Gradually Varied Flow Computation in Cyclic Looped Channel Networks H. Prashanth Reddy 1 and S. Murty Bhallamudi 2 Abstract: A novel and computationally efﬁcient algorithm is presented to compute the water surface proﬁles in steady, gradually varied ﬂows of open channel networks. This algorithm allows calculation of ﬂow depths and discharges at all sections of a cyclic looped open channel network. The algorithm is based on the principles of (1) classifying the computations in an individual channel as an initial value problem or a boundary value problem; (2) determining the path for linking the solutions from individual channels; and (3) an iterative Newton–Raphson technique for obtaining the network solution, starting from initial assumptions for discharges in as few channels as possible. The proposed algorithm is computationally more efﬁcient than the presently available direct method by orders of magnitude because it does not involve costly inversions of large matrices in its formulation. The application of this algorithm is illustrated through an example network. DOI: 10.1061/(ASCE)0733-9437(2004)130:5(424) CE Database subject headings: Gradually varied ﬂow; Steady ﬂow; Open channels; Computation; Hydraulic networks. Introduction In hydraulic engineering applications, it is necessary to compute steady state, gradually varied ﬂows in open channels such as riv- ers, streams, and canals. These computations enable determina- tion of the water surface elevations along the length of a channel for a speciﬁed discharge. An exact knowledge of the water sur- face elevations allows the planning, design, and safe operation of the system. The effect of channel modiﬁcations, on the upstream or the downstream water level, can be assessed through these computations. Steady ﬂow solutions are also required for speci- fying accurate initial conditions for starting unsteady ﬂow com- putations. Also, a steady ﬂow solver may be used as a subroutine in mathematical models for sediment transport or stream water quality, in which the subroutine calls to the ﬂow solver are made thousands of times. Several efﬁcient numerical techniques have become available for gradually varied ﬂow (GVF) computations in the last four decades (Chow 1959; Chaudhry 1993). Classical numerical meth- ods such as the standard step method, which are based on single- step calculations, are well suited for single and series channels, but not for complex open channel networks. However, in many situations, GVF computations may be required for steady ﬂow in open channel networks or a system of interconnected channels. Open channel networks occur in braided river systems, divided shipping channels, interconnected storm water systems, or irriga- tion canal systems (Wylie 1972). The Brahmaputra–Ganga deltaic system in the Indian subcontinent (Mukherjee 1985) and the Mekong river basin in South East Asia (Cunge et al. 1980) are a few examples of interconnected rivers with complex ﬂow situa- tions. Although considerable research has been carried out in recent years with regard to unsteady ﬂows in channel networks (Choi and Molinas 1993; Kutija 1995; Nguyen and Kawano 1995; Sen and Garg 1998), not much attention has been paid to the problem of steady GVF computation in open channel networks. Wylie (1972) developed an algorithm to compute the ﬂow around a group of islands, in which the total length of the channel between two nodes is treated as a single reach to calculate the loss of energy and the node energy is used as a variable. In this method, the channel is not divided into several reaches as in a ﬁnite- difference method. A reach is deﬁned as the portion of the channel between two ﬁnite-difference nodes. Chaudhry and Schulte (1986) presented a ﬁnite difference method for analyzing steady ﬂow in a parallel channel system. Their formulation is in terms of the more commonly used variables, ﬂow depth and discharge. Schulte and Chaudhry (1987) later extended their method for ap- plication to general looped channel networks. In their method (referred to as the Direct Method in the rest of the text), a channel i in the system is divided into several reaches, N i . The continuity and the energy equations can be written in terms of ﬂow depths, and ﬂow rates for all the reaches, resulting in a total of 2 i=1 M N i  equations because there are N i + 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. The system of nonlinear simultaneous equations resulting from the above formulation is solved using the Newton–Raphson iteration technique. This requires inversion of the system Jacobian for every iteration step. In this formulation, the size of the Jacobian increases if the number of reaches in each channel is increased to increase accuracy. Therefore the above method becomes compu- tationally intensive for large unstructured channel networks. 1 Graduate Student, Dept. Of Civil Engineering, Indian Institute of Technology, Madras, India. 2 Professor, Dept. of Civil Engineering, Indian Institute of Technology, Madras, India. E-mail: bsm@civil.iitm.ernet.in Note. Discussion open until March 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be ﬁled with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on March 5, 2003; approved on February 2, 2004. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 130, No. 5, October 1, 2004. ©ASCE, ISSN 0733-9437/2004/5-424–431/ $18.00. 424 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2004