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Experiments on turbulence beneath a free surface in a stationary field generated by a Crump weir: free-surface characteristics and the relevant scales

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Abstract

This work concerns the analysis of experimental instantaneous fluid levels and three-component fluid velocity measurements in a stationary flow field generated by a Crump weir in a laboratory flume using an ultrasonic distance sensor and a three-probe arrangement of an ultrasonic Doppler velocity profiler. The tests are characterised by different and increasing Froude numbers (Fr = 0.10–0.38), with the free surface of the fluid ranging from flat (low Froude number) to almost aerated (high Froude number). The statistics of the free surface are computed, and the relevant length and velocity scales are measured. A free-surface boundary layer was detected having a thickness proportional to the root mean square of the free-surface height series and with a velocity scale that related well to the free-surface elevation time gradient. The mean velocity profiles are presented. There are many indicators that a specific regime occurs with an optimal tuning between free surface and turbulence. In this regime, the length scales are raised.

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Abbreviations

\( \langle \ldots \rangle \) :

Space average operator

\( \overline{ \ldots } \) :

Time average operator

\( \widetilde{ \ldots } \) :

Phasic average operator

δv :

Thickness of the viscous sub-layer

Φ j :

Volume fraction or concentration for the j phase

Λzz :

Integral length scale in the vertical computed on using the vertical fluctuation velocity

ρ:

Mass density

σ:

Surface tension

ν:

Kinematic fluid viscosity

ζ0 :

Abscissa in the beam axis reference system

Θ:

Temperature

a :

Weighting function

A, B:

Matrix for reference transformation

a_umrms, a_dmrms:

Root mean square of the up-midlevel amplitude (crests) and of the down-midlevel amplitude (troughs)

c :

Celerity of propagation of ultrasound

d′ :

Head over the weir crest

d :

Water depth upstream

DNS:

Direct numerical simulation

f co :

Cut-off frequency

f e :

Frequency of the carrier

Frs :

Froude number based on free-surface scales

Frupstream, Fr:

Froude number in the upstream section, in the section of measurement

FS:

Full scale

H, Hrms:

Wave height, root mean square wave height

H1/3, H1/10, …:

Mean value of the first third, of the first tenth, …

h, hmeas, hwave:

Instantaneous filtered value, measured value, value due to potential flow

k :

Coefficient

L 0 :

Distance of the target

PDF:

Probability density function

PIV:

Particle image velocimetry

Q :

Volume discharge

Re, Res :

Reynolds number, based on surface scales

R 2 :

Coefficient of determination

S/N:

Signal to noise ratio

t :

Time

T :

Period of the waves, period of time average

Tmean, T1/3, …:

Period of the waves, mean value, mean value of the first third, …

t prf :

Time between two subsequent pulses

US:

Ultrasound

UVP:

Ultrasonic Doppler velocity profiler

u, v, w:

Streamwise, spanwise, vertical fluid velocity

u′, v′, w′:

Streamwise, spanwise, vertical fluctuating fluid velocity

u′rms, v′rms, w′rms:

Streamwise, spanwise, vertical root mean square value of the fluctuating fluid velocity

u i :

Velocity component along the i beam axis

u s :

Velocity scale

uupstream, ums:

Mean fluid velocity in the upstream section, in the section of measurement

V :

Volume of integration

V s :

Velocity of the surface

Wes :

Weber number, based on surface scales

x, y, z, x i :

Spatial co-ordinates

x, s:

Space vector

X j :

Phasic function for the j phase

z s :

Instantaneous level of the free surface

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Acknowledgments

Support from FIL 2008 is acknowledged. The paper was completed and revised during my sabbatic leaving in CEAMA, Grupo de Dinámica de Flujos Ambientales, University of Granada, Spain, where I was kindly hosted by Miguel A. Losada.

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Authors and Affiliations

  1. Department of Civil Engineering, University of Parma, Viale G.P. Usberti, 181/A, 43100, Parma, Italy

    Sandro Longo

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  1. Sandro Longo
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Correspondence to Sandro Longo.

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Longo, S. Experiments on turbulence beneath a free surface in a stationary field generated by a Crump weir: free-surface characteristics and the relevant scales. Exp Fluids 49, 1325–1338 (2010). https://doi.org/10.1007/s00348-010-0881-5

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