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Double-averaged rough-bed open-channel flow with high Glossosoma (Trichoptera: Glossosomatidae) abundance

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Abstract

Spatially averaged velocity distributions, turbulence characteristics, and stream bed roughness elevations were collected in two streams with rough-bed substrate. Variogram analysis of substrate roughness height yielded characteristic length scales of the stream bed over which bed elevations were correlated from 0.14 to 0.41 m. Temporally and spatially averaged (double-averaged) vertical velocity profiles followed a composite distribution consisting of a linear distribution below the roughness crest height and a power or wake law above the crest. Our double-averaged velocity data demonstrated the applicability of both the wake law and power law to open-channel flow for which a low ratio of flow depth to roughness height does not support the development of the universal logarithmic velocity law. A power-law scaling relationship among spatially averaged Glossosoma density, stream bed roughness characteristics, and double-averaged fluid flow conditions was developed. The density of Glossosoma scaled directly with substrate crest elevation, normalized spatial fluctuation of longitudinal velocity in the proximity of the bed, and inversely with the standard deviation of the crest elevation. The proposed dimensionless scaling relationship explains 84 % of the Glossosoma variability.

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Abbreviations

\(\alpha \) :

Exponent in the power law distribution

\(\upbeta \) :

A parameter in the exponential distribution (1/m)

\(\text{ C}_\mathrm{p}\) :

Parameter for the power law distribution

\(\text{ C}_\mathrm{w}\) :

Parameter for the wake law distribution

\(\Delta \) :

Distance to sill of variogram indicating characteristic length scale of the variable in question (m); subscripts x and y indicates longitudinal and transverse directions, respectively

\(\text{ Fr}=\left\langle \text{ u} \right\rangle _\mathrm{vol} /\sqrt{g\left\langle \text{ H} \right\rangle }\) :

Froude number

\(g\) :

Gravitational constant

\(\text{ G}_\mathrm{density}\) :

Glossosoma spatial density on the stream bed \((\text{ no./m}^{2})\)

\(\text{ G}_\mathrm{mass}\) :

Glossosoma spatial density, expressed in biomass, on the stream bed \((\text{ g/m}^{2})\)

\(\text{ H}\) :

Maximum flow depth (m) from free surface to roughness troughs within the boundaries of a window defined for spatial averaging

\(\text{ I}_{\tilde{\mathrm{u}}} ={\left\langle {\tilde{\text{ u}}} \right\rangle _\mathrm{mb}}/{\left\langle \text{ u} \right\rangle _\mathrm{c}}\) :

The spatial velocity fluctuation at the mean-bed elevation normalized by double-averaged velocity at the crest

\(\upphi =\frac{\text{ A}_\mathrm{f} }{\text{ A}_\mathrm{o}}\) :

Roughness geometry function = area of fluid/total spatial averaging area

\(\upphi _{\mathrm{int}} =\int \limits _{\mathrm{z}_\mathrm{t}}^{\mathrm{z}_\mathrm{c}} \upphi \text{ dz}\) :

Integral of the roughness geometry function in the interfacial sublayer

\(l_\mathrm{c}\) :

Shear length scale characterizing flow dynamics within the roughness layer (m) [1, 33]

\(\kappa \) :

Von Karman’s constant = 0.41

\(\upmu \) :

Dynamic viscosity (kg/(ms))

\(\nu \) :

Kinematic viscosity \((\text{ m}^{2}/\text{ s})\)

\(\Pi \) :

Coles wake strength parameter

\(\text{ Q}=\text{ A}\left\langle \text{ u}\right\rangle _\mathrm{vol}\) :

Bulk discharge \((\text{ m}^{3}/\text{ s})\), where A = cross-sectional area

\(\text{ Re}={\left\langle \text{ u}\right\rangle _\mathrm{vol} \left\langle \text{ H} \right\rangle }/\nu \) :

Reynolds number; a ratio of inertial to viscous forces in the flow

\(\text{ Re}_{*}=0.5\text{ z}_\mathrm{c} \left\langle {\text{ u}_{*}} \right\rangle /\nu \) :

Roughness Reynolds number where \(\text{ z}_\mathrm{c}\) takes the place of substrate grain diameter

\(\upsigma _{\mathrm{z}_\mathrm{k}}\) :

Standard deviation of the roughness heights of each grid cell within a spatial averaging window (m)

\(\left\langle \text{ U}\right\rangle _\mathrm{vol}\) :

Bulk velocity, i.e. triple averaged velocity magnitude: temporally-, horizontally-, then depth-averaged (m/s)

\(\text{ u}, \text{ v}, \text{ w}\) :

Longitudinal, transverse, and vertical velocity components, respectively (m/s)

\(\text{ u}_\mathrm{rms}\) :

Root mean square of the temporal velocity fluctuations (m/s)

\(\left\langle {\text{ u}_{*}} \right\rangle =\left\langle {\left( \overline{\text{ u}^{\prime }\text{ w}^{\prime }}^{2}+\overline{\text{ v}^{\prime }\text{ w}^{\prime }}^{2} \right)^{0.25}} \right\rangle _\mathrm{c}\) :

Shear velocity; the spatially averaged measure of total vertical flux of horizontal momentum (see Stull [47, p. 67]). In this study this is evaluated at or extrapolated to \(\text{ z}_\mathrm{c}\) (m/s)

\(\left\langle \text{ u}\right\rangle _\mathrm{c}\) :

Spatially averaged velocity at the elevation of \(\text{ z}_\mathrm{c}\) (m/s)

\(\left\langle \text{ u}\right\rangle _\mathrm{mb}\) :

Spatially averaged velocity at the elevation of \(\text{ z}_\mathrm{mb}\) (m/s)

\(\left\langle \text{ u}\right\rangle _\mathrm{PIV}\) :

Spatially averaged velocity magnitude estimated from surface particle image velocimetry (PIV) (m/s)

\(\tilde{\text{ u}}\) :

Tilde denotes a spatial fluctuation of a time averaged flow variable; in this case it is the difference between \(\bar{\text{ u}}\) at each point within a spatial-averaging window and \(\left\langle {\bar{\text{ u}}} \right\rangle \), where \(\bar{\text{ u}}\) is the time-average of \(\text{ u}\) (m/s)

\(\text{ z}\) :

Height above \(\text{ z}_\mathrm{t}\) (m)

\(\text{ z}_\mathrm{c}\) :

Maximum \(\text{ z}_\mathrm{k}\) within a spatial averaging window (m)

\(\text{ z}_\mathrm{k}\) :

Bed roughness height; the distance from the roughness peak to the roughness trough within a spatial averaging window grid cell (m)

\(\text{ z}_\mathrm{mb} =\text{ z}_\mathrm{c} -\int \limits _{\mathrm{z}_\mathrm{c}}^{\mathrm{z}_\mathrm{t}} \upphi \text{ dz}\) :

Mean bed elevation following Nikora et al. [5]

\(\text{ z}_\mathrm{r}\) :

Upper boundary of the roughness layer

\(\text{ z}_\mathrm{t}\) :

Trough elevation, or the minimum roughness elevation within an averaging window (m)

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Acknowledgments

This work was supported by the National Science Foundation under grant IGERT: Nonequilibrium Dynamics Across Space and Time: A Common Approach for Engineers, Earth Scientists, and Ecologists (Grant Number DGE-0504195), as well as the National Center for Earth-surface Dynamics (NCED), a Science and Technology Center funded by the office of Integrative Activities of the National Science Foundation (under Agreement Number EAR-0120914). We thank the University of California Natural Reserve System and the Steel family for providing access to streams in the Angelo Coast Range Reserve, and the Belwin Conservancy for providing access to Valley Creek in Minnesota. The authors are grateful to Dr. V. Nikora, University of Aberdeen, for encouragement to integrate the abundance of Glossosoma with double-averaged hydraulic variables.

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  1. Department of Civil Engineering, St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN, 55414, USA

    Mark W. L. Morris & Miki Hondzo

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Morris, M.W.L., Hondzo, M. Double-averaged rough-bed open-channel flow with high Glossosoma (Trichoptera: Glossosomatidae) abundance. Environ Fluid Mech 13, 257–278 (2013). https://doi.org/10.1007/s10652-012-9265-0

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