Liquids are transported from one location to another using natural or constructed conveyance structures. The cross section of these structures may be open or closed at the top. The structures with closed tops are referred to as closed conduits like tunnels and pipes, while those with the top open are called open channels like rivers, streams, estuaries etc. The properties and the analyses of these flows are discussed in this book.
In this chapter, commonly used terminology are first defined including free-surface flows, pressurized flows, hydraulic grade line, piezometric and velocity heads. Following that, the classification of flows is covered beginning with flow classification based on time or distance criterion, as well as the effect of viscous and gravitational forces on flow classification in open channels. Then, the terminology and properties of a channel section are outlined including the cross-sectional flow area, wetted perimeter, hydraulic radius, top width, and hydraulic depth for different open channel cross sections. To account for nonuniform velocity distribution at a channel section, expressions for the energy and momentum coefficients are then developed. A discussion of the pressure distribution in a channel section whether with parallel or curvilinear flow is followed. Reynolds transport theorem is then briefly addressed to simplify the presentation of its applications in later chapters. Finally, the chapter concludes with a brief overview of the most prevalent hydraulic models used in open channel analysis, as well as the use of dimensional analysis.
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In this research, the trade-off between the number of restrictions and the robustness of the primary formulation of entropy models was evaluated. The performance of six hydrodynamic models in open channels was assessed based on 1730 Laser-Doppler anemometry data. It was investigated whether it is better to use an entropy-based model with more restrictions and a weak primary formulation or a model with fewer restrictions, but with a strong formulation. In addition, it was also investigated whether the model performance improves with the insertion of restrictions. Three of the investigated models have a weak formulation (open-channel velocity field represented by Cartesian coordinates); while the other three models have a strong formulation, according to which isovels are represented by curvilinear coordinates. The results indicated that models with two restrictions performed better than those with one restriction, since the additional restriction includes information relevant to the system. Models with three restrictions perform worse than those with two restrictions, because the information lost due to the use of a numerical solution was more substantial than the information gained by the third restriction. In conclusion, a strong primary formulation brought more information to the system than the inclusion of a third constraint.
We investigated the performance of six entropy models designed to simulate open-channel velocity fields. The simulated velocities were compared with accurately-measured laboratory data. Three of the investigated models have a weak primary statement, i.e., they assume that isovels could be well represented by Cartesian coordinates; whereas the remaining three models have a strong statement according to which isovels are better represented by curvilinear coordinates. In this work, data were extracted from the experiments made by Steffler et al. (1985)STEFFLER PM, RAJARATNAM N & PETERSON AW. 1985. LDA measurements in open channel. J Hydraul Eng 111(1): 119-130.: run 1 (hereafter called SRP1), run 2 (SRP2), and run 3 (SRP3). The experiments (see the main characteristics in Table I) were performed at the Thomas Blench Laboratory flume located at the University of Alberta, Canada. The velocities were accurately measured using a Laser-Doppler anemometer. The models performance was assessed with the Nash-Sutcliffe coefficient (NSE) and the root mean square error (RMSE).
Two primary statements were assumed and models with one, two or three constraints used, respectively, in order to maximize the entropy function H (Equation 1), in which u means the longitudinal velocity; p(u) the respective probability density function; and Umax the maximum velocity in the cross section. The six entropy models are divided into two groups: three models admit the Cartesian coordinate system (weak primary statement, Equation 2), whereas the three others admit the curvilinear coordinate system, as described in (Chiu 1988CHIU CL. 1988. Entropy and 2-D velocity distribution in open channels. J Hydraul Eng 114(7): 738-756.) (Figure 1).
Model U1y (one constraint and Cartesian coordinates) is based on Chiu (1987)CHIU CL. 1987. Entropy and probability concepts in hydraulics. J Hydraul Eng 113(5): 583-599., who proposes the primary statement (Equation 2), according to which F(u) is the probability of the longitudinal velocity being less or equal to u at a point located at distance y from the channel bed. In Equation 2, D is the flow depth at the channel.
The strong statement admits that the longitudinal velocity is directly associated with the curvilinear, rather than with Cartesian coordinates Chiu & Chiou (1986)CHIU CL & CHIOU JD.1986. Structure of 3-D flow in rectangular open channels. J Hydraul Eng 112(11): 1050-1067.; and that isovels can be represented by ξ coordinates (Equations 15-17), as proposed by Chiu (1986). The isovel (ξ) shape parameters (δy, ε) and variables (y, z) are defined in Figure 1. Parameter βi characterizes the velocity distribution of the primary flow.
Model U1ξ (one constraint and curvilinear coordinates) is based on Chiu (1988)CHIU CL. 1988. Entropy and 2-D velocity distribution in open channels. J Hydraul Eng 114(7): 738-756., who proposes the strong primary premise (Equation 18), according to which F(u) is directly associated with the isovel (ξ) spatial distribution, with the key parameters ξmax and ξ0, respectively, the maximum and minimum ξ values of the open-channel flow. The entropy function H (Equation 1) was maximized and subjected to one constraint (Equation 3), the same as in model U1y. The result was applied to Equation 18, yielding the velocity-distribution (Equation 19), where λ7 is the Lagrange parameter, estimated by Equation 20.
Model U2ξ (two constraints and curvilinear coordinates), developed by Chiu (1988)CHIU CL. 1988. Entropy and 2-D velocity distribution in open channels. J Hydraul Eng 114(7): 738-756., uses curvilinear coordinates (Equation 15), Equation 18 as primary statement, and two constraints: Equations 3 and 6, the same of model U2y, resulting in Equation 21. The parameters λ8 and λ9 are estimated in an analogous way as in model U2y, as shown in Chiu (1988)CHIU CL. 1988. Entropy and 2-D velocity distribution in open channels. J Hydraul Eng 114(7): 738-756..
The U3ξ model for the velocity field in open channels consists in solving Equations 22 and 23. The system parameters λ10, λ11, and λ12 are estimated analogously as in model U3y, using the Boussinesq coefficient, maximum velocity and average flow velocity, as demonstrated by Barbé et al. (1991)BARBÉ DE, CRUISE JF & SINGH VP. 1991. Solution of three-constraint entropy-based velocity distribution. J Hydraul Eng 117(10): 1389-1396.. 2ff7e9595c
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