Analytical findings for power law cross-sections: Uniform flow depth

Abstract

Analytical results concerning open channel flows are presented, assuming that the cross-section is defined by a power law relationship between the channel width and the channel depth. Explicit equations to compute the normal flow depth are derived by considering the liquid discharge, the channel roughness height and the cross-section geometry (based on knowledge of the power law exponent, the reference width, and the reference depth) as known quantities. Such equations are deduced by writing the physical quantities as a power expansion in the power law exponent and expressing the wetted perimeter using a Gauss hypergeometric function. With the designed procedure, an accurate estimations of the integrals required to invert the uniform flow formula are obtained, at least for cross-sections characterized by aspect ratios of technical interest. Two relationships are proposed between the normal depth and the flow discharge. The first relationship is shown to work well for any discharge, provided that the width to depth ratio is sufficiently large. If this is not the case, the second procedure must be used for non-dimensional discharge larger than a given threshold, while the former procedure remains valid under the threshold.