Direct measurements of the bottom friction factor beneath surface gravity waves
Introduction
Waves propagating over a bed lose energy due to interaction with the bed. One of the most important dissipative mechanisms is associated with bed friction, which causes a thin boundary layer to develop above the ocean bed. The flow in the boundary layer is usually turbulent in field conditions. This turbulence causes turbulent shear stresses which are associated with energy dissipation. Putnam and Jonsson [1] considered monochromatic waves and expressed the rate of energy dissipation, E, as the work done by the wave orbital velocity against shear stress at the bottomwhere ub is the horizontal water particle velocity just outside the boundary layer, which can be derived from potential theory for waves and τ0 is the bottom shear stress, which depends on wave and bottom characteristics.
In the case of a wave spectrum, Hasselmann and Collins [2] showed that waves lose energy by bottom friction according to:where ubk is the orbital bottom velocity of the wave component with wave number k,〈…〉 denotes the ensemble mean and Sbf(k) is the time rate of energy density loss at wave number k. It should be noted that linear wave theory is assumed in the development of the spectral formulation in Eq. (2).
A review of previous research carried out on the bed oscillatory boundary layer and resulting energy dissipation shows that these studies can be categorized into two major groups, (i) those dealing with monochromatic waves and (ii) those dealing with spectral conditions.
The bed boundary layer under monochromatic waves has been studied both analytically and experimentally. Analytical studies in this field are mainly based on solving the linearized momentum equation for the flowwhere ρ is the density of water, z is the vertical coordinate measured upwards from the bed, τ is the shear stress (τ=τ0 at z=0) and u is the instantaneous velocity. Eq. (3) represents one equation in two unknowns, i.e. u and τ. Either a drag law model or an eddy viscosity model can be used to parameterize τ in Eq. (3) and hence obtain closure.
In the drag law model, the maximum bed shear stress, τ0max is expressed aswhere fw is a friction factor and Ub is the maximum near bed velocity.
In the eddy viscosity model the shear stress is parameterized in terms of the velocity gradient in the bottom boundary layer:where νt is the turbulent eddy-viscosity.
Nearly all the analytical solutions use an eddy viscosity model to close the set of equations. The main difference between these solutions is in the way in which they model the eddy viscosity. Some authors have used time-variant eddy viscosity, such as Trowbridge and Madsen [3], [4]. Others have used time invariant eddy viscosity, such as Kajiura [5], Grant and Madsen [6], Brevik [7], Myrhaug [8], You et al. [9] and Hsu and Ou [10]. Time invariant models differ in the way they prescribe the eddy viscosity distribution within the boundary layer. Also, some authors have used higher order turbulent closure schemes, (k−ε closure), for modeling the turbulent field within the wave boundary layer, such as Justesen [11] and Aydin and Shuto [12].
Experimental studies of the bottom boundary layer and bottom dissipation with monochromatic waves, fall into three groups depending on the laboratory devices used. They are: (i) oscillating trays, (ii) oscillating flow tunnels and (iii) wave flumes. Bagnold [13], Kalkanis [14], [59] and Sleath [15] used an oscillating tray, while [16], [17], [18], [19], [20], [21], [22], [60], [23], [58], [24], [25] carried out their tests in oscillating flow tunnels. [26], [27], [28], [29], [30], [34], [31] performed tests in wave flumes their tests were carried out in oscillating flow tunnels [16], [17], [18], [19], [20], [21], [22], [23], [58], [24], [25]. Tests were performed in wave flumes [26], [27], [28], [29], [30], [31]. The experimental works can also be categorized based on the way that the bed shear stress is obtained. A potentially reliable method, used by Kamphuis [18], involved the direct measurement of the shear force exerted on a segment of the bed. Another method widely used by other authors, e.g. [32], [33], is the measurement of the velocity field within the boundary layer and the determination of the bed shear stress by numerically solving the momentum Eq. (3). A brief examination of the literature shows that the bulk of the experimental studies have been performed in oscillating flow tunnels with only a small number in wave flumes. These flume measurements are rather old and generally lack velocity measurements close to the bed. Table A1 shows various expressions for the friction factor suggested by different authors for both mobile and immobile flat and rippled beds.
As can be seen in Table A1, (Appendix A), there are numerous empirical relationships for the estimation of bottom friction under waves. In Fig. 1 some of the best known expressions for the bottom friction factor coefficient are compared. It can be seen that all the curves show a similar trend and cover a narrow region in the friction factor-relative roughness plane, although the logarithmic scaling of the graph masks significant differences in magnitude (factor of 2).
The relationships shown in Fig. 1 were obtained using a variety of methods, both experimental and theoretical. The available experimental data, are shown in Fig. 2.
An examination of the data and the summary curves in Fig. 2 shows some interesting features. It can be seen that the values of friction factor shown by the open circles are much smaller in magnitude than other results. These data were obtained by Sleath [33] through measurement of the Reynolds stress within the boundary layer. The remainder of the data were obtained either through the momentum integral equation, direct measurement of the shear stress or the decay of wave height. This significant difference shows that, unlike in unidirectional flow, the Reynolds stress method does not give realistic values for bottom shear stress. Sleath [33] cites the finite value of the correlation between the stream-wise and vertical components of the bed velocity as a reason for this discrepancy.
The second feature of note is that, generally, friction factors obtained in a wave flume, e.g. [30], [31], [34], are larger than those obtained from water tunnels or oscillating trays. Brevik [7] attributed this difference to the inherent inaccuracy in the wave attenuation method for measuring the bottom friction factor, particularly when the slope of the energy line is small and the side wall effects are significant. Simons et al. [31] cited the uncertainty in velocity measurements as a reason for those data points which show larger friction factors than the empirical relationships. They also pointed out the low Reynolds Numbers of these points and argued that they might be in the transitional smooth-to-rough regime where the Nikuradse roughness may be different from that determined in tests in a current alone.
The third feature in Fig. 2 is that for small values of Ab/ks, the friction factor continues to increase. Malarkey and Davies [35] have found similar results with an inviscid model. These results contradict the assumptions made by Kajiura and Jonsson [5], [36] that the friction factor approaches a constant value as Ab/ks becomes small.
Studies of bottom dissipation under a wave spectrum can also be categorized into two main groups, (i) analytical and (ii) experimental. The analytical works can be further divided into two groups, (i) drag law models, investigated by Hasselmann and Collins [2] and Collins [37] and (ii) eddy viscosity models, investigated by Madsen et al. [38] and Weber [39].
Experimental studies under spectral conditions have mainly been conducted in the field. Bretschneider and Reid [40] made the first such field study, followed by Iwagaki and Kakinuma [41], Ifuku and Kakinuma [42], Lambrakos [43], Myrhaug et al. [44] and Young and Gorman [45]. Simons et al. [46] conducted their experiments in a wave basin. The field measurements either determined the decay of the wave spectrum or the velocity profile near the bed. The former have a fundamental problem because there are invariably other parameters active, in addition to bottom friction, adding uncertainty to the results. The accuracy of the results of the second group is also questionable because the velocity is usually measured only at a limited number of points at rather uncertain levels above the bed. In addition, there is usually no accurate information about the bed forms. The friction coefficients, mainly or partly based on field measurement, (Table 1), vary significantly. In addition, their accuracies have been questioned as a result of measurements made by other authors, such as Bouws and Komen [45], and Young and Gorman [47].
Table 1 summarizes relationships developed for spectral conditions. The friction factors in this table can be used in expression (6) proposed by Luo and Monbaliu [48] and Weber [39], [49] to obtain the bottom friction dissipation,where k is the wave number of the spectral component, h is the water depth, ϕ is the direction of propagation of the spectral component with frequency f,F(f,ϕ) is the two dimensional frequency spectrum, Sbf is the time rate of energy density loss at wave number k and cf is a dissipation coefficient, the value of which is given in Table 1 for different methods.
The terms in Table 1 require explanation:
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cJ is the friction factor proposed in the JONSWAP study.
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cDHC is the friction factor obtained by Hasselmann and Collins [2], using a drag law model. In this expression i and j denote the orthogonal components of the bottom velocity. The value of cf in this expression was obtained from the decay of a wave spectrum in the Gulf of Mexico.
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cDC is the friction factor proposed by Collins [37] which is an approximation of the Hasselmann and Collins [2] expression. The main difference is that in Ref. [37] the bracketed term in cDHC is approximated by the root-mean-square bed velocity of the spectrum.
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cDM is the friction factor proposed by Madsen et al. [38], obtained from an eddy viscosity model. In this model a spectrum of waves is approximated by an equivalent monochromatic wave. Hence, the value of fw can be obtained from existing friction factor diagrams for monochromatic waves. The velocity term in this expression is the root-mean-square bed velocity obtained from the spectrum, and Sub is the near-bottom orbital velocity spectrum.
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cE is the friction factor obtained by Weber [49], using a one layer eddy viscosity model. The function Tk, as stated in her paper, “…expresses our ignorance of the stress and the velocity profile in the boundary layer”. Therefore it is an unknown function. However, uf and Tk can be obtained by an iterative method, detailed in her paper, provided the roughness height is given. In this expression kN represents the roughness height which was proposed to be equal to 4 cm.
A review of the literature reveals that analytical methods are far from being applicable in engineering problems. Also there are insufficient experimental data under spectral conditions to provide certain and reliable empirical expressions. In addition, the limited experimental data under monochromatic waves, which form the basis for the vast majority of the existing empirical relations, e.g. [18], [36], have been obtained from oscillating flow tunnels rather than under more realistic wave conditions. Under real waves the horizontal flow at the bed is nonuniform and can not be exactly simulated in a water tunnel.
In addition, Asano and Iwagaki [50] state that the wave nonlinearity produces significant effects on near bottom water particle velocity and the resultant bed shear stress. They showed that the friction factors estimated by nonlinear calculations can possibly be several times as large as those proposed by existing expressions. Real waves, including those generated in wave flumes are usually non-linear to some extent, while oscillating water tunnels and oscillating trays typically generate a purely monochromatic oscillatory boundary layer. Further, a review of the literature shows that there are little experimental data on bed friction under a spectrum of waves.
In view of the above, two questions are raised. (i) How realistically can water tunnels and oscillating trays model real waves in regard to bed friction? (ii) How can the existing body of knowledge for monochromatic waves be extended to a spectrum of waves? To generate more experimental data to compare the friction factors obtained in wave flumes with those in oscillatory water tunnels, to examine the effects that nonlinearity has on the friction factor, and to find a way to apply the considerable body of knowledge for monochromatic wave conditions to a spectrum of waves, a series of tests in a wave flume have been performed, and the results are presented here.
The arrangement of the paper is as follows. In Section 2 the experimental design and instrumentation is considered, together with the accuracy of each of the systems. This is followed in Section 3 by the determination of the bed roughness for each of the bed conditions examined. Section 4 provides the results for hydraulically smooth and rough beds under the action of monochromatic and spectral wave conditions. Finally, conclusions are presented in Section 5.
Section snippets
Instrumentation and experimental design
To assess the wave energy dissipation due to bottom friction, direct observations have been utilized in this study. The basic parameters measured were:
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Total frictional fluid force on the bottom, using a shear plate.
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Velocity field within the bottom boundary layer, using Laser Doppler Anemometry (LDA) and Acoustic Doppler Velocimetery (ADV) techniques.
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Water surface elevation, using three capacitance wave gauges.
Fig. 3, Fig. 4 illustrate the experimental apparatus used.
As can be seen in these
Bed roughness conditions
Three different bed conditions were used to investigate the bottom friction dissipation under waves in the wave flume. They were: (i) sand grains, (ii) spherical aggregate with diameters between 4 and 9.5 mm and (iii) rib roughness elements with a triangular cross-section. The rib elements were placed 6.5 cm apart in the flow direction (Fig. 7). As described earlier, the area surrounding the shear plate was also covered in these respective roughness elements. Identical bed conditions were also
Bottom friction measurements
The objective of the measurements presented below is the determination of the bottom friction factor under a progressing wave train. Two different measurement techniques were used. For the sand grain material, spurious forces exerted on the edges of the shear plate (see below) made this an unreliable measurement technique. For this case, however, a well formed boundary layer develops and the friction factor can be obtained from measurements of the velocity profile and solution of the momentum
Conclusions
The literature on bottom friction dissipation under oscillatory waves is very large. The vast majority of this literature has, however, been for monochromatic waves and has been obtained in laboratory studies using water tunnels or oscillatory trays. When applying these studies to field applications, two significant questions are raised:
- 1.
How can friction factor values obtained for monochromatic conditions be applied to more typical spectral conditions?
- 2.
Are measurements made in oscillatory water
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