Flow deceleration as a method of determining drag coefficient over ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, C03001, doi:10.1029/2001JC001262, 2004

Flow deceleration as a method of determining drag coefficient over roughened flat beds C. E. L. Thompson and C. L. Amos School of Ocean and Earth Science, University of Southampton, Southampton Oceanography Centre, Southampton, UK

M. Lecouturier L’Institut des Sciences de la Mer de Rimouski (ISMER), Universite´ du Que´bec a` Rimouski, Rimouski, Que´bec, Canada

T. E. R. Jones Department of Mathematics, University of Plymouth, Plymouth, UK Received 11 December 2001; revised 22 December 2003; accepted 16 January 2004; published 2 March 2004.

[1] The drag coefficient (CD) is a fundamental parameter in the determination of shear stress or the drag force between a fixed object and fluid moving over it. In natural settings, conventional methods of defining it are largely impractical and so either smooth-bed CD or constant values are used irrespective of bed roughness or flow strength. This paper deals with the determination of CD over naturally roughened beds. The work was carried out in two annular flumes of known, constant water mass. In an otherwise balanced system, flow deceleration is a manifestation of the total drag force exerted at the rigid boundaries (Newton’s second law). The inversion of this relationship is used to yield the bed drag coefficients. The advantages of this method include its accurate use over rough and irregular beds, as shown by experiments over patchy and homogeneous gravel beds and over a wide range of Reynolds numbers. The value of CD was found to converge to the constant value of 3
103 determined by Sternberg [1968] at intermediate velocities, and a reduction in the drag coefficient occurred at high velocities. Results showed that patch spacing did not influence the shear stress value in the case of one-grainthick gravel patches. A modification of the equipment for field use may give advantages where traditional methods fail due to difficulties in obtaining accurate velocity profile INDEX TERMS: 4294 Oceanography: General: Instruments and techniques; 4558 measurements. Oceanography: Physical: Sediment transport; 4211 Oceanography: General: Benthic boundary layers; KEYWORDS: drag coefficient Citation: Thompson, C. E. L., C. L. Amos, M. Lecouturier, and T. E. R. Jones (2004), Flow deceleration as a method of determining drag coefficient over roughened flat beds, J. Geophys. Res., 109, C03001, doi:10.1029/2001JC001262.

1. Introduction [2] The estimation of fluid forces on coastal and riverine fixed structures is determined by the interaction between the flow and the stationary bed or body over which the flow travels. This takes place across the boundary layer, wherein the velocity gradient is a manifestation of a viscous or inertial drag force being applied to the rigid body by the moving fluid. Defining the drag force of a fluid on a natural bed is fundamental in defining flow and discharge in channels and pipes, or defining the benthic boundary layer of coastal water masses. The bed shear stress (t0) applied by the fluid is of particular importance in bed erosion, or, sediment transport prediction [Allen, 1977]. [3] The drag force (FD) exerted by a moving fluid on a horizontal bed is usually defined from a knowledge of the mean ‘‘free-flow’’ fluid speed (u ), fluid density (r), the area Copyright 2004 by the American Geophysical Union. 0148-0227/04/2001JC001262

u2. of interaction (A), and a drag coefficient (CD): FD = ACD r CD for a given height is known to depend on bed roughness, skin friction, and flow intensity [Shames, 1962]. It is also dependent on turbidity [Gust, 1976], sand transport rates, and wave-current interactions [Grant and Madsen, 1979; Green and McCave, 1995]. Although it is well defined for simple objects or smooth beds [Allen, 1977], it has proved problematic to determine over natural rough beds due to the way CD and natural bed roughness are determined. [4] Shear stress in the case of natural rivers comprises two parts: a shear stress t’ due to the resistance of particles forming the bed (skin friction) and an additional shear stress t’’ due to irregularities of the channel bed and banks, i.e., the bed forms (form drag) [Petit, 1989]. The largest constituent of drag comes from the bed forms, and this can be seen as ripples on sand or bioroughness on mud. The relationship between the shear stress and drag coefficient u2, where it is normal to use is generally defined as t0 = rCD a constant value for CD. There is still some controversy about the importance of energy dissipation due to the

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roughness of the grains of sediment on an irregular bed, and its ability to drive sediment transport. It is probably reasonable to conclude that the roughness of grains of sediments is not the major contributor to energy dissipation when the steepness of the ripples is greater than about 0.16, but that it becomes progressively more important as ripple steepness decreases [Sleath, 1985]. [5] The local boundary shear stresses in the field are usually estimated by some indirect method that relies on theoretical or empirical arguments. The most commonly used are described below. 1.1. Pressure Gradient Method [6] This uses the assumption that t0 can be estimated from the local horizontal component of the pressure gradient. The difference between the pressure recorded by a Preston tube and the undisturbed static pressure is used to determine the friction velocity, which in turn can be used to calculate shear stress [Patel, 1965]. However, it requires that convective accelerations are small; in rivers with complex topography, this is generally not the case [Whiting and Dietrich, 1990]. It also requires accounting for resistance due to bed forms such as bars, dunes, and ripples [Einstein and Barbarossa, 1952]. As it is measured in the water column, it must be assumed that measurements are taken in the boundary layer. This difference in pressure also results in a slope of the water surface, which can be measured and used to determine a value for the shear stress. 1.2. Law of the Wall [7] In this method, several velocity measurements are used to define the gradient of velocity above the bed, and the local boundary shear stress, t0, is calculated from the law of the wall, t0 = r(uzk)2[ln(z/z0)]2, where uz is the velocity at height z above the boundary, k is von Karman’s constant, and z0 is the height above the bed where the velocity is projected to go to zero. Considerable error arises with application of this procedure because boundary shear stress is extremely sensitive to the gradient of velocity and in the estimation of the roughness length (zo). Both accurate velocity measurements and exact elevations above the presumed bed level are difficult to obtain, as are simultaneous measurements of velocity within a profile. Research has shown that repeated velocity profile measurements with long sampling intervals (>10 min) for individual points yield highly variable estimates of boundary shear stress [Whiting and Dietrich, 1990]. 1.3. Inertial Dissipation Method [8] This method makes use of the fact that in the inertial subrange, the flux of energy from low to high wave numbers equals the dissipation rate. In order to estimate the shear stress from this method, two assumptions must be made. These are that (1) there is a balance between the production and dissipation of turbulent energy; and (2) measurements are made within the constant stress layer, in which the local stress equals the bottom stress [Huntley and Hazen, 1988] [Stapleton and Huntley, 1995]. 1.4. Turbulent Kinetic Energy (TKE) Method [9] Making an extension of the principle Reynolds decomposition, an instantaneous velocity is a function of

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the fluctuations due to waves and turbulence [Soulsby, 1983b]. The fluctuations due to turbulence (denoted by the subscript t) can be calculated from the spectrum of a velocity time series [Hennay et al., 1994]. This method can be applied to all three components of flow and the turbulent kinetic energy density, E, calculated from the definition E = 1/2r(u2t þ v2t þ w2t ) [Soulsby, 1983b; Soulsby and Humphrey, 1989]. From this it is possible to calculate the bed shear stress according to Soulsby [1983b], t = 0.19E. 1.5. Reynolds Stress [10] The Reynolds stress method uses turbulent momentum flux to estimate stress at a particular height [Soulsby, 1983b] and is the most direct method of calculating the bed shear stress. However, this method is particularly sensitive to sensor misalignment and can give errors of as much as 156% per degree of misalignment in wave-dominated conditions [Soulsby and Humphrey, 1989]. 1.6. Quadratic Stress Law [11] The Quadratic Stress Law defines the shear stress as t0 ¼ CD r u2 :

ð1Þ

CD for irregular, natural surfaces is difficult to determine. This is because it is conventionally measured by (1) hydrodynamic head loss (only practical in flumes) [Patel, 1965]; (2) shear plates placed on a rigid surface (only practical on flumes with rigid beds); (3) velocity profiling within the wall or bed boundary layer [Sternberg, 1968; Yalin, 1972]; (4) hot film probes relating heat dissipation to momentum flux (shear stress) [Graham et al., 1992]; or (5) measures of turbulence and associated total kinetic energy dissipation or Reynolds stresses near the boundary [Soulsby, 1983a]. These methods are all labor intensive, dependant on strict instrument calibration and (in the case of items 2 to 5) provide local, not total, drag estimates. [12] Drag is usually expressed non-dimensionally with the aid of a friction factor. There are many different coefficients; however, their use requires knowledge of bed roughness as well as mean velocity at a known height. [13] The above relationships hold true for any type of fluid flow relative to a rigid or flexible object. They apply to coastal, lacustrine, or fluvial water masses moving over natural beds, particles settling in a still fluid, movement of flows around complex structures, and forces applied on coastal or fluvial structures.

2. Flow Deceleration as a Manifestation of Drag Force [14] In the enclosed environment of a laboratory flume, Newton’s second law governs the deceleration of an inviscid fluid of constant and known mass. That is, the drag force due to the walls and bed summed over a given time manifests itself as the rate of change in the linear momentum vector quantity (p = mdu dt ) of the flow, where m is the fluid mass, p is the pressure change, and du/dt is the rate of change of velocity with time. For a constant mass, the transfer of momentum from the fluid to the bed through the boundary layer is manifested by flow deceleration which is

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balanced by changes in the vertical velocity gradient as defined by eddy viscosity (e). Thus FD ¼ A

  @ e @u @y @y

;

ð2Þ

where y is the height above a rigid boundary, and A is the wetted area. The loss of momentum (flux) per unit area per unit time is a measure of the bed shear stress (t0). Momentum transfer takes place across a boundary layer characterized by a gradient in velocity from the free-stream value to zero (for a non-slip condition) at the bed. Thus CD ¼

  m du 1 d50 ¼ f ; Re ; h A dt ru2

ð3Þ

where Re is the grain Reynolds number (u* rd50/m), d50 is the median grain size (a roughness length is used herein, determined by Fung [1997] as 0.00045 m for acrylic), h is the water depth, and m is the absolute viscosity. [15] CD has been shown by Nikuradse [1950] to depend solely on the Reynolds number, and relative roughness. The drag coefficient can therefore be calculated purely from the change in velocity with time as a fluid decelerates through friction effects with the walls, which simplifies matters greatly as no knowledge is required of the velocity profile. [16] Experiments were carried out in two annular laboratory flumes of different scales, the Mini Flume and the Lab Carousel. It was assumed that the overall rate of deceleration was constant throughout the flow; this assumption appears justified on the observation that flow ceased synoptically throughout the flume based on measurements taken across the base of the flume during deceleration.

3. Mini Flume [17] The Mini Flume is a laboratory-based, annular flume constructed of two acrylic tubes placed one inside the other to form a channel 0.045 m wide (Figure 1). For the purpose of these experiments, the flume was sealed onto a smooth acrylic base. This configuration is the smooth bed condition and allows for estimation of wall effects. [18] Four paddles arranged equidistantly on the lid of the flume generate a current; their speed is controlled by a Compumotor 1 digital stepping motor, powered by a 1000-watt Xantrex1 power supply. The motor is, in turn, controlled by a PC through a RS232 serial link. An ASCII script file regulates the motor settings and paddle speeds. Current speeds in both the tangential and radial directions are measured using a Marsh McBirney1 electromagnetic current meter (EMCM) at a height of 0.085 m above the smooth bed at center channel. [19] Data are logged from the EMCM to a Campbell Scientific CR10 data logger at a rate of 4 Hz. The time between the two PCs is synchronized at the start of each experiment and was within 1 s. It is known that the EMCM would impart drag into the flow; however this drag is a constant. Its effect was evaluated in the Lab Carousel (described below) where a non-intrusive method of measuring flow was available. [20] The flume was filled with saline water (9 psu), at room temperature (18C). After initial investigations, the

Figure 1. Mini Flume dimensions and instrument positions.

water depth in the channel was reduced to 0.16 m in order to reduce the wall effects of the flume, and the existing paddles were replaced with elongated ones. This resulted in the wall area being reduced to 85% (0.255 m2) of the total area, rather than the typical 90% (0.03511 m2) used in the experiments of Fung [1997] and O’Brien [1998]. The bed area was 0.0407 m2. The bed was lined with gravel of varying sizes to simulate natural roughnesses. Thus changes in bottom roughness took place over 20% of the flume area. The drag component due to the smooth walls was evaluated following Vanoni and Brookes [1957] and subtracted from the total drag to determine the bed effects. [21] The main disadvantage of an annular flume is the curvature of flow. The shear stress distribution across the bed is not uniform in the radial direction, and a secondary flow is generated in the flume cross section [Maa, 1990; Yang et al., 2000; Graham et al., 1992].

4. Lab Carousel [22] The Lab Carousel (Figure 2) is a 2-m-diameter acrylic annular flume, designed as the laboratory version of the Sea Carousel [Amos et al., 1997, 1992]. It consists of a channel 0.15 m wide with a maximum water depth of 0.40 m. The bed and walls were smooth with no protuberances. A rotating lid fitted with eight equidistant paddles generates a current, and the speed of the lid is controlled by an E-track1 AC inverter motor controller. The paddles have been designed to minimize the secondary circulations that usually occur in annular flumes. The motor at the center of rotation of the lid sat upon a hydraulic jack. The lid could

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Figure 2. Lab Carousel instrumental setup and dimensions. thus be lifted out of the water while the paddles were still moving. The flume has been set up with a 1-dimensional class IIIb Helium-Neon 10 mW LDV (Laser Doppler Velocimeter) that measures the tangential component of flow at a height of 0.15 m. These data were logged at 20 Hz to a PC. The laser is a non-intrusive method of measuring the flow velocity at the center of the channel. The flume was filled with fresh water at room temperature (18C), to a water depth of 0.40 m. This gave a wall to bed area ratio of 8:2, which was equivalent to that of the Mini Flume.

5. Calibration [23] The Mini Flume EMCM was calibrated by adding neutrally buoyant Goodyear pliolite1 particles to the water column. The speed of the particles was determined from video recordings of the flow made at five levels of constant flow speed. The flow speeds were measured continuously with the EMCM for 5 min. Zero offsets in the sensor were electronic in origin, not an error of measurement. This calibration differs from earlier ones due to sensor instability over long timescales. Offsets were checked at the beginning and end of each experiment to determine short-term instabilities. [24] The pliolite was prepared by sieving to a size range of 710 mm to 1.4 mm and soaking overnight in a soap solution to ensure that no air was trapped on the uneven surface of the particles. The white particles were filmed against a black gridded background at the height of the EMCM, and their velocity was determined from the time taken for a particle to travel a horizontal distance of 0.05 m across the field of view. The video was focused on those particles in the middle of the channel, so as to eliminate wall effects. Figure 3 shows the calibration of the Mini Flume EMCM. The relationship between the raw EMCM data (in mV) and the actual velocity was of the form u ¼ 0:0011V þ 0:177

r2 ¼ 0:99:

of the hot film probes were recorded for a period of 3 min at a rate of 0.1 Hz (internally averaged from 1 Hz) at mean current speeds of 0.15, 0.28, 0.38, 0.43, 0.57, 0.65, 0.72, 0.91, 1.09, and 1.27 m s1. These values were then converted into shear stresses using the calibration equations: t0 (inner) = ((V2  2.9644)/1.9044)3Pa; t0 (middle) = ((V2  3.4653)/2.2381)3Pa; t0 (outer) = ((V2  3.1363)/1.9281)3Pa; where V is the probe output voltage (mVolts), and the subscripts refer to their relative locations within the flume channel. Figure 4 shows this calibration of flow velocity against stress in both the tangential and radial directions as measured by the probes. The radial stresses were generally found to be less than 1% of the tangential stresses, and in all cases are less than 5%. Figure 5a shows the cessation of flow in the Lab Carousel, confirming that flow ceases synoptically across the base of the flume (see section 1). Figure 5b represents the relationship between the radial and tangential components of flow for different lid speeds and positions within the flume, and Figure 5c demonstrates that the tangential velocity is a good proxy for the total resultant velocity in this flume in terms of the deceleration rates du/dt. This is confirmed by the low resultant bed shear stresses demonstrated in Figure 4, which indicate that the magnitude of the bed shear stress vector in the rotational

ð4Þ

[25] Flow within the Lab Carousel was also measured directly by tracking neutrally buoyant particles within the water column. Additionally, three hot film probes were flush-mounted through the base of the flume at positions of 0.135 m and 0.015 m from the inner wall and at the flume channel center line. The probes were calibrated using the methodology of Graham et al. [1992]. The output voltages

Figure 3. Calibration of the Mini Flume. The velocity values were found from tracking the transport rate of neutrally buoyant particles in the flow.

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homogeneous gravel beds of differing median grain sizes (0.011, 0.016, and 0.022 m) were tested. Last, several patchy beds were investigated. Initially, gravel patches 0.10 m wide, constructed of 0.0016-m-diameter gravel were tested, then the distance between these patches was varied (0.05, 0.10, 0.50, 1.00, 1.50 m). This was done in order to investigate both the effect of grain size and of gap size on the mean bed shear stress. [29] The raw data was put through a first-order low-pass Butterworth filter with a cutoff frequency of half the Nyquist frequency to reduce the scatter encountered at low velocities. [30] The drag coefficient due to the flume and bed roughness was calculated using equation (3). The total drag force on the decelerating fluid was calculated from Figure 4. Calibration of bed shear stress in Lab Carousel using three hot film probes showing tangential and radial bed shear stresses. flow is not significantly larger than it would be in a purely translational flow for the same tangential velocity.

FDðtotalÞ ¼ CD r u2 AðtotalÞ ;

where A(total) is the total wetted area of the flume. The total drag force may be decomposed into a wall component and a bed component, FD(total) = FD(wall) + FD(bed), as suggested by Einstein [1942]. In the smooth bed case where the walls and bed are constructed of the same acrylic substance, CD may be assumed constant across all the surfaces; therefore

6. Experimental Procedure and Analysis [26] The Mini Flume was filled with saline water, to ensure a stable calibration of the EMCM over the duration of the experiments. Data from the EMCM were recorded under still water conditions for 5 min to measure sensor electronic offsets. The flow was then accelerated to 0.23 m s1 and maintained at this speed for 5 min to ensure complete adjustment of the water column and boundary layer. At this time, flow speed was assumed to be constant and the addition of momentum by paddle rotation was assumed to be balanced by the total fluid drag over the wetted area. Then the lid and paddles were lifted from the flow with the result that flow deceleration took place in proportion to the total drag force within the flume. The procedure was replicated three times and the mean value was used to determine wall effects. Thereafter, the bed was artificially roughened to provide five distinct bed types using well-sorted aquarium and natural gravel evenly distributed over the bed. The median diameters (d50) of the gravels were 0.0056, 0.0067, 0.011, 0.016, and 0.022 m. [27] All the beds were one grain thick, and were laid flat (no bed forms). Gravels were chosen in order to provide a wide range of natural roughnesses that could be characterized by a single diameter. No bed forms were generated by the flow, nor was the sediment transported thus eliminating a secondary (more complex) source of drag [Li and Gust, 2000; Nielsen, 1992]. [28] The Lab Carousel was filled with fresh water to 0.40 m and allowed to adjust to room temperature (18C). Data recording began under still water conditions for 5 min, and then the flow speed was increased to 0.38 m s1. The flow was maintained at a constant velocity for 5 min to allow the flow to equilibrate. At this point, the lid was lifted from the water using the hydraulic jack, and the deceleration of the fluid was recorded by the LDV. Several types of bed were investigated in the Lab Carousel. The first was a smooth bed in order to evaluate wall effects. Then, three

ð5Þ

FDðwallÞ ¼ CD r u2 AðwallÞ :

ð6Þ

For the rough bed cases, the value of FD(wall) was calculated using the method described by Vanoni and Brooks [1957] for sidewall correction. [31] The resulting value of FD(wall) was then subtracted from FD(total) to determine FD(bed) for each bed grain size. Once FD(bed) was known, CD(bed) for each bed type was determined using CDðbed;uÞ ¼

FDðbed;uÞ : ru2 AðbedÞ

ð7Þ

The effect of the presence of the EMCM on the total drag in the Mini Flume was investigated in the Lab Carousel using a scaled equivalent, and was calculated to be 40% of the

Figure 5a. Deceleration of fluid across the base of an annular flume.

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Figure 5b. Relationship between the tangential and radial velocities of flow in the Lab Carousel (based on work by Fung [1997]), at three different lid velocities and three different positions within the flume. Included is a line representing 10% of the tangential component of the flow, which coincides with the radial component.

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Figure 5c. A comparison of the decelerations of the tangential and resultant velocities. total drag. This amount was therefore removed from the total drag during the calculations before the drag coefficient was calculated. The drag coefficient was then standardized to a height of 0.10 m above the bed in order to compare results.

7. Results [32] Figure 6 shows a typical deceleration time series for the smooth bed conditions in the Mini Flume and the Lab Carousel. It takes approximately 15 min for the flow in the water column of the Lab Carousel to decelerate from 0.38 m s1 to still water, and 3 min for the Mini Flume to flow decelerate from 0.23 m s1 to zero. [33] As CD varies with velocity (u), the magnitude of the wall drag force will also vary with u. The relationship of the form FDðwallÞ ¼ 64u þ 43u2  0:0006; ðNÞ R2 ¼ 0:98

ð8aÞ

was determined from equation (6) for the Mini Flume, and FD ðwallÞ ¼ 2:5u þ 1002u2 þ 0:3; ðNÞ R2 ¼ 0:89

ð8bÞ

Figure 6. An example of decelerating flow in the Mini Flume and Lab Carousel for smooth bed conditions.

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Figure 7. Relationships between FD(wall) and flow speed for the Mini Flume and the Lab Carousel.

for the Lab Carousel (Figure 7). This relationship differs for each of the rough beds, according to the roughness, although the variation with velocity remains. By subtracting FD(wall) from FD(total), FD(bed) is derived. Thus CD(bed) is evaluated from equation (7) for a given flow velocity. [34] Figure 8 shows the bed drag coefficients for both the Mini Flume and the Lab Carousel. For the total and bed drag coefficients, all the beds produced the same shaped relationships between velocity and drag coefficient. The general pattern is for a steady decrease in CD with increasing u to values as low as 1
104, then a leveling out or slight increase in CD at higher u, and finally a sudden drop in value. This sudden drop according to Hughes and Brighton [1967] is either the onset of motion of some of the grains at high velocities or a shift in the separation point of the boundary layer enveloping the gravel bed. The former was confirmed by observation to occur for the smaller grain sizes (0.0056 and 0.0067 m) in the Mini Flume. This sudden drop is not evident in the Lab Carousel, although there is a degree of leveling off. [35] In both the Mini Flume and Lab Carousel, the CD values for the smooth beds are approximately 2 orders of magnitude lower than those of the gravel beds. The distinction between the different gravel beds is less well defined, however. In the case of the Mini Flume, there appears to be a well-defined increase in CD with increasing grain size between the 0.011, 0.016, and 0.022 m beds, although the 0.0056- and 0.0067-m beds do not follow the same pattern. [36] In the Lab Carousel, there is also an increase in CD with gravel size for the homogeneous beds, although it is small. For the patchy beds, where regular 0.10-m gravel patches are interspersed with increasing amounts of exposed smooth beds, there seems to be very little distinction between different spacings. The mean drag coefficient derived by Sternberg [1968] from field measurements in Puget Sound is also plotted in Figure 8 (3
103). In the Mini Flume, CD appears to converge to this value. In the Lab Carousel the roughened beds also converge to this value at high velocities. The smooth bed case tends toward a lower value of CD, which appears to support the summary of Soulsby [1983b].

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Figure 8. Bed drag coefficient for (a) the Mini Flume, (b) the homogeneous beds in the Lab Carousel, and (c) the patchy beds in the Lab Carousel. Every second point is plotted for clarity.

[37] Figure 9 shows comparisons of our CD data with the previously published results of Hughes and Brighton [1967] and Nikuradse [1950]. As can be seen in the Mini Flume work (Figure 9a), the smooth bed data fits very well with the Fanning smooth bed data up until Re = 1000, and also with Hughes and Brighton’s [1967] work on flat plates thereafter. The cross-over point coincides with the changeover between transitional and rough turbulent flow for the

smooth bed. The Lab Carousel data for a smooth bed also overlap the Fanning and Nikuradse smooth bed work (Figure 9b), although our relationship is steeper. [38] The values of the bed shear stress within the two flumes have been calculated for the range of flows using equation (1). The results of this are shown in Figure 10. The general trend in both the Mini Flume and Lab Carousel data is for an increase in shear stress with increasing velocity. In

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u > 0.3 m s1. Also shown are the stresses as determined using the quadratic stress law and Sternberg’s constant CD = 3
103 [Sternberg, 1968], and from the law of the wall in the Sea Carousel, a field-based equivalent of the Lab Carousel [Amos et al., 1992]. The values found by the deceleration method are lower than those found using the quadratic stress law with a constant CD, and the Law of the Wall in the Sea Carousel. This shows that the use of a constant CD may not be the best choice. Sternberg [1968] showed that the friction factors of tidal channel beds with differing bed roughnesses exhibited a region, corresponding to intermediate and lower values of Re, that is characterized by relatively large scatter in CD; this dispersion decreases with increasing values of u and therefore Re. This pattern, and the consequent change in graph shape can be used as a basis for separating the CD-Re diagrams into regions of ‘‘hydrodynamically transitional’’ and ‘‘hydrodynamically rough’’ flows. Figure 12 shows the estimated boundaries for the 0.0056-m gravel case in the Mini Flume. This critical Re would seem to be approximately 23,000. It is slightly different for each of the different roughnesses (being, for example, 25,100 for 0.011-m gravel). This corresponds well with the work of Sternberg [1968], who found that the boundary between transitional and rough flows occurred at different values of Re for each type of roughness. These boundaries are not as clear in the results of the Lab Carousel where the data seems to follow a much steadier gradient.

8. Discussion

Figure 9. Comparisons of experimental data with other published work for (a) the Mini Flume and (b) the Lab Carousel. Notation; 1, flat plates [Hughes and Brighton, 1967]; 2, Fanning smooth bed [Hughes and Brighton, 1967]; 3, Nikuradse (rough pipes). The dotted line represents Sternberg’s mean CD for turbulent flows (3
103). Every second point is plotted for clarity. the Mini Flume, there seems to be an increase in shear stress with increasing grain size from 0.011 to 0.016 to 0.022 m, although scatter in the results makes this increase difficult to define. The results for the 0.0056- and 0.0067-m gravels are much closer to those of the smooth bed, implying that the roughness elements are not large enough to significantly increase the shear stress of the bed. In the Lab Carousel, there is a similar increase in shear stress with the addition of roughness elements, although the increase is less well defined, especially at low velocities. At higher u, the increase is approximately 1 order of magnitude. For the homogeneous gravel beds, there does appear to be an increase in shear stress with increasing grain size, but in the patchy bed experiments, there seems to be no defined increase or decrease in shear stress with change in gap size. [39] Figure 11 shows a comparison of the shear stress values determined by several different methods in the Lab Carousel. The stresses determined using the hot film probes fit well with the values determined by flow deceleration (by inputting the values of CD into the quadratic stress law) over a smooth bed at low velocities, but are higher at

[40] The accurate determination of CD(bed) is important for a large number of sediment and flow calculations in the coastal zone, lakes, and rivers. The results presented here indicate that the use of the described method of flow deceleration may be very useful in determining the drag coefficients of various bed types in the flume environment. Developments in field deployable annular flumes [Amos et al., 1992] allow this method to be transferred to field applications. The correlation of the smooth bed cases in both flumes with Fanning, and Hughes and Brighton’s [1967] earlier work is very promising. [41] The lack of distinction between the values of CD for the different patch configurations implies that it is the grain size (and therefore skin friction) dominating the drag coefficient, rather than the patchiness, and that instabilities spawned from the gravel dominate the entire flow volume. This may also be due to the low elevation difference between the gaps and patches (one grain diameter), and their step-like nature (rather than ripple shaped). These results agree with those of Sternberg [1968], who concluded that the drag coefficient is not very sensitive to the bed configuration. [42] These graphs confirm that the common practice of assuming a constant drag coefficient for a particular bed is inaccurate: CD varies with typical tidal velocities over 3 orders of magnitude and is not constant in the turbulent rough flow as was often assumed. In light of these investigations, it seems that the grain roughness of the bed makes little difference to the bed shear stress. And in the case of patchy beds, the size of the patch is unimportant in determining the mean bed shear stress. [43] One of the benefits of flow deceleration is that one 15-min deceleration time series can provide a continuous

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Figure 10. Bed shear stresses calculated for (a) the Mini Flume, (b) the Lab Carousel for homogeneous beds, and (c) the Lab Carousel for patchy beds. Every second point is plotted for clarity. locus of data on CD versus u, whereas traditional methods required separate experiments for each value of u: much more labor intensive. This method allows the drag to be determined from simple average velocities in the outer region, eliminating the need to profile within the logarithmic part of the boundary layer (often difficult to define). [44] Owing to the small size of the Mini Flume, it was not possible to use a non-intrusive method of measuring the velocity in the flume. The EMCM imparted some drag on the flow, and so must be taken into account when discussing

the results of this investigation. However, the correlation with the flat bed conditions of other authors implies that this effect was successfully accounted for. The matter is addressed when using the Lab Carousel, where a nonintrusive LDV (laser Doppler velocimeter) was used for measuring the velocity and the EMCM component of drag was removed from the data. [45] The distinction of the boundaries between transitional and rough turbulent flow for the Mini Flume are relatively difficult to define. However, the boundary seems to fall

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between Re = 20000 and Re = 40000 (Figure 12) for roughnesses up to 0.016 m where, in this case, Re is the hydraulic Reynolds number (Re = urHr/m, and Hr is the hydraulic radius of the flume, Hr = cross-sectional area/ wetted perimeter). For the 0.016-m and 0.022-m cases, the exact boundary is less clear, but would seem to indicate a value of approximately Re = 50000. These values compare well with those of Hughes and Brighton [1967], who determined a range of 3000 < Re < 70000 for pipe flow.

9. Conclusions [46] Flow deceleration has proved to be a very useful method to determine bed drag coefficients. The repeatability of the method has proved to be very good, and the ease of computation is valuable. The speed at which a large number of data points can be gathered gives this method an advantage over other approaches. There are fewer assumptions made in using this method as it is based directly on Newton’s second law. The bed roughness does not have to be determined, and only one measurement of velocity is required, doing away with the need for time-consuming profiling which can often be difficult to perform accurately. [47] There is a convergence of the drag coefficient data to Sternberg’s constant CD of 3
103 at intermediate u, implying that in turbulent rough flows, this constant value of CD in calculations of shear stress may be a reasonable approximation; however, there are large variations in CD with u at lower velocities. Figure 11 shows that using a constant CD leads to large errors in stress estimates over a typical range of flow velocities. [48] At high velocities in both flumes, there is a drop in CD (Figures 8a and 8c) similar to that reported by Hughes and Brighton [1967] for circular cylinders. It appears that the patch spacing of a one-grain-thick bed does not affect the bed shear stress. On homogeneous beds, the grain size over the gravel range makes a small difference to the shear stress.

Figure 11. Comparison of shear stress values found using hot film probes (Probes 1-3), the quadratic stress law (tQS) with constant CD as determined by Sternberg [1968], the law of the wall [Amos et al., 1992], and flow deceleration for the Lab Carousel.

Figure 12. Rough-transitional boundaries for 0.0056-m homogeneous gravel in the Mini Flume. [49] Acknowledgments. This work was completed under the EPSRC grant GR/R46311/01 and the NERC grant GR9/4717. The authors would like to thank John Davies and Ray Collins for their continued and essential technical support throughout the investigation.

References Allen, J. R. L. (1977), Physical Processes of Sedimentation, 248 pp., Allen and Unwin, London. Amos, C. L., J. Grant, G. R. Daborn, and K. Black (1992), Sea Carousel-A benthic, annular flume, Estuarine Coastal Shelf Sci., 34, 557 – 577. Amos, C., T. Feeney, T. Sutherland, and J. Luternauer (1997), The stability of fine-grained sediments from the Fraser River Delta, Estuarine Coastal Shelf Sci., 45, 507 – 524. Einstein, H. A. (1942), Formulas for the transportation of bed load, Transactions Am. Soc. Civ. Eng., 107, 561 – 577. Einstein, H. A., and N. L. Barbarossa (1952), River channel roughness, Trans, Am. Soc. Civ. Eng., 117, 1121 – 1132. Fung, A. (1997), Calibration of flow field in mini-flume, report, Geol. Surv. of Can., Atlantic Dartmouth, N. S., Can. Graham, D., P. James, T. Jones, J. Davies, and E. Delo (1992), Measurement and prediction of surface shear stress in annular flume, J. Hydraul. Eng., 118(9), 1270 – 1285. Grant, W. D., and O. S. Madsen (1979), Combined wave and current interaction with a rough bottom, J. Geophys. Res., 84, 1797 – 1808. Green, M., and I. McCave (1995), Seabed drag coefficient under tidal currents in the eastern Irish Sea, J. Geophys. Res., 100, 16,057 – 16,069. Gust, G. (1976), Observations on turbulent drag reduction in a dilute suspension of clay in seawater, J. Fluid Mech., 75, 29 – 47. Hennay, A., J. J. Williams, J. R. West, and L. E. Coates (1994), A field study of wave-current interactions over a rippled sand bed, in Sediment Transport Mechanisms in Coastal Environments and Rivers, edited by M. Belorgey, R. D. Rajaona, and J. F. A. Sleath, pp. 345 – 374, World Sci., River Edge, N. J. Hughes, W. F., and J. A. Brighton (1967), Fluid Dynamics, 265 pp., McGraw-Hill, New York. Huntley, D. A., and D. G. Hazen (1988), Seabed stresses in combined wave and steady flow conditions on the Nova Scotia Continental Shelf: Field measurements and predictions, J. Phys. Oceanogr., 18, 347 – 362. Li, M. Z., and G. Gust (2000), Boundary layer dynamics and drag reduction in flows of high cohesive sediment suspensions, Sedimentology, 47, 71 – 86. Maa, J. P.-Y. (1990), The bed shear stress of an annular sea-bed flume, in Estuarine Water Quality Management: Monitoring, Modelling and Research, edited by W. Michaelis, pp. 271 – 275, Springer-Verlag, New York. Nielsen, P. (1992), Coastal Bottom Boundary Layers and Sediment Transport, 324 pp., World Sci., River Edge, N. J. Nikuradse, J. (1950), Laws of flow in rough pipes, Tech. Memo. 1292, pp. 1 – 62, Natl. Advisory Comm. on Aeronaut., Natl. Aeronaut. and Space Admin., Washington, D. C. O’Brien, D. J. (1998), The sediment dynamics of a microtidal mudflat on varying timescales, Ph.D. thesis, Univ. of Wales, Cardiff, UK. Patel, V. C. (1965), Calibration of the Preston tube and limitations on its use in pressure gradients, J. Fluid Mech., 23, 185 – 208.

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Petit, F. (1989), The evaluation of grain shear stress from experiments in a pebble-bedded flume, Earth Surf. Processes Landforms, 14, 499 – 508. Shames, I. (1962), Mechanics of Fluids, 555 pp., McGraw-Hill, New York. Sleath, J. F. A. (1985), Energy dissipation in oscillatory flow over rippled beds, Coastal Eng., 9(2), 159 – 170. Soulsby, R. (1983a), Dynamics of Marine Sands, Thomas Telford, London. Soulsby, R. L. (1983b), The bottom boundary layer of shelf seas, in Physical Oceanography of Coastal and Shelf Seas, edited by B. Johns, pp. 189 – 266, Elsevier Sci., New York. Soulsby, R. L., and J. D. Humphrey (1989), Field observations of wavecurrent interactions at the sea bed, in NATO Advanced Research Workshop in Water Wave Kinematics, edited by A. Torum, pp. 413 – 428, Kluwer Acad., Norwell, Mass. Stapleton, K. R., and D. A. Huntley (1995), Seabed stress determinations using inertial dissipation method and the turbulent kinetic energy method, Earth Surf. Processes Landforms, 20, 807 – 815. Sternberg, R. W. (1968), Predicting initial motion and bedload transport of sediment particles in the shallow marine environment, in Shelf Sediment Transport, edited by D. J. P. Swift, D. B. Duane, and O. H. Pilkey, pp. 61 – 82, Dowden, Hutchinson and Ross, Inc., Straudsburg, Pa.

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Vanoni, V. A., and N. H. Brooks (1957), Laboratory studies of the roughness and suspended load of alluvial steams, Sed. Lab. Rep. E68, Calif. Inst. of Technol., Pasadena, Calif. Whiting, P., and W. Dietrich (1990), Boundary shear stress and roughness over mobile alluvial beds, J. Hydraul. Eng., 116(12), 1495 – 1510. Yalin, M. S. (1972), Mechanics of Sediment Transport, 290 pp., Pergamon, New York. Yang, Z., A. Baptista, and J. Darland (2000), Numerical modelling of flow characteristics in a rotating annular flume, Dyn. Atmos. Oceans, 31, 271 – 294. 

C. L. Amos and C. E. L. Thompson, School of Ocean and Earth Science, University of Southampton, Southampton Oceanography Centre, European Way, Southampton, Hampshire SO14 3ZH, UK. ([email protected]; [email protected]) T. E. R. Jones, Department of Mathematics, University of Plymouth, Plymouth, Devon, UK. ([email protected]) M. Lecouturier, L’Institut des Sciences de la Mer de Rimouski (ISMER), Universite´ du Que´bec a` Rimouski, Rimouski, Que´bec, Canada. (magali9@ free.fr)

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