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.