KINEMATICS OF WATER PARTICLE MOTION WITHIN THE SURF ZONE Rafael Steer ClbrarK Naval PcKfi^aduate School MontefBy, Califomia 93940 NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS Kinematics of Water Particle Motion Within the Surf Zone by Rafael Steer Advi sor: E . B. Thornton September 1972 ^ App^vzd ijo-i pubtic ^eXezLSe; dOtt/uhLition unlAjnltzd. Kinematics of Water Particle Motion Within the Surf Zone by Rafael Steer Lieutenant J.G. , Colombian Navy- Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1972 Library Naval Postgraduate School Monterey, California 93940 ABSTRACT Simultaneous measurements of sea surface elevation and horizontal and vertical particle velocities at 39 and 69 cm elevations in the column of water of 130 cm total depth were made inside the surf zone. Also, the offshore sea surface elevation at this location was measured for purposes of comparison. The velocities were measured using electro- magnetic flow meters, and the sea surface elevation was measured using pressure wave gauges. Probability density functions, pdf, were determined for each record. The pdf's for the sea surface elevation and particle velocities inside the surf zone were highly skewed. Spectral computations show that the range of significant energy was between 0.05 and 0.6 hertz. The phase angle was compared to linear wave theory and shows a shifting of phase for the horizontal velocity with sea surface elevation from 0 degree at low frequency to 90 degrees at higher frequencies. The energy- density spectra show that the horizontal component is approximately 95% of the total kinetic energy of the surf zone. In the range of significant energy, a coherence of about 0.9 was found for the sea surface elevation and particle velocities which indicates that the particle motion inside the surf zone is for the most part wave-induced. TABLE OF CONTENTS I. INTRODUCTION ' 7 A. REVIEW OF PREVIOUS WORKS 7 B. OBJECTIVE 8 II. NATURE OF THE PROBLEM 9 A. CHARACTERISTICS OF THE SURF ZONE 9 B. STATISTICAL ANALYSIS 10 1. Probability Density Function 10 2. Energy-Density Spectra 11 3. Cross -Spectral Density - 12 4. Phase Angle 14 5 . Coherence Function 14 III. INSTRUMENTATION 16 A. FLOW METER 16 B. WAVE GAGE 24 IV. PRESENTATION OF DATA 27 A. MEASUREMENT TECHNIQUES 27 B. DATA PRE-PROCESSING 29 V. ANALYSIS 34 A. PROBABILITY DENSITY FUNCTIONS 34 1. Sea Surface Elevation 34 2. Vertical and Horizontal Water Particle 34 Velocities B. SPECTRAL ANALYSIS 37 1. Offshore and Inshore Sea Surface Elevation 37 2. Sea Surface Elevation and Water Particle 43 Velocities ' a. Horizontal Particle Velocities 43 b. Vertical Particle Velocities 48 3. Horizontal and Vertical Water Particle 48 Velocities VI. CONCLUSIONS 55 BIBLIOGRAPHY 56 INITIAL DISTRIBUTION LIST 57 DDFORM 1473 60 LIST OF FIGURES 1. Electromagnetic Water Current Meter, EPCO 6130 17 2. Assembly for Dynamic Calibration of the Water Current 19 Meters 3. Measured and Actual Velocities. Current Meter Serial 20 Number 637. (Used at 69 cm Depth) 4. Measured and Actual Velocities . Current Meter Serial ^1 Number 638. (Used at 39 cm Depth) 5. Frequency Response of Water Current Meter SN 637 22 6. Frequency Response of Water Current Meter SN 638 23 7. lEC DP200 Portable Wave Recorder and SDP201 Wave 26 Gage 8. Schematic of Tower used in Taking the Measurements 23 Showing Location of Instruments 9. Water Current Meters Mounted on the Tower at Low Tide 30 10. Beach Profile with Location of Measurement Site 31 11. Strip Chart Record Sample 33 12. Probability Density Function of Sea Surface Elevation 35 Outside Surf Zone 13. Probability Density Function of Sea Surface Elevation 36 Inside Surf Zone 14. Schematic of Sea Surface with Peaked Crests and 38 Elongated Troughs 15. Probability Density Function of Horizontal Water 39 Particle Velocity Inside Surf Zone 16. Probability Density Function of Vertical Water Particle 40 Velocity Inside Surf Zone 17. Spectra of Sea Surface Elevation Offshore and Inside 42 Breaking Zone 18. Spectra of Sea Surface Elevation and Horizontal Particle 44 Velocity, Depth 39 cm 19. Spectra of Sea Surface Elevation and Horizontal Particle 45 Velocity, Depth 69 cm 20. Spectra of Sea Surface Elevation and Vertical Particle 46 Velocity, Depth 39 cm 21. Spectra of Sea Surface Elevation and Vertical Particle 47 Velocity, Depth 69 cm 22. Spectra of Horizontal Particle Velocity at Depth 39 cm 50 and Horizontal Particle Velocity at Depth 69 cm 23. Spectra of Vertical Particle Velocity at Depth 39 cm and 51 Vertical Particle Velocity at Depth 69 cm 24. Spectra of Horizontal and Vertical Particle Velocities at 52 Depth 39 cm 25. Spectra of Horizontal and Vertical Particle Velocities at 53 Depth 69 cm I. INTRODUCTION A. REVIEW OF PREVIOUS WORKS With reference to the surf zone only the more general features and characteristics of the motion of the fluid are at present understood. This is due to the lack of understanding of waves after the breaking point. Most of the knowledge at this time is based on approximations of wave theory or empirical relationships . The difficulties are practical as well as theoretical. The surf zone is a very hostile environment to work in and a very difficult one to reproduce properly in the laboratory. Another major difficulty has been a lack of instrumentation to make velocity measurements for wave- induced motion and turbulence . Only a few experiments have been con- ducted to measure and study the kinematics of water particle motions due to waves in shallow and intermediate water, and almost none for the kinematics of water particle motion inside the surf zone itself. Among the laboratory experiments the most notable is a study by Iversen (1953) in which photographic techniques were used to obtain a Lagrangian description of water particle motion. However, due to the slope of his model beach, spilling breakers were not considered. Field measurements have been made by a number of authors using a variety of instruments. Inman (1956) measured the drag force to infer water particle motion. Walker (1969) used propeller type flow meters. Miller and Zeigler (1964) used meters based on acoustic principles; they compared their measurements with higher order wave theory and found some qualitative agreement. Thornton (1969) used an electro- magnetic flow meter and presented his results in the form of spectra. His results showed a gradual decay of energy across the surf zone and a shifting of energy to the higher frequency turbulent region of the spectrum. Some attempts have been made to theoretically describe the kinematics of the surf zone. One of them by Collins (1970) describes the probability distribution functions of the wave characteristics of wave height and period for the region inside the surf utilizing the hydro- dynamic relationships for shoaling and refraction. B . OBJECTIVE The objective of this research was to make preliminary studies on the kinematics of the water particle motion within the surf zone and within breaking waves. With this purpose in mind, simultaneous measure- ments were made of the instantaneous sea surface elevation and of horizontal and vertical particle velocities at different elevations in the same column of water in the surf zone, and of the offshore sea surface elevation. The probability density functions and spectra of the wave and particle velocity measurements were determined. II. NATURE OF THE PROBLEM A. CHARACTERISTICS OF THE SURF ZONE Theories developed to describe characteristics of waves, such as wave height, period, and particle velocity, can be applied reasonably well to deep water waves, and with less accuracy to shoaling waves up to the point of near-breaking conditions. However, upon breaking the waves lose their ordered character and can no longer be described analytically. Each type of breaker, spilling, plunging, or surging, (depending upon the beach slope and deep water wave steepness) causes a different type of flow and consequently a different distribution of energy in the surf zone. Furthermore, moving boundaries at the bottom and surface of the sea together with the unsteady flow motion, make the surf zone a very difficult place for the application of a theoretical model. However, results of experiments by Thornton (1969) suggest that the kinematics is not as disorganized as one might be led to believe. In order to develop a theoretical model it is necessary to have at least some a priori knowledge of the characteristics of the wave induced motion and the structure of the turbulence. If it is desired to study the details of surf zone flow, it is necessary to have a very large model or to take measurements in situ with appropriate instruments for measuring instantaneous velocities. A limitation inherent in laboratory studies is space. The most commonly occurring breaking wave is of the spilling type which is observed on very flat beaches; this calls for very long wave tanks. 9 Direct field measurements can overcome these problems. A difficulty is that the large scale conditions peculiar only to the place where the data is taken may affect the mean flow in such a way that it is applicable only to a particular set of conditions. However, small scale features, such as turbulent motion in the breaking waves, are not affected directly by the large scale conditions and their study produces better results when done in the field. Furthermore, the process generally can be assumed to be stationary, that is, the mean and variance do not change with time. This is a good assumption for short measurements on the order of 2 0 minutes as done in this research. In general, the surface waves and wave induced water particle motion have the characteristics of random phenomena, therefore it is necessary to utilize statistical procedures to describe them. The basic analyses and their application to the present problem are described in the following sections . B . STATISTICAL ANALYSIS 1 . Probability Density Function The probability density function for random data describes the probability that a particular process will assume a value within some defined range at any instant of time. The probability that the sample time history record x(t) assumes a value between x and (x + A x) will approach an exact probability description as the length of the sample record, T, approaches infinity, and as Ax approaches zero. In equation form: 10 p(x) = Lim Lim AX — >■ 0 T *►« T |_ AxJ where T. is the total amount of time that x(t) falls inside the range (x, X + Ax), and T is the total observation time. The probability density function is always a real, non-negative function. 2 . Energy-Density Spectrum The energy-density spectrum is computed from the Fourier transform of the auto-covariance function of the record time series. The auto-covariance function describes the general dependence of the values of the data at one time on the values at another time. It is expressed as a function of lag time t as t+T The quantity cp, , ( r ) is always a real-valued even-function, with a maximum at t = 0 . A Parzen lag window is applied to the covariance functions to account for the finite length of the record in computing the spectra. The Parzen lag window has the advantage of no negative side lobes and maintains the sample coherence between its theoretical values of 0 and 1. This eliminates any numerical instability problem that may arise in the analysis procedure, particularly when calculating cross-spectra . The Parzen lag window is given in Bendat and Piersol (1966) by the formula 11 P(r) :i 6 (^)^ + 6 i~-)^ m m 1 - 6 i-r—r + 6 i~-r r < m -I 3. m 2 1 - ( — ' ) m 0 r > T 111 ^ _ V ^ T <.T m where T is the total time lag. m The Fourier transform of the modified auto-covariance function is the energy -density spectrum, given by 0 CDii(f)= I I -00 00 P( r )
= T-^"io 7 / -iW-^ft- ' )dt The function ^ ( ^ ) is always a real-valued function, but does not necessarily have a maximum of t = 0, or is an even-function, as was the auto-covariance function. The cross -spectral density function is expressed as 0,,.«= / P( r )
^2^- T )] cos (2 TT f T ) dr and the imaginary part is called the quadrature -spectrum 00 / Q^^{i) = 2 / P( r ) [
n®CD22<«
and has a range of values
0 < Tj(f) < 1
2
In the ideal case of two completely coherent records, ^i?^^^ ^^^ ^^^
maximum value of unity. As the correlation between the records decreases
14
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the value of T , ^ (f ) decreases, and reaches the minimum value of
zero when the records are completely incoherent or statistically
independent.
15
III. INSTRUMENTATION
A. FLOW METER
The water particle velocities were measured using two Engineering-
Physics Company water current meters, type 6130, see Figure 1. The
current meter operation is based on the electromagnetic induction principle,
A symmetrical magnetic field is generated in the water by a driving coil
imbedded in the probe. When water in the vicinity of the probe has a
velocity relative to the magnetic field, an electric field is induced as
expressed by the vector equation
£"= u"xT
where E is the induced electric potential, B is the magnetic field, and
"iT is the water velocity.
The induced electric potential is sensed by two electrodes, in
contact with the water, oriented in the same direction as the induced
field. This produces a dc signal proportional to the magnitude of the
velocity component perpendicular to the electrode's axis. Orthogonal
components of the velocity are measured using two pairs of electrodes
placed with axis perpendicular to each other.
The measureable range of velocities is from 0 to 1.5 m/sec with
a maximum output error of one percent of full scale reading. The in-
strument has two electrical time constant settings, 0.3 or 1.0 seconds.
The output rms noise level is a function of the electrical time constant
and is given by the expression
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during calibration.
The current meter is calibrated under steady flow conditions by the
manufacturer but different characteristics are expected under unsteady
flow conditions due to a change in the boundary layer on the face of the
transducer. The probe was recalibrated by oscillating it in a water tank.
The set-up is shown in Figure 2 . The carriage on which the flow meter
probe was mounted travels back and forth on rails . It is driven by an
electric motor with a constant throw arm and variable rotation velocity.
The peak carriage velocity was calculated from the tangential velocity
of the motor arm. The ratio of carriage velocity, V , to the velocity
measured by the instrument, V , was found for different angular velocities
m
This was done for each channel of the current meter by orienting each
pair of electrodes perpendicular to the direction of flow respectively.
The results are shown in Figures 3 and 4.
The ratio of V /V is also plotted as a function of the frequency
m t
of oscillation in Figures 5 and 6. The range of frequencies measured
ranges from 0 to 0.5 Hz; the latter being the highest frequency which
the carriage was capable of without becoming unstable. It is noticed
that there is a decrease in the velocity ratio with increasing frequency.
A measured response time, or time constant, can be calculated from
the measurements. Using the usual definition, the response time
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Measured and Actual Velocities . Current Meter Serial
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Figure 3
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Measured and Actual Velocities. Current Meter Serial
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Figure 4
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Frequency Response of Water Current Meter
Serial Number 637
Figure 5
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0.5 Hz
Frequency Respone of Water Current Meter
Serial Number 638.
Figure 6
23
corresponds to the time required for the signal to decrease by 3 db from
the true value, that is, when V /V = 0.707. Referring to Figures 5
m t
and 6, the instruments have an average response time of 2 seconds
corresponding to 0 . 5 Hz.
The probe is made of fiberglass material with dimensions 11 inches
in length and 3/4 inches in diameter. The system senses variations in
velocities over an area of approximately two to three probe radii from
the transducer. This limits the spatial resolution to disturbances with
a wave length of 11 cm, corresponding to a deep water wave period of
0.27 seconds .
B. WAVE GAGE
The water surface elevation was measured with an Interstate
Electronics Corporation SDP 201 differential pressure sensor, and a
DP 200 wave recorder, shown in Figure 7. The pressure sensor is a
small, unbonded strain gauge bridge. Direct current exitation for the
bridge is supplied by a voltage regulator located in the transducer
housing. The sea pressure is coupled by a neoprene diaphragm to a
silicone fluid filling the interior. One part of the transducer is exposed
directly to the interior fluid, and the other is connected to a chamber
which is connected to the interior fluid by a length of capillary tubing.
This arrangement acts as a hydraulic filter developing a reference
pressure which is an average value of the external sea pressure. By
the action of this filter, the transducer senses rapid pressure fluctuations
only and slow changes such as tides are lost through the hydraulic filter.
24
The dc signal voltage output has a range of + 2 ,5 volts . In the
wave recorder, the signal is transferred to a chart paper. The recorder
also has an outlet for an external recording system. Maximum dynamic
range of the wave meter is + 20 feet with a linearity of one percent.
In addition, a stilling well type mean water level indicator was
used. This was a two inch diameter clear plastic tubing capped on
both ends with a 1/16 inch diameter hole in the bottom and a 3/8 inch
hole in the top for an air vent. The mean water level was measured by
viewing the water level using a surveyor's transit from the shore and ^
comparing it to a graduation on the side of the tubing.
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IV. PRESENTATION OF DATA
A. MEASUREMENT TECHNIQUES
The experiments were performed at Del Monte Beach in front of
the Naval Postgraduate School beach laboratory, on August 1, 1972. The
waves at this location are generally highly refracted and directionally
filtered, and break almost parallel to the shore line. At the time of
making the measurements the breaking waves generally were of the spilling
type with an occasional plunging type.
It is necessary to mount the instruments on a stable platform that
does not vibrate and can stand the forces of the breaking waves. A
tower was built for this purpose and arranged as shown schematically
in Figure 8. The tower was constructed of 2 inch diameter steel pipe
with a two foot diameter flat circular base to keep it from sinking into
the sand. It was fastened to the ground by four stays tied to screw-type
anchors buried approximately 2 feet in the sand. The tower was tested
for endurance before the actual taking of data for a period of 24 hours.
The entire structure proved to be very stable; the vibration was very
small and was neglected in the calculations.
The transducers were arranged such that they were aligned
vertically in order to get measurements in the same column of water.
The two flow meters were oriented to measure vertical and horizontal
(inshore-offshore) water particle velocities . They were mounted at
the end of a horizontal 3/4 inch diameter pipe that extended approximately
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