Laboratory Study of Surface-Gravity Wave Energy Input.

Ivan Savelyev.

Sponsored by:

Literature review.

Early theoretical works:

•Jeffreys, H., 1924: On the formation of waves by wind. Proc. Roy. Soc., 107A, 189-206.

•Jeffreys, H., 1925: On the formation of waves by wind. II. Proc. Roy. Soc., 110A, 341-347.

Experiments with wind over solid waves:

•Stanton, T. E., D. Marshall, and R. Houghton, 1932: The growth of waves on water due to the action of the wind. Proc. Roy. Soc., 137A, 283-283.

•Thijsse, J. T., 1951: Growth of wind-generated waves and energy transfer. Gravity waves, National Bureau of Standards, Washington Circular 521, 281-287.

Currently used theory:

•Miles, J. W., 1957: On the generation of surface waves by shear flows. Journal of Fluid Mechanics, 3, 185-204.

•Miles, J. W., 1959: On the generation of surface waves by shear flows, Part 2. Journal of Fluid Mechanics, 6, 568-582.

•Miles, J. W., 1960: On the generation of surface waves by turbulent shear flows. Journal of Fluid Mechanics, 7, 469-478.

•Janssen, P. A. E. M., 1991:Quasi-linear theory of wind-wave generation applied to wave forecasting. J. Phys. Oceanogr., 21, 1631-1642.

•Belcher, S. E., and J. C. R. Hunt, 1993: Turbulent shear flow over slowly moving waves. J. Fluid Mech., 251, 109-148.

Recent experimental studies:

Okuda, K., Kawai, S. & Toba, Y. 1977 Measurement of skin friction distribution along the surface of wind waves. J. Oceanogr. Soc. Japan 30,190-198.

Snyder, R. L., F. W. Dobson, J. A. Elliott, and R. B. Long, 1981: Array measurements of atmospheric pressure fluctuations above surface gravity waves. Journal of Fluid Mechanics, 102, 1-59.

Banner, M. and Peirson, W. 1998 Tangential stress beneath wind-driven air-water interfaces. J. Fluid Mech., vol. 364, pp. 115-145.

Donelan, M., Babanin, A., Young, I. & Banner, M. 2006 Wave-Follower Field Measurements of the Wind-Input Spectral Function. Part II: Parameterization of the Wind Input. J. Physical Oceanography, vol 36, pp 1672-1689.

Wave frequency range: f = 1 ÷ 3 Hz,

Significant wave height: Hs = 0 ÷ 9 cm,

Wind speed at 10m: U10 = 0 ÷ 23 m/s.

Experiment Setup

Pressure

Transducers

Water ElevationGauges

Motor ElevationGauge

Analog – Digital Converter

Calibration

Transducer time lag

correction

Data Storage

Elliott tube clog condition

Signal conditioning

Smoothing algorithm

No

Yes

Time lag correction

Linear motor

Data Flow: Real time

Motion

Controller

Wave follower position response to water elevation signal. Left: green – follower position spectrum, blue – water elevation spectrum. Right: blue - water elevation, red – Elliott probe position.

Pressure transducer response to an incoming pressure wave. Time lag due to membrane acceleration and noise filtering electronics – 30ms.

Covered Parameters:

Wave number k = 6 ÷ 40 [1/m]

Wave frequency f = 1 ÷ 3 [Hz]

Wave phase speed Cp = 0.5 ÷ 1.1 [m/s]

Wind speed at 10m height U10 = 0 ÷ 23 [m/s]

Wind speed at L/2 height U(L/2) = 0 ÷ 10 [m/s]

Inverse wave age U10/Cp = 4 ÷ 32

Pressure – slope correlation <Pr*Sl> = -0.0008 ÷ 0.0734

Static air pressure at the surface (blue line) averaged over several hundred periods at each wave phase for four various wind/wave conditions. Error bars show 95% confidence interval. Green dashed line illustrates idealized wave shape. U10 – wind speed at 10m height, U10/Cp – inverse wave age (Cp – wave phase speed), f – dominant frequency, Hs – significant wave height. Wind direction is from right to left.

Pressure – slope correlation dependence on wind speed at 10m height (left) and at L/2 height (right), where L - dominant wave length. Error bars show 95% confidence interval.

Future work:- Compare measured form drag with wave energy growth rates.- Measure the pressure - slope correlation over a range of wave frequencies and wind speeds including strongly forced breaking wave conditions.- Use Particle Image Velocimetry to deduce the viscous drag contribution to the wave growth.