Coastal Engineeringin Japan, Vol. 6,1963

HYDRAULIC STUDY ON

Yoshiro Fukui* Makoto Nakamura** Hidehiko Shiraishi** Yasuo Sasaki**

I.INTRODUCTION Up to present, the visitation of tsunami has inflicted great damageupon our àountry repeatedly.Especially, the fact that the Pacific coast suffered from the serious disaster due to the Chilean Earthquake tsunami. is fresh inour memory. Tsunami is generated in the open as the long with very small steepness and draws toward the shoreline.The coast where the great energy of tsunami is transported without reduction has, generally speaking, fairly deep water depth from offshore to the vicinity of the shore line.The Sanriku Coast, north- eastern coast of Japan, that has been frequently attacked by tsunami has a typi- cal feature as mentioned above.Namely, the submarine ditch with 5,000-8,000 metres of water depth lies along the shore line and prevents the decrement of the energy of tsunami. As far as the two-dimensional deformation of tsunami is concerned, the phenomenon must be considered first.In the case of a tsunami the period of which is nearly equal to the specific period of the bay, the energy of tsunami is conserved in the form of seiche motion.Therefore it is accumulated in making the tsunami height inside the bay larger and larger.Secondly, the reflection and contraction of thunami must be taken into consideration.Owing to these pheno- mena mentioned above, the tsunami concentrates its energy in the most interior part of the bay. And the above phenomena become more conspicuous with the increase of the reflection of tsunami from the bay . On the other hand, inside the bay where the near shore water depth is shal- low, the tsunami is breaking and the reflected and concentrated energy is dis- sipated there.Accordingly the tsunami induces a great force of destruction in this zone. Differing from the , the breaker of tsunami is more akin to bore because ofts very long wave length and large scale.Moreover, in the case that the water depth becomes shallow abruptly, the above tendency is accelerated. On the other hand, the tsunami that approaches to the entrance of bay in the shape of non-breaker transports its energy to the inside of the bay in the shape of flow.In this paper, the tsunami is classified into two types;i.e.the one is the "progressive breaker type" and the other is the "seiche type." There are many difficulties in .establishing the planning of counter-tsunami measures; that is, how to estimate 1) the tsunami run-up height on shore-land and on dike, 2) the quantity of overflow discharge across a dike crown and 3) the tsu-

*Japan Engineering Consultants Co., Ltd. **Agricultural Engineering Research Station, Ministry of Agriculture and Forestry. 68

nami pressure actingon a dike. As to the progressive breaker type oftsunami that is consideredto have the strongest destructivepower against shore land, the authors have analysed itsin. fluence upon dikesexperimentally and theoretically. conventional formulas nd the The authors obtainedthe necessary data to design dikesagainst tsunami. 11. LABORATORY APPARATUS AND TESTPROCEDURE Most of the actual dikesagainst tsunami is zone at very shallow constructed on shore or in inshore water depth.Accordingly, the tsunami those dikes isone of the progressive breaker which strikes against type and may be treatedas a typical bore.The tsunami of thistype has great destructive, tioned.The authors power as Previously men- conducted the experimentalstudy on the tsunami behavior in the shape of bccre.in this experiment size and the other two different water tanks,one is large small, were used inorder to test the scale effect. 1. Laboratory, Apparatusfor Small Scale Tests Thewater tank as shownin Fig. 1 and Photo. 1 one sidewas glassed. was made of steel and its The movable single slopewas set as a dike model and three pressure Plane view gauges were attach. ed on the slope.The bore genera tor of flap gate typewas installed D.M.EtCtnC oscillograph at the other end of thewater tank. / In order to obtain Strain meter many kinds of the length and shapeof bore, the 21.Qm quantity of water stored insi1ethe (c) Sido view (b)Bore generator (a)' flap gate was variedby selecting five positions of theflap gate. The Wave essrire gauges flap gate was fixedvertically by a ,wire rope stretched 'Fig. 1 obliquely to Experimental apparatusfor a small ,ward the storage tank, scale test. and also connectedto two Counterweights through two wires the opposite side ofwater tank in order stretched toward after it is operied. to keep the gate horizontallyin the air V After the waterwas C 1, the flap gate' stored as shown in theside-view of Fig. was opened in a mo- p ment by thecounterweights as soon as the wire fixing thegate was taken off.

2.LaboratoryApparatsgfor Large Scale Tests The water tankshown in ,Fig. 2 and Photos. 2, 3and 4 was used for large scaletests.The system of bore generator,was almost thesame as that for smallscale tests and six pressure gaugeswere attached at intervals of twenty Photo. 1Experimental water tank fora centimeters along small scale test. 69

)lar.eai

Oikm made)

H _en-mailflg omellOgeap Etectnc oscdtognma'

Stesin mate, Stran male,-

Sde nev (a). Bose genta1o. )b) ( I.

3Q.er- 1Wave p,enswe gangeS

I 30.5.' 25.5.- 75.5', FIg. 2 Experimental apparatus for a large Photo. 2Experimental water tank for a scale test. large scale test.

Photo. 3Bore generator for a large scale Photo. 4 Large scale model of dike. test. the slope of dike model. 3.Measuring Apparatus The water stage gauges were arranged as shownin Figs. 1 and 2.' The water stage .gauges installed in frontof the flap gate and the slope recordthe change of bore shape and the velocity ofbore travelling through these measuring points, and that installed inside the flap gaterecords the decrease of water depth in

- r ::

attached on Photo. 5 Recorder of wave gauges (large - Photo. 6 Pressure gauges scale test). the slope (large scale model). 70

the storage tank. The lengthof bore was determined from its twosuc- ceeding troughs.The principle of the water stage gauges isas follows; the change of water levelwas replac- ed with that of electric resistanceof two parallel wires, and thissystem was included in a bridge circuit. A portion of recording system is shown in Photo. 5..Strain gauges wereus- ed as pressure gauges, and Photo.6 shows the reverse side of theslope attatched with pressuregauges. The Photo.7Part of pressure gauges ona recordingsystemof the pressure large scale. gauges for large scale tests is shown inPhoto.7. 4.Experimental Results A sample record of theelectric oscillnr is shown in Fig. 3. aph in the case of small scaletests The curves of (a), (b) and(c) were recorded by thewater stage gauges.The curve (a) of the water ® I2O(g/cm) stage gauge installed inside the flap -1 ,.ReIle, bore 9I(g/crs) gauge gate shows that the water depth t Bore presowe SCa1>*. began to decrease just after &02( the gate r a essu gaug opened and kept low waterlevel sI Pressure gauge until the arrival of the reflexbore. ( Pressure gauge The curve'(b) of thewater stage Cl gauge installed in front of the gate p shows the shape of the initialbore. f Ware gauge The curve (c) of the 'j15 water stage gauge installed, in front of the. slope shows first the shape of theincident Fig. 3 Sample recordsby electricoscillo- graph (small scale test). bore and next theduplication of the incident and reflexbore with

an ts bo in fir in Photo. SIncident cr breaking borein a Photo. 9 small scale tank. Reflection of bore afterrun- w ning up the slope ina small ar sccale test tank. m 71

Photo. 10Reflex bore. Photo. 11Run-up on the slope in a large scale test tank. the lapse of time.Photographies 8, 9 and 10 show the situation mentioned above. Figure 4 is a sample record of water stage gauges by a pen-writing oscillograph in the caseoflarge scale tests and has the same charac- C Sec - I Sec - teristics as in the case of small scale tests.Photographies 11 shows the Fig. 4 Sample records of wave gauges by run-up phenomena of bore on the pen-writing oscillograph(large slope in a large tank.The curves of scale test). (d), (e) and (f) in Fig. 3 and the six - a curves in Fig. 5 were recorded by pressure gauges.Itisconsidered a from these data that bore pressure a operated impulsively on the slope at the moment when the bore struck ,1 the slope, and kept almost constant value continuously.The height of bore run-up was measured with the 0 1 2 3 4 5 naked eye. Fig. 5Sample records of wave pressure byelectricoscillograph(large scale test). III. VELOCITY OF THE BORE TYPE TSUNAMI In order to explain the hydraulic and dynamic characteristics of the tsunami which Is classifiedas the.. bore type, the velocity of bore and induced flow must binvestigated first of all.The symbols as shown. in Fig. 6 are adopted and the two cross-sections, II and between which the head of bore is included, are determined. The following for- mulas are introduced from the continuity condition of flow. UHüh=CC cUHüh in which C and U are shown in thefollowing formulas. C=Hh (2)

H Uand ü in. the above formulas present therespective mean values of U and u. Euler's law of iiiomentum is adoptedin regard to the portion of fluidbetween the two cross-section, I-I and 11-11.The following formula taking inthe direction of flow is led. U2dS+ççpu2dS .-555pUdV+55 PSdV}_cc S2 PdScc rdl (4) =55+52 where p is the density of water, ris the frictional stress of bottom,V is the capacity and S is the area.Each term of Eq. (4) is simplified asfollows: ,i555 pUdV=r1pUr1PCHU

where 1 crc-dV=1, ,c5) U pud V av2 at555v2 where

51pU2dS=i1pU2Si=iiiPU2H 555 ( where iccIU\2dS

55 where

and C 5pdS=jwoH2 5 pdS=_4w0h2

of fluid.Though the last term on theright-hand where wo is the unit weight following formula side in Eq. (4) can not bediscussed easily, the authors lead the using the frictional stress. r=pu*2=pP/(AT+_log)} (9) of' bottom, U" is the friction velocity,k is the where r is the frictional stress determined value showing the degree of roughness onbottom, z is the parameter by the scale of flow (the waterdepth etc.) and A is a coefficientof distribution The area of integral, l, mustbe determined in correlation to of flow velocity. The following formula is h, H and the coefficient of theroughness of bottom, n. led by using the mean value of r inthe distance of 10. 55 tdl4/(Ar+ - log where l0/1 2.3 z\2 1)/ A+ - logT) (10) ean be determinedby using water depth, Therefore, it is considered that 7) This h; height of incident bore, C,and coefficient of roughnessof bottom, n. correlation is shown by

These simplified expressions areput into Eq. (5).Therefore the equation of mo- mentu.m is _0pH1J2 ripCCH_r2puchThPt12 + ij2pi2h =(i!)(H2 h2)

If 1il10=7 is defined, then C(T1LIH_T2uh)(7U2H_112u2b2)(_)(H2_h2)

Equation (1) is put into the aboveequation, and hence the expression ofU is reduced as follows:

h(1+) U2+ (13) 2(iHoC) - I2(jHrjC)) 2H(tiH) From Eq. (1) and (13), the borevelocity is obtained as g2H(H±h)_2hH(T2h+V2C)ui2 hH(T1+T2) )- /( hH(1+2) 2_ 1 1 h;U ± U + 2(71HijC) V 12(1Hi,'C) 2(OHC) (14) reduced from Eqs. (5),(6),(8), If Ti=To=" and 1)21, the following formlas are 74

(13), and (14). hü±/(__/\2 q2(H+h)_2h2 H -v H)+.2H(H) (15) h hHü \ 2gC2H(H+h)_2hH22 (16) 2(H) } Ifij is defined,Uand c can be calculated by using Eqs.(15) and (16), respectively. As for the signs, positive signis used for the case that bore advances inthe direction of flow and negativesign is used, in the opposite direction.From Eqs. (7), (8), (12) and (13),j is defined as follows:

=*(*)8dH_F(n, -) (17) The first term of the righthand side is the value showing thedegree of water disturbance occured by theadvance of bore.Therefore, it may be named the coefficient of disturvance.The second term .is a coefficientshowing the resistance of bottom. The former isnearly equal to zero in case when thewater is disturb. ed sufficiently.That is, if h/H-+ 0, then

1 (18)

The value of the latter' becomeslarger in proportion to the scale ofdisturbance, i.e. as h/H becomes smaller,or it becomes smaller in proportion to the height of bore.That is, if h/H-p 1, then

F(n,4H)-_0 (19) Here, if the coefficient of disturbanceis considered to be nearly equalto 1 and the resistance of bottom is neglected,is also nearly equal to 1. by And C is given

gC2H(JI+h) C=--{Cu±H2ü2+ H2ü2 =u± J-J(H+h) (20)

This formula is identical with theusual formula of bore velocity.Equation (20) is the approximate formula incase ofij 1. Hence, it has the followingcontradiction. lim C -+

This means that Eq. (20) is not appliedto all cases, and hence the limit ofrange r of h/H whithin which Eq. (20) is applicablemust be given.Especially,it is un- reasonable to apply Eq. (20) to ,because the tsunamis have generally small value of h/H.However, in case of h- 0, Eq. (16) is reduced to r t Jim I h-O 2(1v) That is, it has the finite value and is applicable to all cases.Moreover,ij isgiven experimentally. V In this experiment, ü is neligible becauseof ü = 0.Therefore Eq. (15) and (16) may be reduced 1: to the following forms t 75

U-c/g(H+h) -V2H(Hrjc) / gH(H+h) C_V2(H) velocity Equations (15),(16),(21) and (22) are the improved formu1s of bore revised by the authors and are generalexpressions including the usual formulas. The value of i obtained by theexperiment is shown in Fig. 7. The plotted data are obtained by the least square mean method.Figure 8 shows the comparison between

2

0/ 0 013

roughness

8 I

6

5.,, A ( -

Fig. 7Relation between resistance Co. Fig. 8Theoretical and experimental values efficient Tj and h/H. of bore velocity. the experimental data. and the curvescalculated by Eq. (22).The data in the case of b=20 cm are plotted below thecalculated curve. From this fact it may be expected that the influence of the steelside-wall of water tank upon bore velocity becomes larger because of increaseof water depth.This fact is proved by the data of the large scale experimentin which the influence of side-wall isnegligible. Hence, it can he said that thetheoretical formula agrees quite well with theexperi- mental data in general. IV. RUN-UP OF BORE TYPE TSUNAMION DIKES After striking the dike, the bore runs up onits slope. Then the kinetic energy of borechanges itself into the potential energy.If the velocity of water particle was observed microscopically,it may be recognized that the phenomena moment when are very complicated.Especially the motion of .water particle at the bore strikes the slopeis quite complicated.Though it is difficult to observe microscopically the energy of water particle,it may be considered that the veloci- ty of water particle has arelation to the microscopic value of borevelociry, C. Hence the following relation isassumed here: vm=aC (23) where a is a non-dimensional coefficient (a>1).If the flow is perfect fluid, the horizontal kinetic energy of bore strikingthe slope changes itself into the poten- tial energy. The vertical height of run-upfrom still water level, R, is written by 76

R 2q (24) symi and As the energy is dissipated inreality, Eq. (24) is multipled the coefficient k. by a non-dimensional That is, ing I k' k'a2 R=v____C22 2g In this equation, Mor tion k=k'a2 (25) and k is determined II is as an experimental coefficient.Hence R is rewittenas R=jC2 (26) Figure 9 shows the experimentalvalue of k.If Eq. (16) is introducedinto Eq. (26), the following formulais obtained; Whei k R f h -±/(hHü_ \ C2H(H+h)_2hH1j2 head 2g (Hri) U_VH_) + (27) 2(Hr) E -If u=0, R_k gH(H+h) kH(H+h) Zg 2(H) - 4(H) (28) Figure 10 shows the comparison of the experimentaland calculated value of R. The relation between slopeand R dependsupon k of Eq. (25).As k' is a factor introduced owing to the effectof energy dissipation, it is smaller than1 and de- 3 creases in proportion to slope,a is larger than 1 as it is shown by Eq.(23).

1.0 12 14 V. CLAPOTIS HEIGHTOF REFLEX BORE FIg. 9 Relation between k andslope Striking against the slope of dike, of dike, bore begins torun up the slope.In the - wake of this, reflex bore 2' appears and the water depth in frontof the slope in- : : MIII creases in size due to the fact thatthe 6 incident bore fallson the reflex bore. 4 This compositewave advances in the IIfl reverse direction in the shape of bore. $RIP!!iIIRI Since the velocity of reflexbore depends I Otisupon the velocity of incident bore,clap- '.1VIIRRIIIRI0.2 0..'.08' .. .1 .1, ' has a large in R an initial .-. period of the arrival ofincident bore. Fig. 10Relation between C and R. Accordingly, since the longperiod wave such as tsunami keeps height for a fairly long a large clapotis time, a large quantity ofwater overflows crown the height of which is across dike lower than the clapotiscrest height.Here, the 77

symbols as shown in Fig. 11 are adopted foruse and the condition of continuation of flow between the cross-sections I and Ills shown by the follow- ing formulas; OI Coo To Co U0- = H- = - -:--- - Moreover, the equation of momentum for thepor. tion of fluid body between the cross-sections I and II is as follows; Fig. 11Reflex bore.

-j- pUodV- jV2 PUdV} pUo'dS- (Ho2_Jj2)_rdl

Where Vi is the volume from the head of boreto section I and V2, from the head of bore to II. Each term is simplified as follows:

pU0UH0H pUOHS

Co Co pUdV=pUL=pUCOH_PU2H2 8tV2 Co pUo°dS =,iiPtj2Ho=i)jp.U2 ) JH0 where

1.f and irr 'U'° v=:HHU ()dS As for the last term of the right hand side, cc Ddl=sPHouo2=3Pçoo J .100 where /H j8Fn,-j-- Therefore the equation of momentum is simplifiedas follows; PU2HI 2 where

Moreover, the equation is rearranged as follows: pU2{2H2- 7)C0'H2 712CoH WQCO(H,H2)

_2H2 (._1)} =i(___,H0 \rfJ0 tb-) -'} Here, the ratio of (Ho/H)_m is defined and named"the ratio of duplication." The above equation is rewritten by usingm as follows;

o(m_1)} =i(m_1)(m2_1)

gH -j-m(ml)(m2-1)+,j(m-1)+rj2m(m_1)__2m=0 (29) If

2m(m.4)(m2_1)+m2_2m_1O (30)

SinceUand H are the known values, m can be obtained from Eq. (30) and Ho is given by H0 = mH. Figure 12 shows the relation between gH/2C2and m. The the- oretical curve is calculated by Eq. (30), hence the influence of bottom friction is neglect- oSrnall scale test ed.It may be proved easily 0 \ Sao Large scale test 00 that each of Eqs. (29) and (30) , has only one real root under 00 the condition of m>1. V E e VI. PRESSURE OF BORE /m100o,0 TYPE TSUNAMI In thispaper,the bore II pressure is classified into the the "impulsive pressure" and Fig. 12 Calculated and experimental values oftio the"continuouspressure." of duplication. The former is the force which operates impulsively on the slope of dike at the same time when bore strikes againstthe slope and the latter is that which operates continuously and statically whenthe water depth in front of dike rises up due to that the reflex bore fallson the incident bore. 1.ImpulsIve Pressure It is considered that the factor which is closely relatedwith impulsive pres- sure is the velocity of bore.Therefore, the relation between impulsivepressure Pk and C is plotted in a logarithmic section-paper as shown in Fig. 13.Though the data obtained by large scale tests are separated from thatby small scale tests to some extent, the slopes of two straight lines are about 4 incommon with each other. On the basis of this fact,it seems that the exponent of bore velocity which is related with impulsive pressure is about 4.After this is recognized as the experimental facts, the formula forP is led from dimensional analysis.. The 79 maximum impulsive pressure on the slope 100 80 Onelargescrnle of dike is represented bypa and the fac- 60 are selected tors which are related to p,..,, 40 1iieon a smal scale as follows: 12 p=f(C4pgPCT) (31) .20 o0 Hence, 00 6 0

0.1 0.2 0406 1.0 2. Moreover, C' (rn/sect pK0- (32) Fig. 13Relation between C and P.

whereK0 is a non-dimensional coefficient named "thecoefficient of wave press- ure."Figure 14 is obtained by rearranging the data of Fig. 13.In Fig. 14 the difference between the data obtained bysmall 3 scale experiments and that by large scale ex- .2 .O a targe scat1 periments disappears and the theoretical line value 10'OQo a small scat_____ is fit for the experimental data. The of Ka is obtained from Fig. 15. 'S The distribution of impulsive pressureis 'a shown in Fig. 16 and the maximum pressure is appeared nearly at still water level.The formulas of the distribution of impulsive pres- 8 sure are obtained as follows; 0 '8 R, Pa (33) 0. 02 04 0.6080 20 40 6080100 1 A C- Porn

Fig. 14 Theoretical and experi- pa=(1_A_)Pans mental values of impul- sive pressure. where Pa is the impulsive pressure at the point of height R9 above still water level and A is anon-dimensional constant (Co1.4) determined experimentally. From Eqs. (26), (32)and (33), the following formula is led.

02 04 06 0.8 10 2 1.4 Slope -

FIg. 15Relation between ko and slope. 80

2. 2R) 0 PkK0_woC4 A 0 = g2C kC2 o0 0 (34) 0000 : 0 li: 2.Continuous Pressure ° ° g 0 Since the pressure of this case :0008.00 '00 is nearly equal to hydrostatic pres- 04 06 sure of the clapotis height of reflex - 0.0 1.0 bore, the distribution of continuous Fig. 16Vertical distribution of implusive pressure is shown in Fig. 17 by using pressure. the clapotis height of reflex bore, Ca. 2. The formulation of Fig. 17 is

--=1B- (35) WoCa 0 Ca "0 where p, is the continuous pressure 00 . O C 0 0O 0* at the point of height R9 above still 0 0C .. ,.0 0 )0/ :000 0 0oo:. P water level and B is a non-dimen- 0 sional coefficient (=0.75) determined 02 04 06 0.8 1.0 pup1. experimentally.Therefore, Fig. 17Vertical distribution of continuous (36) pressure. Ps=wo(Ca_fRs)

VU. QUANTITY OF OVERFLOW DISCHARGE ACROSS DIKE CROWN As mentioned previously the water depth in front of dike increases consider- ably due to that the reflex bore falls on the incident bore.Here the duration of the increased wave height is taken into consideration.It seems that the lapse of time, T, during which the height of reflex bore, Ca, decreases into the height of incident bore, C, is in proportion to the length of incident bore, 2, and in inverse. proportion to the velocity of reflex bore, C0. That is,

(37) where K is a non-dimensional constant.Moreover the following formula is led by using the relation of Co= UHICO3 in which UH obtained from Eq. (1); ih+CC CaC If ü=O,

Co -CC CaC Introducing the above result into Eq. (37), the following formula is obtained; K2(CaC) TcK2/( CaC) _KT(_1)=KT (38) where K is aQn-4imensiQnal coefficient (0 2/3) determined experimentally. The

.-. .. -.,- .-.- 4 81

Fig. 19 Overflowdischarge ofbore typetsunamiacrossdike crown.

experimental data and the corresponding values cal- culated by using Eq. (38) are plotted in Fig. 18. Next, the quantity of overflow discharge across a dike crown is estimated under the conditionof C 02 04 0.6 0.8 1.0 HHo>H or Ha.(H

where p (1)is a coefficient depending upon the bore shape. The quantity of overflow discharge for dt hours is given by dQ= ./EHw8/2dt where dt=-dH Therefore, the quantity of overflow discharge by a bore per unit length of dike, Q, is calculated by 2 4r-ET H0h/2=2%'2Tc(HoH4)°'2 çwOHS/2dH pCo jo 5p HoH 39

VIII. CONCLUSIONS From the standpoint of preservation of shore structures, the authors classify the tsunamis into two types as previously mentioned and have analysed theoretically and experimentally the behaviour of bore type tsunami.Especially, the scale effect has been studied by using twc kinds of wave tank, one is small size and the other large size. The hydraulic problems on tsunamis are under the rule of Froude's similarity law.The small scale model study has been carried out under the con- cept of Froude's similarity law, and the factors of viscosity and viscosity, 82 which seem to be the important factors next to the gravity and inertia forces, are considered in the form of bottom friction.Although the necessary data can be obtained by the small scale experiment, the large scale .experiment has also been carried out in order to test the scale effect on the air mixing phenomenon at the head of bore and to increase the number of measuring point of bore pressure. Figures presented in this paper show the identity of results of both experiments. Accordingly, it seems that the small scale experiment is satisfactory to solve this kind of hydraulic problems. In the design of actual dikes, nothing can be obtained as the data of actual tsunamis except the traces left behind the tsunami attack.Therefore, it is very important to make a decision whether the traces show the height of incident tsu- nami, or the clapotis height of reflex tsunami, or the height of run-up on shore land, considering the features of fore-shore and hinterland.For instance, on the coast where there is a mountain close to the shore line, the traces show the cia- potis height, while on the coast with a gentle slope the traces show the run-up height.In the case of tsunamis invading the inner part of hinterland, the traces near shore line may show the height of incident tsunami.