivil Eng ine e ring a nd G e o sc ie nc e De p a rtm e nt o f the Unive rsity o f Ne wc a stle Up o n Tyne , Unite d King d o m . The c ha nne l is ho rizo nta l, 36 m lo ng a nd 3.6 m wid e , with a p a rtly tra p e zo id a l c ro ss se c tio n a s ind ic a te d in Fig ure -1. A g a te is lo c a te d b e twe e n two im p e rvio us b lo c ks in o rd e r to sim ula te a b re a c h in a d a m o r a d yke . In this stud y, a ste a d y flo w with a d isc ha rg e o f 95 l/ s wa s sim ula te d thro ug h the b re a c h, b y m a inta ining the g a te o p e n. A sim p lifie d urb a n d istric t is p la c e d in the c ha nne l, with a sta g g e re d la yo ut. The Stric kle r c o e ffic ie nt o f the c ha nne l is e q ua l to 100 m

  255 a p p ro a c h o n unstruc ture d g rid s with a G o d uno v-typ e sc he m e [4, 5]. A o ne - d im e nsio na l Rie m a nn p ro b le m is d e fine d in the lo c a l c o o rd ina te syste m a tta c he d to e a c h inte rfa c e . The flux ve c to r in the d ire c tio n no rm a l to the inte rfa c e is c o m p ute d using the fo llo wing p ro c e d ure . In a first ste p , the Rie m a nn p ro b le m in the g lo b a l c o o rd ina te syste m is tra nsfo rm e d to the lo c a l c o o rd ina te s syste m . In a se c o nd ste p , the lo c a l Rie m a nn p ro b le m is so lve d using a m o d ifie d HLLC Rie m a nn so lve r. The n the flux is tra nsfo rm e d b a c k to the g lo b a l c o o rd ina te syste m . The se num e ric a l tre a tm e nts a re no t d e ta ile d in this p a p e r, se e re fe re nc e [3] fo r m o re d e ta ils.

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  c o e ffic ie nt, o r to use the lo c a l he a d - lo sse s fo rm ula tio n o f Eg . (4). The first so lutio n is no t p hysic a lly a p p ro p ria te b e c a use Stric kle r fo rm ula tio n is va lid fo r turb ule nt she a r stre ss within the b o und a ry la ye r a t the b o tto m a nd wa lls, whe re a s lo c a l he a d -lo sse s a re a ssum e d to b e id e ntic a l o ve r the e ntire flo w c ro ss- se c tio n a nd sho uld the re fo re sim p ly b e p ro p o rtio na l to the sq ua re o f the ve lo c itie s invo lve d [3].

  Ta b le 1. C ha ra c te ristic m e sh, c e lls num b e r a nd c o m p uta tio na l tim e fo r e a c h sim ula tio n.

  De sc rip tio n C R PR PR

  Up stre a m p a rt Do wnstre a m p a rt e xc e p t urb a n a re a Urb a n a re a To ta l c e lls num b e r C o m p uta tio na l tim e (400s sim ula te d )

  0.40 m 0.20 m 0.03 m

  10335 132 m n 0.40 m 0.20 m 0.03 m

  15299 200 m n 0.40 m 0.20 m 0.14 m

  5549 63 m n Fig ure 2. Sim ula te d fre e surfa c e p ro file witho ut lo c a l he a d lo ss a nd d e p ths a lo ng the a xis y = 0

  Fig ure 3. Sim ula te d fre e surfa c e p ro file witho ut lo c a l he a d lo ss a lo ng the a xis y = 0

  Urb a n Flo o d s Mo d e lling a nd Two -Dime nsio na l Sha llo w Wa te r Mo d e l with Po ro sity (Rud i He rma n)

  257

  4.2 Influe nc e o f lo c a l he a d lo sse s Sim ula te d fre e surfa c e p ro file s (C R a nd PC ) w e re c o m p a re d with the e xp e rim e nta l d a ta b ut PC sim ula tio n he re invo lve d lo c a l he a d -lo ss c o e ffic ie nts. The va rio us PC sim ula tio ns will b e re c a lle d b y the c o d e PC _s x -s y in wha t fo llo ws. The c o e ffic ie nt s x o f Eq . (5) ha s b e e n fixe d g re a te r tha n the c o e ffic ie nt sy, b e c a use o nly the tra nsve rsa l stre e ts a re c o ntinuo us. The c o m p uta tio na l tim e o f a ll PC _s x -s y sim ula tio ns is a p p ro xim a te ly the sa m e a s g ive n in the ta b le -1. The fo llo wing p a irs (s x -s y ) ha ve b e e n te ste d : (10-5), (7-4), (5- 3) a nd (2-1). All the re sulting sim ula te d p ro file s a re ve ry c lo se to the C R p ro file a nd e xp e rim e nta l d e p ths, b ut PC _5-3 a p p e a rs to b e the c lo se st (Fig ure 3).

  C o m p a re d to the va rio us p a ra g ra p h re sults, it c a n b e no te d tha t the use o f the lo c a l he a d -lo sse s c o e ffic ie nt le a d s to a c le a r im p ro ve m e nt o f the sim ula te d p ro file . The inc re a se o f the fre e surfa c e p ro file b e fo re the c ity is we ll p ro d uc e d , b ut the d ro p a fte r the c ity is und e r e stim a te d . The thre e PC _10-5, PC _7-4, PC _5-3 sim ula tio ns a re ve ry c lo se to e a c h o the r, so the se nsitivity o f the m o d e l to the two lo c a l he a d -lo sse s c o e ffic ie nts is no t ve ry hig h.

  4.3 C o m p a riso n o f the ve lo c ity fie ld s The C R a nd PC _5-3 sim ula te d ve lo c ity fie ld s a re c o m p a re d to the o ne o b ta ine d b y the d ig ita l im a g ing te c hniq ue o ve r the lim ite d d a ta a c q uisitio ns wind o w. It a p p e a rs tha t e xp e rim e nta l a nd C R ve lo c ity fie ld s a re ve ry sim ila r (Fig ure 4). The m a in flo w skirting the c ity is we ll sim ula te d , a s we ll a s the re -c irc ula tio n zo ne s lo c a te d ne xt to the b uild ing s. The se c o nd a ry flo ws e xisting the c ity thro ug h the tra nsve rsa l stre e ts se e m to b e we ll re p re se nte d to o .

  Diffic ultie s to ha ve tra c e r p a rtic le s in the rig ht lo we r c o rne r o f the a c q uisitio n wind o w a re re sp o nsib le o f the la c k e xp e rim e nta l ve lo c itie s in this zo ne . But it se e m s tha t the e xte nsio n o f the re - c irc ula tio n zo ne b e twe e n the b uild ing no . 2 a nd no .4 is und e r e stim a te d in the C R sim ula tio n.

  Exp e rim e nta l a nd PC _5-3 ve lo c ity fie ld s a re o b vio usly ve ry d iffe re nt insid e the urb a n a re a , b e c a use o f the a b se nc e o f b uild ing s in the PC sim ula tio n (Fig ure -5). But it is inte re sting to se e tha t the use o f p o ro sity a nd lo c a l he a d -lo sse s m a ke s the flo w m a inly e sc a p ing fro m the urb a n a re a .

  The a g re e m e nt b e twe e n the e xp e rim e nta l a nd the C R ve lo c itie s is ve ry g o o d . This c o uld b e use d to va lid a te the PC sim ula tio ns witho ut the ne e d to tra c k the tra c e r p a rtic le s o ve r the e ntire c ha nne l. At a la rg e sc a le , the ve lo c itie s c a n b e c o nsid e re d a s c o rre c tly sim ula te d with the PC _5-3 sim ula tio n.

  Fig ure 4. C R a nd e xp e rim e nta l ve lo c itie s fie ld s. Jurna l SMARTe k, Vo l. 4, No . 4, No p e m b e r 2006: 253 - 259 Fig ure 5. PC _5-3 a nd e xp e rim e nta l ve lo c itie s fie ld s.

6. Re fe re nc e s

  5 . C o nc lusio ns

  A la rg e -sc a le m o d e l ha s b e e n use d to sim ula te a ste a d y flo o d flo w thro ug h a sim p lifie d urb a n a re a . It ha s b e e n c o m p a re d with b o th e xp e rim e nta l d e p th a nd ve lo c ity m e a sure m e nts, a nd c la ssic a l two -d im e nsio na l m o d e l. The la rg e -sc a le m o d e l g ive s a g o o d d e sc rip tio n o f the m a in fe a ture s o f the flo w (e xc e p t insid e the urb a n a re a ), a t a m uc h lo we r c o m p uta tio na l c o st tha n c la ssic a l m o d e ls. This illustra te s the p o ssib le o p e ra tio na l use s o f suc h m o d e ls, (i) la rg e -sc a le sim ula tio n re sults m a y b e use d to p ro vid e b o und a ry c o nd itio ns to lo c a l m o d e ls whe re the d e ta ils o f urb a n a re a s a re re p re se nte d , (ii) the sim ula tio n o f flo o d p la in fe a turing urb a nize d a re a s, b ut whe re the d e ta ils o f the flo w within the urb a n a re a s a re no t o f d ire c t inte re st. Furthe r inve stig a tio ns a re ne e d e d to e xp re ss the c o m p o ne nts o f the lo c a l he a d -lo sse s te nso r a s func tio ns o f the g e o m e tric a l c ha ra c te ristic s o f the urb a n a re a : b uild ing s d e nsity, stre e ts wid th, d ire c tio n a nd slo p e .

  C a p a rt H., Yo ung D.L., Ze c h Y.,2002, “

  Vo ro no i ima g ing me tho d s fo r the me a sure me nts o f g ra nula r flo ws” .

  Exp e rim e nts in Fluid s, Vo l. 32, No .1, 2002, p p 121-135. De vina A., D’ Alp a o s L., Ma ttic hio B.,2004,

  “ A Ne w Se t o f Eq ua tio ns fo r Ve ry

  Sha llo w Wa te r a nd Pa rtia lly Dry Are a s Suita b le to 2D Nume ric a l Do ma in” ’ Pro c e e d ing s Sp e c ia lly

  C o nfe re nc e ‘ Mo d e lling o f Flo d d Pro p a g a tio n O ve r Initia lly Dry Are a s’ Mila no , Ita ly.

  G uino t V., So a re s-Fra za o S., 2006, “ Flux

  a nd So urc e Te rm Disc re tiza tio n in Two Dime nsio na l Sha llo w-Wa te r Mo d e ls With Po ro sity o n Unstruc ture d G rid s” . Int.

  Jo urna l.’ Num e ric a l Me tho d s in Fluid s, Vo l 50, No .3, (2006), p p 309- 345.

  G uino t V., 2003, “ G o d uno v-typ e

  Sc he me s, a n Intro d uc tio n fo r Eng ine e rs” , Else vie r, Am ste rd a m ,

  The Ne the rla nd s.

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  Da mBre a k Flo o d Wa ve Nume ric a l Dyna mic s” , Sp ring e r, Be rlin, Simula tio ns” , Pro c e e d ing o f the G e rm a ny, (1997).

  Inte rna tio na l Se m ina r a nd Wo rksho p o n Re sc ue Ba se d o n Da m b re a k Flo w Ana lysis, Se ina jo ki, Finla nd , (2000).

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