Minimum Specific Energy and Critical Flow Conditions in Open Channels

Transcription

1 Minimum Speifi Energy and Critial Flow Conditions in Open Channels H. Chanson Abstrat: In open hannels, the relationship between the speifi energy and the flow depth exhibits a minimum, and the orresponding flow onditions are alled ritial flow onditions. Herein they are reanalyzed on the basis of the depth-averaged Bernoulli equation. At ritial flow, there is only one possible flow depth, and a new analytial expression of that harateristi depth is developed for ideal-fluid flow situations with nonhydrostati pressure distribution and nonuniform veloity distribution. The results are applied to relevant ritial flow onditions: e.g., at the rest of a spillway. The finding may be applied to predit more aurately the disharge on weir and spillway rests. DOI: 0.06/ ASCE :5 498 CE Database subjet headings: Energy; Critial flow; Open hannel flow; Weirs; Spillways; Coeffiients. Introdution Considering an open hannel flow, the free surfae is always at atmospheri pressure, the driving fore of the fluid motion is gravity, and the fluid is inompressible and Newtonian. Newton s law of motion leads to the Navier Stokes equations. The integration of the Navier Stokes equations along a streamline, assuming that the fluid is fritionless, the volume fore potential i.e., gravity is independent of the time, for a steady flow i.e., V/ t=0 and an inompressible flow i.e., =onstant, yields P + g z + v = onstant where =fluid density; g=gravity aeleration; z=elevation aligned along the vertial diretion and positive upward ; P=pressure; and V=veloity Henderson 966; Liggett 993; Chanson 999, 004. Eq. is the loal form of the Bernoulli equation. In this study, the singularity of the depth-averaged Bernoulli priniple for open hannel flow is detailed, i.e., the ritial flow onditions. Detailed expressions of the ritial flow properties are derived for the general ase of nonhydrostati and nonuniform veloity distributions. The results are then applied to the rating urve of weir rest ating as disharge meter. Appliation to Open Channel Flows In open hannels, it is ommon to use the depth-averaged Bernoulli equation within the frame of relevant assumptions e.g., Liggett 993 Reader, Environmental Fluid Mehanis, Dept. of Civil Engineering, The Univ. of Queensland, Brisbane QLD 407, Australia. h.hanson@uq.edu.au Note. Disussion open until Marh, 007. Separate disussions must be submitted for individual papers. To extend the losing date by one month, a written request must be filed with the ASCE Managing Editor. The manusript for this paper was submitted for review and possible publiation on June, 005; approved on Deember 9, 005. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 3, No. 5, Otober, 006. ASCE, ISSN /006/ / $5.00. H = d d v y 0 g + z y + P y g dy = V g + d + z 0 = onstant where H=depth-averaged total head; z 0 =bottom elevation; d=flow depth; =momentum orretion oeffiient or Boussinesq oeffiient ; and V=depth-averaged veloity V = d v dy 30 y=distane normal to the hannel bed and =pressure orretion oeffiient defined as d = + d P y dy 0 g d 4 For a flat hannel assuming a hydrostati pressure distribution, the pressure orretion oeffiient is unity and the depthaveraged total head H equals H = V g + d + z 0 = E + z 0 5 where E = depth-averaged speifi energy. Critial Flow Conditions Considering a short smooth transition, and assuming a onstant flow rate, the relationship between the speifi energy E=H z 0 and flow depth exhibits a harateristi shape e.g., Fig.. For a given ross-setion shape, the speifi energy is minimum for flow onditions d,v alled the ritial flow onditions. The onept of ritial flow onditions was first developed by Bélanger 88 as the loation where d=q / g A for a flat hannel, where Q=flow rate and A=flow ross-setion area. It was assoiated with the idea of minimum speifi energy by Bakhmeteff 9, 93. Both Bélanger and Bakhmeteff devel- d 498 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 006

2 Fig.. Relationship between dimensionless speifi energy E/d and dimensionless flow depth d/d in smooth retangular hannel assuming hydrostati pressure distribution oped the onept of ritial flow in relation with the singularity of the bakwater equation for d=q / g A i.e., ritial flow onditions. A typial situation with minimum speifi energy is shown in Fig. where ritial flow onditions our at the point of maximum invert elevation: i.e., the weir rest. Assuming a smooth fritionless overflow, the depth-averaged Bernoulli equation states H = z 0 rest + = z 0 rest + rest d + rest V g 6a where H=upstream total head; z 0 rest =rest elevation; =minimum speifi energy Fig. ; and rest and rest =, respetively, momentum and pressure orretion oeffiients at the rest. Note that, at the rest, the y diretion is exatly vertial, the streamlines are urved, the pressure distribution is not hydrostati, and the veloity distribution is not uniform. For a retangular hannel, the ontinuity and Bernoulli equations give two equations in terms of the ritial flow depth and depth-averaged veloity are as follows: Bernoulli equation = rest d + rest V g ontinuity equation q = V d = C D g 3/ min 3 E 6b where q=disharge per unit width and C D =dimensionless disharge oeffiient. 7 If the minimum speifi energy and flow rate per unit width q are known parameters, the ombination of Eqs. 7 and 6b gives a third-order polynomial equation in terms of the dimensionless flow depth at rest i.e., d / d 3 3 =0 8 d 3 + rest rest C D rest Eq. 8 has one, two, or three real solutions depending upon the sign of the disriminant see the Appendix 6 = 3 rest 4 rest C D rest rest C D rest 9 The solution of Eq. 8 gives an expression of the dimensionless ritial depth as a funtion of the pressure orretion oeffiient, momentum orretion oeffiient, and disharge oeffiient. Further the Bernoulli equation implies for an overflow 0 rest d 0 The detailed solutions of Eq. 8 are developed in the Appendix. Meaningful solutions exist only for 0. These solutions are plotted in Fig. 3 as d / rest versus rest C D rest. The analytial results are ompared with the reanalysis of experimental data Fawer 937; Vo 99, flow net analysis Fawer 937, and detailed analytial solution Fawer 937. The experimental flow onditions are listed in Table and one omplete data set is presented in Fig. 4. JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 006 / 499

3 Fig. 3. Dimensionless ritial flow depth at rest of weir: d / rest versus rest C D rest omparison with experimental data Fawer 937, Vo 99, flow net analysis with eight stream tubes Fawer 937, and detailed analytial solution Fawer 937 Fig.. Critial flow onditions at weir rest definition sketh Overall the data Fig. 3 follow losely the solution S3 where Solution S3 d = os /3 3 rest + 3 os /3 os = rest C D rest It is unlear why experimental data do not follow the solution S, although it is oneivable that S might be an unstable solution. For a hydrostati pressure distribution rest = and a uniform veloity distribution rest =, the disharge oeffiient is unity and the flow depth at rest equals d = 3 for example, an ideal fluid flow above a broad-rested weir. Disussion 3 The analysis of Eq. 8 yields basi onlusions. First the produt rest C D rest must be less than or equal to unity rest C D rest 4 As the momentum orretion oeffiient rest is equal to or larger than unity, Eq. 4 implies that disharge oeffiients larger than unity may be obtained only when the rest pressure distribution is less than hydrostati. Seond, for transition from sub- to superritial flow, experimental results Fig. 3 indiate that d rest 5 3 That is, the dimensionless ritial flow depth equals the solution S3 Appendix, and that the streamline urvature usually implies rest at a weir rest. Therefore Eq. 5 does not imply d / /3. The result is well known for overflow irular weirs e.g., Vo 99; Chanson and Montes 998. Third the pressure and veloity distributions at the rest an be predited using ideal fluid flow theory i.e., potential flow theory. Hene rest and rest may be alulated theoretially beause the entire flow field may be predited assuming an ideal fluid with irrotational flow motion. Assuming a two-dimensional flow, the vertial distributions of pressure and veloity may be aurately determined numerially by a omputational method, graphially by a flow net analysis, or analytially for simple geometries e.g., Fawer 937; Rouse 946; Jaeger 956; Vallentine 969. Then the relevant parameters beome d / and C D,orq, d, and. Table. Summary of Reanalyzed Experimental Flow Measurements Referene Configuration Measurements Remarks Fawer 937 Cirular weir R=0.035 m Invert pressure distributions..5 m long m wide upstream flume. Vo 99 Vertial upstream wall 3: downstream slope Weir height: and m Cirular weir R= m Upstream slope: 90, 75, 60 Downstream slope: 75, 60, 45 Vertial distributions of pressure and veloity Pitot tube. Invert pressure distributions. Vertial distributions of pressure and veloity LDV..8 m long 0.54 m wide upstream flume. Also Ramamurthy et al / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 006

4 appliable to long waterway: e.g., undular flow in a ulvert barrel. Aknowledgments Fig. 4. Dimensionless pressure and veloity distributions at rest of weir: P/ g d and V/ g Emin versus d / Fawer 937; d = m, = m, R=0.035 m omparison among experimental data, flow net analysis eight stream tubes, and detailed analytial solution ideal fluid Appliation: Spillway Crest as Disharge Meter If the momentum and pressure orretion oeffiients may be predited theoretially, the ontinuity and Bernoulli equations imply that a spillway rest may be used as an aurate disharge meter using the solution of Eq. 8. In pratie the upstream head above spillway rest i.e., is known and the unknown is the flow rate q. If the flow depth at the rest i.e., the ritial flow depth d is measured, Eq. and Fig. 3 provide the value of the disharge oeffiient C D satisfying Eq. 8, and the flow rate is dedued from Eq. 7. In ontrast with the free overfall, a spillway rest is a better disharge meter beause the orretion oeffiients and the disharge oeffiient are lose to unity. Considering a irular weir R=4.5 m with an upstream vertial weir in a retangular hannel, the upstream head above rest is 0.6 m and the measured depth on the rest is 7.4 m. Compute the flow rate. Fawer 937 derived the flow net solution of this ase. His graphial result based upon eight stream tubes predited: rest =.0 and rest =0.5. For these values, Eq. and Fig. 3 imply that C D =.44. Fawer onduted the orresponding experiment that yielded C D =.4, while his detailed potential flow solution gave C D =.38. Both results are lose to the analytial predition Eq. and Fig. 3. Conlusion Critial flow onditions in open hannel are reanalyzed using the depth-averaged form of the Bernoulli equation. At ritial flow, a new analytial expression of the ritial flow depth is derived for ideal-fluid flows. The result is applied to ritial flow situations with nonhydrostati pressure distribution and nonuniform veloity distribution. It yields pertinent information on the flow properties at a weir rest. The findings may be applied to predit aurately the disharge at the rest at spillways and weirs, by ombining Eq. and Fig. 3 with simple ideal fluid flow theory e.g., flow net analysis. In real-fluid flows, boundary frition indues a flow region affeted by shear and momentum exhange: i.e., a developing boundary layer. Converging and aelerating flow situations e.g., spillway intake have generally thin boundary layers. Present ideal fluid results may be applied to short transitions of real fluids to a satisfatory degree of approximation, but they are not The writer thanks Dr. Sergio Montes The University of Tasmania for many helpful exhanges and disussion. He thanks further Professor A. S. Ramamurthy Conordia University, Canada for providing the original data of his former Ph.D. student Vo 99 and Professor C. J. Apelt The University of Queensland for helpful omments. Appendix: Critial Flow Depth The ontinuity and Bernoulli equations give an expression for the ritial flow depth as the solution of d 3 d 3 3 =0 + rest rest C D rest 6 Eq. 6 has one, two, or three real solutions depending upon the sign of the disriminant = rest rest C D rest For 0, Eq. 6 has only one real solution 3 d rest = rest C D rest rest C D rest 7 rest C D rest + 3 For =0, the following ondition holds: + 6 rest 6 rest 7 8 rest C D rest =0 9 The only physial solution of Eq. 6 is d rest = 3 = 3 rest C D rest 0 For =0, the seond real solution is negative: d / = / 3 rest. For 0 there are three real solutions solution S d = rest 3 + os /3a solution S d rest = 3 os /3 3 os /3 b JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 006 / 50

5 solution S3 d rest 3 = 3 os /3 + 3 os /3 where os = rest C D rest Note that 0 implies: rest C D rest. pressure orretion oeffiient; and water density kg/m 3. Subsript ritial flow onditions i.e., at minimum speifi energy ; rest onditions at rest. Referenes Notation The following symbols are used in this paper: A flow ross-setion area m ; C D disharge oeffiient; d flow depth m measured perpendiular to hannel bottom; d ritial flow depth m : i.e., flow depth at minimum speifi energy; E speifi energy m ; minimum speifi energy m ; g gravity onstant; H depth-averaged total head m ; P pressure Pa ; Q water disharge m 3 /s ; q water disharge per unit width m /s; R radius m or urvature of spillway rest; t time s ; V depth-averaged veloity m/s ; V ritial flow veloity m/s : i.e., depth-averaged flow veloity at minimum speifi energy; v loal veloity m/s ; W hannel width m ; y distane measured perpendiular to hannel bottom m ; z elevation m taken positive upward; z 0 bed elevation m taken positive upward; momentum orretion oeffiient; Bakhmeteff, B. A. 9. O neravnomernom dwijenii jidkosti v otkrytom rusle. Varied Flow in Open Channel. St Petersburg, Russia in Russian. Bakhmeteff, B. A. 93. Hydraulis of open hannels, st Ed., MGraw Hill, New York. Bélanger, J. B. 88. Essai sur la solution numérique de quelques problèmes relatifs au mouvement permanent des eaux ourantes, Essay on the numerial solution of some problems relative to steady flow of water. Carilian-Goeury, Paris in Frenh. Chanson, H The hydraulis of open hannel flows: An introdution, st Ed., Edward Arnold, London. Chanson, H The hydraulis of open hannel flows: An introdution, nd Ed., Butterworth-Heinemann, Oxford, U.K. Chanson, H., and Montes, J. S Overflow harateristis of irular weirs: Effets of inflow onditions. J. Irrig. Drain. Eng., 4 3, 5 6. Fawer, C Etude de quelques eoulements permanents à filets ourbes. Study of some steady flows with urved streamlines. Thesis, Imprimerie La Conorde, Lausanne, Switzerland in Frenh. Henderson, F. M Open hannel flow, MaMillan, New York. Jaeger, C Engineering fluid mehanis, Blakie & Son, Glasow, U.K. Liggett, J. A Critial depth, veloity profiles and averaging. J. Irrig. Drain. Eng., 9, Ramamurthy, A. S., Vo, N.-D., and Vera, G. 99. Momentum model of flow past weir. J. Irrig. Drain. Eng., 8 6, Rouse, H Elementary mehanis of fluids, Wiley, New York. Vallentine, H. R Applied hydrodynamis, SI Ed., Butterworths, London. Vo,N.D. 99. Charateristis of urvilinear flow past irular-rested weirs. Ph.D. thesis, Conordia Univ., Canada. 50 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 006