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Linear and angular momentum conservation in hydraulic jump in diverging channels

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Linear and angular momentum conservation inhydraulic jump in converging channels
Alessandro Valiani
a
, Valerio Caleﬃ
a,
∗
a
Universit`a degli Studi di Ferrara, Dipartimento di Ingegneria, Via Saragat, 1 - 44122 Ferrara, Italy
Abstract
This article is the continuation of a previously published work concerningthe integral conservation of linear and angular momentum in the steady hy-draulic jump in a linearly diverging channel [Valiani and Caleﬃ,
Advances in Water Resources
, 34(2), pp. 227–242]. Here, the same reasoning is applied toa linearly converging channel, and the general framework is shown to remainvalid. The ﬂow is considered to be divided into a mainstream that conveysthe total liquid discharge and a roller in which no average mass transportoccurs. The wall friction is neglected, and the ﬂow can be considered to beaxially symmetric. It is assumed that no macroscopic rheological relationshipholds; consequently, the mass, momentum and angular momentum integralbalances are independent relationships. The normal stresses are assumed tobe hydrostatic on the vertical and cylindrical surfaces. The viscous shearstresses are assumed to be negligible with respect to the turbulent stresses.Assuming that the horizontal velocity distribution in the mainstream is uni-
∗
Corresponding author.
Email addresses:
alessandro.valiani@unife.it
(Alessandro Valiani),
valerio.caleffi@unife.it
(Valerio Caleﬃ)
Preprint submitted to Advances in Water Resources February 17, 2012
form and that the horizontal momentum and angular momentum in the rollerare negligible with respect to their mainstream counterparts, an analyticalsolution is obtained for the free surface proﬁle of the ﬂow. This solution al-lows the determination of the sequent depths and their positions. Thus, thelength of the jump, which is assumed to be equal to the length of the roller,is also found. The mainstream and roller thicknesses can also be derived. Byextension, the volumes of the roller, the mainstream and the whole streambetween the sequent depths can be easily determined. This model may betheoretically used to derive the average shear stress exerted by the roller onthe mainstream and the related power loss per unit weight. This relation-ship gives the well-known classical expression for the total power loss in the jump as an independent result, and it consequently demonstrates that theproposed mechanical model is internally consistent. The initial expansionrate of the mainstream thickness is obtained from an empirical relationshipthat is diﬀerent from that of the diverging case, as can be expected from thediﬀerent geometry of the expanding and contracting jets.
Keywords:
hydraulic jump, radial ﬂow, free surface ﬂow, convergingchannel, linear momentum balance, angular momentum balance
1. Introduction
The importance of the hydraulic jump in environmental hydraulics andhydraulic engineering is well known, as is its relevance to both ﬂuid mechanicsand classical hydraulics. We refer the reader to the literature for an extensivediscussion; see, for example, [1, 2, 3]. This article is a continuation of the work
presented in [3], which concerns the hydraulic jump mechanics in linearly2
diverging channels. The present work concerns the same matter in linearlyconverging channels.The ﬁrst author identiﬁed the problem of unbalanced angular momentumin the hydraulic jump for rectangular channels in [2]; the same problem isshown to exist also for the jump in diverging channel in [3], and the presentwork discusses the problem in converging channels. The latter case is lessused in engineering practice because it is capable of producing an unstableposition [4, 5]. Nevertheless, it is of signiﬁcant importance to both basic ﬂuidmechanics and water works management [6].This paper focuses on fully developed turbulent ﬂow over a smooth bot-tom with negligible surface tension and viscosity eﬀects. This domain is thetypical ﬁeld of hydraulic engineering, while physicists often work on small-scale problems that are dominated by viscosity and surface tension (see, forexample, [7] and the references therein).For an extended review of the classical hydraulic jump we refer to [2,8], while [3] addresses the case of the linearly diverging channel (or radial
jump). Important ideas concerning the physics (including the fundamentalreasoning on the dimensionless parameters that govern the phenomenon) inradial jumps are stimulated by two classical experimental works of the 1960sby Rubatta [4, 9], in which accurate large-scale experimental investigationsare described.When treating a stationary jump or a moving bore at the length scaleof the channel, it must be considered a discontinuity in the water depth.The modern high-resolution techniques of scientiﬁc computing have deeplyanalyzed this problem; see, for example, [10, 11, 12].
3
The present approach enlarges the scale of observation of the phenomenon,using a “hydraulic engineer” approach, in the sense that an exhaustive treat-ment of the turbulence is completely omitted. Turbulence eﬀects are averagedover both a proper time/space scale with the intent of closing the integralmechanical balances.This study strictly follows previous works [2, 3], which were in turn in-
ﬂuenced by the hydraulic research on coastal zone dynamics of the 1990s[13, 14, 15]. The fundamental idea of the study is to treat the rollers of
standing or moving breaking waves as the mechanism of energy dissipationand the mainstream as the means of mass transport.
2. Physical remarks and basic assumptions
Textbooks on open-channel hydraulics (e.g., [1, 16]) suggest that themacroscopic properties of the hydraulic jump in prismatic channels are themass conservation and the total force conservation, while abrupt discontinu-ities occur in the depth, average velocity and speciﬁc energy. The jump iscalled an inviscid shock by gas-dynamics researchers and obeys the “entropycondition”, in which momentum conservation implies energy dissipation.This classical treatment is revisited in [2, 3], in which the following physi-
cal aspects are highlighted: the gradual increase of the water elevation insidethe jump ﬁnite (non-zero) length, which corresponds to a gradual decreasein the mean velocity and speciﬁc energy; the velocity proﬁle inversion insidethe roller; the non-negligibility of the vertical turbulent stresses and verticalvelocity (and consequently of the vertical momentum) in the jump; and theimportance of the previously described ingredients to the determination of a4
proper closure of the integral angular momentum (moment of momentum)balance.In this work, these issues are analyzed for the case of a hydraulic jump inconverging channels. The same matter was analyzed in [3] for diverging chan-nels. The linear (in both the horizontal and vertical directions) and angularmomentum balances are analyzed. The lateral wall friction is neglected, andinstability eﬀects that cause the loss of axial-symmetry are not considered[17, 18]. Thus, the treatment is equivalent to a radial jump with an angularamplitude of 2
π
, provided that the mass ﬂux proceeds from inﬁnity towardsthe pole.Bottom friction is also neglected to facilitate a direct comparison withthe classical results, which preserve the total force in prismatic channelsaccording to the well-known Bakhmeteﬀ relation [19] for sequent ﬂow depthsand a comparison with the radial jump with a mass ﬂow from the pole towardsinﬁnity [3].The scheme is two-dimensional (in each radial vertical plane) and axiallysymmetrical. All of the results can be referred to the unit-angular amplitudechannel. The upstream section is placed at a larger distance from the polethan the downstream section, and both sections are considered to be distin-guishable at the typical level of approximation found in hydraulics textbooks[1], even if the detection of the jump limits is not so easy in practice [4, 9].A further discussion can be found in [3].In [3], it is also explained that the scheme of the inviscid shock in aprismatic channel implies that the shock position is indeterminate, while thesame scheme implies a precise position of the zero-length shock itself in the5

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