IMPACT CHARACTERISTICS OF FREE OVER-FALL IN POOL ZONE WITH UPSTREAM BED SLOPE

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1 476 Jornal of Marine Science an Technology, Vol., No. 4, (4 DOI:.69/JMST IMPACT CHARACTERISTICS OF FREE OVER-FALL IN POOL ZONE WITH UPSTREAM BED SLOPE Shi-I Li, Jen-an Chen, ao-ming Hong, Hng-Shin Hang, an Rajkmar V. Raikar 3 Key wors: oen channel flow, Dro strctres, free over-fall, imact characteristics. ABSTRACT This aer resents the theoretical eations for the imact characteristics of free over-fall (ICFOF in the ool zone with sloing stream e. The varios imact characteristics consiere are the rink eth, the imact osition an imact angle, the tail water eth an the eth of water ool. In orer to overcome the comlexity in etermining the ICFOF, they were exresse in imensionless forms as fnctions of stream e sloe (UBS, stream Froe nmer an Dro nmer. The mltile regression analysis was se to otain the emirical exressions for the ICFOF sing the laoratory exerimental reslts. It was fon that the imensionless rink eth ecreases with an increase in stream Froe nmer an UBS. With the increase in oth the Dro nmer an UBS, the imensionless imact osition increases, while the imact angle ecreases. On the other han, the imensional tail water eth an the imensional eth of water ool vary irectly with the Dro nmer an inversely with the UBS. In aition, the emirical eations roose for horizontal stream sloe were comare with those of revios investigators. Frther, a arameter calle sloe effect ratio is efine to state the effect of UBS on the ICFOF. Moreover, the emirical eation erive in this sty can e se to esign the strctre of ownstream aron ractically. An examle illstrates the calclation of the imact osition/ tail water eth for ownstream aron. Paer smitte //; revise 5/6/3; accete 6/4/3. Athor for corresonence: ao-ming Hong ( leway@m.e.tw. Deartment of Civil Engineering, National Chng Hsing University, Taichng, Taiwan, R.O.C. Deartment of Lanscae Architectre an Environmental Planning, MingDao University, Changha Conty, Taiwan, R.O.C. 3 Deartment of Civil Engineering, K. L. E. S. College of Engineering an Technology, Belgam, Inia. I. INTRODUCTION For the rose of river reglation, sloe ajstment an for water intake stailization, hyralic engineers sally emloy hyralic strctres sch as weirs or check ams in rivers. However, these strctres sally roce a large elevation ifference etween stream an ownstream sies reslting into a free over-fall flow, which may ince scor on the ownstream of the hyralic strctre an amage it eventally. Previosly, many researchers stie the imact characteristics of free over-fall flow. In general, the research toics on free over-fall flow incle shae of scor hole, imact osition, imact force, etc. Stein an Jlien [] showe that the scor volme er nit with at any time can e exresse as twice the sare of the maximm scor eth at the corresoning time for the cohesive an non-cohesive seiments. Chen an Hong [3] se two tyes of sans, which incle niform an grae one, to simlate the e material ownstream of the weir henomena e to free over-fall flows. On the other han, Chamani an Beirami [], Tokyay an iliz [], Hong et al. [7] an Chen et al. [4] investigate the imact characteristics of free over-falls (ICFOF. Chamani an Beirami [] eveloe an emirical eation to estimate the relative energy loss at ros for sercritical aroaching flows. Tokyay an iliz [] se exerimental ata to analyze the characteristics of sercritical flow at a vertical ro in a rectanglar channel. However, Hong et al. [7] resente regression an semi-theoretical moels to reict the ro force an the ro length of a free-falling nae at an aerate straight-ro sillway. On the other han, Chen et al. [4] eveloe the theoretical eations for imact force an imact osition of free over-fall flow on a sloing ownstream sloe. In all the aove-mentione sties, the stream e sloe (UBS of the free over-fall was consiere horizontal, which ischarge horizontal jet. Hence, all the ICFOF flow stie were with horizontal stream e. However, after the constrction of hyralic strctre, the entire e loa an in articlar coarse seiments are not transorte to the

2 S.-I Li et al.: Imact Characteristics of Free Over-Fall in Pool Zone with Ustream Be Sloe 477 φ S vent Brink Deth ( Using the law of momentm conservation etween aroaching section an rink, one can write the following eation: φ S control volme in ro ege ( cos ϕ β β + + g g (a H vent f water ool l l where is the ischarge er nit with, g is the gravitational acceleration, is the velocity in the ro ege ( /, an β an β are the moment correction factor at the aroaching section an at rink, eing eal to. E. (a can e rearrange to otain rink eth as L, f f a θ l l Control volme in channel e Fig.. Schematic iagram of free over-fall flow with UBS. ownstream sie. As a conseence of the accmlation of the seiment on the stream sie of the strctre, the stream channel e sloe rises. This will change the inclination of free over-fall flow into an olie jet, an hence moify the imact osition an imact force of free over-fall flow on the ownstream sie. Therefore, the resent sty aims at the eveloment of emirical eations for the reiction of the ICFOF in ool zone with UBS. Base on the theoretical analysis, the UBS an Dro nmer were selecte as the ineenent variales of emirical eations. A series of laoratory exeriments were concte to evalate the inflencing factors of UBS for ICFOF. Frther, Mltile Regression Analysis (MRA is se to evelo the emirical eations. II. THEORETICAL ANALSIS. Imact Characteristics The schematic iagram of free over-fall with UBS an horizontal ownstream e is eicte in Fig., in which H is the ro height, is the rink eth, L is the imact osition, θ is an imact angle, is the tail water eth an is the ool water eth. De to stee inclination of stream e ϕ, the sercritical flow occrs in the stream channel, an the stream water eth will e smaller than the critical water eth c. The corse of free over-fall is ivie into two control volmes along the water nae as shown in Fig.. Assming the stream energy correction coefficient α i an the momentm correction factor β i eal to, the ICFOF can e erive as follows [8,, 3]: m F cos ϕ + ( where F is the Froe Nmer of stream channel ( / g cos ϕ. Imact Position (L Using Newton s laws of motion, the two imensional trace of water nae can e exresse y X t (a gt sinϕ t ( where X an are the horizontal an vertical coorinates of imact osition of the nae from the ro ege, resectively an t is the time taken y the water nae to travel from the ro ege to the imact osition. Sstitting E. (a into E. (, yiels g SX X cos ( ϕ (c where S is the UBS ( tan ϕ. Assming rectanglar channel, the critical eth of stream channel is c ( /g /3 an the horizontal critical velocity is c (g c /. From E. (c the imact osition L can e exresse as Hg S S S L g S ( + ( where c c. 3 Imact Angle (θ 3 (

3 478 Jornal of Marine Science an Technology, Vol., No. 4 (4 Using the law of energy conservation with UBS, the following eation can e otaine α cos α3 H + Zn + + ϕ y+ (3a g g ( ρ + cosθ ρ ρ (4a m a m a m f where m is the average velocity in the mixing section of nae an reeate rotary flow of water ool. Sstitting f, an rearranging, where Z n is the e elevation aove ro ege. Becase the istance etween measre osition of stream water eth an the ro ege eal to.3 m, Z n.3s, y is the nae eth imacting on ownstream sie of ro ( an α an α 3 are the energy correction coefficient at the aroaching section an ownstream of the rink, eing eal to. Rearranging E. (3a, the imacting velocity of water nae on the ownstream of the ro is written as By continity eation, a cosθ + cosθ f m m a (4 (4c g H +.3S + + (3 g Using the law of horizontal momentm conservation (ΣF x ρg( ρ( cosθ (3c where ρ is the water ensity. Sstitting E. (3 into E. (3c, the imact angle θ can e exresse as.5g( + θ cos g H +.3S + + g (3 4 Tail Water Deth ( Accoring to the exerimental oservation (Fig., the following flow conitions were assme: (a When water nae imacts the ownstream channel e, a ischarge er nit with f from water ool will e mixe into the water nae to increase the nae ischarge into ( + f with corresoning constant velocity m. ( When steay flow occrs, the ischarge flowing into water ool is f, an the ischarge flowing into ownstream is. Therefore, ner the ieal flow conition, there is always a reeate rotary ischarge f occrring in the water ool. Frther, The nae in the water ool is consiere to e ivie into two arts: (i the with of otsie art a with a ischarge er nit of ; an (ii the with of insie art with a reeate rotary ischarge er nit with f. The tail water eth a an f. From the law of momentm conservation at the ownstream e, it yiels Sstitting E. (4 it reslts f cosθ + cosθ (4 Hence, f will e the fnction of. Using momentm eation rior to imact an after imact, the following eation can e otaine cosθ ρβ ρ( + β ρ + + cos β θ 3 f 4 m 4 m (4e where β 3 an β 4 are the moment correction factor efore an after the imact of jet, consiere as. Therefore, m ( + cosθ (4f From continity eation, the velocity in ownstream m. By momentm conservation, one can write c c (4g Finally, the tailwater eth can e otaine y rearranging E. (4g with sstittion of m as ( g g H +.3S g + (4h 5 Deth of Water Pool ( Assming steay flow an static ressre istrition at the ro ege (Fig., the momentm conservation in horizontal irection (ΣF x etween stream section an ownstream

4 S.-I Li et al.: Imact Characteristics of Free Over-Fall in Pool Zone with Ustream Be Sloe 479 of the rink can e written as ρβ ( 3cosθ ρβ+ ρg (5a The momentm eation in the control volme of channel e (Fig. can e exresse as ρβ 3cosθ + ρg ρβ5+ ρ g (5 where β 5 is the moment correction factor ( in the ownstream channel e. Rearranging E. (5 an sing E. (4h, the eth of water ool is ( + (5c g In generally, if the stream Froe nmer, Dro nmer D ( /gh 3 an UBS are known, the ICFOF can e calclate to rovie the reference of strctre esign for free over-fall flow.. Develoment of Dimensionless ICFOF an Emirical Formlas Althogh the ICFOF can e calclate y theoretical eations, the comlexities of theoretical eations will ince the ifficlties in alication. Therefore, simle eations sch as emirical eations otaine y mltile regression analysis (MRA shol e more convenient in alication than theoretical eations. This sty rece the general form of emirical eations as follows: Dimensionless Brink Deth ( From E. (, iviing y c, the imensionless rink eth can e written as ( F cos ϕ + c c As c is constant for articlar ischarge an is eenent on the vale of stream Froe nmer F, the imensionless rink eth can e exresse as the fnction of UBS S an F : (, (6 f S F (7a Assming the existence of either ower or exonential relationshi etween, F an S, can e written as: a F e (7 cs where a, an c are constant. Dimensionless Imact Position ( L Diviing L from E. ( y H, the imensionless imact osition L is otaine as ( ( ( + D S+ S + D + + S L L H S 3 3 where D is Dro nmer ( /gh 3. On the other han, sing Bckingham π-theorem, Hong et al. [7] erive that L is a fnction of stream channel e sloe S an Dro nmer D as (, (8 L f S D (9a an an emirical formla can e exresse as where a, an c are constant. c S L a D e (9 3 Imact Angle (θ Sstitting critical eth c in E. 3( y normal eth, the imact angle θ can e exresse as.5g( + θ cos.3s DH gh H H ( It is also assme the imact angle θ is a fnction of stream channel e sloe S an Dro nmer D. Hence, ( θ f S D (a 3, an the corresoning emirical formla is where a 3, 3 an c 3 are constant. 3 c3s θ a3 D e ( 4 Dimensionless Tail Water Deth ( From E. 4(h, the imensionless tail water eth ( / H is

5 48 Jornal of Marine Science an Technology, Vol., No. 4 (4 H ( DH.3S DH H D H H ( Assming the is a fnction of stream channel e sloe S an Dro nmer D the relationshi can e written as which can e exresse as: ( f S D (3a 4, where a 4, 4 an c 4 are constant c S a D e (3 5 Dimensionless Deth of Water Pool ( The imensionless eth of water ool ( / H can e written from E. 5(c as: High Level Water Tank High Level Water Tank Flow Control Gate One-ste Free Over fall Flow Flow Gie Plate Water-level Gage.3 m Test Area Pressre Gages m (a To View Water-level Gage One-ste Free Over fall Pm Low Level Water Tank Low Level Water Tank ( Sie View Fig.. Schematic iagram of free over-fall flow with UBS. cs δ e (7a cos + DH H H H ϕ (4 Exressing as a fnction of stream channel e sloe S an Dro nmer D or (, f5 S D (5a c S a D e (5 where a 5, 5 an c 5 are constant. All the coefficients a i, i an c i (i to 5 can e etermine y exerimental ata. In general, if the three hysical arameters sch as UBS S, stream Froe nmer F an Dro nmer D are known, all of the ICFOF can e calclate. In aition, for horizontal stream sloe, ICFOF are only the fnctions of Dro nmer D. 3. Sloe Effect Ratio The sloe effect ratio (SER δ may e efine as δ (6 A( S> A( S where A (S > is a imact characteristics with UBS S > an A (S is the same imact characteristics with UBS S. The SER can e se to illstrate the change ratio of imact characteristics e to UBS. Varios SER for imensionless ICFOF are efine as follows. cs δ e (7 L 3 δ e cs (7c θ δ (7 4 e cs cs 5 δ e (7e Es. (7a-(7e show that SER is the fnction of exonential ower of UBS S. A larger vale of UBS S inces a larger SER. III. EXPERIMENTATION Exeriments were carrie ot in a glass-sie laoratory channel m long,.3 m wie an.4 m ee, at the National Chng Hsing University, Taiwan. The ottom sloe of the channel was ajstale. The flow rate in the channel was accrately controlle y a constant hea tank with a varialesee electronic controller. Water is recirclate from a low level water tank to a high level constant hea water tank y a m. An ltrasonic water-level gage, RPS-4A, was se to measre the water eth. Fig. illstrates the exerimental set. The vertical ro moels were mae of high ensity acrylic fire lates. For ro moels of height H.3 m,.5 m,. m an.5 m were se in the resent sty. The air cavity etween the ro wall an ottom of the nae was well ventilate y means of holes mae in the ro wall (see Fig.. The ressre measring system consists of 8 ressre gages monte in the mile of a high ensity acrylic fire

6 S.-I Li et al.: Imact Characteristics of Free Over-Fall in Pool Zone with Ustream Be Sloe 48 Tale. Range of exerimental conitions. Test Conitions Conitions Ustream channel e sloe, S (% (Control gro,, 4, 6 Dro height, H (m.3,.5,.,.5 Unit with flow rate, (m 3 /s/m Ustream Froe nmer, F Dro nmer, D ( F -.57 e -4.68S Davis et al. ( e -.5F S % S % S 4% S 6% Vent Dro Ventilation Pressre transcers Flow irection.3 (m F Fig. 4. Variation of imensionless rink eth with stream Froe nmer for varios UBS. Pressre Meter Flow PCD-3A Data Recorer Data Processor (a Sie View Flow corresoning ranges of stream Froe nmer F an the Dro nmer D were an , resectively. The vales of Froe nmer inicate that the stream flow conition was sercritical flow, which sorte the assmtion mae in the theoretical analysis. IV. ESTABLISHMENT OF EMPIRICAL FORMULAE AND DISCUSSIONS The laoratory exerimental reslts were se to aot the mltile regression analysis (MRA to otain the coefficients of emirical formlae. ( ToView Fig. 3. Layot of ressre measring system. late. The ressre gages were locate along the centerline of the channel an their to levels were ket on the channel e srface in orer to sense the imact ressre of free over-fall. Fig. 3 illstrates the layot of ressre measring system. When free-falling nae imacte the ownstream e srface, the imact ressre sense y the ressre gages (KOWA BP-5GRS was sent to the ata recorer, which col save an recor the real-time imact ressre. The maximm vale of imact ressres otaine y the ressre gages is the imact osition (L. i.e. L is the istance etween the ro wall an the osition of the maximm vale of the imact ressre. The exeriments were concte ner steay flow conition. The measrement osition of imact ressre was locate along the center line of the channel in orer to eliminate the effect of friction e to glass walls. Total 64 exeriments were erforme to otain ICFOF. Tale frnishes the range of the exerimental ata. The UBS S ranges etween -6%, the ro height H ranges from.5 m-.3 m, the nit ischarge is in the range of.79 m 3 /s/m-.48 m 3 /s/m. The. Brink Deth The laoratory exerimental ata for ifferent UBS S -6% are se to otain the mltile regression eation for imensionless rink eth following E. (7. The eation for imensionless rink eth with coefficient of etermination vale R.983 is. 765F e ( S E. (8 inicates that with an increase in Froe nmer F as well as the UBS S, the imensionless rink eth ecreases. Fig. 4 eicts the variation of imensionless rink eth with Froe nmer F for varios UBS S. The eation roose y Davis et al. [6] with UBS S % is also shown in Fig. 4. From Fig. 4, it can e oserve that E. (8 fits very well with the exerimental ata an comares close with eation of Davis et al. [6]. Hence it can e se to etermine the rink eth.. Imact Position The exerimental reslts are se to otain the mltile regression eation for the imensionless imact osition L. Althogh E. (9 exresses the imensionless imact osition as a fnction of Dro nmer D an UBS S, two tyes of

7 48 Jornal of Marine Science an Technology, Vol., No. 4 (4 L S % H.5 m H. m H.5 m H.3 m S % H.5 m H. m H.5 m H.3 m S 4% H.5 m H. m H.5 m H.3 m S 6% H.5 m H. m H.5 m H.3 m Chen ( L.35D.5 e.3s Chanson(995 L.7D.75 L.533D.6 e 6.668S L.533D.6 e 6.668S L.533D.6 e 6.668S L.533D.6 e 6.668S.... D Fig. 5. Variation of imensionless imact osition with Dro nmer for varios UBS. eations are eveloe for the L in the resent sty. Initially, the mltile regression eation for the imensionless imact osition L are otaine y consiering only the Dro nmer D an the eations for ifferent UBS are: (a ( (c ( L.775D for S 6% (9 In Es. (9a-(9, exonent of Dro nmer D is constant an eal to.6 for all vales of UBS, however, the constant a increases with an increase in the UBS to 4%, while it ecreases for S 6%. On the other han, E. ( reresents the mltile regression eation for the imensionless imact osition L as s fnction of Dro nmer D an UBS S S L D e ( E. ( shows that the large vales of Dro nmer D an UBS S ince a greater imensionless imact osition L. The correlation of L with D an S is consieraly as exresse y the correlation coefficient R.94. Fig. 5 resents the variation of imensionless imact osition L with Dro nmer D for ifferent UBS S. The exerimental ata collase on the crve of L otaine y E. (. In aition, the imensionless imact osition L comte y eations of Chanson [] an Chen [5] for UBS S are shown in Fig. 5(a. From Fig. 5(a, it is clear that the L etermine y Chanson [] eation is larger than the exerimental vales, while that from Chen [5] eation is more close to the exerimental reslts. The Chanson [] eation over reicts the L as it oes not incororate the effect of UBS S. However, the Chen s [5] reslts are comarale with the resent sty. 3. Imact Angle The mltile regression eation [following E. (] for imact angles θ otaine y the laoratory exerimental ata (with R.93 is S θ D e ( The variation of imact angles θ with Dro nmers D resente in Fig. 6 shows ecrease in imact angle θ with an increase in the Dro nmer D. Also, for the same ro nmer D, the imact angle θ ecreases with an increase in UBS S. The variations of θ for ifferent UBS comte y E. ( are exresse in E. (a-(..57 θ D for S (a. 6 L. 533D for S (9a.57 θ D for S % (.6 L.747D for S % (9.57 θ 4.865D for S 4% (c.6 L.9948D for S 4% (9c.57 θ 4.798D for S 6% (

8 S.-I Li et al.: Imact Characteristics of Free Over-Fall in Pool Zone with Ustream Be Sloe S % θ D e -.563S S % S % S 4% S 6%.8.5. S % S % S 4% S 6% Chanson (995.65D.44 θ (egree 6.9 S %.3779D.365 e -.78S 5 4 S 6% θ D e -.563S.... D Fig. 6. Variation of imact angle with Dro nmer for varios UBS..6.3 S 6%.3779D.365 e -.78S.... D Fig. 7. Variation of imensionless tail water eth with Dro nmer for varios UBS. 4. Tail Water Deth Following E. (3, the mltile regression eation for imensionless tail water eth is otaine as.36.78s D e ( E. (3 eicts that the imensionless tail water eth increases with an increase in Dro nmer D ecrease in UBS S. The variation of with Dro nmer D an UBS S is shown in Fig. 7. In aition, the relationshi roose y Chanson [] etween imensionless tail water eth an Dro nmer D for S shown in Fig. 7, inicates the over estimation of imensionless tail water eth for ro nmer D >.3. However, E. (3 fits the resent exerimental ata romisingly with R.967. Frther, the relationshis etween imensionless tail water eth an Dro nmer for varios UBS are 5. Deth of Water Pool for D S (4a for % D S (4 D S (4c for 4% D S ( for 6% The mltile regression eation for imensionless eth of water ool can e exresse as.4.57s.34d e (5 From E. (5 it is clear that the imensionless eth of water ool varies irectly with Dro nmer D an inirectly with UBS S. Fig. 8 illstrates the eenency of imensionless eth of water ool on Dro nmer D for ifferent UBS S. In aition, the eations roose y Ran [9] an Chanson [] are also shown in Fig. 8 (a for S, which illstrate the close corresonence of E. (5 with that of Ran [9] an Chanson[]. Frther, E. (5 reicts the imensionless eth of water ool very well having correlation coefficient R.99. Now, for ifferent UBS S, the exressions for imensionless eth of water ool can e exresse as 4..34D for S (6a D for S % ( D for S 4% (6c D for S 6% (6 Generally seaking, in the case of a flat stream e, the horizontally jet fall own into the ownstream e to gravity. However, in the case of a stee stream e, in aition to gravity, the stee ownwar jet accelerates the flow rate in the vertical irection, so as to shorten the imact time/osition. Emirical Es. (6a-(6 confirme the inferences.

9 484 Jornal of Marine Science an Technology, Vol., No. 4 (4 Imact characteristics of free over-fall flow with stream e sloe Dimensionless rink eth Dimensionless imact osition Imact angle θ Dimensionless tail water eth Dimensionless ool eth Tale. Sloe effect ratios (SER for varios ICFOF. Eation of SER δ 4.68S Calclate SER for UBS S % S 4% S 6% Relation e negative 6.67S L e ositive.78s.57s.563s e negative e negative e negative S % H.5 m H. m H.5 m H.3 m Ran (955.D. S % H.5 m H. m H.5 m H.3 m Chanson ( D.5.34D.45 e -.57S (a Deth of Water Pool Using E. (7, the SER for varios ICFOF can e otaine. Tale gives the Eations. of SER an the corresoning vale of SER for varios ICFOF with ifferent UBS S. The varios imact characteristics of free over-fall vary with UBS S. In Tale, two tyes of relationshis for ICFOF are efine: one ositive an another negative. If the SER for a articlar ICFOF increases with an increase in UBS S, sch a relationshi is efine as a ositive relation. On the other han, if a articlar ICFOF ecreases with an increase in UBS S, then it is consiere as negative relation. From Tale, it can e oserve that, excet imensionless imact osition all the ICFOF have negative relationshi with UBS S..3.. S 4% H.5 m H. m H.5 m H.3 m.34d.45 e -.57S ( Alication of Emirical Eations Imact osition an tailwater eth are sefl for the esign of ownstream aron. The aron length/eth can e etermine y the imact osition/ tail water eth. Assming the ro height H 6. m, the with of Sao am m, the esign ischarge rate Q, CMS, an the UBS S %. The ischarge er nit with is Q,. (m /s S 6% H.5 m H. m H.5 m H.3 m.34d.45 e -.57S (c.. The Dro nmer is D gh Using E. (, the imact osition is...34d.45 e -.57S.... D Fig. 8. Variation of imensionless ool eth with Dro nmer for varios UBS. (.6 L.533D ex(6.67 S H ex( (m Using E. (3, the tail water eth is

10 S.-I Li et al.: Imact Characteristics of Free Over-Fall in Pool Zone with Ustream Be Sloe ex(-.78 D S H ex( (m V. CONCLUSION The theoretical eations to reict the imact characteristics of free over-fall (ICFOF in the ool zone with sloing stream e are eveloe. The imact characteristics consiere in the sty comrise of the rink eth, the imact osition an imact angle, the tail water eth an the eth of water ool. These imact characteristics are exresse in imensionless forms. The ata otaine from the series of laoratory exeriments were se to otain the mltile regression eations for the imensionless imact characteristics. The reslts are smmarize as follows:. The imensionless rink eth ecreases with an increase in stream Froe nmer an UBS.. The imensionless imact osition, the imensionless tail water eth an the imensionless eth of water ool increases with an increase in Dro nmer. 3. With the increase in UBS, the imact angle, the imensionless tail water eth an the imensionless eth of water ool ecreases, while the imensional imact osition increases. 4. The emirical relations roose for revios investigators for horizontal stream e were also comare. 5. The sloe effect ratio exresses the effect of stream sloe on the varios imact characteristics. A large sloe effect ratio can ince a large vale of imact characteristics of free over-fall flow. The UBS reslting into a large sloe effect ratio is consiere as ositive relation to imact characteristics of free over-fall flow, otherwise it is negative relation. It was fon that most of UBS have negative relation to imact characteristics of free over-fall flow, excet imensionless imact osition. 6. The emirical eations eveloe in this sty can e alie in the ractical esign of ownstream aron sch as imact osition or tail water eth. ACKNOWLEDGMENTS The athors wol like to thank the research sort from the National Science Concil of the Relic of China (roject nmer -33-B-45- an 99--E NOTATION The following symols are se in this aer: a,, c constants [ ]; D Dro nmer, /gh 3 [ ]; a f with of otsie art of nae [L]; with of insie art nae [L]; eth of water jet eflecte into the water ool [L]; F stream Froe nmer / g [ ]; g gravitational acceleration [LT - ]; H ro height [L]; imact osition [L]; L L imensionless imact osition, L /H [ ]; ischarge er nit with [L T]; f ischarge er nit with from water ool [L T]; S stream channel sloe, tan ϕ [ ]; t time [T]; imacting velocity of water nae on ownstream of ro [LT - ]; velocity at ro ege [LT - ]; c critical velocity [LT - ]; m average velocity in mixing section of nae an reeate rotary flow of water ool [LT - ]; X horizontal coorinate of imact osition of nae from ro ege [L]; vertical coorinate of imact osition of nae from ro ege [L]; stream water eth [L]; tail water eth [L]; imensionless tail water eth, /H [ ]; c rink eth [L]; imensionless rink eth, / c [ ]; critical water eth [L]; ool water eth [L]; imensionless eth of water ool, /H [ ]; y nae eth imacting on ownstream sie of ro [L]; Z n water srface elevation aove ro ege at measre osition of stream water eth [L]; α energy correction coefficient [ ]; β momentm correction factor [ ]; δ sloe effect ratio A( S> A ( S [ ]; ϕ inclination of stream e [ ]; θ imact angle [ ]; an ρ water ensity [ML -3 ]; REFERENCES. Chamani, M. R. an Beirami, M. K., Flow characteristics at ros, Jornal of Hyralic Engineering, Vol. 8, No. 8, (.. Chanson, H., Hyralic Design of Stee Cascaes, Channels, Weirs an Sillways, Pergamon Press, New ork ( Chen, J.. an Hong,. M., Characteristics of check am scor hole y free over-fall flow, Jornal of Chinese Institte Engineering, Vol. 4, No. 6, (. 4. Chen, J.., Hang, H. S., Hong,. M., an Li, S. I., The imact characteristics analysis of free over-fall flow on ownstream channel e, Jornal of Chinese Institte Engineering, Vol. 34, No. 3, (.

11 486 Jornal of Marine Science an Technology, Vol., No. 4 (4 5. Chen, S. W., Exerimental Sty of Rivere Girle Arrangements in the Downstream of Sao Dam, Master Thesis, Deartment of Civil Engineering, National Chng Hsing University, Taichng, Taiwan (. 6. Davis, A. C., Ellett, B. G. S., an Jaco, R. P., Flow measrement in sloing channels with rectanglar free overfall, Jornal of Hyralic Engineering, Vol. 4, No. 7, ( Hong,. M., Hang, H. S., an Wan, S., Dro characteristics of freefalling nae for aerate straight-ro sillway, Jornal of Hyralic Research, Vol. 48, No.,. 5-9 (. 8. Ien, P. J., Engineering Hyralic, John Wiley an Sons, Inc., New ork ( Ran, W., Flow geometry at straight ro sillways, Jornal of Hyralic Engineering, Vol. 8,. -3 (955.. Shi,. H., A Sty of Hyralic Characteristics of Sercritical Free Overfall Imact, Master Thesis, Deartment of Civil Engineering, National Chng Hsing University, Taichng, Taiwan (.. Stein, O. R. an Jlien, P.., Seiment concentration elow free overfall, Jornal of Hyralic Engineering, Vol., (994.. Tokyay, N. D. an iliz, D., Characteristics of free overfall for sercritical flows, Canaa Jornal of Hyralic Engineering, Vol. 34, No., (7. 3. White, M. P., Discssion on energy loss at the ase of a free over-fall, Transactions American Society of Civil Engineering, Vol. 8, (943.