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Experimental study on oscillating grid turbulence and free surface fluctuation

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Abstract

This paper analyses the interaction between the turbulence and free surface. The phenomenon takes place in many natural flows and industrial processes. In the present experiments, turbulence is generated by a vertically oscillating grid moving beneath the free surface. Fluid velocity has been measured through a hot-film anemometer, and the free surface elevation has been measured by an ultrasonic sensor. Integral length scales and several turbulence estimators have been computed. In order to detect the generation of turbulence near the free surface, the correlation between free surface elevation and the underneath flow velocity has been studied, as well as the time lag between turbulence and free surface. The free surface dynamics has been characterized by a velocity scale and a length scale. The kinetic energy associated with the free surface fluctuations increases with the Reynolds number at a rate depending on the frequency of the grid movement. For Reynolds number larger than ≈1000, however, the relationships collapse to a single curve characterized by a lower rate. The present experiments do not achieve the inertial sub-range in the vertical velocity fluctuations, and the estimated spectrum decays with an exponent smaller than −3, which is the typical value for the two-dimensional turbulence in the inertial sub-range. The macro length scale, estimated by using the Taylor’s frozen turbulence hypothesis, experiences a decay away from the grid, which follows reasonably well the profile of Thompson and Turner (J Fluid Mechanics 67: 349–368, 1975). The micro length scale reduces immediately beneath the free surface, which can be interpreted by the increase of dissipation rate in the subsurface layer. The classification diagram by Brocchini and Peregrine (J Fluid Mech 449: 225–254, 2001) indicates that most tests fall in the weak turbulence domain, but some tests fall in the wavy domain. The vertical velocity fluctuations and the free surface level show a significant correlation with a negative phase lag, that is, the free surface fluctuations are ahead of the vertical velocity fluctuations.

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Abbreviations

\( \overline{ \ldots } \) :

Time average operator

\( \widetilde{ \ldots } \) :

Slowly varying term operator

\( \widehat{ \ldots } \) :

Slowly varying term plus fluctuating term operator

\( \otimes \) :

Dyadic product operator

E{…}:

Ensemble average operator

α, αp :

Angle, principal axis of the Reynolds stress tensor angle

κ:

Turbulent kinetic energy

Λ:

Integral length scale

λ:

Taylor length scale

ρ:

Water mass density

σ:

Surface tension

ν:

Kinematic fluid viscosity

χ w′w′ :

Autocorrelation coefficient

τ:

Time lag

ϒ:

Integral time scale

υ:

Taylor time microscale

ψ:

Coherence

a, A :

Coefficient

DAQ:

Data acquisition board

DOA:

Degree of anisotropy

d :

Water depth

f G :

Frequency of the grid

Fr, Fr s :

Froude number, based on free surface scales

g :

Gravity acceleration

H, H rms, H max :

Wave height, root mean square wave height, maximum wave height

H 1/3, H 1/10 :

Highest one-third wave, one-tenth wave, etc.

L :

Length scale

L w , L ∞ :

Integral length scale in the vertical, far field longitudinal integral length scale

LDV:

Laser Doppler velocimetry

M :

Mesh size

P ab :

Power cross-spectral density between the variables a and b

q :

Velocity scale

R :

Water depth above the maximum level of the grid

\( R_{{ \, w^{\prime } w^{\prime } }} \) :

Time correlation function for the vertical fluctuating velocities

R2 :

Coefficient of determination

Re, Re G , Re L , Re s :

Reynolds number, of the grid, based on the grid parameters, based on surface scales

S :

Stroke

s :

Thickness of the grid bar

t :

Time

t ave :

Interval time of average

T mean, T 1/3,…:

Period of the waves, mean value, mean value of the first third, etc.

TKE:

Turbulent kinetic energy

u,w :

Horizontal, vertical fluid velocity

\( u_{\text{rmsHT}}^{\prime } \) :

Hopfinger and Toly (1976) horizontal velocity fluctuation r.m.s. value

u′,w′:

Horizontal, vertical fluctuating fluid velocity

\( u_{\text{rms}}^{\prime } ,w_{\text{rms}}^{\prime } \) :

Horizontal, vertical fluctuating fluid velocity r.m.s. value

\( w_{{{\text{rms}}\infty }}^{\prime } \) :

Far field vertical fluctuating velocity r.m.s. value

u s :

Velocity scale based on interface kinematics

V, V 1, V 2 :

Velocity

\( V^{\prime } \), \( V_{ 1}^{\prime } \), \( V_{ 2}^{\prime } \) :

Fluctuating velocity

W :

Velocity scale

We s :

Weber number, based on surface scales

z :

Vertical spatial coordinate

zt 3 :

Bed stress log layer limit

z G :

Distance of the mean position of the grid from the still water level

z s :

Instantaneous level of the free surface

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Acknowledgments

The experimental activity was mainly carried out by Luca Chiapponi during the preparation of his PhD Thesis in the Laboratory of Hydraulics of DICATeA, University of Parma. Part of the experimental activity by Sandro Longo and Mara Tonelli was carried out in CEAMA, Grupo de Dinámica de Flujos Ambientales, University of Granada, Spain, kindly hosted by Miguel A. Losada. Financial support from CEAMA is gratefully acknowledged. Special thanks are given to Simona Bramato and Christian Mans, who provided great help with experiments. This paper was written while Sandro Longo was visitor at Cambridge University Engineering Department, Cambridge, UK, kindly hosted by Dongfang Liang, who also critically reviewed the manuscript.

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  1. Sandro Longo

    Present address: CUED, Cambridge, UK

Authors and Affiliations

  1. Department of Civil Engineering, University of Parma, DICATeA, Parco Area delle Scienze, 181/A, 43124, Parma, Italy

    Luca Chiapponi & Sandro Longo

  2. Dipartimento di Chimica, Fisica e Ambiente, DCFA, Università degli Studi di Udine, via del Cotonificio, 108, 33100, Udine, Italy

    Mara Tonelli

Authors
  1. Luca Chiapponi
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  2. Sandro Longo
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  3. Mara Tonelli
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Correspondence to Sandro Longo.

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Chiapponi, L., Longo, S. & Tonelli, M. Experimental study on oscillating grid turbulence and free surface fluctuation. Exp Fluids 53, 1515–1531 (2012). https://doi.org/10.1007/s00348-012-1367-4

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  • DOI: https://doi.org/10.1007/s00348-012-1367-4

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