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Slopes of small-scale wind waves and their relation to mass transfer rates

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Abstract

A technique is described which measures the time-varying slope of a wavy air-water interface. New results are presented for statistical and spectral properties of slopes of wind generated waves in a rectangular flow channel. Previous measurements of oxygen absorption in the same facility are reviewed in light of these results in order to relate surface structure to mass transfer rates. Measurements of mass transfer rates and RMS wave slopes are found to agree for facilities with a large range of sizes when comparisons are made at the same friction velocities.

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Abbreviations

a :

wave amplitude

b :

frequency roll-off exponent for the PSD

c :

wave speed

C :

concentration

C(m) :

water height dependent calibration constant

C i :

concentration at the interface

C b :

bulk concentration

C αβ :

constant tensor for Eq. 10

d :

distance from the bottom of the channel to the scattering glass

D :

diffusivity of absorbing species in the liquid

D G :

diffusivity of absorbing species in the gas

E :

cumulative spectral energy

f :

frequency in cycles per second

f Li :

focal length of lens i

f m :

frequency where the phase velocity is at a minimum

f N :

Nyquist frequency

f s :

sampling frequency

f v :

viscous cut-off frequency

G :

subscript denoting gas phase

G xx (f) :

stream wise power spectral density function

G zz (f) :

spanwise power spectral density function

h :

channel height

I :

light intensity scaled 0 to 255

K :

liquid mass transfer coefficient

L :

subscript denoting liquid phase

m :

average water height

M :

magnification

n i :

index of refraction of media i

N :

number of frames; number of measurements in a sample set

N F :

rate of absorption per unit area

P :

number of pixels per frame above the threshold intensity

s :

total wave slope, √s x 2 + s z 2

S i :

image distance

s o :

object distance

s x :

wave slope in the streamwise direction

s z :

wave slope in the spanwise direction

S c :

Schmidt Number

t :

time; channel glass thickness

u :

velocity component in the x direction

U L :

bulk liquid velocity

U G :

bulk gas velocity

v :

velocity component in the y direction

v * :

friction velocity of the liquid at the interface

w :

velocity component in the z direction; laser beam diameter; chanel width

w 0 :

minimum laser beam waist

x :

coordinate in the streamwise direction for the laboratory system

x′ :

coordinate in the streamwise direction for the system embedded in the interface

x 0 :

lateral displacement of laser beam on the scattering glass

x d :

lateral displacement of laser beam on the diode array

X :

streamwise pixel coordinate for the laboratory system

y :

coordinate in the vertical direction for the laboratory system

y′ :

coordinate in the direction normal to the interface

z :

coordinate in the spanwise direction for the laboratory system

z′ :

coordinate in the spanwise direction for the system embedded in the interface

Z :

spanwise pixel coordinate

α :

wave number

β :

term defining the variation of v

y :

ratio of surface tension to fluid density

δ :

thickness of the mass transfer boundary layer

η :

displacement of the interface from mean liquid level

θ i :

angle of light ray deviation from surface normal in medium i

λ :

wavelength

v :

kinematic viscosity

ϱ :

density

τ i :

interfacial shear stress

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Authors and Affiliations

  1. University of Illinois, 61801, Urbana, IL, USA

    S. R. Duke, L. M. Wolff & T. J. Hanratty

Authors
  1. S. R. Duke
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  2. L. M. Wolff
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  3. T. J. Hanratty
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Additional information

This work was supported by the Department of Energy under grant DOE DEF G02 86ER 13556 and the National Science Foundation under grant NSF CTS-09877.

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Duke, S.R., Wolff, L.M. & Hanratty, T.J. Slopes of small-scale wind waves and their relation to mass transfer rates. Experiments in Fluids 19, 280–292 (1995). https://doi.org/10.1007/BF00196477

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  • DOI: https://doi.org/10.1007/BF00196477

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