Curved-streamline transitional flow from mild to steep slopes

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Curved-streamline transitional flow from mild tosteep slopesOscar Castro-Orgaz a & Willi H. Hager ba Instituto de Agricultura Sostenible, Consejo Superior de Investigaciones Cientificas ,Finca Alameda del Obispo., E-14080, Cordoba, Spain E-mail:b VAW, ETH Zürich , CH-8092, Zürich, Switzerland E-mail:Published online: 26 Apr 2010.

To cite this article: Oscar Castro-Orgaz & Willi H. Hager (2009) Curved-streamline transitional flow from mild to steepslopes, Journal of Hydraulic Research, 47:5, 574-584

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Journal of Hydraulic Research Vol. 47, No. 5 (2009), pp. 574–584

doi:10.3826/jhr.2009.3656

© 2009 International Association of Hydraulic Engineering and Research

Curved-streamline transitional flow from mild to steep slopes

Écoulement transitoire curviligne entre une pente douce et une pente raideOSCAR CASTRO-ORGAZ, (IAHR Member), PhD, Research Engineer, Instituto de Agricultura Sostenible, Consejo Superiorde Investigaciones Cientificas, Finca Alameda del Obispo., E-14080, Cordoba, Spain. E-Mail: [email protected]

WILLI H. HAGER, (IAHR Member), Professor, VAW, ETH Zürich, CH-8092, Zürich, Switzerland.E-mail: [email protected]

ABSTRACTFlows from mild to steep slopes are relevant in hydraulic engineering relating to chute spillways and slope changes in irrigation networks. These flowsare associated with transitional flow from sub- to super-critical conditions, smooth curvilinear-streamline feature involving a continuous free surfaceprofile, and a rapid variation of the bottom pressure profile. Herein, the transition from a horizontal to a steeply sloping rectangular channel reach isinvestigated to analyze the application range of the Boussinesq-type equation. Slope breaks with a rounded transition from the brink section to thetailwater channel are considered and compared with the potential flow solution based on the Laplace equation. The Boussinesq-type equation is furthercompared with test data for a steep downstream slope to investigate the strong curvilinear gravity effect. A singular point analysis is also comparedwith the Boussinesq approach. The generalized momentum equation for curvilinear flows is developed and compared with the results pertaining tothe energy concept.

RÉSUMÉLes écoulements passant de pente douce à forte se rencontrent en hydraulique des déversoirs et des réseaux d’irrigation. Ces écoulements font latransition entre un régime fluvial et torrentiel, le caractère curviligne lisse impliquant une surface libre continue, et une variation rapide du profil depression de fond. On étudie ici, la transition entre un canal rectangulaire horizontal et une pente forte pour analyser le champ d’application d’uneéquation de type Bousssinesq. Des ruptures de pente avec une transition arrondie entre la section de bord et le canal d’évacuation sont considérées etcomparées à la solution de l’écoulement potentiel basée sur l’équation de Laplace. L’équation de type Boussinesq est aussi comparée à des essais enpente raide pour étudier l’effet de la pesanteur en forte courbure. Une analyse de point singulier est également comparée à l’approche de Boussinesq.L’équation généralisée des quantités de mouvement pour les écoulements curvilignes est développée et comparée aux résultats relatifs au conceptd’énergie.

Keywords: Curvilinear flow, One-dimensional flow, Open channel, Spillway, Transitional flow, Weirs

1 Introduction

Flows from mild to steep slopes are associated with transitionalflow from sub- to super-critical conditions, smooth curvilinear-streamline feature involving a continuous free surface profile,and a rapid variation of the bottom pressure profile (Fig. 1). Thetransitional flow profile h = h(x) from mild to steep slope wasanalyzed by Massé (1938) using the singular point theory basedon the gradually-varied open channel flow equation. Assuminga hydrostatic pressure distribution this approach does not pre-dict the detailed two-dimensional (2D) flow features in a verticalplane associated with curvilinear-streamline flow; therefore, itmay only be considered an estimate for h(x). The singular pointanalysis relates to a finite free surface slope at the critical depthlocation, contrary to the unrealistic vertical flow profile of stan-dard gradually-varied flow (Chow 1959). Although the singular

Revision received May 19, 2009/Open for discussion until April 30, 2010.

574

point method approximates weakly-curved mild to steep slopetransitions, no comparison with experimental data is availableyet. At an abrupt slope break (Fig. 1(a)), the flow separates atthe bottom kink (Rouse 1932, Weyermuller and Mostafa 1976),a feature beyond the scope of the present research.

The inclusion of streamline curvature effects in the open chan-nel flow equation due to Boussinesq (1877) assumes a linearvelocity distribution normal to the channel bottom resulting in apseudo 2D approach. Similar closure hypotheses were advancede.g. by Fawer (1937), Matthew (1963, 1991), Mandrup Andersen(1975) or Hager and Hutter (1984). Mandrup Andersen (1975)used a Boussinesq-type energy equation for slope breaks of lessthan 5◦ degrees obtaining fair agreement with his own test data.However, his approximation was not compared with severe slopebreaks of say larger than 30◦, involving strong curvilinear effects,as are common in hydraulic engineering. The mathematical

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Journal of Hydraulic Research Vol. 47, No. 5 (2009) Curved-streamline transitional flow from mild to steep slopes 575

Figure 1 Transitional flow from mild to steep slopes (a) Typical viewof experimental test (Rouse 1932) (b) definition sketch

development of Boussinesq-type equations is commonly sub-jected to small streamline curvature (Hager and Hutter 1984).Interestingly, as discussed by Matthew (1995), their range ofapplication may be much higher than expected from the limitedmathematical constraints. Other approximations for flow overcurved channel bottoms using bottom-fitted coordinates includethe perturbation approach of Dressler (1978) and the more recentmodels by Berger and Carey (1998) and Dewals et al. (2006). Inaddition, moment of momentum equations were also formulatedto predict flow over curved bottoms by Khan and Steffler (1996a,1996b).

Herein, transitions from a horizontal to a steeply slopingrectangular channel reach are investigated to analyze the appli-cation range of the Boussinesq-type equation. Slope breaks witha rounded transition from the brink section to the tailwater chan-nel slope are considered and compared with the potential flowsolution based on the Laplace equation (Montes 1992, 1994).The Boussinesq-type equation is further compared with test dataof Hasumi (1931) and Westernacher (1965) for transitions witha large downstream slope to investigate the strong curvilineargravity effect. Massé’s (1938) singular point analysis is furthercompared with the Boussinesq approach. A generalized momen-tum equation for curvilinear flows is developed and comparedwith the results pertaining to the energy concept, to check theaccuracy of the results.

2 Extended Boussinesq energy equation

2.1 Governing equations

Matthew (1991) obtained for 2D, irrotational and incompressiblefree surface flow a second order approximation for the horizontalvelocity u in the x-direction by iterative Picard expansion of the

stream function as

u = q

h

[1 +

(z′′ − 2h′z′

h

) (2η− h

2

)

+(h′′

2h− h′2

h2

) (3η2 − h2

3

)](1)

with q = discharge per unit width, h = flow depth measuredvertically, h′ = dh/dx, h′′ = d2h/dx2, z = bottom elevation,z′ = dz/dx, z′′ = d2z/dx2, y = vertical coordinate and η =y − z. To the same order of approximation, the vertical velocityprofile v(η) varies linearly from the channel bottom to the freesurface as

v = q

h

[z′ + η

hh′

](2)

Similar results to Eqs. (1) and (2) were obtained by Hager andHutter (1984) and Hager (1985) yet by defining the flow depthas the vertical projection of an equi-potential curve, or normal,rather than vertical distance between the bottom and the freesurface, as Matthew (1991) did. At the free surface (η = h) thepressure is atmospheric and the total head H is given by

H = z+ h+ q2

2gh2

(1 + 2hh′′ − h′2

3+ hz′′ + z′2

)(3)

similar to the equations of Fawer (1937), Matthew (1963), Hagerand Hutter (1984) and Montes (1998). As previously, Eq. (3)differs from the earlier results by Hager and Hutter (1984) andHager (1985) in the definition of the flow depth. However, bothresults are correct to the same order of accuracy. Equation (3)is a 2nd order differential equation from which the free surfaceprofile h = h(x) may be obtained. For given H and prescribedboundary conditions at the two extreme channel sections, Eq. (3)may be solved numerically. The velocity distributions u(η) andv(η) are then computed from Eqs. (1) and (2), and the pressurep distribution deduced from the Bernoulli equation as

p

γ= H − z− η− u2 + v2

2g(4)

again with minor differences if compared to Hager and Hutter(1984). The bottom (subscript b) pressure profile pb = p(η = 0)is obtained from Eq. (4), using Eqs. (1) and (2), as

pb

γ= h+ q2

2gh2(2hz′′ + hh′′ − h′2 − 2z′h′) (5)

Note that if z′′ = h′′ = h′ = 0 the pressure distributionis hydrostatic and the pressure head equal to the vertical flowdepth h.

2.2 Boundary conditions

Test data of Hasumi (1931) and Westernacher (1965) indicatethat the critical depth hc = (q2/g)1/3 for parallel-streamline flowestablishes on the horizontal slope, at a distance of around 3hcupstream from the brink section, located at the beginning of

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576 O. Castro-Orgaz and W.H. Hager Journal of Hydraulic Research Vol. 47, No. 5 (2009)

the circular arc transition. At the critical section, a hydrostaticpressure distribution prevails (Westernacher 1965). A Cartesiancoordinate system (x, η) is placed at the brink section, with acircular-shaped transition of radius R connecting the horizontaland the tailwater reaches (Fig. 1b). Thus, the upstream (subscriptu) boundary conditionhu = h(xu = −3hc) = hc is used for com-putational purposes. This critical flow condition at the upstreamboundary section fixes the energy line on the horizontal bottom(z = 0) to H = 3hc/2.

The downstream (subscript d) boundary condition is set wherethe streamlines may be assumed to be nearly parallel to the chan-nel bottom. From Eq. (3), this condition results for a slopingbottom in (Hager 1999)

H = z+ h+ q2

2gh2(1 + z′2) (6)

The downstream flow depth hd thus must satisfy Eq. (6). Basedon test data of Hasumi (1931) and Westernacher (1965) xd ∼=+3hc. Further comments on the accuracy of this approximationare given below.

2.3 Discretization scheme

The computational domain −3 ≤ x/hc ≤ +3 was divided into120 to 180 computational nodes depending on whether the finalselection for xd was +3 or larger, as discussed below. The termsh′ and h′′ in Eq. (3) were estimated with 5-point central finitedifferences as (Abramowitz and Stegun 1972)

h′ = −hi+2 + 8hi+1 − 8hi−1 + hi−2

12�(7)

h′′ = −hi+2 + 16hi+1 − 30hi + 16hi−1 − hi−2

12�2(8)

to reduce truncation errors, with i = computational node indexin the x-direction and � = step length. � was successivelyreduced until the numerical solution had converged. The termsh′ and h′′ for the computational nodes in the extreme boundarysection vicinity were estimated with 3-point central finite dif-ferences, to avoid imaginary nodes outside the computationaldomain. Equations (7) and (8) were substituted using Eq. (3) foreach computational node, resulting in a system of i−2 non-linearimplicit equations for flow depth hi in each computational node.The system of equations was solved iteratively as an optimizationproblem using the SOLVER module of Excel.

2.4 Free surface and bottom pressure profiles

Equation (3) is based on the assumption that the boundary stream-lines, given by the free surface and the bottom profiles, arecontinuous at least up to second order derivatives. The prob-lem treated herein consists of a horizontal channel followed by asteep chute, connected by a circular-arc transition. This bottom

geometry is continuous in the bottom slope but the bottom cur-vature has discontinuities both at the beginning and the end ofthe transition. The bottom profile thus violates the assumptionsof Eq. (3) at two computational nodes.

A first solution of the numerical model of Eq. (3) was made,and the discontinuities were removed with a numerical estimationof bottom curvature from a 5-point central finite difference as

z′′ = −zi+2 + 16zi+1 − 30zi + 16zi−1 − zi−2

12�2(9)

thereby numerically determining the free surface profile h =h(x). Once h = h(x) was found, pb = pb(x) was computedfrom Eq. (5) using the numerical results. The curvature term atthe downstream end of the circular arc generated an abrupt peakwith this method. Surprisingly, however, this resulted in excellentcomputed free surface profiles h = h(x), whereas the pressurehead profile pb = pb(x) was poor (Fig. 2), with a peak at theend of the circular arc transition not observed in experiments(Fig. 3 below). Thus, a smoothed curve for the bottom profilein the circular arc transition was added to provide a continuoustransition of bottom curvature, and to improve pb = pb(x). A5th degree polynomial was used to approximate the circular arcprofile z/R = −1 + [1 − (x/R)2]. At the upstream extreme ofthe smoothed interval the conditions were z′ = z′′ = 0, whereasz′′ = 0 and z′ = −So were imposed at the downstream end.The beginning and end of the smoothed interval were consid-ered iteratively to reduce the differences between the circulararc and the polynomial fits. The up- and down-stream smoothedinterval limits were x/hc = −0.1 and 1.05, respectively, forSo = 1, and x/hc = 0 and 1.1 for So = 1.73. The free sur-face profile h = h(x) was numerically determined using thesmoothed transitional curve resulting in almost the same resultas computed previously from Eq. (9), with imperceptible devia-tions, but significantly improving pb = pb(x) (Fig. 2), implying

Figure 2 Effect of smoothed transitional bottom profile on free surface(h+ z)/hc and bottom pressure (pb/(γ)+ z)/hc profiles for R/hc = 1and So = 1

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Figure 3 Comparison of computed (h + z)/hc and (pb/(γ) + z)/hc distributions from Eqs. (3) and (5) with test data of Hasumi (1931) for[R/hc; So] = (a) [1.59;1], (b) [1;1], (c) [0.76;1], (d) [1.59;1.732], (e) [1;1.732], (f) [0.76;1.732]

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that h = h(x) is independent of the lower boundary conditionz = z(x), whereas pb = pb(x) is not.

The numerical results of Eq. (3) are compared in Fig. 3 withexperimental data of Hasumi (1931) for So = 1 and 1.732,and R/hc = 1.59, 1 and 0.76. These compare favourably withobservations, even for R/hc = 0.76 and So = 1.732 (60◦) corre-sponding to highly curvilinear flow. The model results are furthercompared in Fig. 4 with the free surface profiles of Westernacher(1965) from potential flow nets for So = 1.5 and R/hc = 1.6,and 2.57, resulting again in excellent agreement. As mentionedabove, the downstream boundary condition Eq. (6) was generallyset at xd ∼= +3hc.

However, computational results indicated that the boundarysection had to be moved to xd ∼= +3.5hc in certain simulations.If the downstream boundary section was located too close to

Figure 4 Comparison of computed profiles (h + z)/hc from Eq. (3)with test data of Westernacher (1965) for [R/hc; So] = (a) [1.6;1.5],(b) [2.57;1.5]

the brink section, an abrupt drawdown of the flow profile h =h(x) resulted near xd in an unrealistic, sub-atmospheric pressurepeak pb = pb(x) close to xd due to the Boussinesq terms h′ andh′′ in Eq. (5). This erroneous mathematical effect was simplyremoved by increasing xd until the entire profile pb = pb(x)

on the steep slope was nearly parallel to the channel bottom.If xd is located too close to the brink, the profile h = h(x) ofnecessity generates an abrupt drawdown to satisfy the parallelflow boundary condition imposed at xd . This drawdown impliesboth h′ and h′′ < 0, resulting that pb close to xd reduces aboveits correct value for nearly parallel-streamline flow (Eq. (5)). Theposition of the downstream boundary condition has to satisfy theoriginal hypothesis of nearly parallel-streamline flow, allowingto determine xd by iteration, until the stable solution is obtained.

The brink depth hb = h(x = 0) as a function of So forR/hc =1 using Eq. (3) (Fig. 5(a)) is compared in Fig. 5(b) with thesolution of the inverse form of the Laplace equation (Montes1994)

∂2y

∂x2

(∂y

∂ψ

)2

+ ∂2y

∂ψ2

[1 +

(∂y

∂x

)2]

− 2∂2y

∂x∂ψ

∂y

∂x

∂y

∂ψ= 0

(10)

where ψ = stream function. The Laplace equation predicts anearly constant valuehb ≈ 0.70 forSo > 1, whereas the extendedBoussinesq equation yields hb ≈ 0.68. Given the high numer-ical accuracy obtained with the present discretization scheme,this difference appears to be a consequence of the vertical veloc-ity profile implicit in Eq. (3), although irrelevant for practicalpurposes. Based on Fig. 5(b) the potential flow solution fromEq. (3) is in excellent agreement with the more accurate resultsof the Laplace equation. Test data of Mandrup Andersen (1975)and Weyermuller and Mostafa (1976) for small downstreamslopes also corroborate the results of the Boussinesq model, withimperceptible differences from the Laplace equation.

The results for the free surface slope h′b at the brink section

are successfully compared with the Laplace equation in Fig. 5(c),predicting a limiting value of h′

b = −0.27 (15.1◦), whereas theBoussinesq energy equation yields h′

b = −0.31 (17.22◦). Testdata of Mandrup Andersen (1975), Weyermuller and Mostafa(1976) and Hasumi (1931) also agree with Eq. (3). The equationfor the free surface slope at a free overfall (Matthew 1995)

h′2 = 3

(1 − hb

hc

)3

(11)

was also inserted in Fig. 5(c), using the computed values for hbin Eq. (3), resulting in excellent agreement with the numericalresults. Accordingly, there is a strong analogy between the flowsover a free overfall and the transition from mild to steep slopes.

The model results for hb are compared in Fig. 5(d) with testdata of Westernacher (1965) for So = 1.5 as a function of dimen-sionless discharge qo = q/(gR3)1/2. The computed curve fromEq. (3) is slightly below the test data, because of a small vis-cous effect. The brink depth results based on potential flow netsby Westernacher (1965) are also included, resulting in fair agree-ment with Eq. (3). Based on Fig. 5 the transition from mild to steepslope may be proposed as a simple and accurate flow measuring

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Figure 5 Brink depth results (a) definition sketch, (b) hb/hc(So) from(—) Eq. (3) for R/hc = 1, (c) h′

b(So) from (—) Eq. (3) for R/hc = 1,(- - -) Eq. (11), (d) hb/hc(qo) from (—) Eq. (3) for So = 1.5. (◦)Laplace equation (Montes 1994), test data of (�) Weyermuller andMostafa (1976), (�) Mandrup Andersen (1975), (•) Hasumi (1931),Westernacher (1965) (�) potential flow solution, (◦) test data.

device, similar to a free overfall. As shown in Fig. 5(b), the brinkdepth ratio hb/hc varies only with R/hc, or qo = (R/hc)

−3/2 ifSo > 1. Figure 5(d) relating to So = 1.5 indicates the generalrelationshiphb/hc(qo)without any chute slope effect. Measuringhb, q may be assumed, hc = (q2/g)1/3 is determined, resultingin qo = (R/hc)

−3/2. Estimating hb/hc, qo may be obtained withFig. 5(d), to be compared with the value previously assumed. Thisiterative sequence is repeated until sufficient convergence. Forpractical purposes the curve of Fig. 5(d) may be approximated bythe empirical equation hb/hc = 0.70q−0.06

o for 0.01 < qo < 0.6such that q is directly obtained for a given hb.

2.5 Velocity distribution

As discussed by Montes (1994) using the numerical solution ofEq. (10), the linear vertical velocity distribution implicit in Eq. (2)is generally not reached in zones of highly curvilinear flow, suchas along the brink depth reach. However, this does not necessarilyimply that the Boussinesq solution fails.

Herein, it was previously shown that the profiles h = h(x)

andpb = pb(x)may be accurately predicted with the Boussinesqapproach, as they are not significantly affected by the approxi-mation in Eq. (2). Figure 6 further compares the present resultsfor the dimensionless horizontal velocity distribution u/Uc(νc)computed from Eq. (1) with these of Montes (1994), in whichνc = η/hc and Uc = q/hc. Note that the results from Eq. (1)are in excellent agreement with the results of the Laplace equa-tion for So = 0.087 and So = 0.176 (Fig. 6(a, b)). For highchute slopes So = 1.5 and 1.732, the profile u/Uc(νc) remainsunaffected, as was also observed by Montes (1994). Despite devi-ations between Eq. (1) and the results from Eq. (10) of Montes

Figure 6 Dimensionless horizontal velocity distribution u/Uc(νc) from(—) Eq. (1), (◦) Laplace equation (Montes 1994) for R/hc = 1 andSo = (a) 0.087, (b) 0.176, (c) both 1.5 and 1.732

(1994) shown in Fig. 6(c) for So = 1.5 and 1.732, the results arereasonably accurate for practical purposes.

3 Longitudinal momentum balance

3.1 Vertically-integrated momentum equation

The continuity equation for 2D, irrotational and incompressiblefree surface flow reads

∂u

∂x+ ∂v

∂y= 0 (12)

and the Euler momentum equation in the x-direction is with ρ =water density (Jaeger 1956, Montes 1998)

u∂u

∂x+ v

∂v

∂y= − 1

ρ

∂p

∂x(13)

Combining Eqs. (12) and (13), and integrating verticallyleads to∫ h+z

z

∂u2

∂xdy +

∫ h+z

z

∂ (uv)

∂ydy = − 1

ρ

∫ h+z

z

∂p

∂xdy (14)

Using the Leibniz rule, and the boundary conditions v(η = 0) =z′q/h and v(η = h) = (z′ + h′)q/h yields

d

dx

∫ h+z

z

(p

γ+ u2

g

)dy = −z′pb

γ(15)

The left side integral of Eq. (15) is referred to as specific momen-tum S, resulting in the generalized longitudinal momentum

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balance for potential curved-streamline flow

dS

dx= −z′pb

γ(16)

Equation (16) implies that for potential flow over a sloping bot-tom H is conserved whereas S is not. The specific momentumis unbalanced due to a pressure force resultant acting on thechannel bottom in the horizontal direction. It is noteworthy torelate Eq. (16) to the standard gradually-varied flow equation, forwhich the pressure distribution is hydrostatic, i.e. pb = γh andS = h2/2 + q2/(gh), resulting for potential parallel-streamlineflow in the well-known statement

dS

dx= −z′h (17)

3.2 S-H relationship

The Bernoulli Eq. (4) for potential flow may be combined with

S =∫ h+z

z

(p

γ+ u2

g

)dy (18)

to eliminate the pressure term, resulting in (Matthew 1995)

S =(H − z− h

2

)h+

∫ h+z

z

(u2 − v2

2g

)dy (19)

Inserting the velocity vector components u and v from Eqs. (1)and (2), integration of Eq. (19) yields by retaining first orderterms

S =(H − z− h

2

)h+ q2

2gh

(1 − z′2 − z′h′ − h′2

3

)(20)

which is the relationship between S and H for curved stream-line potential flow. Once the free surface profile h = h(x) isdetermined from Eq. (3), the profile S = S(x) is available.

3.3 Bottom pressure profile

From the profile S = S(x) the term dS/dx was obtained from a5-point central finite difference scheme as

dS

dx= −Si+2 + 8Si+1 − 8Si−1 + Si−2

12�(21)

to compute the bottom pressure distribution pb = pb(x) fromEq. (16). Obviously, along the horizontal reach, z′ = 0 andS = const = 3h2

c/2. Figure 7 comparespb = pb(x) from Eqs. (5)and (16) for x/hc > 0. The results are in excellent agreement,with imperceptible differences in the major part of the compu-tational domain, indicating that the numerical results obtainedherein for h = h(x) satisfy both the energy and momentum bal-ances, and may be considered accurate for potential flows in mildto steep slope transitions. Near x = 0 both z′ and dS/dx ≈ 0,such that Eq. (16) may be not accurate. This was observed inthe simulations for So = 1.732, where a peak in pb = pb(x)

resulted near x = 0 (Fig. 7(d)). Note that Eq. (16) was usedto compute pb = pb(x); near x = 0, pb is determined by theratio (dS/dx)/(dz/dx), whose terms are nearly zero in this zone.The simplification used herein for the bottom profile, using a

smoothed polynomial, results in the discrepancies observed. Thisfurther proves the sensitivity of the bottom pressure profile to thebottom geometry. For x < 0 conservation of momentum waspreserved, with numerical values for S in each computationalnode of the order of S/h2

c = 1.499. Figure 8 successfully com-pares the numerical results for the pressure distributions on thehorizontal bottom for So = 1 with these from the Laplace equa-tion by Montes (1994). It may therefore be concluded that themomentum balance and the pressure distribution in the domainx < 0 are correctly reproduced with the Boussinesq equation.

Using the results of Fig. 7, the minimum bottom pressurehead pM/(γhc) is plotted in Fig. 9(a) versus R/hc for So = 1and 1.732. The theoretical results are compared with Hasumi’sdata (1931), resulting in excellent agreement for So = 1, as wasalso observed in Fig. 3. For So = 1.732 the Boussinesq modelresults in lower values for pM/(γhc), although the profile shapepb = pb(x) is well predicted (Fig. 3). Again, the small deviationsmay be attributed to the approximate transition geometry usedherein. The x-coordinate of the minimum pressure section xMis plotted in Fig. 9(b) versus R/hc for So = 1 and 1.732. Thetheoretical results again compare well with Hasumi’s data (1931)for So = 1 and So = 1.732. The Boussinesq model locatesthe minimum pressure near the brink section, yet the results arereasonable for practical purposes.

4 Critical depth theory

4.1 Free surface control points

In potential flows the energy head is constant, i.e. dH/dx = 0.Differentiation of Eq. (3) results in

1 = q2

gh3

(1 + 2hh′′ − h′2

3− h2h′′′

3h′

)(22)

for the horizontal slope reach. Equation (22) is equivalent to theminimum specific energy condition (Hager 1985, Castro-Orgazet al. 2008), associated with critical flow. However, for the caseconsidered herein, there is no extreme in the channel bottomprofile, as for weirs where z′ = 0 at the crest section. This impliesthat Eq. (22) does not apply to any particular channel section, asin curvilinear critical flow, and is a relation describing the entirefree surface profile in a horizontal channel. The upstream criticaldepth section with parallel streamline flow is a particular case ofEq. (22), where h′ = h′′ = h′′′, i.e. q2/(gh3) = 1. A critical flowapproach with curvilinear streamlines does not provide a usefulsolution for the transition from mild to steep slopes, given theabsence of a fixed channel section for critical point computations.

4.2 Gradually-varied singular point method

Based on Eq. (6), the gradually-varied flow equation for potentialflow in a sloping channel is

dh

dx= −z′

1 −[q2

gh3(1 + z′2)

] (23)

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Figure 7 Comparison of computed pressure head profiles pb/(γhc)[x/hc] from (—) Eq. (5) and (◦) Eq. (16) for [R/hc; So] = (a) [1.59;1], (b) [1;1],(c) [0.76;1], (d) [1.59;1.732], (e) [1;1.732], (f) [0.76;1.732]

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Figure 8 Comparison of computed pressure head profilespb/(γhc)[x/hc] from (—) Eq. (5) and (◦) Montes (1994) forx < 0, R/hc = 1 and So = 1

At the brink section there is a critical point based on Eq. (23).Using Massé’s (1938) singular point method, the free surfaceslope at the critical point is

dh

dx= −

(−hc

3z′′

)1/2

(24)

Equation (23) was solved numerically using a standard 4th orderRunge-Kutta scheme, starting at the critical point with Eq. (24)for the free surface slope. Obviously, for the horizontal bottomreach (z′ = 0) Eq. (23) yields dh/dx = 0, i.e. h = const =hc. The results are compared in Fig. 10 with Eq. (3), resultingin excellent agreement for x/hc > 1. The 1D and 2D results

Figure 9 Minimum pressure features, Comparison of computed minimum bottom pressure (a) head pM/(γhc) versus R/hc from (- - -, —) Eq. (5)and (◦,�) Hasumi (1931) for So = 1 and 1.732, respectively, (b) position xM/(γhc) versus R/hc from (- - -, —) solution of Eq. (3) and (◦,�) data of(Hasumi 1931) for So = 1 and 1.732, respectively

therefore practically agree, and the streamline curvature effect isabsent. For hypercritical flow with F = q/(gh3)1/2 > 3 (Castro-Orgaz 2009) Eq. (23) may be simplified to

dh

dx= −z′

−[q2

gh3 (1 + z′2)] (25)

whose general solution with So = −z′ is, using the boundarycondition h(x = 0) = hc

h

hc=

(1 + 2So

1 + S2o

x

hc

)−1/2

(26)

Figure 10 shows that Eq. (26) yields almost the same resultas Eqs. (23) and (3) for x/hc > 1, resulting in an accu-rate approximation for the chute flow portion. There are onlyminor differences between Eqs. (23) and (25) in the domain0 < x/hc < 1, where the results of Eq. (23) collapse withthe improved results from Eq. (3). The hypercritical approachderived from the gradually-varied flow theory may therefore beused on the supercritical slope reach.

5 Conclusions

The extended Boussinesq type energy equation was applied tothe curvilinear flow zone in slope breaks with a rounded transi-tion from the brink section to the downstream chute slope. Theresults compare favourably with the potential flow solution basedon the Laplace equation, and the experimental data of mild tosteep slope transitions, resulting in excellent predictions for thefree surface and bottom pressure profiles. This indicates that theBoussinesq approach may be used in zones of high curvilineareffects as a reasonable approximation to the more complete solu-tion based on the Laplace equation. Its practical applicability

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Figure 10 Comparison of computed profiles (h + z)/hc[x/hc] fromEq. (3) and Eq. (23) starting at the critical point (•) and Eq. (26) forR/hc = 1 and So = (a) 0.5, (b) 1

is, therefore, much wider than expected from the smallstreamline curvature constraint assumed for its mathematicaldevelopment.

Massé’s (1938) singular point analysis was compared withthe Boussinesq approach for potential flow, resulting in excellentagreement for the chute flow zone. However, this method is basedon the gradually-varied flow equation and thus neither predicts thestrong variation of the bottom pressure profile below the hydro-static value, nor the correct free surface profile at and upstreamof the brink domain. The generalised momentum equation forcurvilinear flows was developed and compared with the resultspertaining to the energy concept, indicating that the potential flowsolution presented herein satisfies both conservation principles ofenergy and momentum.

Acknowledgements

The first author was supported by a contract of modality JAE-DOC of the program “Junta para la Ampliación de Estudios”,CSIC, National Research Council of Spain, co-financed by theFSE.

Notation

g=Acceleration of gravity (m/s2)F = Froude number (-)H = Total energy head (m)h= Flow depth measured vertically (m)h′ = dh/dx (-)h′′ = d2h/dx2 (m−1)

h′′′ = d3h/dx3 (m−2)

hc = Critical depth for parallel-streamline flow (m) = (q2/g)1/3

y=Vertical elevation (m)p= Pressure (N/m2)

pb = Bottom pressure (N/m2)

q= Unit discharge (m2/s)qo = Dimensionless discharge (-)R= Radius of circular-arc transition (m)S= Specific momentum (m2)

So = Channel bottom slope (-)u=Velocity in x-direction (m/s)U = Mean flow velocity (m/s) = q/hv=Velocity in y-direction (m/s)x= Horizontal distance (m)z= Elevation of channel bottom (m)z′ = dz/dx (-)z′′ = d2z/dx2 (m−1)

η=Vertical coordinate above channel bottom (m)i= Computational node index in x-direction (-)�= Step length (-)γ = Specific weight of water (N/m3)

ρ=Water density (N/m3)

νc = Dimensionless vertical coordinate above channel bottom (-)ψ= Stream function (m2/s)

Subscripts

b= Relative to brink sectionc= Relative to critical flowd= Relative to downstream boundary conditionM= Relative to minimum bottom pressureu= Relative to upstream boundary condition

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Documents

Curved-streamline transitional flow from mild to steep slopes