Numerical simulations of overland floods in urban areas using a conservative Godunov-type scheme

The scheme is based on unstructured computational meshes, in generaỊ to deal with complicated urban geometries.. The modcl has been applied to studying tvvo cxperiments of [r]

VNU Ịournal of Science, E arth Sciences 25 (2010) 168-176

Numerical simulations o f overland íloods in urban areas

using a conservative G odunov-type scheme

Nguyen Tat Thang*

Im titu te o f M echanics, Vietnam Academ y o f Science a n d Technology, VAST

Received 14 September 2009; received in reviscd form 24 September 2009

Abstract Floođs in urban areas due to lcvee overtopping and/or breaking may causc a lot of sevcre damage o f property and lost of liuman lives In case of rivcr đike and/or dam break, thc problem is characterized by the overland propagation of discontinuity íronts or hydraulic jumps It

is o f immense imporlance that urban planners and personnel havc tools to assist in predicting and

evaluating beíorehand the flood process in such incidents Recentỉy, witlì the rapid development of Com puter rcsources and n u m e rica l methods, numerical m odels basetl on mathematical m o d cls for simulation o f flood scenarios become highly useĩul A model for thc simulation o f tvvo dimensional (2D) overland íloods in urban areas has therefore bcen devclopcd A íìnite volumc Godunov-type numericaỉ scheme is applied in the model This numerical scheme has some important advantages It is a conservativc scherne and able to model more accuratcly hydraulic

shockvvave propagation The scheme is based on unstructured computational meshes, in generaỊ to deal with complicated urban geometries The modcl has been applied to studying tvvo cxperiments

of overỉand ĩloods These experiments were carricd out in research institutions in Japan and Italy

The computed results show general agreemcnt with thc mcasured ones The model is prospective for analyzing overland ílood process in practical cases

Keywords: Numerical simulation; overland ílood; godunov-type scheme; Un-structured meshes.

1 I n t r o d u c t i o n

M athem atical m o d e ls for the numerical

solution o f the 2-D S aint V en an t equations have

long been developed A pplications o f such

m ođels, vvhich are b ased o n advanced

n u m e r i c a l' techniques, to the sim ulations o f

overland (loods in urban areas have attracted

m uch attention recently [1, 2, 3, 4, 5] These

m odels are highly useful to urban planners to

evaluate the impact o f urban dev elo p m en t to

postulated ílood events T h e re ío re numerical

* Tel.: (+44) 01224 273519

Email: thang.tat.nguyen@abdn.ac.uk

168

m odels for sim ulations o f o verland floods arc urgently needed T h e d e v e lo p m e n t o f numerical

m ethods for the solution o f the 2 D shallovv vvater eq u a tio n s o riginally started vvith the traditional finite d iffe re n c e m e thods, then vvith the íinite ele m e n t m e th o d s and n o w vvith the finite v o lu m e o n e s [6] T h a n k s to the rapid pro & re ss o f t h e C o m p u t e r t e c h n o l o g y ,

c o m p u tin g ability increascs incredibly Ít

en h an c es iireatly the d e v e lo p m c n t o f new, com p lica ted 2 D flood sim ulation m odels Such advanced m o d e l s usually based on the ílexible irregular 2D c o m p u tatio n al meshes (unstruclured m esh) In add itio n , the Godunov

N.T Thang / VN U Ịoum al o f Science, Earth Sciences 25 (2009) 168-176 169

m ethod, w hich is originated in aerodynam ics

and very e íĩic ie n t in dealing vvith pro b lem s vvith

discontinuities, has recently been applied to

fluid d y nam ics [7] M o reo v er unstructured

niesh g eneration te c h n iq u es and m odels have

reccntly been m u c h develo p ed and more

pow erful T a k in g th e se advantages, w e have

develo p ed a C om puter m odel to study 2D

overland íloods in urban areas Such overland

íloods are ty pically 2D , an d usually o c c u r in

v ery com plica ted geom etries T he m odel uses

unstructured m e sh es so th a t it can accurately

deal w ith g eo m etric ally c o m p le x 2 D dom ains

T he unstructured m e sh es used consist o f a set

o f co n n ected-convex po ly g o n s w ith an arbitrary

n um ber o f sides ỉn fact, d u e to the limited

ability o f m e sh g enera tion packages, the typical

meshes used usually are triangular meshes O ur

model is based on the G o d u n o v method This

method is c o n serv ativ e and able to simulate

unsteady flow s w ith the p resence o f hydraulic

discontinuities O n e o f the im portant difficulties

arising in th e im plem entation o f the

discretization s c h e m e is the trea tm e n t o f the

vvet-dry fronts [8] S uch fronts are inner

boundaries, i.e boundaries inside

com putational d o m a in s T h e y vary durin g the

flood process T h is situation is a very com m on

in overland íloods A special te chnique has

been ap p lied b ased on the o n e m entioned in

published literature [8] T he m odel is written in

Com paq F O R T R A N 6.6 p rogram m ing

language T w o ex p e rim e n ts o f the overland

íloods in urban areas [9, 10] hav e been studied

num erically using the m odel T h e com puted

results are c o m p a re d w ith the m e asured ones

Acceptable ag re e m e n ts are o btained T he study

shows th a t the m o d e l is able to deal w ell with

vvet-dry m o v in g boundaries

This p ap er b rie íly presents the numerical

model in Part II C o m p u te d results and

co itp a ris o n s for the experim en ts in Japan are

given in Part III T h o s e o f th e experim ent in

Italy are presented in Part III C o nclusions are

m entioned in Part IV Finally a list o f references is provided at the en d o f this paper

2 N u m e r ic a l m o d e l f o r t h e s o lu tio n o f th e 2D

s h a llo w w a t e r e q u a t ỉ o n s

2.7 The sy ste m o f eq u atio ns

T h e m odel is based on the 2 D system o f the unsteady Saint V en an t equations vvritten in conservative form as shovvn below [4]:

ỡ ư t ÔF(U) t ỔG(U) +

+

õy = S ( x , y , U ) (1)

\vhere u =

' h N

q* (conservative variable),

F =

qy

q* g h 2

, G = q«qy

q y = v h ; h is the vvater depth; g is the gravìty

acceleration; (w ,v ) are th e X a n d y

com p o n en ts o f the depth averaged velocity respectively; s is the so u rce term E quation (1) can be rew ritten in th e follow ing form:

^ + V E (U ) = S ( x , y , U )

w here E = / F N

v G ,

T h e u n k n o w n s th a t need to be c o m p u ted are

h , q x and q y o r h , u h a n d v h

2.2 N u m e ric a ỉ technique

For a fĩxed control volu m e Q as sh o w n in

F i g l , the integral form o f (2) is w ritten as:

170 N.T Thang / V N U Ịournaỉ o f Science, Earth Sciences 25 (2009) 168-176

& n + J v E ( U ) d Q = J s ( x , y , U ) d n (3)

A pplying the G a u s s ’s theorem , (3) can be

rewritten in the folIow ing form

Ệ - | u d Q + c f (E n)ds = J s d Q (4)

w here ÕQ d enotes the boundary surface o f the

2D volum e Q , a n d n is the unit outw ard

normal vector (F ig l)

U "*1 = U n

Fig 1 A control volume (eỉement or cell) in tvvo

dimensions (NS: number o f sides; dsk: the length of

the side k)

Since equation (4) is w ritten for each

individual control volu m e (an e le m e n t o r cell o f

the com putational m eshes), the discretization

technique is applied to each elem ent Denoting

by U t the av erage (or discrete) value o f

x o nservative variable o v er the volu m e Q ,

using equation (4), the follow ing conservation

equation can be w ritten for e a c h cell i:

Aj + cỊ (E n )d s = J s d Q (5)

õt

w here A , is the area o f the 2 D v o lu m e Q [4].

A pplying the m id-point rule to

approxim ating the c o n to u r integral in (5) and a

sim ple approxim ation for the tim e derivative, a

finite difference like form o f (5) is vvritten as:

X E * k.nk.ds, + AtS*" (6)

T h e id e a s o f the G o d u n o v m e th o d and the

R o e ’s a p p r o x im a te R ie m an n so lv e r [11], w hich

a re o rig in a te d in aero d y n a m ics, are applied to

the a p p r o x im a tio n o f the E \ flux [7].

A ll d e ta ils o f the system o f equations and

d is cretiza tio n s c h e m e sho ald be referred to [4]

A s for b o u n d a r y co nditions, the model uses

th re e ty p c s o f b o u n d a ry co nditions Each o f

th o se is u s e d w h e re relevant T h e first one is the

co n d itio n o f the river w ater discharges from riv er o u tle ts flow ing into the simulation

d o m a in T h e s e c o n d o n e is the reilective and

n o -s lip b o u n d a r y conditio n applied to rigid

b o u n d a rie s A n d the last o n e is the free flow

co n d itio n at o p c n sea bo u n d arie s [4]

T h e n u m e ri c a l sch em e shovvn here, for

u n s tru c tu rc d m e sh e s in g e n e ra l, is highly effic icn t fo r the solution o f the propagation o f

w av es in spatial d o m a in s o f com plicated

g e o m e try [7] T h ere fo re it will be applicd in

th is study

3 N u m e r ic a l stu d y o f th e o v erla n d ílood

e x p e r im c n ts

3.1 E x p e r im e n ta / m o d e l o f a d ik e brecik

in d u c e d o v e r la n d f lo o d (Japan)

E x p e r im e n ta ì m o d e ỉ d e s c r ip tio n :

T h e e x p e r i m e n t o f a d ik e b reak induced

o v erla n d flo o d in a city area vvas pcrformed in

D PR I ( D is a s te r Prevention R esearc h Institute),

K y o to U n iv e r s it y in Japan T he experimcnt

a im e d to s im u l a te o verland íloods, vvhich is

c a u s e d b y a w a te r flow o v e rto p p in g the river

b ank into th e city (F ig.3), in a rcal site chosen

as s h o w n in Fig.2 This is a highly urbanized area o f the a n c ie n t city o f K y o to , Japan The

N.T Thang / V N U lournaỉ o f Science, Earth Sciences 25 (2009) 168-176 171

sitc c o v e r s a square arca o f l k m X 2 k m T he

e x p e rim e n ta l m odcl sitc is re d u cc d to a sm aller

scale o f lOm X 2 0 m [9] Fig.3 s h o w s positions

n u m b e re d from 1 to 8 vvhere th e vvatcr depth

was m easured du rin g the e x p e rim e n t The

M a n n in g rou g h n ess cocfficient d e te rin in e d in

the e x p e rim e n t is calculated to be 0.01 T he

\vhol«e experim ental site is dry j u s t bcforc the

e x p e rim e n t begins

K y o to

G o s h o

u s model site

G S model site

J R Tokaỉdo Lỉne

Fig 2 The rcal experimental site

T h e av erage slope o f the site (dovvnvvard to

the South direction) is a b o u t 0.005 The

experim ental m odel assu m es th a t there is no

vvater in vading into residential a n d building

areas so tliat flood w a te r o n ly flovvs in the

c o m p l i c a t e d S tr e e t n etw ork in t h e m o d c l e d s ite

(Fig.3) Fig.4 shovvs the ex p e rim e n ta l m odel set

up in the H ydraulic Laboratory o f D PR I The

d ischarge o f the vvater flow ing th ro u g h the dike

break point is C o m p u t e r co n tro lled a n d sh o w n

in Fig.5

Fig 4 The experimenlal model

Q b r e a k

- Q m Fig 5 The inflow dischargc

172 N.T Thang i VNƯ lourttal o f Sáence, Enrth Sciences 25 (2009) 168-176

N u m e r ic a ì m o d e l:

T h e data structurc o f the com putationa!

m e sh e s and g e o m e try needed for the numerical

m o d e l d e v e lo p c d here is com p lete ly the sam e as

th e o n e d e s c rib e d and used in the model

m e n tio n e d in [5] S o m e im portant features are

ab strac teđ here: th e n u m b e r o f unstructured

m e s h e s is 4 9 9 6 ; the m e sh es o f streets are very

fin e b u t th o se o f building blocks are kept coarse

t o save the tim e n eed e d for m esh generation and

fo r num erical sim ulation This is

stra ig h tfo rw a rd since vvater d o es not penctrate

into th e se b lo c k s du rin g the experim ents It is

n o tc d here ihat th e com putational m e sh es can

b e v ery Aexible and irregular (unstructured

m e sh cs)

Fig 7 The computed result o f the vvater depth

distribution after 5 minutes

Fig.7 show s the distribution o f the w ater depth com puted in the area and the

d ev elo p m en t o f the o verland flood in the area

a íter 5 minutes

C o m p a riso n s betyveen th e c o m p u te d resuìts

a n d th e m e a s u re d o n e s :

W a te r depths are m e asured a t the points (N o l to N o 8) m e n tio n e d in Fig.3 T he data is

p ro v id ed by the H ydraulic R esearch G ro u p in DPRI T h e s e results are c o m p ared vvith the oncs cornputed by the num crical model The

c o m p ariso n s o f thc vvater depths are shovvn in from Fig.8 to F ie l 1 below

Fig 6 T he com putational meshes

N.T Thang / V N U Ịournaì o f Science, Earth Sciences 25 (2009) ĩ 68-176 173

Fig 10 Comparison o f the vvater depth at point

No.6

Fig 11 Comparison o f the water depth at point

No.8

S o m e r e m a r k s :

- T he model develo p ed in th is study has been successfully applied to the s im u latio n o f the overland ílood process in the experim ent

- T he c o m p u ted r e s u l t s s h o w acceptable agreem ent w ith the m e asured ones S om e

d iíĩeren c es are assum ed to be d u e to the nature o f to o shallow depth o f th e advan c in g fonts o f vvatcr (vvet-dry m o v in g b o u n d arie s) in the ex perim ent T he depth o f th o se íronts is o f the o rd e r o f less than lm m T h e re fo re the suríace rou g h n ess w o u ld not bc the same

e v ery w h erc (a constant value o f the rou g h n css coeíĩic ien t is used in the num erical sim ulation) T h is problem w o u ld need a theoretical trcatm ent in the n u m c rical m odel,

o r need to u s c different v alu es o f the M anning roughness co efficient at the a d v a n c in g front

P ropcr trea tm e n t o f the pro b lem is the subject

o f furthcr study

- T h e d e v e lo p m e n t o f the flood in the area during the experim en t is a ls o c o m p a re d with the o b scrv ed one T he ex te n sio n o f the flooded a r e a in the n u m e rical sim ulation agrees vvcll vvith that in the experim ent

- T he n um erical m odel deals vvell w ith very irregular g eom etry an d vvet-dry

m oving /v ary in g boundaries

3.2 E xp e rim e n t o f a flo o c ỉ in ío a City a r e a in the

fr a m e w o r k o f th e C A D A M (E u ro p ea n

C o n c e rte d A c íio n o n D a m -B re a k M o d elin g )

p r ọ ịe c ỉ (exp e rim en t p e r fo r m e d in Iíaỉy')

D e sc rip tio n o f th e e x p e rim e n ta l m odel:

T he experim en tal model set u p rep ro d u ces a 5km reach o f the T o c e R iv e r in Italy (F ig 12)

T here are íloodplains, reservoir, structures, and buildings etc in this area T h e scale betvveen the experim en tal m odel and the real site is 1:100 T he scale o f the ex p e rim e n ta l m odel is

5 5 m x l 3 m [10] F ig l2 shovvs th e overvievv o f the m odel g eom etry and to p o g rap h y T he experim ent sim ulated a flo o d causcd by a

174 N.T Thang / V N U Ịournal o f Science, Earth Sciences 25 (2009) 168-176

reservoir dam break in the upstream area o f the

modeled site (left hand side in F ig l2 ) T he

flood vvater tlovvs into the m ode led site through

the A D boundary (Fig 15)

*

Fig 12 A n im ag e o f th e ex p e rim en tal m odel taken

from a D TM (D igital T errain M o d el) (F ig u re from

[10])

In Fig 13, the gauge positions for measuring

w ater depth in the experim en t are shovvn The

M anning coefficient in the experim en t is

determ ined to be 0.0162 T he experim ent starts

vvith the dry bcd condition in the w hole area

T he discharge o f the ílood vvater ílovving into

the area is also C o m p u te r controlled as the

previous experim ent in Japan T he discharge

curve is presented in Fig 14

Fig 13 G auge p o sitio n s fo r m casu rin g the vvater

depth (F ig u re íìo m [10])

F ig l4 show s the discharge o f the flood

w ater flowing into the experim ental model site

during the experim ent A total am o u n t o f about

18.4 m3 o f w ater ílovvs into the area during the

experiment

D isg h a rg e

- D.scharge T im e [s]

Fig 14 T he d ischarge o f t h e flood vvater invading

into th e experim en tal m odel site

N u m e r ic a l m o d e l:

T h e experim en tal area is div id ed into 14651 quadrilateral e lem en ts (com pu tatio n al meshes) and 15000 n odes (the total n u m b e r o f all vertexes o f the quadrilatcral elements) The elem ent size is 1 4 c m x l4 c m In this case, the

to p o g ra p h y is not too co m p lic a te d so that, for

co nvenience, w e used quadrilateral elements A structured-curvilinear m esh genera to r package

C C H E M esh G en erato r [12] is uscd to generate the co m putational m eshes T h e m e sh es can be generated as fine as w e want Ít can be seen in Fig 15 th a t the m e sh es generated are really fine

so that they can reconstruct vvell the com p lica tcd topog rap h y o f the experimental area

X

Fig 15 T h e co m p u tatio n aỉ m esh es g en e rated using

the C C H E m esh g en erato r

In F ig l5 , A D is the inflow boundary; AB and C D a re the rigid boundaries and BC is the free outf!ow boundary

C o n ip u íe d resulís:

T h e co m p u te d results o f the w ater depth are

c o m p ared vvith the m e asurcd o n es provided by

C A D A M prọịect T h e results o f the

c o m p ariso n s are shovvn in from Fig 16 to F ig l9 below

ỡeptn lem)

A <

• • » v *• ’ - V

- s * * ' * : ■ ■*> - • •

_ • h P 4 m s r — — h P 4 c c r o p ị

Fig 16 C o m p ariso n o f th e w a te r depth at point

N o P 4

N T Thang / V N U Ịournaỉ o f Science, Earth Sciences 25 (2009) 168-176 175

n S G O m s r h S G O c o r n p l

Fig 17 C o m p ariso n o f th c w a te r depth at point

N0 S6D

Fig 18 Comparison o f the water depth at point

N0.S8D.

Some remarks:

- T h e co m p ariso n s shovv th at th e com puted

vvater d ep th s ag ree q u ite w ell w ith the

m easured o n es M o reo v er th e arriv al tim es o f

thc ad v a n c in g íro n ts (d isco n tin u ities) are

m odeled fairly ex ac tly T h is sh o w s the

a d v an tag eo u s featu re o f th e G o d u n o v -ty p e

schem e

- T he d ev elo p m en t o f t h e ílo o d o v e r w e t - d r y

bcd w ith com p licated to p o g rap h y has been

reproduced

- U sin g th e m o del, o v erla n d flo o d s cau sed by

dam /dike break o r o v erto p p in g into areas vvith

differen t ty p es o f stru ctu re s c an be m odeled

p ro p erly

4 C o n clu sio n s

A C o m p u t e r m o d e l f o r t h e s i m u l a t i o n o f

overland ílo o d s in c i t y / u r b a n a r e a s w ith

co m plicated to p o g rap h y /g eo m etry h a s been developcd A nevv d iscrctizatio n tech n iq u e has been ap p lied in the m odcl T he m odel exploits adv an tageo us features o f a G odunov-type num erical schem e an d th e R o e’s ap prox im ate

R iem ann so lv er vvhich is o rig in ated in aero dy nam ics T his sch em e deals w ell vvith hyd rau lic d isco n tin u itics in o v erlan d flood flovvs vvhich are caused by dike o r d a m breaks

T h e m odcl uses ílex ib le co m p utatio nal m eshes vvhich are u nstru ctured m eshes T h e re íò re thc

m odel can be ap p lied to problem s vvith irregular geom etries T he m odel has been ap plied to sim u lation s o f tw o ex p erim cn ts o f overland íloods in city areas in Japan and Italy The

co m p uted results agree vvell w ith the m easurcd ones T he treatm en t o f vvet-dry and m oving bou nd aries im plem ented in the m odel does vvork properly T h e m odel is highly prospective

fo r stud ying o verland floods in practical cascs

in real city areas

A ck n o w led g em en ts

T he au th o r is g rate íu l to the H ydraulic

R esearch G ro u p in DPR1 for p ro viding their exp erim ental results T h e au th o r also thanks Prof N g u y en V an D iep at the In stitu te o f

M ech an ics, V A S T , vvho has been actively leading the research on d am /d ik e b reak and

o v erlan d flood p ro b lcm s in Institute o f

M echanics, for p ro v id in g th e experim ental resu lts from C A D A M project

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http://www ncche.olemiss.edu/softwarc/downloa

ds, National Centcr for Computational

H ydroscience and Engineering, T he University

of Mississippi