A theoretical investigation of boundary layer flow and bottom shear stress for smooth, transitional, and rough flow under waves

Abstract

Velocity and shear stress distributions and the relationship between maximum near‐bottom orbital velocity uom and maximum shear velocity u*m in an oscillatory boundary layer are computed for hydraulically smooth, transitional, and rough turbulent flow using unsteady boundary layer theory and a single, continuous expression for eddy viscosity K. Velocity profiles over a half wave cycle calculated using a time‐independent form of K compare favorably with available measured profiles in smooth, transitional, and rough turbulent flows; computed shear stress profiles agree reasonably well with measured stress profiles. Use of a time‐dependent eddy viscosity generally improves the agreement between measured and computed velocity and shear stress near the bed but not in the outer boundary layer. Maximum computed bottom shear stress, however, does not differ significantly from values calculated with a time‐independent K owing to the choice of u*m as the turbulent velocity scale in K. The ratio of maximum orbital velocity to maximum shear velocity is computed as a function of two nondimensional parameters, a Reynolds number Re*=u*mδw/ν and an inverse Rossby number ξ0=ωz0/u*m; δW is wave boundary layer thickness, ω is wave frequency, ν is kinematic viscosity, and z0 is the bottom roughness parameter. These independent variables of the nondimensional unsteady boundary layer equation can be related to the more commonly used wave boundary layer parameters which are expressed in terms of orbital velocity instead of shear velocity. At the fully rough and fully smooth turbulent flow limits, u*m/uom is given by a single curve as a function of ξ0 or Re*, respectively. These curves compare favorably with available measurements and expressions for wave friction factor. The nondimensional equations also yield the dependence of u*m/uom on ξ0 and Re* for transitionally rough turbulent flow.