Velocity Distributions in Open Channels and the Calculation of Discharge

Abstract

The accurate representation and integration of velocity measurements in open channels is important in irrigation and river engineering. The traditional approach for velocity is to use an approximate physical theory, giving the well-known logarithmic formulas, plus less well-known correction formulas in terms of mathematical functions. The approach is criticized here as being too prescriptive and not capable of systematic improvement or generalization. A different paradigm is suggested, oriented toward practice and numerical solution. The velocity is written as a polynomial, a series of monomial terms, in terms of the relative height of a point above the bed. In the first contribution it is raised to a fractional power, mimicking the actual shear flow in a stream where velocity goes to zero on the bed but with a large gradient. Polynomials with just two extra terms can describe well a number of laboratory and field measurements. It is computationally better, however, to use the monomials rearranged as Chebyshev polynomials. This is simply done and can be used as a means of approximating several measurements at arbitrary points to give an accurate depth-averaged velocity. Using the polynomial approximation, the accuracy of standard hydrographic and hydrometric methods is then examined. The well-known two-point 0.2/0.8 method of integration is surprisingly proved to be accurate to within 1% for any smoothly varying quantity. Such high accuracy has been found experimentally; what is noteworthy is its general theoretical validity—and its simplicity. Procedures for integrating across a stream are then considered and it is shown that a common approach, the mean section method, is not correct. Then the polynomial approximation method is generalized to two dimensions to give a method for the calculation of discharge also for arbitrary distributions of velocity measurement points in general cross sections. The velocity distribution in an open channel can be simply and accurately approximated by a polynomial in terms of the local height above the bed expressed as a fraction of the total depth. The first polynomial term is a simple fractional power of that quantity, which mimics the well-known behavior that as the velocity goes to zero on the bed, the gradient becomes large. It is computationally better for other polynomial terms, instead of just integer powers, to use combinations of them in the form of special functions, for which formulas are given. Then, if several velocity measurements are made on a vertical line, they can be approximated using optimization methods. To calculate discharge accurately, such a procedure is actually not necessary. A surprising and general proof is given that any smoothly varying physical quantity can be integrated to within 1% accuracy using just the mean of values at 20% and 80% of the domain. This has been determined empirically in hydrography and hydrometry and is standard practice. The simplicity and accuracy of the procedure is most fortunate. Finally, the polynomial method is extended to two dimensions, enabling the approximation of arbitrary measurements of the velocity field and the determination of discharge in any river or canal.