Effect of Cross-Sectional Shape on Free-Surface Instability

Abstract

The effect of cross sectional shape on free-surface hydrodynamic instability has been analyzed. The characterizing parameter is the dimensionless relative kinematic wave celerity c drk = β - 1, in which p is the exponent of the normal discharge-flow area rating. Two generic types of cross-sectional shapes are identified: (1) those of constant c drk , and (2) those of variable c drk . Three cross-sectional shapes of constant c drk are: (1) hydraulically wide, with c drkM = 2/3, (2) triangular, with c drkM = 1/3, and (3) inherently stable, with c drk = 0 (Manning or Chezy). (The subscript M refers to Manning friction). Cross-sectional shapes of variable c drk include trapezoidal, rectangular, and circular shapes. Two asymptotic cross-sectional shapes are identified: (1) hydraulically wide, and (2) hydraulically narrow. These theoretical cross-sections set limits to the range of variation of c drk for trapezoidal and rectangular shapes. The hydraulically wide channel sets the upper limit, with c drkM → 2/3; the hydraulically narrow channel sets the lower limit, with c drk → 0. Two types of stable channels are identified: (1) inherently stable, and (2) stable. An inherently stable cross section is such that the Vedernikov number V is identically zero for all Froude numbers. A stable cross section is such that V ≤ 1 for Froude numbers in the range F ≤ F ns , in which F ns is a design neutral-stability Froude number. A stable cross-sectional shape is designed by setting c drk and the related cross-sectional parameter δ to match a certain choice of F ns . The resulting stable cross-sectional shape is much narrower that the comparable inherently stable shape.