ABRUPT DEFLECTED SUPERCRITICAL WATER FLOW IN SLOPED CHANNELS

293Available online at www.sciencedirect.comScienceDirectITDJoumal of HydrodynamicsEL SEVIER2008,20(3):293-298www. sciencedirect.com/science/journal/10016058ABRUPT DEFLECTED SUPERCRITICAL WATER FLOW IN SLOPEDCHANNELS"LIU Ya-kun, NI Han-genState Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024,China,E-mail:liuyakun@dlut.edu.cn(Received July 2, 2006, Revised March 26, 2007)Abstract: The efect of the bottom slope on abrupt deflected supercritical water flow was experimentally and theoretically studied.Model tests were conducted in a flume of 1.2 m wide and 2.6 m long with sloped bottom at an angle 35.54, its length of delectorwas 0.2 m and the deflection angles were 159 and 30*. An approximate method for calculatjng the shock wave angle and depth ratioof the abrupt deflected supercritical water flow was suggested, and a correction coefficient for the hydrodynamic pressure wasintroduced to generalize the momentum equation in the direction perpendicular to the shock front. It must be noticed that in thesloped channel the shock wave angle and the depth ratio are no longer constant as those in the horizontal channels, but slowly changealong the shock front. The calculated results are in good agreement with measured data.Key words: shock wave, abrupt deflection of supercritical water flow, slopping channel1. Introductionobtained a simple explicit expression for the shockIn hydraulic engineering all the impact-typewave height ratio with the consideration of the effectenergy dissipators， slit-type buckets and theof the non-hydrostatic pressure distribution. But allspillway with flaring piersh are used with the rapidthe work mentioned above involved exclusively theincrement of flow height caused by the abruptcases of horizontal bottom. Up to the date, there is nodeflection of the supercritical flow to form spray flow.simple and reliable method to determine the shockIn principle, the Ippen theory of shock wavello.n maywave in the sloped channel. This article offers tobe invoked to calculate the shock parameters for thedetermine the effect of the bed slope on the abruptabrupt deflected supercritical flow. Since Ippendeflected supercritical water flow.developed the basic theory of shock wave someresearchers have studied the characteristics of shockwave from various viewpoints,. such as Charles2. Approximate analysis of abrupt deflectedHager et al."3), Zhang and Wul4) and Liu and Nil5].supercritical water flow in sloped channelFor higher Froude number, larger deflection angle and2.1 Basic relationships of shock waveshorter deflector, the assumption of hydrostaticThe abrupt deflected supercritical water flow in apressure distribution which was adopted in the Ippensloped channel is shown in Fig.l, in which中is thetheory is not valid. In Ref.[16]， a correctionangle of the bottom slope, a is the deflection anglecoefficient of the hydrodynamic pressure wasin the A-A plane，o' is the deflection angle in theintroduced to generalize the momentum equation andhorizontal plane respectively, β is the shock waveangle in the A-A plane and β' is the shock wave* Project supported by the Natural Science Foundation ofangle中国煤化工. respectively. TheLiaoning Province (Grant No. 20062177).relati=and β' areBiography: LIU Ya-kun (1968-), Female, Ph. D., AssociateMYHCNMHGProfessor294d' = tan^tan a(1a)The momentum equations in the normal and the( cos平parallel directions for each element and the continuityequation may be written asβ'= tan^tan P(1b) Pg cos,+ phV3.: 5pgh cosy +Cos4)2The relationship between the deflector lengthprojected on various plane may be written ash =4 cosa(2a)V=Va .(4)l = l cosY =4 cosarcosY(2b)hYn =hVn2(5)4_ 4 cosa cos4where P and g denote the water density and the4=-.;(2c)gravitational acceleration, respectively, and ξ is thecosa'cos a'correction coefficient of the non- hydrostatic pressureUnder the influence of the bed slope, the shockdistribution'"'s. Other notations are shown in Fig.1.Upon lttingfront is no longer a straight line as in the case ofhorizontal bed but a curve. So, it is necessary to dividethe shock front into some elements. And followingy=名assumptions are adopted that after the water passhthrough any element of the shock front the flow isindependent of others and the flow direction is paralleland substituting Eq.(5) into Eq. (3), we can obtainto the deflector.(7)+05-1-=1(-51)Figure 1 shows thatV,sinβ =-(8)Plan viewrom Eqs.7) and (8) the following equation isobtained:ξy>-Y(1+2F2 sin' β)+2F?2sin2 β=0/bahawithSide viewF=V(9)√gh cosWIt can be seen from Eq.(9) that the value of Y will besmaller than unity if ξ<0. This is not consistent4-A vicw中国煤化工| the valueof ξ islargc二inted out.Fig.1 Sketch of shock wave in sloped channelMHC N M H Gm Fig.1(c) that foreach element of the shock front, the following relation295are still available as in the case of horizontal bottom:From Eqs.(13) and (14) we getV(10a)tan β第--E弱2g=0(16)Va2(10b)Vr2=tann(β-a)We define a new variable, η, asEqFrom Eqs.(4), (5)and (10) we geth=.:os 4(17)tan(β-a)(11)Using Eq. (17) turns Eq (16) intoη°-η'+G= 0(18)Substituting Eq.(1 1) into Eq(9) yieldswhertan(β-a)」tan(β-a)G=g cos2出(19)2gE(1+2F sin? B)+2F2 sin2 β=0(12)Then the solution of Eq.(18) h, is obtained aGiven the values of 5, F and a, we can get twofollows:solutions of Eq.(12)，and the smaller valueof β isneeded. Then with Eq.(11) the value of Y can becos 4calculated.But due to the sloped bottom, the elevation of theelement and the coming flow depth decrease, thecos~(1-13.5G)]}coming flow velocity and the Froude number of thecoming flow increasc along the shock front. Therefore,After h2j has been determined, using Eqs.(14) and (9)the values of F and a in Eq.(12) are different forgives directly the valuesof V」and Fx.different elements.2.2 Froude mumber of coming flow for each element2.3 Correction cofficient of the hydrodynamicpressure ξIt is reasonable to neglect the energy loss aheadhe shock front and under such condition we alwaysIn the consered case, many factors affect thehavecorrection coefficient of hydrodynamic pressure 5 .As an approximate estimation, we adopt the followingempirical expression:hy cos4 +;+(zo-z.)=h, cos4 +- (13)(21)的Vo1 =4Vx(14)where hn, Vo| and z。are the coming flow depth,ξ(e)= 0.65\0.45(22)velocity and bottom elevation at 0-0 sectionrespectively, and h2, V: and z, are those near thes-s section.(see Fig.1l().where 5(x) denotes the correction coefficient ofUpon denotingnon-hydrostatic pressure at a distance x from the 0-0secti中国煤化工En= h, cos4 +-+(zo-z.) .(15)YHCNMHG2963. Experimental facilities and test results4. Comparison between measured and calculatedTests were conducted in a flume of 2.6 m long,results1.2 m wide and 0.2 m high flume with sloped bottomFor 6 test cases the measured and calculatedat an angle of 35.54°，and its length of deflector wasparameters of the shock wave are compared in Figs. 40.2 m and the deflection angles were 159 and 30° . Theand 5, respectively. In these tables, the values of h,origin of the deflector was 1.0 m downstream of theare calculated with Eq.(20) except for the measuredflume entrance where a hump-type weir was installed.的，all the values of hm are measured,The flow patterms for Fo =4.67, a= 30° andfor Fo =5.33, a=15° are shown in Fig.2 and Fig.3,_=2m, B and ξ are computed using Eq.(1)respectively.and (12) respectively,与are calculated with Eq.(21),Y。are calculated using Eqs.(12) and (11), and finally,using Eq.(6) h。 are calculated. In general, thedifferences between hm and h。 are not scremarkable and fall in an acceptable range. A problemcan be seen from Figs.4 and 5 that for x=0 thevalues of 5 are larger than unity, it means thatbehind the shock wave at x= 0 the bydrodynamicpressure are larger than hydrostatic pressure.0.20e 0.15Fig2 Flow pattern under the conditionsof a= 30*,“0.10↑F。=4.67，h。 = 35mm0.0500.1 0.15 0.20.25Fig.4 Comparison between the measured and calculatedresults for a= 30°0.12r0.100.080.040.C一■9#h,7h。.7h。0.0.10.15 0.2/mFig.3 Flow pattern under the conditionsof ax=15° ,Fo =5.33，ho= 31mmFig.5 Comparison between the measured and calculatedresults for a= 15°The flow depths directly measured along thedeflector for a = 30°and a=15° at different5. ConclusionsFroude number of coming flow at 0-0 section are(1) An approximate method has been worked outlisted in Tables 1 and 2, respectively.It must be noticed that the flow depths directlyto calculate the parameters of the abrupt deflectedmeasured along the deflector are the maximum depthssupercritical water flow in the sloped channels. Underthe influence of the, hed slone the shock front is noof the spray flow at each section.long中国煤化工ided into someTHCNMHG297Table 1 Flow depth along the deflector for a = 30°X(mm)Test No.Frohno(mm)51001502005.9710406C861051105.451:65821041241325.402(7091181391555.0424731081301704.8630)01491854.673:14560182Table 2 Flow depth along the deflector for a= 159FIoho(mm)0506.2120I5535.9464255585.855(087887’5.525:7515.338095elements in the present method. The governingequations are established at the elements and aReferencescorrection coefficient of the non-hydrostatic pressureis introduced. Due to bottom slope, the elevation of1] BOES Robert M., HAGER Willi H. Hydraulic design ofthe element decreases and the Froude number of thestepped spillway[J]. Journal of Hydraulic Engineering,coming flow increases along the shock front. AASCE, 2003, 129(9): 671-679.method for determining the Froude number at each[2]HELLER Valentin, HAGER Willi H. and MINOR.Hans-Erwin. Ski jump hydraulics[J]. 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