# 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.

## Keywords

Free Surface Flow Field Particle Image Velocimetry Froude Number Peregrine## List of symbols

- \( \langle \ldots \rangle \)
Space average operator

- \( \overline{ \ldots } \)
Time average operator

- \( \widetilde{ \ldots } \)
Phasic average operator

- δ
_{v} Thickness of the viscous sub-layer

- Φ
_{j} Volume fraction or concentration for the

*j*phase- Λ
_{zz} Integral length scale in the vertical computed on using the vertical fluctuation velocity

- ρ
Mass density

- σ
Surface tension

- ν
Kinematic fluid viscosity

- ζ
_{0} Abscissa in the beam axis reference system

- Θ
Temperature

*a*Weighting function

**A**,**B**Matrix for reference transformation

*a_um*_{rms},*a_dm*_{rms}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

- Fr
_{s} Froude number based on free-surface scales

- Fr
_{upstream}, Fr Froude number in the upstream section, in the section of measurement

- FS
Full scale

*H*,*H*_{rms}Wave height, root mean square wave height

*H*_{1/3},*H*_{1/10}, …Mean value of the first third, of the first tenth, …

*h*,*h*_{meas},*h*_{wave}Instantaneous filtered value, measured value, value due to potential flow

*k*Coefficient

*L*_{0}Distance of the target

Probability density function

- PIV
Particle image velocimetry

*Q*Volume discharge

- Re, Re
_{s} 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

*T*_{mean},*T*_{1/3}, …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

*u*_{upstream},*u*_{ms}Mean fluid velocity in the upstream section, in the section of measurement

*V*Volume of integration

*V*_{s}Velocity of the surface

- We
_{s} 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

## Notes

### 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.

## References

- Amini A (2009) Velocity profiles and interface instability in a two-phase fluid: investigations using ultrasonic velocity profiler. Exp Fluids 46:683–692CrossRefGoogle Scholar
- Brocchini M, Peregrine DH (2001a) The dynamics of strong turbulence at free surfaces. Part 1. Description. J Fluid Mech 449:225–254zbMATHCrossRefMathSciNetGoogle Scholar
- Brocchini M, Peregrine DH (2001b) The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions. J Fluid Mech 449:255–290CrossRefMathSciNetGoogle Scholar
- Calmet I, Magnaudet J (2003) Statistical structure of high-Reynolds-number turbulence close to the free surface of an open channel flow. J Fluid Mech 474:355–378zbMATHCrossRefGoogle Scholar
- Chanson H, Brattberg T (2000) Experimental study of the air-water shear flow in a hydraulic jump. Intl J Multiphase Flow 26(4):583–607zbMATHCrossRefGoogle Scholar
- Dabiri D, Gharib M (2001) Simultaneous free-surface deformation and near-surface velocity measurements. Exp Fluids 30:381–390CrossRefGoogle Scholar
- De Cesare G, Boillat J-L (2006) Flow velocity measurements using ultrasound Doppler method—10 years experience in hydraulic modeling. In Proceedings of the 5th international symposium on ultrasonic Doppler methods for fluid mechanics and fluid engineering ISUD, pp 113–116Google Scholar
- Dean RG (1965) Stream function representation of nonlinear ocean waves. J Geoph Res 70(18):4561–4572CrossRefGoogle Scholar
- Doering JC, Donelan MA (1997) Acoustic measurements of the velocity field beneath shoaling and breaking waves. Coas Eng 32(4):321–330CrossRefGoogle Scholar
- Eckert S, Gerbeth G (2002) Velocity measurements in liquid sodium by means of ultrasound Doppler velocimetry. Exp Fluids 32:542–546CrossRefGoogle Scholar
- Goda Y (2000) Random sea waves and engineering application, World Scientific PublishingGoogle Scholar
- Gordon L, Oltman J (2000) Surf zone observations with a Nortek vector velocimeter and a Sontek ADV. Nortek technical note no. 014Google Scholar
- Hong W-L, Walker DT (2000) Reynolds-averaged equations for free-surface flows with application to high-Froude number jet spreading. J Fluid Mech 417:183–209zbMATHCrossRefGoogle Scholar
- Hurther D (2001) 3D Acoustic Doppler velocimetry and turbulence in open-channel flow. Ph.D dissertation no. 2395, Ecole polytechnique fédérale de Lausanne (EPFL), Lausanne, SwitzerlandGoogle Scholar
- Hurther D, Lemmin U (2001) A correction method of mean turbulence measurements with a three-dimensional acoustic Doppler velocity profile. J Atmos Ocean Tech 18:446–458CrossRefGoogle Scholar
- Kantoush SA, De Cesare G, Boillat JL, Schleiss AJ (2008) Flow field investigation in a rectangular shallow reservoir using UVP, LSPIV and numerical modelling. Flow Meas Instr 19:139–144CrossRefGoogle Scholar
- Khader MHA, Elango K (1974) Turbulent pressure field beneath a hydraulic jump. J Hydraul Res 12(4):469–489CrossRefGoogle Scholar
- Kikura H, Yamanaka G, Aritomi M (2004) Effect of measurement volume size on turbulent flow measurement using ultrasonic Doppler method. Exp Fluids 36:187–196CrossRefGoogle Scholar
- Komori S, Murakami Y, Ueda H (1989) The relationship between surface-renewal and bursting motions in an open-channel flow. J Fluid Mech 203:102–123CrossRefGoogle Scholar
- Kraus NC, Lohrmann A, Cabrera R (1994) New acoustic meter for measuring 3D laboratory flows. J Hydr Eng 122(7):406–412CrossRefGoogle Scholar
- Lemmin U, Rolland T (1997) Acoustic velocity profiler for laboratory and field studies. J Hydr Eng 123(12):1089CrossRefGoogle Scholar
- Longo S (2006) The effects of air bubbles on ultrasound velocity measurements. Exp Fluids 41(4):593–602CrossRefGoogle Scholar
- Longo S, Losada IJ, Petti M, Pasotti N, Lara J (2001) Measurements of breaking waves and bores through a USD velocity profiler. Technical report UPR/UCa_01_2001, University of Parma, University of SantanderGoogle Scholar
- Miles J, Ganderton P, Elliot J (2002) Vector data in the swash zone. Report. http://www.nortek-as.com/hardware/vector_swash.php
- Misra SK, Kirby JT, Brocchini M, Veron F, Thomas M, Kambhamettu C (2008) The mean turbulent flow structure of a weak hydraulic jump. Phys Fluids 20, 035106, 21 ppGoogle Scholar
- Nadaoka K (1986) A fundamental study on shoaling and velocity field structure of water waves in the nearshore zone. Technical report no. 36, Dept Civ Eng, Tokyo Inst Tech, 125 ppGoogle Scholar
- Quiao HB, Duncan JH (2001) Gentle spilling breakers: crest flow-field evolution. J Fluid Mech 439:57–85Google Scholar
- Reinauer R, Hager WH (1995) Non-breaking undular hydraulic jump. J Hydraul Res 33(5):683–698CrossRefGoogle Scholar
- Savelsberg R, Holten A, van de Water W (2006) Measurement of the gradient field of a turbulent free surface. Exp Fluids 41(4):629–640CrossRefGoogle Scholar
- Settles G (2001) Schlieren and Shadowgraph techniques, visualizing phenomena in transparent media. Springer, BerlinzbMATHGoogle Scholar
- Shen C, Lemmin U (1997) Ultrasonic scattering in highly turbulent clear water flow. Ultras 35:57–64CrossRefGoogle Scholar
- Shen L, Yue DKP (2001) Large-eddy simulation of free-surface turbulence. J Fluid Mech 440:75–116zbMATHCrossRefGoogle Scholar
- Shen L, Zhang X, Yue DKP, Triantafyllou GS (1999) The surface layer for free-surface turbulent flows. J Fluid Mech 386:167–212zbMATHCrossRefMathSciNetGoogle Scholar
- Shen L, Triantafyllou GS, Yue DKP (2000) Turbulent diffusion near a free surface. J Fluid Mech 407:145–166zbMATHCrossRefMathSciNetGoogle Scholar
- Takeda Y (1999a) Ultrasonic Doppler method for velocity profile measurement in fluid dynamics and fluid engineering. Exp Fluids 26:177–178CrossRefGoogle Scholar
- Takeda Y (1999b) Quasi periodic state and transition to turbulence in a rotating Couette system. J Fluid Mech 389:81–89zbMATHCrossRefGoogle Scholar
- Thornton EB (1979) Energetics of breaking waves in the surf zone. J Geophys Res 84:4931–4938CrossRefGoogle Scholar
- Weingand A (1996) Simultaneous mapping of the velocity and deformation field at a free surface. Exp Fluids 20:358–364Google Scholar