DSpace at VNU: Finite volume method for long wave runup: 1D model

A nalytical ap p ro ach p rovides exact solution for idealized situ atio n of geom etry settings suitable for practical applications [9]... C om parisons are carried out betw een sim ula

V N U Jo u rn a l of Science, E arth Sciences 24 (2008) 10-15

Finite volume method for long wave runup: 1D model

Phung Dang Hieu*

Centerỷor Marine and Ocean-Atmosphere ỉnteraction Research Vietnam Institute of Meteorologỵ, Hydrology and Environment

R cceived 20 D ecem ber 2007; receivcd in re v isc d form 15 F e b ru a ry 2008

A b stra c t A n u m e ric a l m o d cl u sin g the 1D shallovv vvater e q u a tio n s vvas d ev o lo p cd for th e

sim u la tio n o f lo n g vvave p ro p a g a tio n a n d ru n u p T he d e v c lo p e d m o d el is b a sc d o n th c F inite

V o lu m c M c th o d (FVM ) vvith an ap p lica tio n o í G o d u n o v - ty p e sc h e m e of socond o rd e r of accuracy

T he m o d el u s e s th e HLL a p p ro x im a te R iem ann solver for th e d e te rm in a tio n of n u m e ric a l ílu x cs at

cell in teríac cs T h e m o d el w a s a p p lie d to th c sim u latio n of long w a v c p ro p a g a tio n a n d ru n u p o n a

p la n e b each a n d sim u la tc d re su lts w ere co m p ared w ith th e p u b lish c d c x p e rim e n ta l d ata T he

c o m p a riso n s h o w s th a t th e p re se n t m odel h as a povver of sim u la tio n o f long vvave p ro p a g a tio n a n d

r u n u p o n b ea ch cs.

Keỵivords: P in ite V oỉu m e M ethod; Shallovv W ater M odel; YVavc R u n u p

1 In tro d u c tio n

Long w ave ru n u p on beaches is one of tho

hot challenging topics recently, for the ocean

related to the sim u latio n or d eterm ination of

w ave ru n u p in gcneral, and long vvave ru n u p

in p articular for practical purposes, such as

design of sea w all/ Coastal structures, etc

Thereíore, d e v e lo p m e n t of a good m odel

capable of sim u latio n of w ave ru n u p is vvorth

for practical u sa g e as vvell as for indoor

researches

analytical and n u m erical m odels based on the

dep th in teg rated shallovv vvater equations to

* Tel.: 84-4-7733090

E-mail: phungdanghieu@ \'kttv.edu.vn

analytical resu its include tho one-dim ensional solution of C arrier and G reenspan (1958) for periodic vvave reílection from a plane beach [1] an d the asym m etric solution by Thacker (1981) [6] for vvave resonance in a circular

p ro v id ed v alu ab le experim ental data of long

w ave ru n u p on a plane beach, w hich then vvere well k n o w n am o n g Coastal engineering com m unity, w h o do the job related w ith num erical m o d elin g oí Coastal h y d ro d y n am ic processes A nalytical ap p ro ach p rovides exact solution for idealized situ atio n of geom etry

settings suitable for practical applications [9] Hovvever, the m ain challenge lies in the tre atm e n t of the m oving vvaterline and flow

d iscontinuity vvhon the w ater clim bs u p and dovvn on beaches

10

Phung Dang tìieu / V N U Ịounuìl o f Science, Earth Sciences 24 (2008) 10-15 11

So far, m any researchers have dcveloped

m odels for the sim ulation of w ave ru n u p

Shuto and Goto (1978) [4] u sed íinite

diííerence m ethod w ith a staggered schem e

and a Lagrangian description of the m oving

shoreline; Liu et al (1995) m odelcd ru n u p

through ílooding and d ry in g of the cells in

response to adịacent w ater level ch anges [3]

Titov and Synolakis (1995, 1998) [7, 8]

proposed VTCS-2 m odel using the splitting

technique and characteristic line m eth o d Hu

et al (2000) [2] developed an 1D m odel using

FVM vvith a G odun o v -ty p e u p w in d schem e

to sim ulate the w ave o v erto p p in g o í seawall

VVei el al (2006) p resen ted a m odcl for long

w ave ru n u p using exact R icm ann solver [9]

In this study, a num erical m odel is

approxim ate Riem ann solver HLL (H arten,

Lax and van Leer) for the sim ulation of long

w ave ru n u p on a beach The m odel is veriíied

for the case of experim ent p ro p o sed by

Synolakis (1987) C om parisons are carried out

betw een sim ulated results an d experim ental

data (Synolakis, 1987) [5] The details of this

stu d y are given belovv

2 N um erical m odel

2.1 Governing equation

dim ensional (1D) d ep th -in teg rated Shallovv

w ater equations in the C artesian coordinate

system ( x ,t ) The conservation form of the

1D non-linear shallovv w ater eq u atio n s is

vvritten as

as follows:

a u ỔF e

— + — = s

where u is the vector of conserved variables;

F is the ílux vectors; and s is the source term

The explicit íorm of these vectors is explained

H iỉ+ ịg rí1 , s =

0

g H - A

& p.

(2)

w here g : gravitational acceleration; p : w ater density; h : still vvater d epth; H : total w ater depth, H = /i + ^ in vvhich Tj(x,t) is the

displacem ent of w ater suríace ÍTom the still

w here n: M anning coefficient for the bed

roughness

2.2 Numerical scheme

T he íinite volum e ío rm u latio n im poses conservation law s in a control volum e Integration of Eq (1) over a cell vvith the applieation of the G reen's theorem , gives:

w here í ì : cell dom ain; r : b o u n d a ry of Q ; n : norm al o u tw ard vector of the bou n d ary

T aking the tim e integration of Eq (4) over

+ jdf Ị F-nár = |áíJ^SáQ

C onsidering the case of one-dim ensional

m odel w ith cell size of A x , from Eq (5) we

can deduce:

— í ư ( x , / 2> ừ — — [ U ( x ,f ,V ic

AxAt i v AxAt l v

x < " 2 x ' 2

(6 )

Ax

1 '» * *

=— — ịdt f Sdx

ố x ầ t Ị L

' 2

12 Phung Dang Hieu / VN U Ịoumal o f Science, Eữrth Sciences 24 (2008) 10-15

A x

N ote that th e integral —-— í u ịx ,L )d x

AxAt JAl v 2 r

t2, d iv id ed by A t The p rescnt m odel uscs

u n iío rm cells w ith dim ension Ax, thus, the

integrated g o v ern in g cq uations (6) vvith a

tim e step At can be ap p ro x im ated vvith a half

tim e step av erag e for the interíace íluxes and

source term to becom e ị :

2 Uí*1 = u ? - ■ £ [ ! ? : / « -I? _ v « ] + A ts r ,,í (7)

w here i is index at thc cell center; k denotes

the cu rren t tim e step; the halí indices i + 1 /2

and i - 1 / 2 in d icate thc cell interíaces; and

k + 1 /2 d en o tes the average w ithin a tim e

step betw een k an d k + 1 N otc that, in Eq

(7) the variables u and source term s are

cell-averaged v alu es (vve use this m eaning

from now on)

To solve th e eq u atio n (7), w e necd to

estim ate th c n u m c ric a l íluxes FấVi/22 and F*-V/22

at the interíaccs In this study, w e use the

G o d u n o v -ty p e schem c íor this purpose

A ccording to the G o d u n o v -ty p e scheme, the

num erical íluxes at a cell interíace could be

obtained by so lving a local R icm ann problem

at the interíace The G o d u n o v schem e can be

expressed as:

(8)

x / t = 0

w here F( ) rep resen ts the num erical flux at

the cell interíace obtained by solving a local

Riem ann problem u sin g the data Uj;1/2 and

Uf+1/2 on each sid e of the cell interíace There

are a n u m b e r of ap p ro x im ate Riem ann

solvers p ro p o sed by d iíícren t authors, such as

O sher, Roe, etc In this study, w e use the HLL

approxim ate R iem ann solver The íorm ula for

the solver is given as:

SL = mi nỊ wL - C L , U “ C #|

SR = max|MR +CR,U + C ’ Ị

u, +u

Q* - t-

2

(9)

(10)

(11)

(12)

(13)

w h ere F* denotes the HLL approxim ate

R iem ann solver; UL and UR are respectively

the d e p th averaged velocities of w ater flow at

left a n d right side of the cell interíace; CL an d

CR are the shallow w ater vvave speeds at left

and right side oí the interíace

In this study, vve used three regions of

w ave specd to estim ate the cell interíace íluxes as follows:

F, SL z 0

SR <: 0

To get a second o rd er of accuracy for the

num erical m odel, U ^ 1/2 and U *l/2/ UL and

UR , CL and CR are in t e r po l a t e d by using a linear reconstruction m cthod based on the averaged values at ccll centers vvith the usage

of the TV D-type iim iter, w hich is the average

of M in-m ode lim itcr and Roe lim iter For the

w et and dry cell treatm ent, w e use a

m inim um vvet dcpth, the cell is assum ed to be dry w hen its vvater d ep th less than the

m in im u m vvet d e p th (in this stu d y w e choose

m in im u m vvet d e p th of 10 5m)

3 S im u la tio n resu lts an d d iscu ssio n

3.1 Experimental coĩĩditiotĩ

A num erical experim ent is carried out for the condition sim ilar to the experim ent done

Phung Dang tìieu / VhlU Ịournal o f Science, Earth Sciences 24 (2008) 10-15 13

by Synolakis (1987) In this experim ent, there

vvas a beach h av in g a slope of 1:19.85

co n n ected to a h o riz o n ta l bottom w ith vvater

d e p th of h = lm The toe of the beach located

at distance x2/h = 19.85 and shoreline vvas at

/ h = 0.3 w as generated at x }/h = 24.42

Corning to the beach from the p a rt of constant

w ater d ep th The ex p erim en t p ro v id e d vvith experim ental d ata of w ater su ríace pro íile at different time Fig 1 show s the sketch of the experim ent

Fig 1 Sketch of S y n o lak is's ex p erim en t.

For the num erical sim ulation, the initial

solitary vvave is sim ulated by the solitary

w ave íorm ula as:

7(x ,0) = 7-sech

h

Ỉ M r

t e ( x - x • u(x,0) = 7 ỹ(x,0 )j£

(15)

(16) The co m p u tatio n do m ain is discretized

into cells in a reg u lar m esh w ith space step

Ax = 0 1 m a n d th e sim ulation is carried out

w ith the initial co ndition given by equations

(15) and (16) S im ulated results of vvater

suríace proíile are recorded for com paring

w ith the experim ental data

3.2 Results and discussion

Fig 2 shovvs the initial free suríace

sim ulated by the num erical m odel

Fig 2 Initial free su ría c e of th e sim u la tio n

05

04

0 3

1 02

h 01

0

-01

0 Eỵọ data (Synoialaỉ, 1987)

— N liti Rssdbt

h

Fig 3 C o m p ariso n vvith ex p e rim e n ta l d ata: n ea r

b re a k in g location.

14 Phung Dang Hieu / VN U Ịoumaỉ of Science, Earth Sciences 24 (2008) 10-15

1 02

/l 01

0

■01

0bp đata {SynoỉalQS 1987]

— Nm Resits

h

Fig 4 C o m p a riso n vvith e x p e rim c n ta l d ata: ru n u p

05

04

03

5 02

h 01

0

•0.1

p h a s e

0 Ejụ đata (Synoỉatas 1967)1

Ị\ i * Ị ị Ĩ Ì - U 1— Njĩ\ResJs

\

0 000 0 ^ 0 0^0 coco oa c

\\J o c

u V Ể L - —»

h

Fig 5 C o m p a riso n w ith ex p e rim e n ta l d ata: ru n d o w n

p h ase

Fig 3 shovvs the com parison betw een sim u lated results and experim cntal d a ta of free su ríace proíile near the brcaking location

It is seen that sim u lated results have som e

d iscrepancy at the vvave crest com pared to the ex p erim en tal data This could be d u e to the lim itatio n of the shallovv vvater equation itselí in sim ulation of vvave dispersion and breaking A íter that, in side the su rf zone,

co m p u ted results agree very vvell w ith the

ex p erim en tal data, especially d u rin g the

ru n u p process on the beach (see Fig 4 at

no rm alized tim e 25, 35, 45) The highest

ru n u p attain s at norm alized tim e of 45 and the h ig h est ru n u p is of 0.5m This result is

ab o u t 1.6 tim es of the initial w ave height The

ag recm en t betvveen sim u lated results and

ex p erim en tal d ata d u rin g the tim c of ru n u p process could be explained as d u e to

e n su re d in the present m odel using the

co n serv ed FVM

For the sim ulation of long vvave ru n u p on beaches, in practice, the m ost im p o rtan t thing

is correctly sim ulated ru n u p process and the

h ig h est clim b u p oí vvater íront A lthough sim u latin g the vvave proíile in the breaking zonc is n ot vvell, the p resen t m odel is still capable of sim ulation of w ave ru n u p process

on the beach, specially the highest ru n u p could be well sim ulated by the m odel This is one of the practical purposes

A t the stage of ru n d o w n (sce Fig 5 at the

n o rm alized tim e of 55), the vvatcr including the position of shoreline and in u n d atio n

d e p th on the beach is still vvell sim ulated

T hus, the dev elo p ed m odel w ith the FVM

p ro p o sed in this stu d y has a povver of

ex p an sio n to a tvvo-dim ensional m odel and is also cap ab le of sim ulation of non-linear w ave

ru n u p , ru n d o w n processes including the

p red ictio n of highest ru n u p of water

Phung Dang Hieu / VN U Ịoum al of Science, Earth Sciences 24 (2008) 20-15 15

4 C onclusions

A FVM based num erical m odel has been

successíully d ev elo p cd for the sim u latio n of

long vvave p ro p ag atio n and ru n u p This

m odel specially well sim ulates the highest

ru n u p of w ater a n d in u n d atio n d e p th on the

beach d u rin g ru n u p and rundovvn processes

sim ulated results and experim ental data

reveals that the m odel has a potential for

practical uses and should be stu d ie d íu rth er

in order to ex p an d to a tvvo-dim ensional

model íor various purposes in praetice, such

inundation on Coastal areas, ílooding d u e to

storm surge, etc

A cknovvledgem ents

This pap er w as com pleted p artly u n d er

íinancial su p p o rt of F u n d am en tal Research

Project 304006 íu n d c d by V ictnam M inistry of

Science and Technology

R eíerences

[1] G.E C arrier, H p G re en sp an , W a tc r vvaves of

íin ite a m p litu d c on a slo p in g bcach, Ịournal o f

Fluid Mechanics 4 (1958) 97.

[2] K H u , C.G M ingham , D.M C au so n , N u m crical sim u la tio n of w a v c o v o rto p p in g of Coastal stru c tu re s u sin g th e n o n -lin e a r sh allo w vvater

eq u a tio n s, Coastal Engineering, Eỉsevier 41 (2000)

433.

[3] P.L-F Liu et al., R u n u p of so lita ry w a v e on a

circd lar island, Ịoum al o f Fỉuid Mechanics 302

(1995)259.

|4] N Shuto, c G oto, N um ericaỊ sim u la tio n of

tsu n a m i ru n u p , Coastal Engineering Ịournaỉ, Ịapan

21 (1978) 13 [5j C.E Synolakis, T h e r u n u p of so lita ry w aves

Ịoum aì o f Fỉui(i Mechanics 185 (1987) 523.

[6] w.c Thackcr, S om e cxact so lu tio n s to n o n lin e a r

sh a llo w -w a tc r cq u a tio n s, Ịournal o f ĩỉu id Mecanics 107 (1981) 499.

[7] v v Titov, C.E Synolakis, M o d clin g of b re ak in g

a n d n o n -b rcak in g long vvave cv o lu tio n a n d

ru n u p u sin g VTCS-2, Ịournal o f Waterwaỵ, Port,

Coastal and Ocenn Engineeríng 121 (1995)308.

[8] v v Titov, C.E S ynolakis, N u m cric al m o d elin g

of t i d a l vvave r u n u p , lournaỉ o f W aterw ay, Port,

Coastal and Ocean Engineering 124 (1998) 157.

[9] Y W ei, x z M ao, K.F C h c u n g , W ell-balanced

F inite V olum e M odel for L ong w a v e ru n u p ,

Ịournal o f W atenvay, Port, Coastal and Ocean Engineering 132 (2006) 114.