Numerical Simulation of Wave Propagation Using the Shallow Water Equations

Transcription

1 umercal Smulaton of Wave Propagaton Usng the Shallow Water Equatons Junbo Par Harve udd College 6th Aprl 007 Abstract The shallow water equatons SWE were used to model water wave propagaton n one dmenson and two dmensons. SWE were appromated b usng fnte dfference method. One dmensonal SWE were tested usng varous ntal condtons. and SWE were used to model tsunam wave propagaton near coast lne. model correctl predcts the behavor of tsunam water wave. model develops numercal oscllaton once the wave reaches shallow area. A real basn data from Gulf of eco has been tested to smulate the wave propagatng n the area. The shallow water equatons descrbe propagaton of water wave whose wavelength s much longer than the depth of water. Tsunams are eamples of such waves. Wavelength of a tpcal tsunam can eceed 00 m. Ths s to be compared wth the average depth of ocean whch s appromatel 5 m []. Therefore tsunam wave propagaton can be modeled usng shallow water equatons. In ths report we use a numercal method to smulate wave propagaton n one and two dmensons. The same scheme s used to model tsunam wave propagaton near a shore. Shallow Water Equatons The dervaton of the shallow water equatons follows from the conservaton of mass and momentum. In dervng the equatons however we must mae an assumpton about the water that s beng modeled. We assume that there s no pressure varaton n the vertcal drecton []. Therefore t cannot correctl model the behavor of a water fall. Ths naturall forces the wavelength of the water wave to be larger than the depth of the water. Furthermore n ths report we neglected frcton between the flud and the basn. However frcton can actuall be accounted for Imamura et al. have successfull modeled the frctonal contrbuton [3]. One dmensonal Shallow Water Equatons The shallow water equatons n one dmenson are = 0 t g = 0. t.a.b

2 where g s the gravtatonal constant s the total thcness of water at and s a quantt defned as product of depth averaged veloct and water veloct n the drecton [4]. Equaton.a and.b are manfestaton of mass and momentum conservaton law respectvel. As mentoned earler frctonal terms have been gnored n our dscusson of SWE. 0.5 Undsturbed Water Surface h basn as Fg. Water level profle of an eample wave The shallow water equatons are coupled frst order dfferental equatons that can be uncoupled to produce two second-order dfferental equatons. t = g h h t = g h g t h where h s the depth of the water the relaton between and h s = h. The form of equaton.a and.b s ver smlar to that of a wave equaton. The speed of and propagaton at gven s therefore determned b total thcness of water c s g = gh..a.b umercal Scheme for Shallow Water Equatons To solve the shallow water equatons numercall we frst dscretzed space and tme. We used fnte dfferences to appromate the spatal and tme dervatves. The tme dervatves were appromated b usng the second-order centered dfferences followng the Cran-colson method. The spatal dervatves were appromated b usng the second-order centered dfferences. For the one dmensonal SWE we used a sem-mplct and mplct method to compare the numercal stablt of soluton. For all smulatons we used spatal grd sze = /8 m and tme step sze t = /3000 sec.

3 t Fg. Spatal and tme dscretzaton. Spatal doman s dscretzed usng ponts = L/. The boundar condton mposed on = 0 and = L are gven as = 0 = ; = 0. 3 where s an appromaton to t. Equaton 3 sets -veloct of water to be zero at boundar. As a consequence / = 0 at the boundar. Sem-Implct ethod Usng fnte dfference method we can appromate equaton as = / cg h = c / 4 c / 4 * * 4.a 4.b where c = t/ the rato between tme step and spatal step. s an appromaton to t. Gven an ntal condton at tme t = 0 ths method can be used to compute and at subsequent tme steps. However ths numercal scheme s not an eplct method but rather a sem-mplct method. To compute we use s from prevous tme step. To compute however we use and whch s obtaned b solvng equaton 4.a for frst. * * Fg 3. Stencl of Sem-Implct method for computng.

4 Implct ethod To ncrease the numercal stablt of the soluton we have mplemented an mplct method on the one dmensonal shallow water equatons. However equaton.b s non-lnear snce = h. In order to mplement an mplct scheme.b had to be separated nto lnear part and non-lnear part. Applng Cran-colson method and fnte dfference method we obtan followng equaton: / 4cgh c / 4 = = c / 4 / cg / 4cgh 5 Left hand sdes of equaton 5 are and terms at tme step. The rght hand sdes are terms at tme step. Values of and at tme step can be computed b solvng a lnear sstem of equatons. Test of Shallow Water Equatons The shallow water equatons n one dmenson were tested wth three dfferent ntal condtons. In all cases the ntal veloct of the water was set to be zero water was at rest at t = 0 and therefore = 0. We vared t = 0 to eamne the results of numercal smulatons. The frst row n Fgure 4 llustrates the three dfferent t = 0 s. The solutons at later tme t are plotted n the second row. The solutons obtaned b usng sem-mplct method and eplct method are plotted on the same graph for each plot. ote that the solutons almost overlap each other. We observe that the soluton to ntal condton n Fgure 4a and 4b develop numercal nstabltes. However these numercal nstabltes are not nherent to shallow water equaton. Equaton s a dfferental form of shallow water equaton whch assumes that the water profle s smooth. Yet the ntal condtons gven are not necessarl smooth. Ths can be understood f we eamne equaton. From equaton whch resembles a wave equaton we can deduce that the soluton to shallow water equaton can be appromated wth superposton of snusods. Therefore the ntal profle of can be thought of as a superposton of snusods. The step functon however can be epressed wth ver hgh frequenc snusods. Those hgh frequenc components develop nto numercal oscllatons. Yet the orgnal shallow water equaton allows for dscontnuous solutons such as a bore [5]. To solve problems wth non-smooth profle fnte volume method should be used.

5 The numercal oscllaton we observe does not arse from nstablt of sem-mplct method ether. Implct method does not mprove the results. The solutons are nearl eactl the same. The numercal nstablt arses from the lmts of fnte dfference method t=0 9.6 t=0 9.6 t= Soluton to Shallow water equaton 0 Soluton to Shallow water equaton 0 Soluton to Shallow water equaton t= t= t= a b c F g 4. Three dfferent ntal condtons: step functon slope snusod are plotted n the frst row. The second row s plots of at later tme step. Results from usng eplct method and mplct method have been plotted. The mamum and mnmum value of t = 0 are 8m and 0m respectvel. The basn depth s 0m for all three cases. a t = 0 and t = b t = 0 and t = c t = 0 and t =. Fnte dfference method assumes that dervatves est at all ponts. Therefore t cannot handle dscontnuous ntal condton such as step functon. It also cannot produce stable solutons for ntal condtons descrbed n fgure 4b snce the dervatve of the ntal condton s dscontnuous. As observed n fgure 4a and 4b numercal nstablt ensues f the ntal condton profle s not twce dfferentable. When gven a snusod as an ntal condton the soluton remans stable for much longer perod. Tsunam wave n The shallow water equatons were used to model wave propagaton near the shore. The basn profle has been appromated b a hperbolc tangent gven as 70 h = tanh 8 0 m 00 m 6 such that the depth would range from 5m to 95m. We made the depth of water to be postve at all

6 snce ver hgh numercal error occurs f we let depth profle be negatve ths would correspond to above sea level. The ntal condton was gven as a Gaussan for and 0 = 0.5ep 0 0 = 50ep 0 /8; /8 = 00 0; 7 was gven a non-zero value to mae the wave propagate to drecton a b Fg 5. a Plot of h the depth profle used n the smulaton. b Intal condton for and 0.0. The result of ths smulaton s plotted n Fgure 6. As the wave approaches the shore and enters the shallower area the wave slows down. Ths s what was epected snce the wave propagaton speed s appromatel c s = g see equaton. As the wave slows down the wdth of the Gaussan wave pacet s reduced. Ths results n ncreased ampltude of the water wave. Soluton to Shallow water equaton Soluton to Shallow water equaton Soluton to Shallow water equaton.5 t=0.5 t= t= depth profle/00m - depth profle/00m - depth profle/00m a b c Fg 6. at varous tme step. The depth profle has been scaled down b a factor of 00 a t = 0. b t =.37 a t = Two mensonal SWE Shallow water equatons n are etensons to equatons. The set of equatons are gven as

7 8.a here s product of. Equaton 8.a s derved om mass conservaton law. Equaton 8.b and 8.c are derved from momentum conservaton l SWE E was ver smlar to that of one a grd. The sze of the spatal doman t me dervatves were 9.a 9.b 9.c alculatng and frst usng equaton 9.b and 9.c. 8.b 8.c 0 g g 0 g g 0 = = = h t h t t w and depth averaged veloct n drecton [6] fr law. Frcton has been gnored n these equatons. umercal Scheme for Two mensona The numercal method used for two dmensonal SW dmensonal case. Space and tme have been dscretzed nto was determned to be a 00m b 00m. The spacng between each grd pont and was se to be m. The sze of the tme step was set to be t = /4000 sec. We used fnte dfference method to appromate SWE. The spatal dervatves were appromated b usng the second-order centered dfferences. The t appromated b usng the second-order centered dfferences followng the Cran-colson method. Equaton 8 therefore can be appromated b c / g c / g / c / c / g c / g / c / 4c / 4c c / h h * 4c / * 4 / = = = Agan the * n equaton 9.a ndcates that the values of and at tme step are from c The boundar condtons are smlar to that of case. and are zero on the

8 - boundares ths s also nown as no slp condton. Water cannot move nto or out of a boundar. It also cannot move along a boundar. Fg 7. Laout of dscretzed grd. The value of at the boundar ponts blue dot are alwas equated to the value of nsde the boundar red dot at each tme step. Smple odel: Hperbolc Tangent Basn The shallow water equatons were used to smulated water wave near a shore. The sea basn was modeled as a hperbolc tangent. The depth profle was gven as 50 h = tanh 0 The ntal condton was gven as 30 0 = 0.5ep = The result of numercal smulaton s shown n fgure 8. The ntal Gaussan profle pans out nto the - plane n all drecton. Bul of the water however travels toward drecton. There are features that are common to the and results. The waves slows down a b

9 a b c Fg 8. umercal smulaton of water propagaton near coast lne. a Soluton at t = 0. b Soluton at t = sec. c Soluton at t = 3 sec as t approaches the shore. ote that the wave front barel moved between t = and t = 3. Also the ampltude of the water ncrease as the wave pacet approaches the shore. Unle the case however we see more severe numercal oscllaton occurrng as the wave approaches the shore. umercal oscllaton s vsble n soluton at t = and t becomes more pronounced as the wave front approaches shallower parts of the shore at t = 3. Implct method has not been mplemented to chec f ths was an artfact of sem-mplct scheme that we used to solve two dmensonal SWE. However we now from the results of SWE smulatons that usng mplct method does not guarantee a stable soluton. One possble source of numercal oscllaton ma be the dscrepanc between analtcal doman of dependence and numercal doman of dependence. In a hperbolc equaton nformaton travels at the speed of wave propagaton. Fgure 9 llustrates the doman of dependence of two nd: analtc and numercal. umercal doman of dependence s determned b the grd spacng and the method used to appromate the dervatves. Analtc doman of dependence s determned b the speed of the wave propagaton c s and the tme step sze. When two doman of dependence dffer from another greatl numercal error s nduced. In hperbolc equatons ths leads to dspersve behavor of the numercal soluton [7]. Waves wth dfferent wavelength travel at dfferent speed.

10 t t c t F g 9. Schematc dagram of analtc doman of dependence and numercal doman of dependence. In ths case analtcal doman of dependence s narrower than the numercal doman of dependence. However as equaton suggests shallow water equatons have the form of a hperbolc equaton. Furthermore c pth of the water: c s s determned b the de s = g. Therefore as the waves approach the shallower regon the analtcal doman of dependence becomes narrower than numercal doman of dependence. Ths results n dspersve behavor whch maes waves wth dfferent wave length travel at dfferent speed. Snce a Gaussan wave pacet s a superposton of man snusods the wave breas down nto dscrete waves travelng at dfferent speed. The result s numercal oscllaton whch s a manfestaton of nterference between varous snusodal waves. Gulf of eco The two dmensonal shallow water equatons were further tested usng a real basn data whch can be obtaned from atonal Geophscal ata Center webste [8]. We obtaned the depth profle of sea basn near the Gulf of eco. The bathmetrc profle was sampled at dscrete ponts separated b appromatel 4000m. The sze of the basn area was 400 m b 400 m. The depth ranged from 50m to m. The raw bathmetr data however could not be used because the profle was not smooth enough. The agged terrans whch s a product of coarse resoluton of grd cell = 4 m was a maor source of numercal nstablt. In order to smooth out the terran and et retan the orgnal profle we appled convoluton to each data pont. h * = S l l where h * s the smoothed depth profle at = and =. S s gven as h l = ep. π 4 S 3

11 Equaton s a dscrete form of convoluton. The result of ths s shown n Fgure 0. a b Fg 0. a Raw depth profle. b Product obtaned after applng convoluton The ntal condton for ths smulaton was gven as = 0.5ep = The results of numercal smulaton are plotted n Fgure. a b

12 c Fg. Wave propagaton n Gulf of eco a Intal condton and sea basn b Soluton at t = 400 sec. c Soluton at t = 550 sec. The lateral rpples are not due to numercal oscllaton. The rpples arse from the shape of the basn. Compare ths wth fgure 8c. The lateral rpples observed n Fgure c are due to the shape of basn. Ths tpe of rpples does not develop when the depth profle s a smple hperbolc tangent gven n equaton 0. Such rpples are not observed n Fgure 8c. The curves observed n Fgure 8c s due to waves that reflected from the = 0 and = 00m boundar. The soluton however qucl becomes unstable as soon as the waves reflected from = 0 catches up wth the orgnal wave n Fgure b the orgnal wave s labeled wth red crcle and the reflected wave s labeled wth blue crcle. Interference between two waves produces numercal oscllatons. Acnowledgement I would le to than Professor Yong for provdng me wth resources and gudng me through out the proect. I than Sean eenehan for gudng me to fnd resources about bathmetr data. I than c Alger for hs dea of applng convoluton to raw bathmetrc data.

13 References [] [] Kevoran J. Partal fferental Equatons: Analtc Soluton Technques. Pacfc Grove Calf. c990. pp 0-9. [3] Imamura F. Yalcner A.C. Tsunam odelng anual raft pp [4] ImamuraF. et al. pp 9 [5] Kervoran pp. -3. [6] ImamuraF. et al. pp 6-7 [7] Yong. Prvate Communcaton Scentfc Computng Lecture 007. [8]