computation of gradually varied flow in compound open
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DESCRIPTIONAbstract. Although, natural channels are rarely rectangular or trapezoidal in crosssection, these cross sections are assumed for the computation of steady, gradually variedflow in open channel networks. The accuracy of the computed results, therefore,becomes questionable due to differences in the hydraulic and geometric characteristicsof the main channel and floodplains. To overcome these limitations, an algorithmis presented in this paper to compute steady, gradually varied flow in an open-channelnetwork with compound cross sections. As compared to the presently available methods,the methodology is more general and suitable for application to compound andtrapezoidal channel cross sections in series channels, tree-type or looped networks. Inthis method, the energy and continuity equations are solved for steady, gradually variedflow by the Newton–Raphson method and the proposed methodology is appliedto tree-type and looped-channel networks. An algorithm is presented to determinemultiple critical depths in a compound channel. Modifications in channel geometryare presented to avoid the occurrence of multiple critical depths. The occurrenceof only one critical depth in a compound cross section with modified geometry isdemonstrated for a tree-type channel network.
Sadhana Vol. 39, Part 6, December 2014, pp. 15231545. c Indian Academy of Sciences
Computation of gradually varied flow in compound openchannel networks
H PRASHANTH REDDY1,, M HANIF CHAUDHRY2 andJASIM IMRAN2
1Department of Civil Engineering, Indian Institute of Technology Kharagpur,Kharagpur 721302, India2Department of Civil and Environmental Engineering, University of South Carolina,Columbia, SC 29201, USAe-mail: email@example.com; firstname.lastname@example.org; email@example.com
MS received 27 April 2014; revised 23 July 2014; accepted 6 August 2014
Abstract. Although, natural channels are rarely rectangular or trapezoidal in crosssection, these cross sections are assumed for the computation of steady, gradually var-ied flow in open channel networks. The accuracy of the computed results, therefore,becomes questionable due to differences in the hydraulic and geometric characteris-tics of the main channel and floodplains. To overcome these limitations, an algorithmis presented in this paper to compute steady, gradually varied flow in an open-channelnetwork with compound cross sections. As compared to the presently available meth-ods, the methodology is more general and suitable for application to compound andtrapezoidal channel cross sections in series channels, tree-type or looped networks. Inthis method, the energy and continuity equations are solved for steady, gradually var-ied flow by the NewtonRaphson method and the proposed methodology is appliedto tree-type and looped-channel networks. An algorithm is presented to determinemultiple critical depths in a compound channel. Modifications in channel geome-try are presented to avoid the occurrence of multiple critical depths. The occurrenceof only one critical depth in a compound cross section with modified geometry isdemonstrated for a tree-type channel network.
Keywords. Gradually varied flow; open channel flow; compound channel networks;critical depth; steady flow; subcritical flows.
Channel networks are ubiquitous in nature, such as braided river channels, divided shippingchannels, interconnected storm water drainage systems and irrigation canals fed from differentsources. The flow of water in an open channel can be treated as steady, gradually varied flow for
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many practical purposes and a number of numerical techniques are available for the computa-tion of these flows (Chow 1959; Chaudhry 2008). The standard step method based on single-stepcalculation is well-suited for a series channel system. The gradually varied flow computationsare also required for the channel network or system of interconnected channels, such as theIndus Basin irrigation system, where the rivers are inter-connected through a number of largelink canals. A number of researchers have studied unsteady flows in channel networks (Kutija1995). More recently, Reddy & Bhallamudi (2004) presented an algorithm for the computationof steady gradually varied flow in cyclic looped channel networks. The algorithm is based oncomputation in individual channel as initial value or boundary value problem and then deter-mining a pair of linking channels by using the iterative NewtonRaphson technique for channelnetwork.
Wylie (1972) developed an algorithm to compute the flow around a group of islands, in whichthe total length of the channel between two nodes is treated as a single reach to calculate the lossof energy and the node energy is used as a variable. In this method, the channel is not dividedinto several reaches as is done in a finite difference method. A reach is defined as the lengthof the channel between two finite-difference nodes. Chaudhry & Schulte (1986a, b) presenteda finite-difference method for the analysis of steady flow in a parallel channel system. Theirformulation is in terms of the more commonly used variables, i.e., flow depths and discharges.Schulte & Chaudhry (1987) later extended their method for application to general looped channelnetworks. In their method, a channel i in the system is divided into several reaches, Ni . Thecontinuity and energy equations can be written in terms of flow depths and flow rates for allthe reaches, resulting in a total of 2
i=Mi=1 Ni equations for Ni + 1 nodes in any channel i and
there are M channels in the system. Additional 2M equations, required for closing the system,are obtained from the boundary conditions and the compatibility conditions at the junctions. Thesystem of nonlinear simultaneous equations resulting from this formulation is solved by using theNewtonRaphson iteration technique. This requires inversion of the system Jacobian for everyiteration step. In this formulation, the size of the Jacobian increases if the number of reaches ineach channel is increased to improve accuracy. Sen & Garg (2002) and Zhang & Shen (2007)developed an efficient solution technique to compute GVF profiles for one-dimensional, steadyand unsteady flow in a general channel network system with trapezoidal cross sections. Channelswith trapezoidal cross-sections were considered in their studies by Schulte & Chaudhry (1987),Reddy & Bhallamudi (2004), Naidu et al (1997) to compute gradually varied flow in channelnetworks. However, if the channels have compound cross sections in both man-made and naturalsystems as in this case, the differences in hydraulic and geometric characteristics between themain channel and the overbank area needs to be accounted for.
The HEC-RAS model released by US Army Corps of Engineering can handle a single riverreach, a dendritic network of channel system to model subcritical, super critical and mixed flowregime water surface profiles. However, the current version of steady flow model within HEC-RAS does not determine the amount of flow going to each reach at a flow split. Therefore,HEC-RAS cannot be used to networks where flow split is unknown at the junctions. Computa-tion of critical depth is very important for the analysis of open channel flow because (i) graduallyvaried flow calculations are unstable near the critical depth, (ii) critical depth is a boundary con-dition at control points and (iii) critical depth is also required to verify the proper boundarycondition for correct flow regime. A simple channel cross-section has only one critical depth fora given flowrate. However, multiple critical depths are associated with compound cross sections(Chaudhry & Bhallamudi 1988). In compound channels, subcritical, critical and supercriticalflows occur simultaneously. In this mixed flow regime, subcritical flow in main channel andcritical or supercritical flows in flood plain is observed over a depth range (Kordi et al 2009).
Computation of GVF profiles in compound open channel networks 1525
The entire cross section with mixed flow regime cannot be considered as a control section andresulting computations of water surface profile and designing of any controlling structure is verycomplicated. The occurrence of flow at critical regime of an open channel is highly unstable(Kordi et al 2009; Bhattacharjya 2006). Flow can be unstable if multiple critical depths occur ina channel section and it is important to design compound channels such that there is only onecritical depth for a given discharge (Chaudhry & Bhallamudi 1988). Two-dimensional flow fea-tures were observed in the flow between multiple critical depths (Blalock & Sturm 1981, 1983;Sturm & Sadiq 1996). Several authors (Knight & Demetriou 1983; Ackers 1993; Bousmar &Zech 1999; Kordi et al 2009) proposed momentum transfer and lateral flow between main chan-nel and flood plains are significant and necessary to be considered in hydraulic computations.These effects are ambiguous and the results of the works of researchers have not yet yielded anycomprehensive method for design and study about complex compound cross section and not yetbeen implemented in any commercial packages. Nevertheless, Sturm & Sadiq (1996) obtainedgood agreement of computed water profiles with experimental data, for depths quite close to thecritical ones by direct integration of the gradually varied flow equation and using a correctedFroude number.
The objective of the present research is to develop an efficient algorithm for computing grad-ually varied flow in open channel networks consisting of compound cross sections and developcriteria for design of compound channels in networks with a single critical depth. In this study,steady gradually varied flow in open channel networks with symmetric compound channel cross-sections is considered. The methodology derived here is based on solving the continuity and theenergy equations by using the NewtonRaphson method. The main advantages are its robustnessand suitability for complex open channel networks.
The following form of the energy equation can be applied between the two ends of a reach tocompute water surface profile (Chaudhry 2008). Schematic of a compound channel cross sectionand reach between two cross sections are shown in figures 1 and 2.
z1 + y1 +1 Q2t12gA2t1
+ he = z2 + y2 +2 Q2t22gA2t2
where z1, z2 are elevations of the main channel invert levels; y1, y2 are depths of water at crosssections; 1, 2 are energy coefficients; Qt1,Qt2 are total discharges at sections 1 and 2; At1and At2 are total flow areas at sections 1 and 2; he is the head loss between two sections.
By neglecting expansion and contraction losses, the energy loss can be expressed as
he = xSF . (2)Average friction slope between two cross sections can be written as
SF = SF1 + SF22 , (3)
where x = reach length; SF = average friction slope between two sections, SF1 and SF2 arefriction slopes at sections 1 and 2.
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