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Alternating skimming flow over a stepped spillway

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Abstract

The study of stepped spillways in laboratory scales has been essentially focused on two separated sub-regimes within skimming flow. In this paper we investigate the appearance of an unclassified alternating skimming flow regime in a 0.5 m wide stepped spillway which does not fit on these earlier definitions, and which does not occur in a 0.3 m wide spillway. Our aim is to explain the genesis of this unclassified flow which is visualised in the physical stepped spillway, by using 3D numerical modelling. Flow depths and velocities are measured using an ultrasonic sensor and Bubble Image Velocimetry in the wider flume (0.5 m). The numerical model is validated with the experimental data from the 0.5 m wide spillway. After validation, the channel width of the same numerical model is reduced to 0.3 m wide spillway in order to characterise (compare) the case without (with) alternating skimming flow. Both cases are solved using Reynolds-Averaged Navier–Stokes equations together with the Volume-of-Fluid technique and SST k-\(\omega\) turbulence model. The experimental results reveal that the alternating skimming flow regime is characterised by an evident seesaw pattern of flow properties over consecutive steps. In turn, the numerical modelling clarified that this seesaw pattern is due to the presence of a complex system of cross waves along the spillway. These cross waves are also responsible for a mass and momentum exchange in the transversal direction and for the formation of the alternating skimming flow in the spillway.

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Abbreviations

\(C_\alpha\) :

Binary coefficient (-)

\(f_x\) :

Fluid x, where \(x=\{1,2\}\) (-)

l :

Step length (m)

H :

Total drop height (m)

\(H_u\) :

Vertical drop height needed to get a uniform flow (m)

\(h_c\) :

Critical flow depth (m)

\(h_{w,US}\) :

Flow depth measured with the Ultrasonic Sensor (m)

\(h_{w,N}\) :

Flow depth calculated with Numerical model (m)

k :

Turbulent kinetic energy (\(m^2s^{-2}\))

p* :

Modified pressure (Pa)

S :

Step number (-)

SE :

Step edge number (-)

SN :

Step niche number (-)

s :

Step heigh (m)

\(\bar{u}\) :

Mean velocity (m s−1)

\(\bar{\mathbf{u }}\) :

Mean velocity vector (m s−1)

\(\bar{\mathbf{u }}_c\) :

Mean compressive velocity (m s−1)

w :

Step/channel width (m)

X :

Distance to the spillway crest in flow direction (m)

Y :

Transversal distance (m)

\(Y_{max}\) :

Maximum transversal distance = w / 2 (m)

Z :

Elevation above pseudo-bottom (m)

\(\alpha\) :

Volume fraction of fluid 1 (–)

\(\kappa\) :

Surface curvature (m−1)

\(\mu\) :

Dynamic viscosity (kg m−1s−2)

\(\varphi\) :

Chute angle (°)

\(\rho\) :

Fluid density (kg m−3)

\(\sigma\) :

Surface tension (kg s−2)

\(\tau\) :

Shear stress tensor (Pa)

f1 :

Fluid 1

f2 :

Fluid 2

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Acknowledgments

Pedro Lopes would like to acknowledge the facilities provided during the 3 months in 2014 as visiting student at FH-Aachen, Germany, from which the experimental results were obtained. All the numerical results here showed were performed on the Centaurus Cluster of the Laboratory for Advanced Computing of University of Coimbra, Portugal. This study had the support of FCT (Portuguese Foundation for Science and Technology) through the Projects UID/MAR/04292/2013 and Grant SFRH/BD/85783/2012, financed by MEC (Portuguese Ministry of Education and Science) and FSE (European Social Fund), under the programs POPH/QREN (Human Potential Operational Programme from National Strategic Reference Framework) and POCH (Human Capital Operational Programme) from Portugal2020.

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Authors and Affiliations

  1. MARE - Marine and Environmental Sciences Centre, Faculty of Sciences and Technology, University of Coimbra, 3004-517, Coimbra, Portugal

    Pedro Lopes, Jorge Leandro & Rita F. Carvalho

  2. Department of Civil Engineering, University of Coimbra, Rua Luís Reis Santos, 3030-788, Coimbra, Portugal

    Pedro Lopes, Jorge Leandro & Rita F. Carvalho

  3. Chair of Hydrology and River Basin Management, Technical University of Munich, Arcisstrasse 21, 80333, Munich, Germany

    Jorge Leandro

  4. Hydraulic Engineering Section, FH Aachen, University of Applied Sciences, Bayernallee 9, 52066, Aachen, Germany

    Daniel B. Bung

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  1. Pedro Lopes
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Correspondence to Pedro Lopes.

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Lopes, P., Leandro, J., Carvalho, R.F. et al. Alternating skimming flow over a stepped spillway. Environ Fluid Mech 17, 303–322 (2017). https://doi.org/10.1007/s10652-016-9484-x

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