Mathematical theory and experimental verification of approximate linear weirs and self-basing linear weirs

Abstract

The analysis of the thesis emphasizes the possibility of using geometrically simple weirs formed by straight lines and/or circular quadrants as effective, sufficiently accurate linear weirs in irrigation and other allied fields. First, an inverted Vnotch (IVN) with possibly the simplest profile shape after the rectangular weir, is shown to possess a nearlinear head-discharge relationship. For flow through this weir of halfwidth W and altitude d, for depths above 0.22d, discharge is proportional to the depth of flow measured above a reference plane situated at 0.08d for all heads in the range 0.22d < h < 0.94d, with a maximum percentage deviation of +1.5% from the theoretical discharge. The analysis is based on two numerical optimization procedures, viz., the rangeofpoints method and the tangent method. Nearly 75% of the depth of the inverted Vnotch can be used for the measuring range. Experiments on four different weirs give an average coefficient of discharge of 0.61. Second, it is shown that the IVN weir can be modified into a chimney weir to increase its linearity range (by more than 200%) by providing a pair of vertical parallel straight lines at a depth of 0.735d above the weir crest, without losing the essential geometric simplicity and accuracy of discharge computations. Significantly, the same linear head-discharge relationship that governs the flow through the IVN is also valid for the chimney weir in the enhanced range, i.e., 0.22d < h < 2.43d, so that the linearity range of the chimney weir is 2.21d as against 0.74d of the IVN. Experiments with three typical weirs are in remarkable agreement with the theory, giving a constant average coefficient of discharge in the neighbourhood of 0.61. Third, it is shown that the bellmouth (BM) weir, formed by the quadrants of a circle, can be used as a practical linear weir over a certain fixed range of head within permissible limits of error in discharge computation. The BM weir has a slightly better linear flow characteristic than the IVN weir, in that the analysis is based on a +1% maximum permissible error as against +1.5% in the case of the IVN. It is found that a top width of 0.2R gives optimum design of the BM weir by giving maximum linearity range. The linearity range of the BM weir can be significantly increased by transforming it into an extended bellmouth (EBM) weir by extending the tangents at the terminal points of the BM weir. For the EBM weir also, a top width of 0.2R gives optimum design of the weir by giving minimum baseflow depth with again a linearity range of nearly 140% over the corresponding BM weir. Further, the EBM weir with a top width of 0.2R has a significant second linearity range which extends beyond the first to a depth equal to 4R, so that the total linearity range is increased by nearly 375%. EBM weirs with top widths of 0.5R and 0.1R have reference planes lying below their crest, rendering them highly suitable as gritchamber outlet weirs. Experiments with three typical weirs fully confirm the theory by giving a constant average coefficient of discharge in the neighbourhood of 0.61. The thesis introduces a new class of linear weirs (Fig. 6.1), “The SelfBasing Linear Weirs.” A rational theory to design these weirs is developed using the exact solutions of certain quadratic weirs which have the significant property of being very good approximate solutions to a particular Fredholm integral equation. These weirs can pass a discharge proportional to the head measured above a fixed reference plane for all flows above a threshold depth d above the weir crest in the range d < h < , with the accuracy increasing rapidly with head. It is shown that the existing linear weirs with bases can be modified into selfbasing linear weirs by the addition of a simple correcting function of the form a/h (a and b are constants). The theory can be used to design special weirs having practical applications in irrigation and environmental engineering. A few theorems on the general properties of linear, quadratic and selfbasing linear weirs, useful in the design of selfbasing linear weirs, are enunciated and proved. A new classification of weirs is introduced. It is hoped that the three geometrically simple linear weirs developed in the thesis, along with the theory and design, will find practical application in irrigation and allied fields. The theory of selfbasing linear weirs, apart from contributing to the fundamental study of hydraulics, is hoped to generate interest towards further research in this field.