Chanson H. Jean-Baptiste Charles Joseph Belanger (1790-1874), the Backwater Equation and the Belanger Equation (796976), страница 3
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3, App. B).Jean-Baptiste BÉLANGER retired in 1864 (HAGER 2003). He died on 8 May 1874 at Neuilly-sur-Seine,and his tomb is today in the old cemetery of Neuilly-sur-Seine (cimetière ancien, 5ème division).Fig. 4 - Jean-Baptiste BÉLANGER among his academic peers at the Ecole Centrale des Arts et Manufactures(Courtesy of the Bibliothèque de l'Ecole Centrale de Paris) - BÉLANGER stands in the middle row, fifthfrom the right (with white hairs)3. The analysis of the hydraulic jump: the "Bélanger equation"From 1821, Jean-Baptiste BÉLANGER worked as a practicing engineer on a solution of gradually-variedopen channel flows. He published a preliminary report in 1823 (2) but he felt that the work lacked theoreticalfoundations: "il a senti de lui-même le désir de l'améliorer" ('he felt himself the need to improve it'). Hedeveloped new ideas in 1826 and completed his report in 1827 (BÉLANGER 1849, p.
90). His reviseddocument was successfully examined by the Commission des Ponts et Chaussées et des Mines on 21 JulyPolytechnique 1931).2in the "Journal des Mines".41827 (3) and published in 1828 (BÉLANGER 1828) (Fig. 5). The reader will find the correspondencebetween the original notations of BÉLANGER and modern hydraulic engineering notations in Table 1.BÉLANGER (1828, pp. 31-36) considered the hydraulic jump as a rapidly-varied flow, across which thegradually-varied flow equation could not be applied. Based upon the experimental observations of BIDONE(1819), he treated the flow singularity (Fig. 6) by applying the energy principle using a formulation derivedfrom a "Traité Spécial" published in 1819 by Gustave Gaspard CORIOLIS (1792-1843) (4): "je me sers duthéorème de Mécanique connu sous le nom d'équation des forces vives" ('I use the Mechanics theoremknown as the equation of conservation of energy').Jean-Baptiste BÉLANGER considered the general case of a hydraulic jump in a sloping channel of irregularsection.
For the particular case of a flat, rectangular, prismatic channel (Fig. 6), he derived the energyequation:V2⎛d2⎞d 2 − d1 = 1 ⎜1 − 1 2 ⎟2 g ⎜⎝ d 2 ⎟⎠(3)Equation (3) corresponds to BÉLANGER's equation [59] (BÉLANGER 1828, p. 35).BÉLANGER's derivation is nothing more than the solution of the energy equation in terms of the specificenergy for a rectangular horizontal channel (Eq. (3)). It would give a reasonable approximation to thehydraulic jump solution for undular and weak jumps since there is very little energy loss in the jump forFroude numbers slightly greater than unity (MONTES 1986,1998), but the development is basicallyincorrect. Equation (3) may be rewritten in a dimensionless form as:⎛ ⎛dd21= 1 + Fr12 ⎜⎜1 − ⎜⎜ 2d12⎜ ⎝ d1⎝⎞⎟⎟⎠−2 ⎞⎟⎟⎟⎠(4)This result, compared to Equation (1), is obviously wrong as illustrated in Figure 7 because it neglected thedissipation of kinetic energy. While BÉLANGER's results matched well the experimental observations forBIDONE (1819) for low Froude numbers, Equation (4) diverges from the theoretical solution (Eq.
(1)) andexperimental observations at larger inflow Froude numbers because the rate of energy dissipation wasignored (Fig. 7). Figure 7 presents a comparison between Equations (1) and (4), and physical measurements.The latters include the data of BIDONE (1819) used by BÉLANGER to check his results as well as newexperimental observations in a 0.5 m wide rectangular channel at the University of Queensland shown inFigure 1B. Simply BÉLANGER (1828) applied incorrectly the Bernoulli principle to the hydraulic jump.3The examination report stated : "la commission est d'avis que le travail de M. Bélanger est fait avec beaucoup detalent, et qu'il peut être fort utile; en conséquence, elle pense qu'il doit mériter à son auteur des témoignages desatisfaction et d'encouragement" ('the committee advises that the study of M.
Bélanger is talentuous and that it isuseful; therefore it believes that his author deserves congratulations').4Gustave Gaspard CORIOLIS studied at the Ecole Polytechnique with Jean-Baptise BÉLANGER, and he was anotherIngénieur du Corps des Ponts et Chaussées. He introduced the kinetic energy correction coefficient (CORIOLIS 1836)and he is well known for his works on rotating bodies.5Jean-Baptiste BÉLANGER found his error in 1838: "de nouvelles réflexions m'ont conduit en 1838 àreconnaitre que cette hypothèse n'était pas admissible" ('new thoughts led me in 1838 to acknowledge thatthe assumption was incorrect') (BÉLANGER 1849, p. 91).
BÉLANGER (1841) solved the momentumequation for a hydraulic jump in a flat channel. For a rectangular channel and neglecting the friction force, heobtained :d211=− ++ 2 α' Fr12d124(5a)where α' is a velocity correction coefficient (5). This reasoning became commonly accepted thereafter(BÉLANGER 1849, BRESSE 1860). For example, BRESSE (1860, p. 251) presented the same result in theform:d211=− ++ 2 Fr12d124(5b)Equations (5a) and (5b) are mere rewritings of Equation (1).Fig. 5 - Cover page of the manuscript BÉLANGER (1828)5Based upon BÉLANGER's (1841, pp.
88-89; 1849, pp. 82-86) development, α' should be the momentum correctioncoefficient, or Boussinesq coefficient, but BÉLANGER (1841,1849) gave a definition corresponding to the Coriolis6Fig. 6 - BÉLANGER's (1828) original sketch of a hydraulic jump: "coupe longitudinale du courant auxenvirons du ressaut"10Momentum Eq.BELANGERBIDONE dataUQ data987d2/d165432111.522.533.544.555.566.577.5Fr1Fig. 7 - Ratio of conjugate depths for a hydraulic jump in a rectangular, horizontal, prismatic channel Comparison between Equation (1) (black dashed lie), Equation (4) by BÉLANGER (1828) (red solid line),experimental data by BIDONE (1819) and data in a 0.5 m wide channel at the University of Queenslandcoefficient.7Table 1 - Notations used by Jean-Baptiste BÉLANGERDefinitionWater depthFlow velocityDischargeWetted perimeterCross-section areaAngle between the invert andthe horizontalPressureLongitudinal distanceDistance normal to the invertDepth below the free-surfaceInvert slopeTotal head line slope (energyslope)Normal depthRectangular channel widthTotal head above weir crestModern notation (+)hv-χωγBÉLANGER(1841,1849)hUQχΩ--psy-i---s-xiIPxyd-ySo = sinθSfHλ--H2LZdoBHBÉLANGER (1828)dVQPwAθNote: (+) CHANSON (1999,2004).DiscussionDespite his erroneous treatment, BÉLANGER (1828) demonstrated some seminal features of the hydraulicjump.
He highlighted the significance of the inflow Froude number Fr1 = V1 / g d1 , showing that ahydraulic jump occurs only for Fr1 > 1: "selon qu'on aura h < v 2 / g ou h > v 2 / g , la formule [du ressaut]sera applicable ou ne le sera pas" ('depending whether h < v 2 / g or h > v 2 / g , the hydraulic jump formulawill be applicable or not'). He also showed the existence of critical flow conditions in a rectangularhorizontal channel for V2 = g d. This was twenty-four and forty-four years respectively before thepublications of Ferdinand REECH (1852) and William FROUDE (1872) who were both credited with theintroduction of the Reech-Froude number V / g d .Jean-Baptiste BÉLANGER applied successfully the backwater equation upstream and downstream of thehydraulic jump, and pointed out that it cannot be applied across the jump itself (BÉLANGER 1828).
Heshowed also how to estimate the jump location by combining the backwater calculations, upstream anddownstream of the jump, with the hydraulic jump equation.84. Gradually-varied flow calculations: the backwater equationBÉLANGER (1828) aimed to calculate the free-surface profiles of gradually-varied open channel flows. Hedeveloped the backwater equation within a series of basic assumptions. These were: (a) a steady flow, (b) anone-dimensional flow motion, (c) a gradual variation of the wetted surface with distance x along the channel,(d) friction losses that are the same as for an uniform equilibrium flow for the same depth and discharge, and(e) a hydrostatic pressure distribution.Within the above assumptions, BÉLANGER (1828, pp. 1-11) derived the backwater equation frommomentum considerations and he obtained :PQ2∂A = 0sin θ ∂x − cos θ ∂d − w (a V + b V 2 ) +Ag A3(6)where θ is the angle between the bed and the horizontal, x is the longitudinal distance positive downstream,d is the flow depth measured normal to the invert, A is the cross-section area, Pw is the wetted perimeter, Q isthe discharge.
Equation (6) corresponds to Equation [16] in BÉLANGER (1828, p. 9). It may be rewritten ina more conventional form as a differential equation:sin θ − cos θ∂d PwQ 2 ∂A−(a V + b V 2 ) +=0∂x Ag A 3 ∂x(7)In Equations (6) and (7), BÉLANGER (1828) estimated the friction losses using the Prony formula (6) :−(∂H4=a V + b V2∂x D H)(8)where H is the total head, DH is the hydraulic diameter: DH = 4 A/Pw, and a and b are constant. Several valueswere proposed for the coefficients a and b (App. C). BÉLANGER (1828) used a = 4.44499 10-5 and b =3.093140 10-4 (in SI units) that were estimated by Johann EYTELWEIN (1764-1848).Equation (8) may be compared with modern expressions in terms of the Darcy-Weisbach friction factor:−()f V2∂H4=a V + b V2 =DH 2 g∂x D H(8b)Denoting Sf the friction slope: Sf = -∂H/∂x, and So the bed slope: So = sinθ, BÉLANGER's backwaterequation (6) may be combined with the continuity equation to yield:∂ ⎛⎜V 2 ⎞⎟d cos θ += So − Sf∂x ⎜⎝2 g ⎟⎠(9)Equation (9) is essentially identical to modern expressions of the backwater equation (HENDERSON 1966,MONTES 1998, CHANSON 2004).
For example, CHANSON (1999) expressed the backwater equation inits most general form as:cos θ6∂d∂θQ 2 ∂A− d sin θ−α= So − Sf∂x∂xg × A 3 ∂x(10)Interestingly BÉLANGER (1849, p. 54) was aware of the work of H.P.G. DARCY (1803-1858) in pipe flows, but hecontinued to use the Prony formula for its simplicity.9where α is the kinetic energy correction coefficient, or Coriolis coefficient. The main differences betweenBÉLANGER's equation (9) and Equation (10) are the Coriolis coefficient α and the non-constant bed slopeterm d sin θ ∂θ / ∂x .
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