Finding the Bed Shear Stress on a Rough Bed Using the Log Law
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 148, Issue 4
Abstract
The procedure commonly used to determine the friction velocity from a measured velocity profile in turbulent flow over a hydraulically rough bed is to fit the log law to the velocity profile and adjust the displacement height to obtain a best-fit line to as many data points as possible in the inner layer. In practice, the process can be subjective and produce large uncertainty in the bed shear stress estimates and/or inconsistent results for the equivalent roughness height. In oscillatory flows, a temporal variation in the equivalent grain roughness is unrealistic because the roughness height should remain constant if the boundary Reynolds number is sufficiently large. An alternative method is presented in this study, in which the equivalent grain roughness is held constant, and the displacement height is varied until the value of the von Kármán constant obtained from the best-fit line is equal to the universally accepted value of about 0.4. The iterative process converges rapidly and is easier to apply than the traditional method, which requires the displacement height to be found by trial and error. The method was tested in steady, shallow uniform flows over a fixed bed of fine gravel. The channel slope was varied, and the velocity profile was measured using the particle image velocimetry (PIV) technique. Good agreement was found between the bed shear stress estimates obtained using the new method and the values calculated from the measured flow depth and channel slope when the ks/d90 ratio was taken from the literature for small values of the h/d90 ratio, where h is the flow depth, ks is the equivalent roughness height, and d90 is the grain diameter with 90% of finer particles, therefore verifying that the new method produced results consistent with published data. The method was then applied to velocity measurements under a solitary wave to obtain the temporal variation of bed shear stress on a gravel bed near the point of incipient wave breaking.
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Data Availability Statement
All data presented in this paper are available from the corresponding author upon reasonable request.
Acknowledgments
Funding for this study was provided by the United States Department of Transportation (USDOT) to the Mountain-Plains Consortium (MPC). Additional funding was provided by the National Science Foundation (NSF) through Grant No. OCE-2049293. The support of the MPC and the NSF is gratefully acknowledged.
Notation
The following symbols are used in this paper:
- A, B
- integration constants in the log law;
- a, b
- slope and y-intercept of the best-fit line;
- b
- channel width [Eq. (7)];
- D
- representative grain size or diameter of uniform grains;
- dr
- grain diameter with r% of finer particles;
- F
- Froude number;
- f
- bulk friction factor;
- fb
- bed-related friction factor;
- fw
- wall friction factor;
- g
- acceleration of gravity;
- h
- flow depth;
- he
- effective flow depth;
- ks
- equivalent grain roughness;
- n
- porosity;
- p
- pressure;
- R
- hydraulic radius;
- Rb
- bed-related hydraulic radius;
- Re
- Reynolds number;
- boundary Reynolds number;
- S
- channel slope;
- t
- time;
- u
- mean velocity;
- u0
- free-stream velocity;
- friction velocity;
- V
- cross-sectional average velocity;
- V1
- depth-averaged velocity;
- y
- vertical coordinate;
- y0
- elevation of the theoretical bottom;
- y1
- top of the gravel bed;
- y2
- water surface elevation;
- α
- the multiplying factor;
- δ
- boundary-layer thickness;
- η
- wave elevation;
- κ
- von Kármán constant;
- ν
- kinematic viscosity;
- ρ
- fluid density; and
- τb
- bed shear stress.
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Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 148 • Issue 4 • July 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: May 30, 2021
Accepted: Feb 1, 2022
Published online: Apr 29, 2022
Published in print: Jul 1, 2022
Discussion open until: Sep 29, 2022
Authors
Appendix. Side Wall Correction of Bed Shear Stress
The bed shear stress in a steady flow is computed using the procedure described in Vanoni and Brooks (1975) and improved by Cheng (2011). For a uniform flow in a rectangular channel with smooth side walls, the bed shear stress τb can be computed using the following equations:where ρ = fluid density; g = acceleration of gravity; Rb = bed-related hydraulic radius; S = channel slope; f = bulk friction factor; fb = bed friction factor; fw = wall friction factor; he= effective flow depth; b = channel width; R(=bhe/(b + 2he)) = hydraulic radius; V(=Q/(bhe) = cross-sectional average velocity; Re(=V(4R)/ν) = Reynolds number; and ν = kinematic viscosity. The effective flow depth he is computed as follows (Ferreira et al. 2012):where h = flow depth measured from the acrylic sheet to which the gravel was adhered; n = porosity of the gravel; and D = thickness of the gravel layer. Values of n = 0.5 and D = d50 are assumed in this study.
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Cited by
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