Chanson H. Jean-Baptiste Charles Joseph Belanger (1790-1874), the Backwater Equation and the Belanger Equation (796976), страница 4
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But Jean-Baptiste BÉLANGER made no further assumption and his development(BÉLANGER 1828, p. 9) is basically identical to the modern forms of the backwater equation used bytoday's hydraulic engineers. BÉLANGER introduced the kinetic energy correction coefficient in a laterdevelopment of the backwater equation (BÉLANGER 1841, p. 78; 1849, p. 74).Equation (6) was tested for a non-prismatic smooth drop inlet (Fig. 8). Figure 8A shows the experimentalfacility and Figure 8B compares the experimental observations with Equation (6) in which the flowresistance was calculated using the Prony formula (Eq.
(8)), with Equation (9) in which the friction slopewas calculated in terms of the Darcy friction factor, and with Equation (10). All the calculations wereperformed using the step method, distance calculated from depth. The experimental data (Symbols [*]) areplotted together with the bed elevation zo and sidewall profiles, and they agree well with the computations(Fig. 8B).
The results show basically very little differences between data and calculations, despite thechallenging geometry and the crude nature of the Prony formula. BÉLANGER's (1828) calculations giveidentical results to modern estimates. But Jean-Baptiste BÉLANGER had neither computer nor calculator,nor even slide rule, to integrate the backwater equation. All the calculations were performed manually (7),and this explains the common usage of PRONY's simplified formula at the time (BROWN 2002).Another comparison is presented in Figure 9 for a long prismatic channel. Figure 9A shows somemeasurements performed by DARCY and BAZIN (1865) in a prismatic, rectangular channel down a steepslope with a downstream control gate.
An undular hydraulic jump was observed at x = 125 m. Figure 9Bpresents the experimental canal along the Canal de Bourgogne and Figure 9C illustrates the measurementtechnique. In Figure 9A, the free-surface measurements (symbols [*]) are compared with Equation (6) inwhich the flow resistance was calculated using the Prony formula (Eq.
(8)), and with Equation (10) in whichthe friction slope was calculated in terms of the Darcy-Weisbach friction factor. The location of thehydraulic jump was derived from the application of the momentum principle neglecting the effects of bedslope (Eq. (1)). The results (Fig. 9A) show little differences between Equations (6) and (10). Again,BÉLANGER's (1828) calculations based upon the Prony resistance formula give results close to modernestimates.CommentsJean-Baptiste BÉLANGER integrated the backwater equation by selecting known water depths andcalculating manually the distance in between: "il s'agit d'intégrer entre deux limites h" ('the integration takesplace between two [water depth] limits h') (BÉLANGER 1828, pp.
11-13). Today this technique is called thestep method distance calculated from depth (HENDERSON 1966, CHANSON 2004) or the direct stepmethod.7Let us remember that the modern slide rule was introduced in 1859, 31 years later, by the French artillery officerAmédée MANNHEIM (1831-1906).10(A) Photograph of the smooth drop inlet experiment - Flow from bottom right to top left0.20.4Left sidewall0.18d (m) datazo (m)d (m) Eq. (10)d (m) Eq. (9)0.16d (m) Eq. (6)Right wallLeft wall0.140.320.240.16d, zo (m)0.120.080.100.08-0.080.06-0.160.04-0.240.02Sidewall (m)Invert-0.32Righ sidewall000.10.20.30.40.50.60.70.80.91-0.41.1x (m)(B) Comparison between experimental data and backwater calculations - Backwater calculations includeEquation (6) (BÉLANGER & PRONY), Equation (9) (BÉLANGER & DARCY-WEISBACH) and Equation(10)Fig.
8 - Free-surface profile in a smooth drop inlet structure for Q = 0.010 m3/s110.8Q=1.03 m3/sData Darcy & Bazin 1865Eq. (10)Eq. (6)0.720.64Water depth (m)0.560.480.40.320.240.160.080708090100110120130140x (m)150160170180190200(A) Comparison between backwater calculations and experimental data by DARCY and BAZIN (1865):experimental channel along the Canal de Bourgogne, Q = 1.03 m3/s, θ = 0.281°, B = 1.99 m, planed boards,series 89 - Backwater calculations include Equation (6) (BÉLANGER & PRONY) and Equation (10)B) Old photograph of the experimental channel along the Canal de Bourgogne at La Colombière, Dijon(France) (Courtesy of the Centre de Culture Scientifique et Technique de Bourgogne)12(C) Free-surface measurement (DARCY and BAZIN 1865, Plate iv, Fig.
8)Fig. 9 - Free-surface profile in a long prismatic channelBÉLANGER further investigated the two singularities of the backwater equation. One corresponded to theuniform equilibrium flow conditions So = Sf, for which the flow depth equals the normal depth. BÉLANGER(1828, p. 10) obtained the normal depth expression of PRONY (1804):(a V + b V 2 )= sin θDH4(11)The second singularity of the backwater equation corresponded to ∂x/∂d = 0 and it yielded the condition:Q2∂A=1g cos θ A ∂d(12)3that corresponds to the critical flow conditions in a channel of irregular cross-section with hydrostaticpressure distribution.
In the particular case of a prismatic rectangular open channel, Equation (12) yields theclassical result: V2 = g d cosθ (LIGGETT 1993, CHANSON 2006). BÉLANGER (1828, p. 29) did not usethe term "critical flow" but he highlighted explicitly the flow singularity: "un cas peu ordinaire" ('a specialcase'). He stressed further the physical impossibility to observe ∂d/∂x = +∞ for this 'special case'.5. DiscussionTwenty one years after this original essay, BÉLANGER (1841) expanded his treatment in the form of aseries of lecture notes for the Ecole des Ponts et Chaussées (8) for the session 1841-1842. His notes formed a8The lecture notes were used at the Ecole des Ponts et Chaussées and Ecole Centrale des Arts et Manufactures, andavailable at the Ecole Polytechnique et Ecole des Mines de Paris.13comprehensive treatise in hydraulic engineering, and they were re-edited several times, while Jean-BaptisteBÉLANGER was lecturing, with relatively small to moderate differences between the various editions: e.g.,BÉLANGER (1841,1849) for the university sessions 1841-1842 and 1849-1850 at the Ecole des Ponts etChaussées respectively.On the hydraulic jump, Jean-Baptiste BÉLANGER (1849) indicated that he found his error in 1838.
Hecorrected his treatment of the hydraulic jump and applied correctly the momentum principle, "le théorèmerelatif à l'accroissement de la quantité de mouvement" ('the theorem related to the rate of increase inmomentum') (BÉLANGER 1841, p. 87). In his derivation, he stated : "l'accroissement algébrique de laquantité de mouvement [...] est égale à la somme des impulsions des forces extérieures, projetéesparallèlement au mouvement" ('the increase in momentum [...] is equal to the sum of external forcesprojected in the flow direction') (BÉLANGER 1849, p. 85).
His newer result yielded the "modern" form ofthe Bélanger equation :d211=− ++ 2 α' Fr12d124(5a)that is basically a rewriting of Equation (1). For a hydraulic jump in a horizontal, rectangular and prismaticchannel, BÉLANGER (1849, p. 88) calculated the loss in kinetic energy head :V12 V2 2 (d1 + d 2 ) 2−=(d 2 − d 1 )2g2g4 d 1d 2(13)that may be rewritten in terms of the head loss in the hydraulic jump:ΔH =(d 2 − d1 ) 34 d 1d 2(14)Equation (14) is a well-known result for a hydraulic jump in a horizontal rectangular channel(HENDERSON 1966, MONTES 1998, CHANSON 2004).In the same treatise, BELANGER (1841,1849) presented explicitly a number of basic features of openchannel flows. He developed an expression of the uniform equilibrium flow depth (normal depth) that wasderived from energy considerations. He further developed the calculations of the normal depth for acomposite channel (Fig. 10), showing accurately that the total discharge is the sum of the flow rates in themain channel and in the flood plain, and that the friction slope is identical for both channel sections, but withdifferent friction coefficients.Fig.
10 - Composite open channel cross-section (BÉLANGER 1849, p. 60, section [156])14BELANGER (1841,1849) showed that, in a rectangular channel, the discharge per unit width is maximum atcritical flow conditions for a given specific energy E: "entre ces deux valeurs de ξ, il y en une pour laquelleQ devient maximum [...] quand on a ζ = 1/3 Z" ('between these two values if ζ, there is one for which Q ismaximum [...] when ξ = 1/3 Z'). He derived the expression of the critical depth dc :dc =2E3(15)Equation (15) was obtained in section [85] (BELANGER 1849, p.
33) as part of a discussion of the overflowon a broad-crested weir (Fig. 11). His treatment of the broad crested weir yielded further the classicalexpression of the flow rate Q:Q⎛2 ⎞= g ⎜ H⎟B⎝3 ⎠3/ 2(16)for a rectangular channel of width B, where H is the total head above the crest invert.All these results are common knowledge today (HENDERSON 1966, CHANSON 2004), but were new andimportant developments in the 1840s.It is worth noting that BÉLANGER (1841,1849) used both momentum and energy considerations in asomewhat inconsistent manner.
Such inconsistencies were discussed by YEN (2002) and CHANSON(1999,2004) in a broader context. In his treatment of the hydraulic jump, BÉLANGER (1841,1849) usedcorrectly the momentum principle, but introduced a kinetic energy correction coefficient. Similarly he solvedthe uniform equilibrium flow based upon energy considerations, although modern treatments are derivedfrom the momentum equation. Late Professor Ben YEN pointed accurately a number of similar discrepanciesin recent studies by some hydraulic engineers (YEN 2002).Fig. 11 - Flow over a broad-crested weir (BÉLANGER 1849, p. 33, section [85])6. ConclusionIn the 1820s, Jean-Baptiste BÉLANGER (1790-1874) worked on a method to calculate gradually-variedopen channel flow properties for steady flow conditions.
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