Laboratory measurement of bottom shear stress

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C7, PAGES 17,011-17,019, JULY 15, 2000 Laboratory measurement of bottom shear stress on a movable bed Kelly L. Rankin and Richard I. Hires Davidson Laboratory, Department of Civil, Environmental and Ocean Engineering, Stevens Institute of Technology Hoboken, New Jersey Abstract. A shear plate was developed to obtain direct measurements of bottom shear stress under nonbreaking surface gravity waves on a movable sand bed. The experiments were conducted under regular wave conditions in a large wave tank where time histories of bottom shear stress and surface elevation were obtained from a shear plate and wave gauges. Measurements were made at scales approaching those encountered in the field. These data were obtained under steep, vortex ripples where form drag dominated over skin friction for all data presented. Wave friction factors were calculated from measurements and were compared with existing theories as well as with previous laboratory studies. Values of the wave friction factor obtained using the shear plate exhibited reasonable agreement with existing theory. Close agreement between values of the wave friction factor obtained from the shear plate and those measured by previous studies demonstrates that use of a shear plate is a reliable additional method for the measurement of bottom shear stress over a movable bed in the laboratory. 1. Introduction A laboratory study has been conducted involving the development and implementation of a shear plate to obtain direct measurements of bottom shear stress under surface gravity waves on a movable sand bed characterized by vortex ripples. This technique for shear stress measurement does not require information about the flow characteristics in the boundary layer above the bed and is well suited for complex near-bottom flow regimes such as those encountered in conjunction with bed forms. Wave friction factors were calculated using measurements of bed shear stress and surface elevation. Previous studies of wave-induced turbulent flows have fo- Copyright 2000 by the American Geophysical Union. Paper number 2000JC /00/2000JC factor were reported. Rosengaus [1987] and Mathisen [1989] estimated the wave friction factor by relating the time-averaged rate of energy dissipation in the boundary layer to the bottom shear stress [see Kajiura, 1968] in a wave flume. They calculated energy dissipation from wave attenuation measurements. Grant and Madsen [1979] derived a theoretical expression for the bottom shear stress using a linear eddy viscosity model and a quadratic relationship between the bottom shear stress and the near-bottom velocity to determine an expression for the wave friction factor. Trowbridge and Madsen [1984] used a timedependent eddy viscosity to determine the wave friction factor. The purpose of the present work is the development of a novel experimental approach to obtain direct measurements of cused on determining the bed shear stress and energy dissipa- the total bottom shear stress for a movable sand bed. Since the tion associated with purely oscillatory motion over a fixed bed. measurement of the bottom shear stress is direct, it is not Jonsson [1966] and Jonsson and Carlsen [1976] related the necessary to determine the rate of wave attenuation or to bottom shear stress to the near-bottom velocity for oscillatory determine the velocity profile over the bed, which can be probflow in the laboratory using the quadratic drag law and an lematic during sediment suspension. The experiment was conempirical friction factor. The resulting relationship neglects ducted in a large wave tank where bottom Reynolds numbers the phase shift between the maximum bottom shear stress and are of the order of 1 x x 104. Measurements were made the external flow. Kamphuis [1975] used a shear plate rough- at scales that approach those encountered in field conditions. ened with fixed sand grains to develop a similar empirical Results are compared with the previous laboratory studies to relationship between the bottom shear stress and the near-bottom verify this experimental method. velocity using direct measurements of bottom shear stress. Simons This paper is organized as follows. Section 2 describes the et al. [1996] also used a shear cell to determine bottom shear experimental setup, the instruments employed, and the calistress for combined wave and current flow over a fixed bed. bration procedures used. The experimental protocol is out- Laboratory studies quantifying the wave friction factor over lined, and the test matrix is presented. Section 3 describes the a movable sand bed through the simultaneous measurement of measurements obtained. Section 4 discusses the procedure for ripple geometry and energy dissipation under oscillatory flow estimating wave friction factors. The wave friction factors obconditions include the study by Carstens et al. [1969], who tained in this study are compared with existing theory and measured energy dissipation and bottom bed form geometry previous laboratory measurements under movable bed condifor a movable bed in an oscillatory flow tunnel. Lofquist [1986] tions. Section 5 summarizes conclusions. performed a similar study, measuring the instantaneou shear stress on the test bed through all phases of the flow. Both the instantaneous friction factor and the time-averaged friction 17, Experimental Setup and Procedure The experimental work was conducted in Davidson Laboratory's high-speed towing tank. The concrete-sided tank is 95 m long and 3.7 m wide and accommodates water depths up to

2 17,012 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED Aluminum Plate (extends to 87.5 cm length) Drag Balance PVC Block Aluminum Plate (bolt to tank floor) Figure 1. Schematic of shear plate design. 1.8 m. Water depth in the tank for this experiment was 1.46 m. A wooden wave-absorbing beach with a slope of i on 5 is located at one end of the tank, and a double-flap articulated wavemaker designed to generate both regular and irregular waves is located at the opposite end. The wavemaker is computer-controlled and can produce wave heights up to 60 cm and wave periods up to 6 s. The procedure for measuring bottom shear stress using a shear plate involves removing a section of the bed and replacing it with an element that is free to deflect in the direction of the tangential force. The shear plate was designed for use under oscillatory flow conditions where the periods of motion range from 1.75 to 6 s and the range of shear forces are from 0.08 to 84 N m-2. The shear plate was m long by m wide (0.543 m 2 area) and was constructed by mounting a 0.63 cm thick aluminum plate on top of a drag balance (Figure 1). The drag balance was instrumented with a waterproofed linear variable differential transformer, which produced a voltage that varied linearly with horizontal deflection of the plate. The entire apparatus was cm in height. The length of the shear plate was limited to one half the wavelength of the shortest wave in the test matrix so that thcrc would not be opposing shearing forces on the plate at all phases of the wave cycle. The marlmum allowable length of thc shcar plate was used so that the force measured due to shear ovcr the sediment bed would bc large relative to secondary forces caused by the pressure disturbances at the edges of the shear plate. These secondary forces are a function of the pressure gradient across the ends of the plate, which are related to the frontal area of the plate and the gap size at the ends. Resolution of the shear plate was to _+0.08 N m --2 The shear plate was bolted to the concrete floor of the wave tank. A false bottom was constructed coplanar with and surrounding the PVC plate. The gap between the shear plate and the coplanar false bottom was fixed conservatively 0.6 cm to ensure that sediment grains would not become trapped or wedged in the gap (Figure 2). Static calibrations were carried out over a range of _+28.90øN in both directions (toward and away from the wavemaker) in 4.40øN increments. A subset between ø and 22.24øN was calibrated in 0.44øN increments. Both calibrations were repeatable, and hysteresis was not detected. The calibration was linear and precise to _+0.04øN. The shear plate was loaded with 222.4øN at its center and again at its edge and was found to be insensitive to both vertical loading and moments. Surface elevation records were obtained using three resistance-type gauges mounted on the sill of the wave tank. The gauges were installed 17.5, 28.2 (over the center of the shear platc), and 38.8 m from the wavemaker. The gauges resolve _+ 1 mm changes in stirface elevation and have an upper frequency Wavemaker IA6m Figure 2. Experimental setup (not to scale).

3 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED 1'7,013 response of 10 Hz. All of the gauges were calibrated over a range of +_15.2 cm. A cm thick layer of cohesionless quartz sand (d so = 0.23 mm) was placed on top of the shear plate and the surrounding false bottom. The layer of sand was not confined to the surface of the plate with side and end walls for several reasons. Primarily, we wished to reduce the frontal area presented to the flow while allowing the bed to taper naturally at the ends because of the angle of repose of the sediment. Second, we wanted to avoid the scour that would inevitably occur in the space between the end walls and the sediment bed. Furthermore, we wished to allow the generation of a natural rippled bed under action of the surface waves. Profiles of the sediment bed were taken to determine ripple height and length after each wave event. To obtain a profile, a clean strip of acetate was woven through the metal wires of a flat rake-like device and coated with a thin layer of petroleum gel before being lowered into the sediment bed. After removal a thin layer of sand having the same profile as the wave ripples on the sediment bed remained on the acetate strip. The wave ripple profile was traced onto a piece of paper, and the ripple height and length were measured from this tracing. Table 1 summarizes the test matrix. The experiments were conducted according to an exact protocol where the following procedure was performed for each run. 1. The wavemaker was programmed with the desired wave height and period according to the test matrix. 2. The wavemaker was turned on for 30 s intervals (to avoid influences by wave reflections from the wooden beach) until equilibrium was achieved between the bed form geometry and the wave characteristics. Ripple measurements were taken with the ripple profiler until no significant changes in ripple height or ripple length were noted. 3. Once the ripple measurements confirmed that the bed form geometry was in equilibrium with the wave characteristics, the wavemaker was started, and the acquisition of resistance data from the shear plate and surface elevation data from the wave gauges began. 4. At 30 s into the wave event the wavemaker and all data acquisition were stopped. 5. The profile of the sediment bed was obtained. After all of the data were acquired for the entire test matrix with the sediment bed in place, the experiment was repeated with the shear plate and surrounding false bottom cleared of sand (Table 1) Measurements 3.1. Resistance and Surface Elevation Time Histories Sample time histories for the resistance on the shear plate with sand, resistance on the shear plate without sand, and the surface elevation are shown in Figure 3. The time history for the resistance on the shear plate without sand contained a conspicuous departure from a sine wave because of a relatively large second-order harmonic signal. This signal was also present in the time history for the resistance on the shear plate with sand. However, the amplitude of the first-order harmonic was over 2 orders of magnitude larger than the amplitude for the second-order harmonic, and the nonlinearity was not easily discerned. These second-order harmonics were eliminated us- ing the harmonic analysis detailed in the following. Harmonic analysis was performed on the resistance and surface elevation time histories to determine the amplitude

4 17,014 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED ResiStance Time $ertes 20,3cm 2 sec wave $url'ace Elevation Time 20.3cm 2.25 ec wave I I!! I --t,,,i,, I IA time #) Figure 3. Time history of measured force on shear plate and of surface elevation. and phase of the harmonic components of the signal. The analysis program reads the input file and determines the fundamental period by counting the number of zero crossings. y,(t)=aio+ : Amcos n -t + ]B,nsin n -t, (1) n=l n=l Next, it estimates the amplitude and phase of the signal by using a least squares method. The analysis assumes that the where n = 1, 2, 3,..., Aio is the mean of the time series, A signal is of the form is the harmonic coefficient, B,is the harmonic coefficient, T is

5 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED 17,015 Table 2. Ripple Geometry Characteristics H, T,, cm h, cm /h, /h, IM,, A/A,, cm s (Mean) (Mean) (Measured) (Predicted) (Measured) (Predicted) H, wave height; T, wave period; r, ripple height; X, ripple length; r /X, ripple steepness; A,, bottom excursion amplitude of the water parcel under a wave. the fundamental period, and NH is the order of harmonics. bed that were of the same order of magnitude as the secondary The program calculates the harmonic coefficients A i,, and Bin to minimize the sum of the squares of the difference between forces (Table 1). Therefore, instead of monitoring the pressure in the gap to estimate the secondary forces, we were able to the recorded value and the estimate. The cosine and sine measure them directly by working through the test matrix with coefficients A in and B i were changed to the amplitude and phase as the plate cleared of sand. Additional forces due to the freestream pressure gradients also arose because of the thickness of the sand bed. An example of the time history of surface elevation and resistance for a run with sand and without sand y,(t) = C,o + C,n cos n -t- qb,n (2) n=l where Cio is the mean of the time series, Ci is the amplitude of the n th harmonic of the time series, and qbi, is the phase of the n th harmonic of the time series. Amplitudes of the firstorder harmonics are reported in Table 1. The amplitudes of the second-order harmonics were nearly identical for runs with sand and without sand and had relatively small magnitudes that ranged from 0.05 ø to 0.10øN. It is believed that the secondorder harmonics are related to the mechanical properties of the instrumentation Ripple Profile Measurements reveals that the resistance time history for the runs with and without sand are in phase and lead the surface elevation time history by 90 ø for all cases (Figure 3). A residual force Fre s was calculated by subtracting the first-order amplitude of the resistance time history for the runs without sand and the resistance contribution Fp due to the free-stream pressure gradient integrated over the sand bed thickness beneath the ripples from the first-order amplitude of the resistance time history for the runs with sand (Table 1). The force component due to the free-stream pressure gradient was calculated as WpgA Fr--k cosh (kh) sinh [k(d + z )] - sinh (kz ) The profile of the ripple geometry taken from the sand bed was traced from the acetate strip on the profiler to a sheet of ß [cos(-cot) - cos (kl - cot)], (4) translucent paper from which the ripple height r was measured where W is the width of the plate (0.62 m), D is the depth of (Table 2). The apparent roughness k, was estimated using the ripple height as the sand bed measured from the base of the plate to the trough level of the ripples, p is the density of the water, A is the wave amplitude, k is the wave number, z is the elevation of the k = 4r (3) shear plate from the bottom of the wave tank (0.1 m), h is the as recommended by Wikramanayake and Madsen [1994]. water depth, co is the wave frequency, and l is the length of the shear plate (0.875 m). Dividing Frcs by the plate area gives the amplitude of the spatially averaged shear stress %m on the 4. Estimates of Wave Friction Factors sand bed (Table 1) Shear Stress Measurements An additional force related to the pressure differential between the two ends of the sand deposit on the shear plate is The shear stress on the sediment bed measured by the shear caused by the interstitial flow through the porous sand bed. plate is an integrated shear stress measurement over the area This flow, in turn, transfers shear through the bed to the top of the shear plate rather than a point measurement of instantaneous shear stress. The shear stress distribution is assumed surface of the plate. An analysis using Darcy's law was performed to estimate the magnitude of the velocity through the uniform in the cross-tank direction perpendicular to wave sand bed and the resulting shearing force on the plate. The travel. The total shearing force measured on the plate is equal following form was used: to the spatially integrated shear stress %m plus the secondary forces discussed in section 2. Secondary forces are usually quantified and eliminated from the total force measured by the shear plate by monitoring the pressure in the gaps between the k dp q = v dx' (5) plate and the false bottom [Riedel and Kamphuis, 1973; Brown and Joubert, 1969]. For this study the relatively large spatial area of the shear plate produced shear stresses over the sand where k is the permeability parameter, estimated for sand to be of the order of 10-6 cm 2, v is the kinematic viscosity of water, and dp/dx is the horizontal dynamic pressure due to the

6 17,016 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED loo ROUGH TURBULENT TRANSITION / * b ee!! ß! ß o E+O2! 1.00E E E+05 Reb 1.00E+06 Figure 4. Inverse relative roughness versus bottom Reynolds number [after Jonsson, 1996]. wave. The velocity through the porous bed was estimated to be of the order of 10-3 cm s -. The Stokes equation for bottom shear stress (_Re O(10-3) for all tests), Re = --. (10) Examination of Figure 4 indicates that our data fall within in the rough turbulent region with the exception of one run (wave Tbm--p lgbmcos(-- ), (6) height = 29 cm, wave period = 1.25 s), which is in the transition region. Integrating over the area of the plate, S, gives the shear where Ubm is the maximum near-bottom wave orbital velocity, was used to estimate the resultant shear stress on the plate, which is of the order of l0-4 N m -2. The total force due to the interstitial flow was calculated to be of the order of 10 -s N, which is 5 orders of magnitude less than the shear stress on the sand bed at the fluid/sediment interface. The additional force on the plate due to interstitial flows was therefore neglected The Wave Friction Factor and Near-Bottom Velocity The quadratic drag law was used to convert measurements of bottom shear stress to values of the wave friction factor from Jonsson's [1966] equation rh,,, = 5/ 'P It. /12 /... (7) where f, is the wave friction factor to be determined experimentally. Similarly, the quadratic drag law can be rewritten for all phases of the wave cycle as I rh: 5f,,pu;, (8) where % is a point measurement of bottom shear stress under a wave and u is a point measurement of near-bottom velocity outside the bottom bounda layer. The wave friction factor is a function of the relative roughness k,m for rough turbulent flows where the bottom excursion amplitude A is defined as stress on the sediment bed measured by the shear plate at one moment in time: (11) Dividing the amplitude of the shear stress time history by the area of the shear plate and, similarly, dividing the corresponding spatially integrated near-bottom velocities by the area of the shear plate produce a maximum, spatially averaged bottom shear stress: ß,,,,,: 5L, p(u;,,,), (12) where hm 2 is the spatially averaged maximum of the squared near-bottom velocities on the plate at one instant in time. Employing (12) with %,, and u m from Table 1, we find f,, the wave friction factor (Table 1). To compare effectively our values for f, with previous results, which employed a theoretical u,,, [see Rosengaus, 1987; Mathisen, 1989], we did the same using the amplitude and frequency of the wave gauge time history and linear dispersion to calculate the near-bottom maximum horizontal velocity A = (9) Following Jonsson [1966], we present in Figure 4 a plot of the inverse relative roughness A /k,using measured values of A and k, versus the bottom Reynolds number: A(.o Ubm--- sinh (kh) ' (13) Since the near-bottom velocity varies spatially as a sinusoidal function for the linear wave theory, the maximum spatially

7 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED 17,017 ' distance from leading edõe of plate (cm) Figure 5. Detail of ripple profile. averaged value for the squared near-bottom velocities over the length of the shear plate at one instant in time can be written as 2U}mll z sin(k/)] ' (14) Employing (12), f,, is found using u,, the spatially averaged maximum of the squared near-bottom velocities calculated prising since our measurements of shear stress had a relatively large form drag component because of the ripple height. For with linear wave theory (Table 1). large values of k,,/a b a considerable part of the bottom shear 4.3. Apparent Roughness and Ripple Geometry stress is due to the pressure gradient over individual roughness elements. This stress leads the skin friction component (the The apparent roughness k, was specified as k, = 4 r/ as component of the stress responsible for moving sand grains) by suggested by Wikramanayake and Madsen [1994] after their 90 ø and therefore does not contribute to sediment motion. The extensive review and analysis of field and laboratory studies. A grain roughness Shields parameter 02.5, which is dependent on general example of the profile of the rippled sand bed and of skin friction only, was calculated to determine the intensity of the bottom roughness is illustrated in Figure 5. We calculated sediment transport at the bed. Ripple generation and maxithe ripple height r/as the average difference between the peak mum ripple steepness occur for values of 02.5 < 0.2. At values and the trough of the linearly detrended bed surface elevation. of 02.5 > 0.2, ripples tend to flatten as sand is removed from For wave-dominated environments, ripple length scales with ripple crests, and at 02.5 > 1.0, ripples flatten completely. wave excursion amplitude (X 0.75A b) and the ripple steep- Values for the grain roughness Shields parameter for these ness r//x are nearly constant between 0.15 and 0.17 for equiexperiments typically ranged from 0.04 to 0.17 with a single librium conditions [Wiberg and Harris, 1994]. The results premaximum value of 0.22, thus confirming that intense sediment sented here are in close agreement with Wiberg and Harris transport did not occur during the experiment. This was in fact [1994] for ripple steepnesses between 0.13 and 0.19 with the confirmed by measuring bed profiles before and after individexception of two runs with ripple steepnesses of 0.26 and ual runs via scuba. The maximum bed level change was mea- Excluding these two values, we calculate an average wave sured to be 4.3%, which was not assumed to be significant steepness of 0.17, indicating that equilibrium conditions were (Table 1). achieved (Table 2). Ripple steepness can be predicted using The apparent roughness was also calculated directly from the grain roughness Shields parameter 02.5 as [Nielsen, 1992] the measurements using the theoretical result found by Grant : (15) within 40% of the measured value (Table 2) where [Nielsen, 1992] O:.s = (s- 1)#d (16) I ( 2'5d50i 0'194 ] f2.s = exp Ab ] (17) The ripple length can be related to the bottom excursion amplitude by [Nielsen, 1992] = ø'34, (18) Ab where ß is the mobility number defined as (Am) 2 : (s- 1)gd' (19) Predicted values for X/At, using (18) were within a factor of 2 of measured values (Table 2). Although high values of shear stress and correspondingly high values of the Shields parameter 0 (average 0.9) were obtained for these tests, visual observations indicated that in- tense sediment motion did not occur. This result is not sur- and Madsen [1979]: 1 pkl,rnl bm r m = ker2 2 x + kei2 2 ' (20) where the mahmum shear velociff is defined as U,m = bm/p, U m is the maximum spatially averaged near-bottom velocity, ker and kei are Kelvin functions of the zeroeth-order, and o is a nondimensional distance from the bed defined as where Z o = k /30 is the vertical distance from the bed where the velocity goes to zero. Similarly, apparent roughnesses were calculated directly from the measurements by substituting (12) into the semiempirical equation Zo 4 + log 0 = log (22) found by Jonsson [1966]. Results are presented in Table 1 and Figure 6. The values of k calculated by Grant and Ma&en's '

8 17,018 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED Grant and Madsen [1979] - Jonsson [1966] ß data Figure 6. Measured values of wave friction factor versus Grant and Madsen's [1979] and Jonsson's [1966] equations. kn/ab [1979] theory differed from those measured directly by a factor varying from 4 for small values of relative roughness to 2 for larger values. Conversely, values of kn calculated by Jonsson's [1966] equation differed from measurements by a factor of 6 for small values of relative roughness to 2 for larger values. is often considered to be equal to fc [Nielsen, 1992], and we do that here. Values of the wave friction factor f,, obtained by using the shear plate were slightly less than values of the wave energy dissipation factor fe obtained in previous studies (Figure 7) Comparison of Wave Friction Factor With Theory 5. Conclusions and Previous Studies Values for the wave friction factor obtained from this study A laboratory study has been conducted involving the develwere compared with the semiempirical formulation of Jonsson opment and implementation of a shear plate to obtain direct [1966, equation (22)] and the theoretical result found by Grant measurements of bottom shear stress under surface gravity waves on a movable sand bed. The entire data set was obtained and Madsen [1979]: under steep vortex ripples, and form drag dominated over skin 0.08 friction for all the data presented. Values of the wave friction factor f,, were calculated using Jonsson's [1966] equation, fw- ker 2 + kei 2 x/ o' (23) Measured values of fw were in agreement with the formula proposed by Grant and Madsen [1979] for all measured values of relative roughness within a factor of 2 (Figure 6). Values for fw were also compared with measurements of the wave energy dissipation factor fe from previous studies using movable beds [Carstens et al., 1969; Lofquist, 1986; Rosengaus, 1987; Mathisen, 1989]. The method for obtaining f,. involves measuring the energy dissipation in the wave boundary layer and requires the simultaneous measurement of wave conditions and bed form geometry. The wave energy dissipation factor fe is related to the wave friction factor f,, by fe = fw COS &. (24) The phase lead of the bottom shear stress, &, over the freestream velocity is 45 ø for laminar flow and slightly less for turbulent flows [Jensen et al., 1989]. The average phase lead for the bottom shear stress measured by Lofquist [1986] was 40 ø, which would result in values offw that are greater than f by 25 %. Because of the uncertainty of the exact value of which relates the wave friction factor to the maximum bottom shear stress. Conclusions are summarized as follows. 1. The shear plate offers a relatively simple method for the direct measurement of bottom shear stress in movable bed environments under a wide range of flow regimes. If secondary forces can be reduced, this instrument will be useful for future studies in which the bottom shear stress is to be measured under complex flow regimes such as in regions where suspended sediment transport is occurring or in regions of complex bottom topography. 2. The experimental design is naturally subject to forces associated with the free-stream pressure gradient, which was confirmed by the 90 ø lead of the resistance time history over the surface elevation for all tests. This deficiency may overestimate values off,, if the total force cannot be accounted for by integrating the pressure force over the sediment bed. 3. Values of wave friction factor fw found in this study are in agreement with Grant and Madsen's [1979] theory by a factor of 2. In general, values of f,, were slightly less than the values predicted by theory.

9 RANKIN AND HIRES: MEASUREMENT OF SHEAR STRESS ON A MOVABLE BED 17,019 Grant and Madsen [1979] - donsson [1966] ß data + Mathisen [1989] x Rosengaus [1987] x I_ofquist [1986] = Carstens et al [1969] +ß + ß knlab Figure 7. Comparison among measured values of wave friction factor with those obtained from previous studies assuming fw fe' 4. Perhaps the true value of this study lies in the verifica- Kajiura, K., A model of the bottom boundary layer in water waves, Bull. Earthquake Res. Inst. Univ. Tokyo, 46, , tion of previous estimates of wave friction factors over a range Kamphuis, J. W., Friction factor under oscillatory waves, J. Waterways of relative roughness values using a completely different and Harbors Coastal Eng. Div. Am. Soc. Civ. Eng., WW2, , independent method. Li, M. Z., Direct skin friction measurements and stress partitioning over movable sand ripples, J. Geophys. Res., 99, , Lofquist, K. E. B., Drag on naturally rippled beds under oscillatory Acknowledgments. The authors thank Michael Bruno for assisting flows, Rep. MP-86-13, U.S. Army Corps of Eng., Coastal Eng. Res. in the experimental work and for useful discussions. We also thank Cent., Vicksburg, Miss., Daniel Cox and Nobu Kobayashi for valuable suggestions. We would Mathisen, P. P., Experimental study on the response of fine sediments also like to thank the reviewers for their useful and constructive com- to wave agitation and associated wave attenuation, M.S. thesis, ments. This work was sponsored by the National Science Foundation Mass. Inst. of Technol., Cambridge, under grant OCE and by the Mid-Atlantic Bight National Nielsen, P., Coastal Bottom Boundary Layers and Sediment Transport, Undersea Research Center contract Adv. Ser. Ocean Eng., vol. 4, World Sci., River Edge, N.J., Riedel, P. H., and J. W. Kamphuis, A shear plate for use in oscillatory flow, J. Hydraul. Res., 11, , References Rosengaus, M., Experimental study on wave generated bedforms and resulting wave attenuation, Ph.D. thesis, Mass. Inst. of Technol., Bagnold, R. A., Motion of waves in shallowater: Interaction between Cambridge, waves and sand bottoms, Proc. R. Soc. London, Ser. A, 187, 1-15, Simons, R. R., R. D. Maclver, and W. M. Saleh, Kinematics and shear stresses from combined waves and longshore currents in the UK Brown, K. C., and P. N. Joubert, The measurement of skin friction in coastal research facility, paper presented at 25th Coastal Engineerturbulent boundary layers with adverse pressure gradients, J. Fluid ing Conference, Am. Soc. of Civ. Eng., Orlando, Fla., Mech., 35, , Trowbridge, J., and O. S. Madsen, Turbulent wave boundary layers, 1, Carstens, M. R., F. M. Neilson, and H. D. Altinbilek, Bed forms Model formulation and first-order solutions, J. Geophys. Res., 89, generated in the laboratory under an oscillatory flow: Analytical and , experimental study, Tech. Memo. TM-28, U.S. Army Corps of Eng., Wiberg, P. L., and C. K. Harris, Ripple geometry in wave-dominated Coastal Eng. Res. Cent., Fort Belvoir, Va., environments, J. Geophys. Res., 99, , Grant, W. D., and O. S. Madsen, Combined wave and current inter- Wikramanayake, P. N., and O. S. Madsen, Calculation of movable bed action with a rough bottom, J. Geophys. Res., 84, , friction factors, Rep. DRP-94-5, U.S. Army Corps of Eng., Coastal Grant, W. D., and O. S. Madsen, Movable bed roughness in unsteady Eng. Res. Cent., Vicksburg, Miss., oscillatory flow, J. Geophys. Res., 87, , Jensen, B. L., B. M. Sumer, and J. Fredsoe, Turbulent oscillatory R. I. Hires and K. L. Rankin, Davidson Laboratory, Department of boundary layers at high Reynolds numbers, J. Fluid Mech., 206, Civil, Environmental and Ocean Engineering, Stevens Institute of , Technology, Hoboken, NJ (krankin@stevens-tech.edu) Jonsson, I. G., Wave boundary layers and friction factors, paper presented at 10th Coastal Engineering Conference, Am. Soc. of Civ. Eng., Tokyo, Jonsson, I. G., and N. A. Carlsen, Experimental and theoretical investigations in an oscillatory turbulent boundary layer, J. Hydraul. Res., (Received December 23, 1997; revised February 3, 2000; 14, 45-60, accepted March 9, 2000.)