A generalized log law formulation for a wide range of boundary roughness typically encountered in natural streams

Authors


Abstract

[1] Applying fluid mechanics principles to mobile bed conditions is simplified by demonstrating that fitting log profiles to streamwise velocity measurements using the plane of zero velocity is applicable to a larger range of roughness conditions than previously expected. Four sets of detailed velocity measurements obtained under the supervision of the authors were analyzed encompassing sediment-scale roughness elements, roughness caused by rigid vegetation, and large-scale roughness elements comprised of mobile bed forms. The analyses resulted in similar normalized zero-plane displacements for all roughness types considered. The zero-plane displacements, dh, normalized by roughness height, ks, were 0.20 and 0.26 for the sediment- and vegetation-scale experiments, respectively. The results for two experiments with bed form dominated roughness were 0.41 and 0.34. Estimates of dh/ks ranging from 0.2 to 0.4 are therefore recommended for a range of roughness types with the higher end of the range more appropriate for the larger, bed form-scale roughness elements, and the lower end for the sediment-scale roughness elements. In addition, it is demonstrated that the location of the plane of zero velocity is temporally constant even when the bed height is not. The findings can be applied to field velocity measurements under mobile bed conditions facilitating the calculation of turbulence parameters such as shear velocity by using point measurements and providing guidelines for the estimation of an appropriate value for zero-plane displacement.

1 Introduction

[2] The location of the boundary in flows over flat, smooth surfaces is apparent and undisputable. However, for hydraulically rough boundaries common in natural rivers and atmospheric flows, the problem of defining a representative boundary is much more complicated. For streams, the problem arises from the irregularity of boundaries due to variable roughness element sizes, shapes, and arrangement [Church et al., 1987], the existence of a bed load layer and saltating particles, or bed forms and their migration [Smart, 1999]. To simplify the problem, an arbitrary boundary is often considered corresponding to the asymptote of an assumed logarithmic velocity profile.

[3] It is a well-accepted practice to quantitatively describe turbulent flows by assuming the existence of a logarithmic distribution of mean streamwise velocity, u, over a certain distance from the boundary, z, known as the log law [Schlichting and Gersten, 2000]

display math(1)

for fully rough boundaries in uniform flow, where uτ is shear velocity (uτ = (τ0/ρ)0.5 and where τ0 is bed shear stress and ρ is the fluid density), ks is a representative roughness height, κ is von Karman's constant (0.4), and Γ is a constant. For fully rough boundaries, the constant Γ is generally assumed to be 8.5 [Schlichting and Gersten, 2000]. A boundary is classified as fully rough if ks > 70∙ν/uτ where ν is the kinematic viscosity of the fluid [Schlichting and Gersten, 2000]. The difficulty is in defining a reference elevation for the logarithmic distribution since roughness elements are frequently mobile, constantly shifting due to forces imposed by the flow above and around them. Often during mobile bed conditions, there are large-scale fluctuations in bed elevation due to bed forms, thereby causing roughness to be dominated by continually migrating bed forms instead of individual sediment particles. It is also possible to have significant sediment-scale roughness elements superimposed on top of bed form-scale roughness. The presence of a roughness sublayer further complicates the problem of boundary layer theory where the individual roughness elements interact with the flow near the boundary, displacing it upwards [Raupach et al., 1991] and causing complex, not well-understood patterns of shear and turbulence due to separation of flow around upstream roughness elements [Recking, 2009].

[4] Defining a flow's boundary is important because it is the location at which the fluid-sediment interaction takes place. It is also important because of the spatial dependence of many aspects of fluid mechanics, such as the location of boundary layers. Much research has been performed on defining boundaries with static roughness elements, but at present, limited information is available discussing how to define the boundary at a location in a stream where large-scale fluctuations in bed elevation occur due to active bed form migration.

[5] This paper examines whether using the common method of locating the plane of zero velocity is appropriate for a wide range of boundary roughness conditions by employing four separate data sets with varying roughness height scales. The roughnesses are representative of sediment particles, submerged vegetation, and bed form roughness. It is also demonstrated that global flow parameters can be obtained from using point measurements, instead of spatially averaged quantities, within the inner flow region. Generally speaking, spatially averaged quantities represent a channel's overall flow characteristics and are used to describe the global flow parameters. However, it is shown that under certain flow conditions, point measurements can be substituted for spatially averaged quantities in obtaining global flow parameters. The usage of spatially averaged velocity profiles over bed form flows was studied by McLean et al. [1999]. They determined that spatially averaged velocity profiles may not be used to completely describe the bed roughness characteristics, although results did indicate that using a spatial average was more accurate than using a local velocity profile. This paper attempts to show opposing results in that point measurements may be acceptable under certain conditions. Results of the analyses contained herein will utilize spatially averaged velocity profiles unless otherwise noted.

2 Background

[6] For flows with rough, static boundaries, it has been verified numerous times that a representative boundary is located a certain distance below the top of the roughness elements (Figure 1). Einstein and El-Samni [1949] were some of the first to investigate the location of the theoretical boundary by performing flume experiments with roughness elements of identical plastic spheres and again with a bed of natural gravel. For each experiment, the location of the zero-velocity plane was selected such that it caused the velocity distribution to appear as a straight line on a semilog plot when considering the logarithmic region to extend over the entire flow depth. The flow depth for these experiments was not reported, although no measurements were obtained above 18 cm. Given the particle size of 6.9 cm, and assuming the flow depth to be approximately 18 cm, the ratio of flow depth to particle size is relatively small so that under such conditions, it is appropriate to consider the log law valid over the entire depth. For the experiment with uniform, spherical roughness elements, their results indicated that the datum should be located 0.2∙d below the top of the roughness elements where d is the sphere diameter. For the second experiment using a bed of natural gravel, the origin producing the best fit of the log profile to the data was located 1.2 cm below the top of the gravel layer, which equates to 0.2∙d67 (d67 = 6 cm) where d67 is the sieve size at which 67% of the bed material is finer by weight [Einstein and El-Samni, 1949]. It should be noted that their results are only comparable to conclusions drawn here if the natural sediments were placed in the flume in a random way and were not allowed to be rearranged by the flow. It is unclear whether or not this was the case.

Figure 1.

Sketch of idealized, exaggerated velocity profile over bed form roughness elements.

[7] It is important to understand that sieve diameter is not necessarily representative of a roughness element's height. Generally, for flow-worked sediments, the height of a natural sediment particle corresponds to the particle's c-axis, where the c-axis is the smallest dimension of a three-dimensional sediment particle having an ellipsoidal shape and the a- and b-axes are the largest and intermediate axes, respectively. Sediment will typically arrange itself such that the particle's a-b plane lies parallel to the bed and the c-axis perpendicular to the bed, resulting in the maximum resistance to motion. Natural gravel has been shown to have a shape factor, SF, of 0.7 [Chang, 1988] where

display math(2)

and a, b, and c are the particle's largest, intermediate, and smallest dimensions, respectively. Considering the following relationship between sieve dimension, D, and a particle's c and b dimensions for ellipsoidal particles [Church et al., 1987]

display math(3)

it is possible to obtain the following by combining and manipulating equations (2) and (3) to read

display math(4)

[8] Using equation (4), it can be shown that the possible ratios for D/c range from 1.00 to 1.23 by considering the two limiting cases a/b of unity (b/c = 1.43) and b/c of unity (a/b = 2.04). Therefore, a sieve diameter d67 = D = 6.1 cm reported by Einstein and El-Samni [1949] corresponds to a roughness height, or sediment c-axis dimension, ranging from 4.9 cm to 6.1 cm. Assuming the c-axis dimension is representative of roughness height (ks = c), which may not be applicable in certain coarse bed streams, the ratio of zero-plane displacement to roughness height could range from 0.20 to 0.24 for the gravel experiment outlined in Einstein and El-Samni [1949]. The resulting value of dh/ks is remarkably similar to the values reported in the literature for other fixed, roughness boundaries and found in Table 1.

Table 1. Summary of Previous Research Determining the Location of the Zero-Velocity Planea
AuthorRoughness TypeRoughness Sizeksλdh/ksNotes
  1. a

    All experiments were conducted using water unless otherwise noted.

  2. b

    Little was included about the experimental setup, only that uniform spheres were used in one experiment and natural gravel in the other; no description of the particle arrangement was given.

  3. c

    z0 (equation (6)) was considered instead of ks (equation (1)), common practice in atmospheric fluid mechanics.

  4. d

    dm = mean grain diameter.

  5. e

    No information regarding the roughness particles' arrangements was included.

  6. f

    Values shown are averages of six separate experiments.

  7. g

    λ calculated using LAI/2 where LAI is the “Leaf Area Index”.

  8. h

    z0 reported instead of ks; ks was calculated using ks = z0∙e−3.4 which assumes Γ = 8.5.

  9. i

    10 mm × 1 mm vertical aluminum strips 60 mm tall.

Einstein and El-Samni [1949]

Plastic spheres and natural graveld = 69 mm, d50 = 69 mmdNot reportedb0.2Considered velocities over entire flow depth

Perry and Joubert [1963]

Horizontal square barsh = 3.2 mmd0.250.8Velocity-determined inner region; wind data

Jackson [1981]

n/an/az0c0 to 0.330.7Theoretical study

Raupach et al. [1991]

n/an/az0cn/a0.51Theoretical study; atmospheric flows

Kironoto and Graf [1995]

Crushed grains and natural graveldmd =2.8 mm, d50 = 23 mmdm, d50Not reportede0.2Considered velocities within bottom 20% of depth

Kanellopoulos [1998]

Lead spheresd = 8 mmd500.70.5Used iterative technique; κ = 0.4 as objective

Smart [1999]

Natural gravel23 mm < d50 < 85 mmz0cMobile gravel bed rivers0.04Velocity-determined inner region; found that dh/z0 = 1.1

Afzalimehr and Rennie [2009]

Natural gravel14 mm < d50 < 23 mmd84Gravel bed rivers0.25Used objective of maximum R2; assumed Γ = 4.4; nonuniform flow
Jarvis et al. [1976]fForest4.5 m < h < 23 mz0c3.7g0.60hAtmospheric flows; values of dh provided in Raupach et al. [1991]

Raupach [1988]

Wheath = 47 mmz0c0.470.29Atmospheric flows; values of dh provided in Raupach et al. [1991]

Raupach et al. [1986]

Aluminum stripsih = 60 mmz0c0.230.16Air flows; values of dh provided in Raupach et al. [1991]

[9] Kironoto and Graf [1995] performed two sets of experiments similar to those of Einstein and El-Samni [1949], one with a rough plate made of fastened crushed grains (ks = dm = 4.8 mm where dm is the mean grain diameter) and one with natural gravel (ks = d50 = 23 mm). Recall that Einstein and El-Samni [1949] considered ks = d67, whereas Kironoto and Graf [1995] considered ks = d50. They ultimately concluded that the log law described their data well in the inner region when assuming dh = 0.2∙d50, or dh/ks = 0.2, by noticing the clear agreement between their data and the log law (equation (1)).

[10] Perry and Joubert [1963] performed experiments in a wind tunnel with 3.2 mm square bar roughness elements spanning the entire width of the tunnel and spaced at 12.7 mm. The location of the zero-velocity plane was found by plotting multiple profiles with different origin locations until a range of zero-plane displacements were found that matched the measured data well. Measured points that clearly did not follow the straight line trend on a semilog plot were omitted, presumably because they were outside of the inner region; no well-defined limit to the inner region, such as 20% of the flow depth, was used. According to their results, the location of the origin was, on average, 2.54 mm below the top of the roughness elements, which results in dh/ks of 0.8 using ks = h where h is the height of the roughness elements.

[11] The location of the origin has been examined theoretically by Jackson [1981]. From the derivation of the log law, he concluded that the location of the origin is where the mean surface shear appears to act and that the zero-plane displacement adjusts the reference to that location. Similar conclusions were drawn by Raupach et al. [1991] suggesting that for static, regular roughness elements, 0 < dh < h since the shear forces must act on some portion of the roughness elements of height h. Jackson [1981] also discussed the effects of roughness density, or packing density, λ, on the zero-plane displacement and indicated that there may exist a threshold for λ above which packing density has virtually no effect on zero-plane displacement. Packing density is defined as the ratio of the roughness element area projected onto a plane perpendicular to the flow direction to the unit ground area surrounding the roughness element (in general, frontal area per unit ground area). Below the threshold, zero-plane displacement is linearly dependent on packing density. He gave no clear expression describing the relationship, nor any indication of the threshold value for λ, only that one exists.

[12] One way to explain this behavior is to consider rod or bar-like roughness elements. As λ increases, the flow becomes less capable of penetrating down into the voids due to the decrease in spacing between elements. As λ increases and the boundary approaches a smooth condition, dh will approach zero. Defining penetration depth, δe, as the distance that Kelvin-Helmholtz vortices penetrate vertically into a vegetated canopy of height h, Nepf and Ghisalberti [2008] found the direct relationship between dh and δe (equation (5)) by considering z = hdh to be the location of the centroid of momentum absorption and that momentum absorption is constant for hδe < z < h

display math(5)

where dh and δe are measured from the top of the roughness elements and z is measured from the bed. Therefore, as λ increases and penetration depth decreases, zero-plane displacement decreases. It is important to note that this is in conflict with the conclusion drawn by Jackson [1981] that for large packing densities, above the threshold, zero-plane displacement is constant. The conflicting conclusions could be explained by the fact that a smooth boundary is not reached for large, sphere-like roughness elements at the maximum packing density.

[13] Log profiles were fit to data from five gravel bed rivers in New Zealand, under both static and mobile bed conditions, in the form of equation (6), similar to that found in Smart [1999],

display math(6)

where ui is the instantaneous streamwise velocity, z is the elevation above the top of the roughness elements, dh is the origin offset and z0 is another measure of boundary roughness height (z0 = 0.033∙ks if Γ = 8.5 in equation (1)). He assumed there was little skewness in the distribution of ui such that the instantaneous value is likely to be equal to the average value (<u>). Shear velocity, uτ, and roughness height, z0, were used as regression parameters and only the portion of the data that appeared as a straight line on a semilog plot of ui vs. z were considered in the profile fitting. The origin displacement, dh, was then determined by varying its value until a maximum correlation coefficient was obtained for the regression. Results indicated that there was no correlation between dh and the size of the bed material for the mobile bed conditions; however, a relationship between dh and z0 was evident, dh/z0 = 1.1 [Smart, 1999]. The bed type was not made clear, although it is probable that bed forms were present. Considering the values of z0 reported and assuming ks = 30∙z0 [Smart, 1999], the average roughness coefficient equates to 2.6 m. The average of reported ks values being an order of magnitude larger than d90 (0.09 m < d90 < 0.16 m) indicates that the roughness was bed form dominated. This seems to indicate that the zero-plane displacement is not proportional to bed material size for mobile beds, as indicated by Smart [1999], but could instead be proportional to the bed form height.

[14] Other methods have been proposed for determining the location of the theoretical origin. Kanellopoulos [1998] calculated bed shear using τ0 = -ρ∙<uw′> where u′ and w′ are the instantaneous fluctuations of streamwise and vertical velocity components, respectively, about the mean. Knowing the bed shear allowed the shear velocity, uτ, to be calculated leaving κ and Γ in equation (1) unknown and used as regression parameters. The location of the zero-velocity plane was varied until the expected value of κ = 0.4 was reached; the value of Γ was not constrained. Data obtained from Kanellopoulos [1998] will be analyzed and further elaborated on in subsequent sections.

[15] More recently, Afzalimehr and Rennie [2009] used an equation in the form of

display math(7)

where p is a proportionality constant between the zero-plane displacement and a representative roughness height, or p = dh/ks. They considered an origin offset equal to dh/ks = 0.25 by maximizing the coefficient of determination between u and ln[(z + pks)/ks] for data collected in four different gravel bed rivers in Iran and Canada under nonuniform flow conditions. Because of the greater influence on roughness by the larger grains, they considered ks = d84. They concluded that for a bed of uniform, spherical roughness elements, the mean surface elevation is 0.18∙h, where h is the sphere diameter, below the tops of the spheres, similar to the conclusion drawn by Einstein and El-Samni [1949] of dh = 0.2∙ks. A 20% difference in bed shear estimates from the log law when considering dh/ks ranging from 0.1 to 0.25 was also reported. Furthermore, Afzalimehr and Rennie [2009] found that their data followed the log law for z < 0.68∙H, where H is the flow depth.

[16] To clarify the notation used herein, zero-plane displacement is defined here as the distance from the top of the roughness elements to the zero-velocity plane. The zero-velocity plane is the elevation that the logarithmic velocity profile asymptotes to zero. It is sometimes referred to as the plane of zero velocity and is often considered the representative/theoretical boundary. Clarification of terms may be found in Figure 1.

3 Summary of Experiments

[17] Velocity measurements from four different experiments were analyzed to study the effects of three different roughness scales on the flow field within the inner flow region. All experiments were either performed by the authors or performed previously to study different objectives under the supervision of at least one author. Spherical roughness elements at maximum packing density are first used to outline the method that has been shown numerous times to be applicable under such conditions. In the next case, cylindrical, rod-like, vertical roughness elements simulating rigid vegetation at a fixed packing density were studied. Roughness elements simulating mobile bed forms under two separate experimental conditions, one in a straight flume and one in a controlled, meandering, sandy bed channel, were also examined. Descriptions of the experiments considered are summarized in Table 2. Two of the four experiments were performed by previous researchers, and their descriptions can be found in detail in Kanellopoulos [1998] and Liu et al. [2008]. The two bed form-scale experiments were conducted by the authors and will be explained in detail here since their discussion is limited in the currently available literature.

Table 2. Summary of Experiments Discussed
AuthorRoughness TypeNo. of Profilesh (cm)H/hSlopeDischarge (m3/s)U (m/s)
  1. a

    Many parameters were not recorded because of the initial nature of the experiment.

  2. b

    Flow depth and bed form heights used are both rough estimates.

  3. c

    Depth average velocities varied throughout the meander.

Kanellopoulos [1998]

Uniform spheres10.8120.075%0.0200.35

Liu et al. [2008]

Rigid dowels47.621.50 to 1.570.3%0.0110.32 to 0.33
CurrentBed form; flume12.34n/aan/aan/aan/aa
CurrentBed form; outdoor505b4.2b0.7%0.28n/ac

[18] The sediment-scale experiment with a static bed consisting of 8 mm diameter spheres arranged at a maximum packing density is detailed in Kanellopoulos [1998]. The experiment was conducted in a rectangular, tilting-bed flume that is 0.6 m wide and 0.3 m deep. Detailed, streamwise velocity measurements were measured simultaneously in the experimental section of the flume with fully-developed flow using a two-dimensional laser Doppler velocimeter (LDV) system with a high measurement density near the bed and decreasing density as distance from the bed increased.

[19] For the vegetation-scale experiment, roughness representative of submerged rigid vegetation was created using static, rigid, acrylic dowels 6.35 mm in diameter, d, and 7.62 cm in height, h, attached to a plexiglass bed in a staggered arrangement (Figure 2). The data set contains four different experimental setups, labeled L1, L2, L3, and L4, with the only differences being small-scale surface roughnesses applied to both the dowels and the channel bed. Water-resistant fine and coarse sandpaper was placed on the bed and around each dowel like a sleeve. Profile L1 contained no skin roughnesses, L2 contained skin roughness on the bed and fine skin roughness on the dowels, L3 contained skin roughness on the bed and coarse skin roughness on the dowels, and L4 contained skin roughness on the bed and no skin roughness on the dowels. Liu et al. [2008] determined that the additional roughness had little influence on the flow's behavior so that all four experimental setups can be considered nearly equivalent. The roughness packing density, λ, for all four data sets was 0.12. The flume was 0.3 m wide with a slope of 0.3% and a flow rate of 0.0114 m3/s with flow depths ranging from 11 cm to 12 cm. Detailed streamwise velocity measurements were taken with a one-dimensional LDV system. Measurements were taken for 20 s at a rate of 75 Hz at points both within and above the dowel array. For more information regarding the experimental setup, see Liu et al. [2008].

Figure 2.

Top view of dowel arrangement for vegetation-scale experiment found in Liu et al. [2008]. Not to scale. Circles represent dowels. Diamonds represent locations of velocity profile measurements.

[20] Data were collected in a 15 m long, 0.9 m wide, and 0.6 m deep tilting flume at the University of Minnesota's St. Anthony Falls Laboratory (SAFL) in Minneapolis, Minnesota for bed form-scale flume experiment. The experiment was initially conducted as an equipment test; however, the data were of high quality and well suited for the discussion here. Due to the purpose of the experiment, flume slope and the flow depth were not recorded. Water was routed into the flume and through a series of flow straighteners at the flume entrance. Flume sediment consisted of coarse sand with a d50 of approximately 1.8 mm and a d90 of 2.4 mm. During the 120 min recording period, the bed elevation changed by approximately 2.3 cm due to the migration of a single, scour-induced bed form through the flume. The 400 s segment encompassing the large change in bed elevation due to the bed form migration is shown in Figure 3, beginning at t = 4400 s and ending at t = 4800 s. The single bed form was caused by the scouring of material around a rock structure installed upstream of the measurement location. Three-dimensional velocity data were collected continuously at one location, for 120 min using a Nortek Vectrino II profiling acoustic Doppler velocimeter (ADV) at a rate of 100 Hz. The instrument takes 30 instantaneous measurements, vertically spaced at 1 mm, giving it the ability to measure an instantaneous, 3 cm velocity profile. Of the 30 measurements, only the ten top-most measurements exhibited no interference from the rising bed and are shown in Figure 3. Bed elevation was also measured with the ADV at a rate of 1 Hz.

Figure 3.

Location of 10 velocity records relative to the changing bed elevation (top-most and bottom-most of the ten velocity records are shown) in the bed form-scale flume experiment. Black horizontal lines represent the elevation of the ten records. ● – Average bed elevation measurements; ▬ – velocity record samples for the lowest and highest of the 10 velocity records considered.

[21] The bed form-scale outdoor experiment was conducted in the Outdoor StreamLab also at SAFL. The meandering channel (Figure 4), with sinuosity of 1.3, was run at bankfull conditions with an average depth of 21 cm, an average width of 230 cm, and a slope of approximately 0.7%. Flow was diverted from the Mississippi River into the channel via an abandoned spillway and maintained at a relatively constant 280 L/s using a manual gate valve, then released back into the river at the tail of the stream (Figure 4). The poorly sorted channel sediment (d50 = 0.7 mm and d90 = 2.0 mm) was collected at the end of the stream and recirculated to a storage tank to be fed back into the head of the stream at a constant rate using an electronically controlled auger. The sediment feed rate regulates the amount of bed load transport within the flume and therefore the equilibrium conditions. Naturally created, mobile bed forms were approximately 1 m in length and up to 0.5 m in height (Figure 5). Velocity measurements were taken at nine permanent cross sections spaced at approximately 2 m (Figure 4) for two separate, but virtually identical, channel and flow conditions. The point bar experienced significant changes between experiments; however, the stream was allowed to run sufficiently long to reach the same quasi-equilibrium state that was present during the first experiment before any measurements were taken during the second experiment. The presence of quasi-equilibrium conditions was established when there were no longer observable changes to the size of the point bar. On the order of 35 velocity measurements were taken per cross section. Each cross section consisted of four to eight vertical profiles spaced at 16 cm to 52 cm depending on the width of the stream. Within each profile, measurement points were spaced vertically from 3 cm to 5 cm. Velocity measurements were taken with a Nortek Vectrino 200 Hz ADV. Velocity samples were taken for 4 min, encompassing over 700 integral time scales for the given flow conditions. The sample time exceeds the maximum length of record necessary for describing turbulence according to Buffin-Belanger and Roy [2005]. Furthermore, since the sample time was approximately twice as long as the bed form wave period, it is safe to consider the local, point velocity measurements as equivalent to the spatially averaged velocity at that location.

Figure 4.

Top view of the meandering channel in the OSL and location of nine permanent cross sections (numbered and shown in red). Flow in the channel is from left to right.

Figure 5.

Bed forms in the Outdoor StreamLab under low flow conditions. Bed forms were created during bankfull flows and remained unchanged as the flow was lowered.

[22] Detailed bathymetry was collected in the OSL using a high-resolution data acquisition carriage (DAQ) equipped with an ultrasonic transducer capable of measuring underwater channel bathymetry at 1 cm horizontal and 1 mm vertical resolutions. The DAQ is equipped with a Keyence LKG laser that is capable of mapping bank topography above the water at 1 mm horizontal and submillimeter vertical resolutions. Up to eight consecutive, five min long bathymetry scans were repeated at every location in the stream, allowing for the measurement of the temporal evolution of the bed. The repeat scans allowed for the extraction of the maximum bed elevation at the location of each vertical velocity profile.

[23] Both bed form-scale experiments' velocity data sets were filtered using a modified phase-space threshold (PST) technique proposed by Parsheh et al. [2010], a modification to the original PST technique outlined in Goring and Nikora [2002]. The Parsheh et al. [2010] method insures that points near the probability density function (PDF) peak, but far from the PDF tail, are not erroneously flagged and remain unchanged.

4 Analysis and Results

[24] It is hypothesized here that in flows containing large-scale variations in bed elevation, such as those due to bed form migration, the same zero-plane displacement concept originally proposed by Einstein and El-Samni [1949] for static, rough beds could be employed and that the difference between the measured bed elevation and the location of the zero-velocity plane would be strongly related to the roughness, or bed form, height. In other words, the method is applicable as long as a relevant value of roughness height, ks, is chosen to represent the total roughness, or dominant height. It will be shown that ks = h may be an appropriate relationship, where h is the roughness element height whether it be grain diameter, vegetation height, or bed form height. It is understood that this approximation of ks using the average bed form height differs slightly from the traditional method used when studying static, grain roughnesses, namely considering ks proportional to the larger grains (ks = d90 or d84 for example). Considering the larger bed forms would not change the results significantly as it would also increase the resulting zero-plane displacement values. For bed forms, it is acknowledged that roughness is related to the bed form amplitude, and wavelength [Smith and McLean, 1977]; however, here it is assumed to be solely dependent on amplitude, more specifically the average bed form height estimated using bathymetry data. It is reasonable to expect that for large wavelengths, the sediment scale roughness elements dominate, whereas for short wavelengths, the bed forms themselves dominate. This relationship may not be the most accurate as it does not account for the effects of roughness element orientation, spacing, or shape, but was chosen for consistency and because the actual, relationship between ks, h, and λ is complicated and not well understood. Note that the ratio of ks/h = 1 is also within the range of acceptable values determined by Schlichting and Gersten [2000] for spherical, hemispherical, and conical roughness elements. The average results for all four data sets are summarized in Table 3.

Table 3. Average Zero-Plane Displacements Normalized by Roughness Coefficient for Four Data Sets With Constant Roughness Packing Density
Data SetRoughness TypePacking Densityks (cm)Average dh (cm)Average dh/ksComment
  1. a

    Calculated using λ.

  2. b

    Calculated as the height to spacing ratio (assuming a bed form height of 0.5 m and wavelength of 1 m).

  3. c

    Only one bed form was present.

Fully packed [Kanellopoulos, 1998]Sediment0.7a0.80.160.20ks = sphere diameter
Vegetation [Liu et al., 2008]Vegetation0.12a7.622.000.26ks = dowel height
Mobile bed – flumeBed form0.5b2.340.800.34ks = bed form height
Mobile bed – outdoorBed formN/Ac51.510.41ks = avg bed form height

4.1 Sediment-Scale Experiment (Static Bed)

[25] High-quality velocity profiles obtained by Kanellopoulos [1998] above a bed made of uniform, 8 mm diameter spheres and a bed slope of 0.075% will be used as an experimental control. The data set will also be used to outline the method employed for determining the zero-plane displacement and to show its validity.

[26] Each data point contains the elevation above an arbitrary datum, generally chosen as the top of the roughness elements, and a matching streamwise velocity measurement. Only data within the bottom 20% of the flow depth were considered to be within the logarithmic inner region in this analysis [Kironoto and Graf, 1995; Detert et al., 2010; Dancey and Diplas, 2008]. A linear regression was fit to the data (following equation (1)) with the natural logarithm of the elevations as the abscissa and the original velocity measurements as the ordinate and with no zero-plane displacement considered (dh = 0). The regression slope, mr = κ/uτ, and regression intercept, br = -Γκ + ln(ks), are representations of the regression parameters shear velocity and Γ, respectively. The variables κ (equal to 0.4) and ks (equal to particle diameter) are both constants. Next, each data point's elevation was simultaneously, iteratively increased, which effectively changes the reference elevation, until the constant Γ takes on the expected value of 8.5 for fully rough boundaries [Schlichting, 2000]. The zero-plane displacement is the distance that each data point elevation was ultimately increased by.

[27] After performing the analysis described above to the data from the sediment-scale experiment [Kanellopoulos, 1998], and assuming ks = d = 8 mm and κ = 0.4, the resulting zero-velocity plane is 0.16 cm below the top of the roughness elements (dh = 0.16 cm) with uτ = 2.6 cm/s and R2 = 0.996 (Figure 6). The resulting ratio of dh/ks is 0.20, virtually identical to the results obtained by Einstein and El-Samni [1949] and Afzalimehr and Rennie [2009]. Recall that they utilized data over the entire flow depth (ratio of particle size to flow depth of 0.4), whereas here data were only considered in the bottom 20% of the flow depth (ratio of particle size to flow depth of 0.08).

Figure 6.

Measured velocity profile and fitted regression for sediment-scale experiment. The measurements' origin (z = 0) is the top of the roughness elements. Solid symbols represent those considered within the logarithmic inner region; empty symbols within the outer region. Raw data obtained from Kanellopoulos [1998].

4.2 Vegetation-Scale Experiment (Static Bed)

[28] The same method was employed to locate the origin of four profiles of spatially averaged velocity in flow through an array of rigid, 7.62 cm tall acrylic dowels with the exception that the regression was fit to all data points above the dowel array (z > 7.62 cm). The inner layer considered spanned the entire flow depth above the roughness elements due to the small ratio of flow depth to roughness height. For consistency with the control experiment, the roughness coefficient, ks, was assumed to be equal to the dowel height (ks = h = 7.62 cm). The dh/ks ratios were 0.26, 0.25, 0.24, and 0.30 for experiments L1, L2, L3, and L4, respectively, with an average value of dh/ks of 0.26. These are similar to the results from the control experiment and values reported in the literature (Table 1). The average R2 for the four profiles (Figure 7) was 0.994 with a minimum of 0.988.

Figure 7.

Regressions used to determine dh for four data sets in vegetation-scale experiment. Solid symbols represent those considered within the log region; empty symbols within the suspected roughness sublayer. Note that here, the datum (z = 0 cm) is taken as the flume bed and not the top of the roughness elements as is the case in other velocity profile figures, although dh is measured with respect to the top of the roughness elements. Raw data obtained from Liu et al. [2008].

[29] In an attempt to examine the relationship between zero-plane displacement and penetration depth, δe, the expression δe = 0.23/(CDλ) proposed by Nepf and Ghisalberti [2008] was used to calculate the penetration depth. The drag coefficient was calculated using CD = 2∙uτ2/(λuh2) where uh is the measured streamwise velocity at z = h. The resulting zero-plane displacements were 1.50 cm, 1.48 cm, 1.40 cm, and 1.72 cm for experiments L1, L2, L3, and L4, respectively, assuming that dh = δe/2 [Nepf and Ghisalberti, 2008]. The resulting ratios of dh/ks are 0.20, 0.19, 0.18, and 0.23 with an average value of 0.20. These results are on the order of 0.2 as has been the case for the same measurements using the iterative technique and the control data set.

4.3 Bed Form-Scale Flume Experiment (Mobile Bed)

[30] During moderate to severe floods, sediment is entrained by the flow resulting in a moving boundary, such as in the case of downstream migrating bed forms. It is therefore important to test the validity of the method described for locating the plane of zero velocity for static beds under mobile bed conditions and determine the corresponding dh/ks ratios to conclude whether or not the method is applicable under such flow conditions. In the presence of bed forms, the dominant bed roughness length scale is the bed form height, h, instead of sediment size such that ks = h. Only velocity records that exhibited no influence from the rising boundary during the entire 400 s period were analyzed (Figure 4). The 20 lower bins that were at or below the height of the bed form, causing them to be influenced by the approaching bed and exhibiting unreasonable values, were excluded. Roughness height chosen was representative of the bed form height (ks = 2.34 cm). Using the same method as before, the resulting plane of zero velocity was 0.80 cm below the maximum bed elevation, or the bed form crest (dh = 0.80 cm) and is shown in Figure 8 along with the time-averaged streamwise velocities. The ratio dh/ks is 0.34, only slightly higher than the values of 0.20 and 0.26 found for the two static bed experiments and within the range of values reported in the literature (Table 1).

Figure 8.

Velocity measurements along with the regression used to determined dh for bed form-scale flume experiment. The datum (z = 0) is taken as the maximum bed elevation, or bed form crest. Note that the water surface elevation is unknown.

[31] Recall that Perry and Joubert [1963] performed experiments under similar conditions with square bars spanning the entire width of a wind tunnel with a height to spacing ratio of 4. Their results indicated a ratio of zero-plane displacement to roughness height of 0.8 for width-spanning roughness elements similar to bed forms. The difference in dh/ks could be partially attributed to the fact that the log-profile fit to data here was within a 1 cm portion of the flow due to instrumentation and experiment limitations, whereas Perry and Joubert [1963] considered data over the entire inner region. Differences could also be attributed to the fact that measurements here were obtained around a single roughness element and Perry and Joubert [1963] had multiple bars (i.e., an entire rough surface).

4.4 Bed Form-Scale Outdoor Experiment (Mobile Bed)

[32] Before demonstrating the validity of the method for determining the zero-plane displacement, the need for considering a zero-plane displacement will be presented by comparing results using the original datum (considering dh = 0) and the velocity profile asymptote as the reference. When considering no zero-plane displacement, there is a large spread of Γ values (Figure 9) with a mean of 6.69 and a standard deviation of 5.31. Recall that the expected value for fully rough boundaries is 8.5 [Schlichting and Gersten, 2000]. Similarly, there is a large spread in shear velocities when considering no zero-plane displacement (Figure 10). The shear velocity distribution with no zero-plane displacement had a mean of 0.12 m/s and standard deviation of 0.080 m/s.

Figure 9.

Values of Γ in equation (1) for bed form-scale outdoor experiment using original boundary elevation (dh = 0). The expected value for fully rough boundaries of Γ = 8.5 is denoted by a dashed line. Note that y/H is the transverse (y axis) distance from either bank that the vertical profile is located normalized by the stream depth and that profiles within two flow depths of either bank were not included to remove the boundary's influence on the profiles.

Figure 10.

Values of uτ in equation (1) for bed form-scale outdoor experiment when using no zero-plane displacement and when considering dh. Note that y/H is the transverse (y axis) distance from the nearest bank normalized by the stream depth. Profiles within two flow depths of either bank were not included to remove the boundary's influence on the profiles.

[33] The undesirable spread in Γ and uτ when considering no zero-plane displacement indicated that a separate formulation was necessary to change the reference elevation. After changing the reference datum to the velocity profile asymptote, by forcing Γ = 8.5 using the method outlined in this paper, the resulting shear velocities appear to collapse to a much tighter distribution (Figure 10), with a mean of 0.080 m/s and standard deviation of 0.024 m/s. The average shear velocity is only slightly less than the reach average value of 0.12 m/s calculated using uτ = (gHS)0.5 where g is the gravitational constant, S is bed slope (0.007), and H is the average flow depth (21 cm). Considering the reduction in spread and that the values match closely with the expected reach average, the method appears applicable under such conditions.

[34] It is worth noting that when Reynolds stress profiles were analyzed, the results indicated similar shear velocities ranging from 0.034 m/s to 0.18 m/s and an average value of 0.072 m/s, which is closer to the results from the log law (average value of 0.08 m/s) than the reach average value (0.12 m/s). Therefore, the selection of ks equal to the roughness height resulted in adequate estimations of shear velocity providing more confidence in the formulation used.

[35] For the 50 log-profiles fit to the data in the outdoor channel, the average zero-plane displacement was 1.51 cm below the maximum bed elevation, or bed form crest, corresponding to an average ratio of dh/ks of 0.41 (Figure 11) when using ks = 5 cm which is representative of the spatially averaged bed form height. The amount of scatter in Figure 11 could be explained if the mean value is considered a reach average while understanding that the actual value may vary from point to point in the stream. Bed form heights varied both in the streamwise and lateral directions with larger bed forms located along the outer bank, across from the point bar. A rough estimation of the spatial average bed form height was obtained by subtracting the maximum and minimum bed elevations, bed form crests and troughs, respectively, over approximately 40 min using the precise DAQ bathymetry measurements. The slight discrepancy of the mean from the expected value of 0.2 could be attributed to the fact that flow through the meander was generally three dimensional in nature, violating one assumption necessary when applying the log law; however, the logarithmic profiles did fit the measured streamwise velocities well when considering only profiles within two flow depths of either bank (y/H < 2.0). The presence of three-dimensional flow may also explain some of the scatter in Figure 11.

Figure 11.

Zero-plane displacement, dh, normalized by roughness height, ks, bed form-scale outdoor experiment when considering the zero-velocity plane as the reference elevation. Dotted line represents the average ratio of dh/ks. Note that y/H is the transverse (y axis) distance from the nearest bank normalized by the stream depth. Profiles within two flow depths of either bank were not included to remove the boundary's influence on the profiles.

4.5 Summary

[36] The resulting zero-plane displacement for all four experiments discussed is between 20 and 41% of the respective roughness height. The results indicate that the smaller, sediment-scale roughness elements make up the lower end of the range, while the larger, stream-spanning bed form roughness elements may encompass the higher end. This similarity in ratios of zero-plane displacement to roughness height for static beds and for mobile beds helps confirm the hypothesis that the method described in the literature for locating a zero-velocity plane in the presence of small, sediment-scale roughness elements may also be applicable for defining a zero-velocity plane in the presence of larger roughness elements such as large-scale fluctuations in bed elevation caused by bed forms. It is also suggested that the analysis be kept to the inner flow region when possible, although adequate results may be obtained from considering a larger portion of the flow, even the entire flow depth in some cases, when the log law is shown to be valid.

5 Discussion

5.1 Relationship Between Zero-Velocity Plane and Original Datum

[37] Since it is rarely known at what stage of the oscillation the bed is in when taking measurements in the field, it is important to explore the relationship between the zero-velocity plane and the reference datum. Using data from the flume bed form-scale experiment, the plane of zero velocity was located using the minimum, maximum, and the average bed elevations as the original reference elevations. Instrumentation was able to measure bed elevation and velocity simultaneously so that bed elevation was known with high confidence at the same time velocity was measured. The location of the velocity zero-velocity plane was 0.80 cm below the maximum bed elevation, 0.37 cm above the average bed elevation, and 1.54 cm above the minimum bed elevation, or bed form trough (Table 4). For all three reference bed elevations selected (minimum, maximum, and average), the origin displacement was located at nearly the same elevation (4.47 cm). This indicates that the zero-velocity plane location is constant; remaining the same, no matter at what stage in the fluctuation the actual bed is located. However, it should be noted that the velocity record should be sufficient to capture at least one full oscillation in bed elevation so as to have a minimally biased velocity measurement that is representative of the long-term average.

Table 4. Zero-Plane Displacement (dh) and Zero-Velocity Plane Location Above an Arbitrary Datum for Mobile Bed Flume Experiment Using Three Different Reference Bed Elevations
Bed ReferenceReference Bed Elevation (cm)dh (cm)Plane of Zero-Velocity Elevation (cm)b
  1. a

    Negative zero-plane displacements indicate that the plane of zero velocity is located above the reference bed elevation.

  2. b

    Zero-velocity plane elevation is equal to the difference between the reference elevation and dh.

Average4.1 cm−0.37a4.47
Minimum2.9 cm−1.54a4.44
Maximum5.3 cm0.804.50

5.2 Differences Between Profiles of Spatial Averages and Point Measurements

[38] It is necessary to comment on the differences in considering spatially averaged velocities instead of point measurements. Spatially averaged quantities give a better representation of a channel's overall flow characteristics and allow for the calculation of global flow parameters, such as zero-plane displacement and shear velocity.

[39] Heterogeneous flow conditions are often encountered above a bed of fully packed spheres in a prismatic channel. Longitudinal heterogeneity was demonstrated by Dancey et al. [2000] in 25 measurements of the local average vertical velocity component located 0.51 mm above four closely packed, 8 mm diameter spherical roughness elements (Figure 12) within the bed matrix. Their results indicate that flow penetrates down into the void between the upstream roughness element and the two adjacent roughness elements (Figure 12) and continues to penetrate until reaching the upstream face of the final sphere where it is forced upward to presumably repeat the process on the downstream face. Spatially averaged quantities should be necessary in order to account for the inhomogeneity, where the local values vary quickly and may not describe the behavior of the entire flow. Spatial averages allow for the calculation of global flow parameters, which describe that macroscale behavior. Local values based on near bed velocity measurements may not describe the overall flow pattern well and therefore should not adequately describe the global parameters.

Figure 12.

Location of velocity measurements in Dancey et al. [2000]. Black circles represent spherical roughness elements. Red diamonds represent locations of velocity measurements 0.51 mm above the top of the layer of spheres.

[40] It has also been postulated that the velocity profile over large-scale, sinusoidal roughness elements on a sandy bed is composed of two separate regions controlled by the different roughness elements present [Smith and McLean, 1977]. Near the bed, within the inner flow region, the profile is governed by the smaller roughness scale composed of individual sediment grains (skin friction). Further away from the bed, in the outer flow region, the profile is governed by the larger roughness elements made of wave-like structures (overall boundary shear) described by spatially averaged quantities.

[41] Using results obtained from the vegetation-scale experiment with λ = 0.12 [Liu et al., 2008], it is shown that spatially averaged quantities are not always required. Profiles of point measurements are shown in Figure 13a, while profiles of spatially averaged velocities are shown in Figure 13b. The profiles of point velocities are scattered below the top of the dowels; however, they collapse to a single profile above the dowel array where the flow is unaffected by the individual roughness elements (Figure 13a). On the other hand, the profiles of spatially averaged velocities (Figure 13b) are similar at all elevations, supporting the aforementioned assumption that flow conditions are identical for experiments L1, L2, L3, and L4. Since the methodology discussed in this paper applies to the inner flow region (above the dowels), it is the case that the results obtained from using point measurements will be similar to those obtained from using spatial averages (Table 5). For this reason, it is safe to assume that one profile of velocity point measurements may be representative of the spatial average and used to determine global flow parameters when dealing with the inner flow region if individual roughness elements lose their own distinct identity and instead behave as parts of a whole. These results are in conflict with those from McLean et al. [1999] who indicated that point measurements produce poor results. It is worth noting that in the case of the mobile bed experiments, the dunes are not spaced so closely that each individual dune loses its own identity; therefore, it is still necessary to consider spatial averages under those conditions. Also of note is that the log law is dependent on many factors, i.e., channel aspect ratio and bed geometry, such that its validity should be carefully considered before its application. This assumption may be utilized to cut down on the length of experiments, especially when global flow parameters are of interest.

Figure 13.

(a) Velocity profiles of point measurements at all six locations for data set L1 in vegetation-scale experiment. Notice that all profiles collapse to a single profile above the top of the dowel array, denoted by the dotted line. (b) Profiles of spatially averaged velocities for all four data sets L1, L2, L3, and L4 in vegetation-scale experiment. Notice how all profiles are nearly identical at all depths. Vertical, dashed lines represent top of dowel array. Raw data obtained from Liu et al. [2008].

Table 5. Normalized Zero-Plane Displacement and Shear Velocity for Vegetation-scale Experiments Found in Liu et al. [2008]a
dh/ks for Point Measurementsdh/ks for Spatial AveragesPercent Differenceuτ (m/s) for Point Measurementsuτ (m/s) for Spatial AveragesPercent Difference
  1. a

    Values obtained for point measurements are from each of the six individual profiles for each experiment.

 
Experiment L1
0.250.263.20.0720.0720.0
0.3015.40.0693.1
0.247.30.0720.0
0.2214.10.0754.0
0.239.50.0732.4
0.286.60.0720.3
 
Experiment L2
0.280.259.80.0700.0690.8
0.250.60.0681.6
0.2212.40.0713.1
0.239.10.0712.6
0.244.90.0690.2
0.276.60.0681.9
 
Experiment L3
0.270.2410.80.0720.0720.5
0.2015.40.0731.0
0.254.00.0743.4
0.267.00.0702.7
0.226.50.0710.8
0.234.70.0720.3
 
Experiment L4
0.330.309.60.0690.0690.2
0.278.10.0700.6
0.301.60.0681.2
0.2710.00.0712.2
0.300.70.0690.3
0.287.50.0712.8

6 Conclusion

[42] Data from four separate experiments were used in order to assess the validity of the methodology typically used to determine the zero-plane displacement for small, static roughness elements. It was determined that the concept is valid for all roughness types, even mobile bed forms. Results suggest that zero-plane displacements ranging from 0.2 to 0.4 times the roughness height are good estimates for all roughness types and sizes. The lower end of the range is suitable for smaller, sediment-scale roughness types and the higher end for larger, channel-spanning roughness elements. The method should be applied to the inner flow region; however, it has been shown in the past that the logarithmic law is applicable for as much as the entire flow depth. It was also determined that the zero-velocity plane location is independent of the reference bed elevation chosen. For the purpose of determining the location of the plane of zero velocity, it does not matter whether data are being collected over the crest or trough or in between.

[43] It also may not be necessary to consider spatially averaged velocities when fitting profiles within the inner flow region. It was shown that only minor differences in the values of zero-plane displacement and shear velocity occurred when using profiles made of spatially averaged velocities and profiles of single point velocities under certain conditions, namely beds comprised of spherical roughness elements and rigid dowel roughness elements. Since it requires more time and effort to measure spatially averaged quantities, the ability to use point measurements to obtain the same results could drastically reduce the amount of experimental resources required.

Notation
a

largest dimension of a spheroid roughness element, m

b

intermediate dimension of a spheroid roughness element, m

br

intercept of linear regression

c

smallest dimension of a spheroid roughness element, m

CD

drag coefficient, m−1

d

diameter of a sphere, m

dh

zero-plane displacement, m

dm

mean grain diameter, m

dX

grain size at which X% of material is finer by weight, m

D

sieve dimension, m

g

gravitational constant, 9.81 m/s2

h

characteristic size of roughness elements (generally height or diameter), m

H

flow depth, m

ks

representative roughness height, m

mr

slope of linear regression

p

proportionality constant between zero-plane displacement and roughness height

R2

coefficient of determination for a curve of best fit

S

energy slope, m/m

SF

shape factor

u

time averaged streamwise velocity component, m/s

u

fluctuation of streamwise velocity from the mean, m/s

<u>

time-average streamwise velocity, m/s

uh

streamwise velocity at height z = h, m/s

ui

instantaneous streamwise velocity, m/s

uτ

shear velocity, m/s

w

vertical velocity component, m/s

w

fluctuation of vertical velocity from the mean, m/s

y

distance from the river right bank in the transverse direction, m

z

vertical distance from the bed, m

z0

measure of boundary roughness height often used in atmospheric fluid mechanics

Greek
Γ

constant of integration for log law

δe

penetration depth, m

θu

median absolute deviation of the streamwise velocity, m/s

κ

von Karman's constant (0.4)

λ

roughness element packing density

uτ

shear velocity, m/s

τ0

bed shear stress, Pa

ρ

fluid density, kg/m3

ν

fluid kinematic viscosity, m2/s

Acknowledgments

[44] We would like to acknowledge NCHRP for providing the impetus to conduct and synthesize the research discussed. We would also like to acknowledge Glenn Moglen for advisement, as well as Kris Guentzel for assistance collecting OSL data and moral support. And Seokkoo Kang for experimental planning and data analysis support. And finally, June Sayers, Tom Wangerin, Anna Dooley, Chris Millren, and Jeff Heur for keeping the OSL operating and making each experiment interesting and enjoyable.

Ancillary