The Logarithmic Law of the Wall in Flows over Mobile Lattice-Arranged Granular Beds

Federica Antico,Rui Ferreira,Ana Ricardo

doi:10.3390/w11061166

Abstract

The purpose of the present paper is to provide further insights on the definition of the parameters of the log-law in open-channel flows with rough mobile granular beds. Emphasis is placed in the study of flows over cohesionless granular beds composed of monosized spherical particles in simple lattice arrangements. Potentially influencing factors such as grain size distribution, grain shape and density or cohesion are not addressed in this study. This allows for a preliminary discussion of the amount of complexity needed to obtain the log-law features observed in more realistic open-channel flows. Data collection included instantaneous streamwise and bed-normal flow velocities, acquired with a two-dimensional and two-component (2D2C) Particle Image Velocimetry (PIV) system. The issue of the non uniqueness of the definition of the parameters of the log-law is addressed by testing several hypotheses. In what concerns the von Kármán parameter, κ , it is considered as flow-independent or flow-dependent (a fitting parameter). As for the geometric roughness scale, k s , it results from a best fit procedure or is computed from a roughness function. In the latter case, the parameter B is imposed as 8.5 or is calculated from the best fit estimate. The analysis of the results reveals that a flow dependent von Kármán parameter, lower than the constant κ = 0.40 , should be preferred. Forcing κ = 0.40 leads to non-physical values of k s and would imply extending the inner layer up about 50% of the flow depth which is physically difficult to explain. Considering a flow dependent von Kármán parameter allows for coherent explanations for the values of the remaining parameters (the geometric roughness scale k s , the displacement height Δ , the roughness height z 0 ). In particular, for the same transport rate, the roughness height obtained in a natural sediment bed is much greater than in the case of bed made of monosized glass spheres, underlining the influence of the bed surface complexity (texture and self-organized bed forms, in the water-worked cases) on the definition of the log-law parameters.

Highlights

The classical idealization of flows over smooth and rough boundaries, successfully extended to mobile boundaries, comprises a logarithmic distribution of the longitudinal velocity in the wall-normal direction
The images acquired by the Particle Image Velocimetry (PIV) system were post-processed by masking the areas in the field of view not occupied by fluid: the band above the free surface and the region occupied by the bed particles
The channel beds in these two cases are substantially different: while the current study adopted a simple lattice-arranged granular bed with no relevant morphological features even at moderate bedload discharges, the bed surface of Ferreira et al [2] exhibits a complex micro-topography, with clusters around larger particles, in the case of the armoured beds, and low amplitude bedload sheets, in the case of the sand-gravel mixture at high values of the Shields parameter

Summary

Introduction

The classical idealization of flows over smooth and rough boundaries, successfully extended to mobile boundaries, comprises a logarithmic distribution of the longitudinal velocity in the wall-normal direction. This log-law should be valid in the overlapping layer between inner and outer flow regions (see, e.g., [1]) when (i) gradients in the longitudinal direction are small, in particular the pressure gradient; (ii) the channel aspect ratio is high so that the mean flow far from the banks or. The characteristic length scale should be of the order of magnitude of the bed amplitude, δ (the wall-normal distance between the planes of the troughs and of the crests) but should be dependent on the type of granular bed and on the Shields parameter [2]

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