Journal of Geophysical Research: Earth Surface

A wind tunnel examination of shear stress partitioning for an assortment of surface roughness distributions

Authors


Abstract

[1] Surface roughness aids to ameliorate wind erosion by extracting a portion of the wind's momentum thereby reducing the quantity of stress on the surface. This paper evaluates the effect of different spatial arrangements of surface roughness on the partition of average drag forces and distribution of stress at the surface. A new tiered force balance was used in a wind tunnel to independently and simultaneously measure the drag on arrays of roughness elements and the drag on the intervening surface. In addition, Irwin sensors recorded point measurements of surface shear stress within the arrays. Roughness arrays consisted of small cylinders in four different spatial arrangements, one being staggered and three being simplifications of natural roughness configurations, at four roughness densities. Results from the tiered force balance and Irwin sensors indicate that the roughness configuration has a small impact on the average (R) and maximum (R″) drag partition. The protection of the surface increased with roughness density regardless of the roughness arrangement. Point measurements of shear stress revealed that the roughness configuration had a small impact on the distribution of shear stress at the surface, and that the maximum shear stress scaled to the average shear stress. Drag partition measurements were compared to the ratios predicted by the Raupach et al. (1993) model and a good degree of agreement was found for all configurations when using measured values of the β and m parameter.

1. Introduction

[2] Roughness elements, such as vegetation, boulders, or crop stubble, attenuate wind erosion by physically covering a portion of the surface and by extracting a portion of the wind's momentum [Wolfe and Nickling, 1993]. However, small spatial densities of roughness elements may lead to an increase in erosion around the roughness elements because of the development and shedding of eddies [Logie, 1982]. However, as the roughness density increases beyond some critical level, erosion tends to decrease. In order to develop a better understanding of the processes associated with aeolian sediment transport, methods are needed to predict the degree of sheltering afforded by roughness, and these methods should utilize parameters that can be measured by conventional or remote sensing methods [Musick and Gillette, 1990]. Accomplishment of this requires knowledge of the relationship between wind erosion and the spatial density, arrangement, and geometric characteristics of roughness elements. Raupach et al. [1993] developed a theoretical model to quantify the effects of roughness elements on sediment entrainment by relating the sheltering effect of obstacles to the partitioning of shear stress between the surface and the roughness element. With further testing, this model has the capability of becoming an important means of enumerating the effect of roughness elements on aeolian processes for surfaces on the Earth, as well as for planets where the atmospheric winds can entrain and transport sediment.

[3] However, essentially all natural surfaces are heterogeneous, yet there are few studies examining the effect of roughness configuration on shear stress partitioning. Results from this study improve the understanding of the relationship between roughness configuration and shear stress partitioning.

2. Background

[4] Several researchers have attempted to develop empirical and physically based models that can account for the effect that increased surface roughness has on reducing wind erosion [Lettau, 1969; Arya, 1975; Raupach, 1992; Marticorena and Bergametti, 1995]. Several of these models are directly or indirectly based on the shear stress partitioning theory presented by Schlichting [1936], who derived the concept while studying the shear stress generated on rivets on the hulls of ships. The theory states that the total drag force imparted upon a surface with roughness elements (F) can be separated into the force exerted on the surface (FS) and the force on the roughness elements (FR):

equation image

Dividing by the surface area affected, the components of the total shear stress can be resolved:

equation image

where τ is the total stress, τR is the shear stress on the roughness elements, and τS is the shear stress on the underlying surface in the absence of roughness elements.

[5] The stress partition problem can be stated as finding the dependence of the ratios τR/τ or τS/τ on the density of roughness elements (λ) and the element shape [Raupach et al., 1993]. The density of roughness elements is defined as:

equation image

and is a function of the breadth (b) and height (h) of the roughness elements, the number of elements (n), and the ground area that they are occupying (S). This definition is based on the reasoning that the momentum absorption by roughness is controlled primarily by the total frontal area of roughness elements [Marticorena and Bergametti, 1995]. Marshall [1971] undertook extensive wind tunnel experiments examining the partition of shear stresses between roughness elements and the surface in the context of wind erosion. He measured τ and τR, for arrays of solid roughness elements with densities and spatial arrangements representative of vegetation in arid regions, leading to a relationship between λ and the stress partition. Marshall's [1971] results provided the first comprehensive data set on the drag partitioning, and set the framework for investigating the effect of nonerodible roughness elements on wind erosion [Gillette and Stockton, 1989; Raupach, 1992].

[6] Stockton and Gillette [1990] expressed the partitioning of shear stress in terms of the threshold friction velocity (u*t), which is the friction velocity (u*) at which sediments become entrained for a given surface. Higher densities of roughness tend to increase u*t [Lancaster and Baas, 1998] from the value of u*t that exists for a similar bare surface. Therefore, the amount of force dissipated by the roughness elements can be specified as:

equation image

where u*tS represents the u*t for a bare, erodible surface, and u*tR is the value for the same surface with nonerodible roughness elements present [Stockton and Gillette, 1990].

[7] Field and laboratory studies have verified the decrease in values of Rt with increasing λ [Musick and Gillette, 1990; Musick et al., 1996; Lancaster and Baas, 1998]. Raupach [1992] modeled the shear stress partition, R, by relating the reduction in τS to the characteristics of wakes generated in the lee of roughness elements based on the geometric and drag properties of cylinders in turbulent flow, along with λ. Raupach et al. [1993] then linked the Raupach [1992] model to aeolian sediment entrainment by associating it with the friction velocity ratio, such that:

equation image

with τS defined as the shear stress acting on the exposed intervening surface, β defined as the ratio of the drag coefficient for an individual roughness element to the drag coefficient of the unobstructed surface (CR/CS). The ratio of roughness element basal area to frontal area (σ) is incorporated in the model such that σλ is the basal area per unit ground area, which is calculated as σ = πb/4h for cylinders. The drag partition prediction agrees well with R from wind measurements of individual drag forces in wind tunnels [Marshall, 1971; Crawley and Nickling, 2003], field studies measuring drag forces [Wolfe and Nickling, 1996; Wyatt and Nickling, 1997; Gillies et al., 2007], wind tunnel and field studies where u*tS/u*tR was measured [Lyles and Allison, 1976; Musick and Gillette, 1990; Lancaster and Baas, 1998], and computer simulations [Li and Shao, 2003].

[8] Because of the spatially and temporally variable distribution of shear stress over rough surfaces, equation (5) was modified by Raupach et al. [1993] to account for the fact that the threshold of particle movement is determined by the maximum stress acting on any point of the surface (τS), not by the spatially averaged stress on the surface. Consequently, an additional parameter (m) was included to account for the spatial averaging of shear stress:

equation image

By Raupach et al.'s [1993] definition, the m parameter (0 < m < 1) scales the maximum shear stress to the shear stress present on a surface with a lower roughness density, such that:

equation image

Raupach et al. [1993] suggested m = 0.5 for flat, erodible, homogeneous surfaces. Evaluation of the model by Wyatt and Nickling [1997] in a field study with shrubs as roughness elements resulted in a lower m value than expected (m = 0.16). They argued that the derived low value of m might have been a consequence of inadequate spatial measurements of τS. Crawley and Nickling [2003] evaluated the m parameter using equation (7) and found that τS was a multiple of τS; however, solving for m resulted in a gross overestimation of R″ when compared to the measured ratios. Therefore, the parameter definition of Raupach et al. [1993] was rejected, leaving no means of defining m independently.

[9] Overall, the model provides good drag partition predictions for staggered arrays [e.g., Marshall, 1971; Crawley and Nickling, 2003]; however, natural surfaces rarely display this configuration. It is uncertain if the drag partitioning of nonstaggered roughness configurations will behave in the same manner as staggered arrays. Sparsely distributed roughness elements under isolated roughness flow will dissipate a portion of the wind's momentum; however, accelerated flow around individual roughness elements within the array may act to increase the shear stress on the surface. Even at higher roughness densities, some distributions of roughness may act to increase local wind shear on the intervening surface, whereas more uniform distributions may affect the wind more evenly over an area [Gillies et al., 2000]. A recent study by King et al. [2008] examining the drag partition for staggered arrays with different sized elements and an unvarying λ found that τS increased with the width of the roughness elements. In addition, results from a field study by Bryant [2004] where the roughness element size was held constant for various values of λ established that the spacing between the elements has a significant role in how stress is partitioned between roughness elements and the surface. These findings suggest that the distribution of stress, as determined by the spacing of the roughness elements, has an effect on the drag partition and this would subsequently influence aeolian sediment entrainment.

[10] The effect of different spatial patterns of surface roughness on the partition of shear stress remains unclear. Therefore, the goal of this study is to evaluate the effect of different spatial arrangements, and the related variation in the distribution of shear stress on the surface, to the partition of drag forces in arrays of roughness elements. A tiered force balance was developed to measure the stress on roughness elements (τR) and the intervening surface (τS) independently and simultaneously in a wind tunnel. This instrumentation measured stresses on arrays of roughness elements with different spatial arrangements with the same value of λ. Point measurements of τS were made within the arrays using Irwin sensors [Irwin, 1980] to examine the distribution of stress partitioning on the surface and to evaluate the m parameter.

3. Methodology

[11] All tests were carried out in the Wind Erosion Laboratory's recirculating wind tunnel in the Department of Geography, University of Guelph [Gillies et al., 2002; Crawley and Nickling, 2003]. The working section is 8.0 m long, 0.76 m high, and 0.92 m wide with a smooth, plywood floor.

[12] Several previous studies have used drag balances to make direct measurements of the stress of the wind on various objects and surfaces [e.g., Marshall, 1971; Grant and Nickling, 1998; Gillies et al., 2000; Nemoto and Nishimura, 2001; Crawley and Nickling, 2003; Gillies et al., 2006, 2007]. In the present study a unique tiered drag balance was used to measure spatially and temporally averaged τS and τR. The individual drag balances used in this study consisted of a 0.499 kg (1.1 lb) SMT Interface force transducer inserted between a free-moving arm and a fixed metal cylinder. Two of these instruments were used in a tiered drag balance similar to that developed by Crawley and Nickling [2003] for the independent and simultaneous measurements of stresses (Figures 1 and 2) . It consisted of two drag plates: the roughness plate to measure τR and the surface plate to measure τS. Plates were constructed from 6.35 mm (1/4 inch) thick varathaned plywood sheets, 0.60 m wide and 1.8 m long. The plywood sheets were attached to aluminum frames that allow for the translational and independent movements of the shear plates. Two force balances are attached to each end of the frame with the upwind translational arms holding the load cells while the rear arms move unrestricted (Figure 2). Each plate was calibrated individually by pulling in a horizontal direction with a fine flexible wire that passed over a pulley to which a series of hanging weights were added incrementally. Correlation coefficients between the applied force and mV output from the transducer ranged from 0.995 to 0.999 for both the surface and roughness plates with the slope coefficients varying by <2.5% for all plates. CR was measured using a single roughness element on the drag balance and CS was measured using the surface drag plate in the absence of any roughness elements.

Figure 1.

Tiered force balance outside the wind tunnel.

Figure 2.

A diagram of the tiered force balance.

[13] For independent measurements of the two drag forces, the roughness elements were required to be situated on top of the surface plate and to connect with the roughness plate below while avoiding contact with the surface. Roughness elements were constructed using film canisters (h = 4.95 mm, b = 31.25 mm) filled with commercial gap sealant that held 75 mm lengths of 6.35 mm threaded rod. These roughness elements passed through 12.7 mm holes in the surface plate and were fixed to the underlying roughness plate. The diameter of the hole was the minimum amount of space required to allow the roughness elements to move freely yet small enough to minimize jetting effects. Since the static pressure in the tunnel as measured with an inclined manometer was the same during operation with the instrumentation in place as without, we can assume that “jetting effects” were negligible.

[14] A space of approximately 2 mm existed between the bottom of the roughness elements and the surface. The tiered force balance instrumentation was located 5.9 m downwind of the beginning of the wind tunnel working area and inserted in a 0.625 m by 1.808 m hole in the wind tunnel floor as shown in Figure 3.

Figure 3.

Tiered force balance in place in the wind tunnel.

[15] Irwin sensors [Irwin, 1980] were also used for measurement of the average and instantaneous shear stress on the intervening surface. These sensors measure the skin friction imposed by the wind close to the surface by measuring a near-surface pressure differential, which enables estimation of surface shear stress in complex, nonuniform airflows [Walker and Nickling, 2003]. Irwin sensors have been used successfully to estimate near surface shear stress in several laboratory wind tunnel and field studies [Monteiro and Viegas, 1996; Wyatt and Nickling, 1997; Crawley and Nickling, 2003; McKenna Neuman, 2003; Gillies et al., 2006, 2007]. The instrument consists of a 12.5 mm diameter brass cylinder with a 2.57 mm center tap operating as a surface pressure port. A second port of 16-gauge stainless steel tubing centered within the inset port measures the pressure at 1.75 mm above the surface. The differential is transmitted to a pressure transducer (ThermBrandt Ltd., 12 mm differential pressure transducer; Model DPT 32S12-0.5; precision ±0.25% full scale) through 0.60 m lengths of 1.59 mm Tygon™ tubing, which translates it to a voltage that is recorded by a datalogger. Each of the 24 sensors was calibrated against the wind profile-derived friction velocity on a plywood surface with no roughness elements present.

[16] Cylindrical film canisters were also used as roughness elements for the wind tunnel floor. Cylinders were used in this study in that they produce a clearly defined and well-known lee-side wake pattern [Iversen et al., 1991] and have been used in previous studies of drag partitioning [Marshall, 1971; Crawley and Nickling, 2003; Bryant, 2004]. The shape and size of the roughness elements remained constant making the variation in the roughness density related only to the number of elements used within each configuration type. Four values of λ (λ = 0.0200, 0.0342, 0.0585, and 0.1100) were examined. These values lie within the range of density values examined in field studies of drag partitioning using natural roughness elements [e.g., Wyatt and Nickling, 1997; Lancaster and Baas, 1998; Lancaster, 2004] and artificial roughness elements [Bryant, 2004; Gillies et al., 2006, 2007] and fall within the range of λ used in laboratory studies [e.g., Marshall, 1971; Musick et al., 1996; Crawley and Nickling, 2003], making the results of this study comparable to previous drag partition research. The number of roughness elements on the wind tunnel floor was 94, 161, 276, and 512 for each respective value of λ.

[17] The use of several patterns of roughness elements facilitated the investigation of different spatial distributions of shear stress. Figures 4a–4d shows the configurations of roughness elements as positioned on segments of wind tunnel floor for each value of λ. The entire length of the wind tunnel was covered with the predetermined densities of roughness elements. The drag partitions for staggered arrays (Figure 4a) were examined since the drag partition prediction of the Raupach et al. [1993] model for this arrangement has been extensively verified [e.g., Marshall, 1971; Gillette and Stockton, 1989; Crawley and Nickling, 2003; King et al., 2005; Gillies et al., 2007].

Figure 4.

Roughness arrays in the wind tunnel for the (a) staggered (λ = 0.0200), (b) streets (λ = 0.0342), (c) clumped (λ = 0.0585), and (d) random (λ = 0.1100) configurations.

[18] Patterns of natural vegetation observed in arid and semiarid locations provided inspiration for the other roughness configurations as they exhibit differences in spacing between roughness elements. The design of the roughness patterns was not intended to reproduce a specific surface or represent vegetation but rather to characterize patterns familiar in the aeolian literature. One arrangement consisted of quasi-randomly spaced roughness elements similar to the “streets” of vegetation observed in arid regions of the U.S. Southwest (Figure 4b). This pattern arises when mesquite shrubs align themselves with the prevailing wind direction [Okin and Gillette, 2001]. Okin and Gillette [2001] hypothesized that “streets have characteristic lengths and are superimposed on a landscape with a more or less random placement.” To stylize this pattern using a set number of cylinders, the pseudo random number function in Microsoft Excel was used to generate coordinates for the location of the streets. Examination of the locations of mesquite dunes in the orthophoto of the Jornada Experimental Range, New Mexico, provided by Okin and Gillette [2001, p. 9675] showed that approximately 20 streets existed in an area that roughly scales to the wind tunnel floor. Three sizes of streets were chosen: 0.12 m × 0.3 m, 0.15 m × 0.45 m, and 0.18 m × 0.60 m. Pseudo random coordinates were generated for the street locations and for the placement of the roughness elements surrounding the streets. The locations of the streets remained the same for each value of λ.

[19] Clumped groups of vegetated roughness in arid regions are often observed [Couteron and Kokou, 1997; Maestra and Cortina, 2002]; therefore, another idealized configuration representing this type of roughness distribution was used (Figure 4c). This type of roughness element arrangement can occur when locations on a surface with slightly higher moisture or nutrient contents produce higher concentrations of vegetation than surrounding soils [Schlesinger et al., 1996; Cross and Schlesinger, 1999]. Coordinates for the clumps were made by generating pseudo random coordinates for the locations of the “central” roughness elements, and then three pseudo randomly placed elements were distributed within a 0.06 m radius of the central roughness element. The number of clumps increased with increasing λ. In addition, some varieties of vegetation found in arid areas are distributed randomly [Fowler, 1986; Haase et al., 1997]. For these arrangements, sets of coordinates generated from the Excel random number function were used to determine the placement of the roughness elements (Figure 4d).

[20] Twenty four Irwin sensors were embedded in the surface plate to make point measurements of shear stress and examine how the spatial distribution of shear stress is affected by the spatial roughness pattern. Measurements of the instantaneous maximum shear stress (τS) were also made in order to calculate R″. For the staggered arrays, the Irwin sensors were placed using a grid system similar to that used by Crawley and Nickling [2003]. Sensors for the nonstaggered arrays were installed at various locations behind, in front of, and adjacent to roughness elements for the nonstaggered configurations. An inestimable number of Irwin sensors would be required to describe the distribution of stress; however, with the limited number of sensors available, several locations on each plate were chosen based on previous estimates of where the maximum shear stress would occur for the evaluation of R″ [Crawley, 2000]. Placements for the remaining sensors were balanced between areas of relatively “open” and “closed” areas.

[21] Measurements of τR, τS, and the distribution of shear stress were taken for each array to evaluate the effect of the spatial distribution of roughness elements on drag partitioning. Each of the 16 surfaces were subject to five freestream velocities measured at 0.37 m above the intervening surface (Uf = 6.58, 8.41, 10.92, 12.92, 14.87 m s−1) to characterize the degree of protection afforded by roughness over various wind conditions. Runs lasted 357 s and stress measurements were made at a rate of 16 Hz.

4. Results

4.1. Tiered Force Balance Measurements

[22] The exposed area of the surface plate and the frontal area of the roughness elements normalized the force measurements from the surface and roughness plates of the tiered force balance. Averages of τR and τS were calculated over the duration of each run. Drag on the roughness elements increased appreciably with increasing Uf for all roughness arrays and strong power relationships were observed between the measured τR and Uf for each configuration and λ (r2 = 0.994 to 1.0). The slopes of these relationships remained relatively constant for all arrays (average slope = 2.25 ± 0.13). Stress on the surface also increased with increasing Uf, with correlation coefficients from power regressions ranging from 0.991 to 1.0. The rate of change in τS with Uf was also relatively constant and lower than the τR relationships (average slope = 1.99 ± 0.17).

[23] The stress on the roughness increased approximately log-linearly with increasing λ for each roughness arrangement, and this relationship is displayed for a representative Uf in Figure 5a. A negative log-linear relationship was found between τS and λ, with τS decreasing because of the increasing shelter provided by the roughness elements (Figure 5b). The total shear stress was calculated using Schlichting's drag partitioning theory (equation (2)). The increase in τ with λ for all roughness arrangements followed closely with the increase in τR with λ (Figure 5c), confirming that the chief effect of an increase in roughness is to increase τ for equivalent Uf values [Raupach et al., 1993; Musick et al., 1996]. The extremely high correlation between both drag forces and Uf measured by the drag plates indicates that the instrumentation is well suited to measure the forces on individual components of a surface. The intervening surface and roughness shear plates responded in a predictable manner to increases in λ; therefore, the new force balance design can be considered appropriate for quantifying the drag partition of roughness arrays. Results from these measurements also aid to establish that drag forces involved in wind erosion processes are highly orderly and predictable.

Figure 5.

Graphs of (a) τR measured by the roughness shear plate, (b) τS measured by the surface shear plate, and (c) the total shear stress (τ = τR + τS), all plotted versus log(λ) for a representative freestream velocity (Uf = 12.96 m s−1).

[24] The shear stress ratio (R = (τS/τ)0.5) represents the proportion of the total momentum flux applied to the surface [Wolfe and Nickling, 1996] and can be used to describe the extent of sheltering afforded by obstacles on surfaces. This drag partition was calculated for each array and Uf combination using the force balance measurements. The results are plotted against λ in Figure 6. Values of R were slightly scattered at each value of λ, which is the result of a slight dependency of R on Uf. R decreased with increasing Uf, as τR and τS increased with freestream velocity at different rates.

Figure 6.

Measured values of R for all roughness configurations and all freestream velocities plot as a function of roughness density.

[25] Aside from the random arrays examined by Marshall [1971], this study has provided the first data set detailing direct measurements of the drag partition for nonstaggered arrays to evaluate the effect of roughness configuration on the drag partition. In Figure 6, it can be seen that the measured drag partitions for each array at all λ's are essentially overlapping one another, suggesting that the arrangement of roughness elements has little effect on R.

[26] A two-way ANOVA test was performed on the dataset to evaluate the small visual differences in R between the configurations. The test confirmed that an interaction does exist between the roughness configuration and the value of λ (p < 0.05), thus the differences between the configurations were examined. Any minor differences between configurations at λ = 0.0200 and 0.1100 were not significant at the 95% confidence level (0.09 < p < 0.92). Drag partitions for the random and staggered array were not significantly different at any λ, nor did any significant differences in R values exist between the clumped and street arrays for any λ. However, it was found that the drag partitions for the staggered arrays at λ = 0.0342 and 0.0585 were significantly different from the clumped and street arrays at those roughness densities (p < 0.05).

[27] At the intermediate λ, τ for the staggered array was the highest, but the amount of stress partitioned to the surface was less than the other configurations. The random arrays had slightly lower values of τ and τS; therefore, even though the quantities of stresses differ between the surfaces, the overall drag partition is similar. The clumped and street arrays generated τ similar to those of the staggered and random arrays but were less effective at partitioning stress from the surface.

4.2. Point Measurements of Surface Shear Stress

[28] Aeolian researchers have generally assumed that the arrangement of roughness elements on an erodible surface will have an effect on the distribution, frequency, and magnitude of stresses at the surface [e.g., Gillette, 1999; Okin and Gillette, 2001; Okin, 2005]. To evaluate this, the Irwin sensor data were used to provide information on the shear stress distribution within the different roughness configurations, along with τS. The magnitude of τS, calculated as the spatial and temporal average of all 24 sensors, and τS, the maximum value of shear stress at one point, increased with Uf as a power function for all roughness configurations and λ (r2 = 0.992 to 1.0). Maximum shear stress values were on average 1.9 times greater than the average values. All configurations displayed a log-linear decrease in τS and τS with increasing roughness λ, as shown in Figure 7 for a representative Uf. No significant differences in τS were found between the configurations (p = 0.48), suggesting that the roughness configuration has little effect on the value of τS.

Figure 7.

Average and maximum values of surface shear stress measured by the Irwin sensors for a representative freestream velocity (Uf = 12.92 m s−1).

[29] The maximum drag partition was calculated from these data, such that R″ = (τS/τ)0.5. Maximum shear stress ratios were on average 1.27 times greater than the R values measured by the force balance. These ratios also decreased log-linearly with increasing λ (Figure 8) and followed the general shape of the Raupach et al. [1993] model. A greater degree of scatter was observed for the maximum shear stress ratios than the mean shear stress ratio as R″ was found to be slightly more dependent on Uf than R is. The ratios for each configuration essentially coincided with one another when plotted as a function of λ. A two-way ANOVA test confirmed that the R″ values between configurations were not significantly different (p = 0.48).

Figure 8.

Plot of the maximum shear stress ratios versus log (λ) for all roughness configurations and freestream velocities.

[30] Frequency distributions of all the shear stress measurements from all 24 Irwin sensors were also generated to visualize the occurrence of higher-magnitude forces at the surface and any differences between λ and configurations. Examples of the frequency distributions of surface shear stress from the Irwin sensor array for the different roughness configurations for one free stream wind speed are shown in Figure 9. The overall form of the distributions approximates the normal. Sheltering effects on individual Irwin sensors and incomplete coverage of the surface cause the departure from a smooth form. The statistics for the spatial measurement of τS (mean, standard deviation, and skewness) for all the test runs are provided in Table 1. Skewness generally increased with λ for all roughness arrays, reflecting the increasing degree of sheltering of the surface. Otherwise there were no consistent differences in the observed frequency distributions of τS between the different configurations at each λ.

Figure 9.

Histograms generated from spatial measurements of τS from the array of Irwin sensors for λ = 0.0200 at a representative freestream velocity (Uf = 12.92 m s−1).

Table 1. Mean, Standard Deviation and Skewness Statistics for τS for Each Roughness Configuration and Freestream Velocity Test
Roughness ConfigurationUf (m/s)λ = 0.0200λ = 0.0342λ = 0.0585λ = 0.1100
Mean τ′s (N/m2)Std. Dev. (N/m2)SkewnessMean τ′s (N/m2)Std. Dev. (N/m2)SkewnessMean τ′s (N/m2)Std. Dev. (N/m2)SkewnessMean τ′s (N/m2)Std. Dev. (N/m2)Skewness
Staggered6.580.060.02−0.310.050.020.200.040.020.530.030.010.77
 8.710.100.03−0.540.080.030.180.060.020.450.050.020.59
 10.920.140.04−0.660.110.030.130.090.030.370.060.030.50
 12.920.180.05−0.710.140.040.040.120.040.330.080.030.38
 14.870.230.06−0.700.180.05−0.060.140.050.290.100.040.32
Clumped6.580.060.03−0.050.040.020.040.040.020.070.030.010.52
 8.710.100.04−0.240.070.03−0.010.070.03−0.130.050.020.37
 10.920.140.05−0.310.100.05−0.090.100.04−0.220.070.030.31
 12.920.180.07−0.340.130.06−0.070.130.05−0.260.090.040.22
 14.870.220.08−0.340.160.07−0.070.160.06−0.230.110.040.18
Random6.580.050.020.060.050.02−0.010.040.020.210.030.010.79
 8.710.090.030.050.070.03−0.220.060.020.020.040.020.53
 10.920.120.050.050.110.04−0.320.090.03−0.090.060.020.38
 12.920.160.060.000.140.05−0.340.110.04−0.140.080.030.29
 14.870.200.07−0.020.170.06−0.330.140.04−0.120.100.030.23
Streets6.580.060.02−0.060.050.020.140.040.020.170.030.020.60
 8.710.090.03−0.300.080.03−0.080.070.030.140.050.020.55
 10.920.140.04−0.400.110.04−0.200.100.040.100.070.030.53
 12.920.170.05−0.460.140.04−0.240.130.050.060.100.040.52
 14.870.220.06−0.450.180.06−0.280.160.06−0.010.120.050.49

5. Analysis and Discussion

5.1. Drag Partitions

[31] The limited amount of research on the interaction between the wind and irregular rough surfaces has led to uncertainty regarding how natural patterns of roughness elements affect wind erosion. Verification of the “shear stress partitioning problem,” as stated by Raupach [1992], for a range of nonstaggered arrays of roughness elements is an essential first step in any endeavor to apply drag partitioning models to various terrains. Results from the tiered force balance measurements clearly show a log-linear decrease in R with increasing λ, thus demonstrating the increased sheltering effect of all arrangements of roughness elements (Figure 6). The lack of difference in R between configurations supports the assertion of Marshall [1971] and Raupach [1992] that the arrangement of roughness elements has little effect on drag partitioning. To illustrate this finding, values of R from this study were compared with those measured with similar instrumentation using staggered arrays by Marshall [1971] and Crawley and Nickling [2003] in Figure 10. Drag ratios for all configurations fall within the range of previously measured values of R. From Figure 7, it is apparent that the values of R measured in earlier studies for various staggered arrays are somewhat scattered as well.

Figure 10.

Values of R measured with the tiered force for all roughness configurations and Uf's plot as a function of log (λ), along with values measured by Crawley and Nickling [2003] and Marshall [1971].

[32] However, the slightly higher R values measured for the street and clumped arrays at the intermediate λ suggests that these configurations were less effective at dissipating the stress that was generated than the staggered or random arrays. An examination of drag measurements for these types of arrays using a finer series of intermediate λ would aid in determining the range of λ where roughness configuration has a minor effect on drag partitioning.

[33] Maximum drag partitions were very similar for all configurations at all values of λ, indicating that R″ was not affected by roughness configuration. The measured values of R″ and those available from the literature are plotted against λ in Figure 11 to aid in verifying this result. R″ values measured in this study for all configurations adhered to the same trend as reported by Crawley and Nickling [2003] who measured R″ for staggered arrays with instrumentation similar to that used in this study, although their values are slightly lower than those observed in this study. Small discrepancies could be attributed to differences in cylinder aspect ratio and different values of Uf. Measured values of R″ from the present study were also compared against ratios where the actual threshold friction velocity ratio was measured [Musick and Gillette, 1990; Musick et al., 1996; Lancaster and Baas, 1998]. Values from this study followed very closely to those of Musick et al. [1996], where the Rt was measured for staggered arrays of cylinders. The close adherence of these measurements of R″ to those estimated from measurements of actual sediment movement emphasizes the importance of using a maximum shear stress ratio as opposed to an average.

Figure 11.

Measured values of R″ from the current study plotted as a function of log λ with previously measured maximum shear stress ratios.

5.2. Raupach et al. [1993] Model

[34] Results from earlier studies have been used to verify the predictions of the Raupach et al. [1993] model [e.g., Marshall, 1971; Musick et al., 1996; Crawley and Nickling, 2003] and the acceptance of the robust qualities of the model is apparent in the literature [e.g., King et al., 2005; Okin, 2005; Shao and Yang, 2005]. However, the drag partition predictions for nonstaggered arrays had yet to be evaluated. Because the majority of roughness features on natural surfaces are not staggered, this knowledge gap creates uncertainty about the applicability of the results of these studies to natural surfaces and the validity of drag partition predictions for nonstaggered arrays.

[35] Proper parameter values are crucial because significant over or underestimation of R″ may result from incorrect values of m or β. Previous research involving the parameterization of the Raupach et al. [1993] drag partition model produced varying values of β and m specific to each study [Wolfe and Nickling, 1996; Wyatt and Nickling, 1997; Luttmer, 2002; Crawley and Nickling, 2003]. The independent measurements of CS and CR from this study allowed for a direct calculation of β. Measured values of β from the current study increased linearly (r2 = 0.976) from 158 to 248 over the range of Uf's as a result of the decrease in CS and increase in CR with wind velocity. Values of β are presented in Table 2. Dependency of β on Uf has not been reported in previous studies; however, different β values for each Uf resulted in a good degree of agreement between measured drag partitions and predictions by the Raupach et al. [1993] model. This dependency arises from the fact that the calculated CS values were decreasing with increasing u* as the flow over this surface (varathaned plywood) was dynamically smooth. Borrmann and Jaenicke [1987] measured the aerodynamic roughness length (zo) of a similar smooth plywood surface to be 2.7 μm, indicating surfaces of this type are exceedingly smooth. Although the physical roughness of the smooth plywood surface was not measured we can safely assume that it is less than 0.1 mm. The roughness Reynolds number (ReR = u*h/ν, where h is a geometrical surface roughness parameter and ν is kinematic viscosity), which is used to define the state of the flow regime [Kondo and Yamazawa, 1986], was less than 5 for all flow conditions, assuming an h value ≤0.1 mm, for which CS was measured indicating smooth flow conditions. The scaling of β observed in this study is less likely to occur in wind flow over natural terrestrial surfaces due to their physically rough nature, which creates conditions for dynamically rough flow even at low wind speeds.

Table 2. Measured Values of β for Each Freestream Velocity and Values of m for Each Uf and λ
Uf (m s−1)βm
λ = 0.0200λ = 0.0342λ = 0.0585λ = 0.1100
6.58158.40.490.430.440.37
8.71190.40.540.480.480.41
10.92203.70.570.510.500.43
12.92220.20.580.530.510.45
14.87248.30.580.550.510.46

[36] To evaluate the predictive capacity of the Raupach et al. [1993] model for nonstaggered arrays, the values of R from the tiered force balance were compared against those predicted by equation (5). The measured values of R are plotted against λ in Figure 12 with the prediction lines of the Raupach et al. [1993] model. The m parameter was set at 1, as specified by Raupach et al. [1993], and the measured β values were used in the model calculation. Overall, there was a good agreement between the calculated and measured values when a different β is used for each Uf, and no significant differences were found for any configuration. Least squares regressions with forced-zero intercepts were carried out to further compare the model prediction to the measured R. Generally, all configurations displayed similar correlation coefficients (average r2 = 0.96 ± 0.03) and slopes (average slope = 0.95 ± 0.04). King et al. [2005] evaluated the predictive capacity of the Raupach et al. [1993] model by comparing model predictions to measured values of R from the literature and found similar results in that a good degree of agreement was found between the measured and predicted values (r2 = 0.89, slope = 0.94 ± 0.07). These comparisons reveal that the model slightly underpredicts the drag partition, which may be a result of an overapproximation of the effective shelter area of the roughness elements. The size and shape of wakes behind solid obstacles in a turbulent freestream are generally understood [Raupach et al., 1991; Yang and Shao, 2005]; however, the complexities of the flow within a roughness array may compromise the assumptions of the Raupach [1992] model. Further investigation into the behavior of wakes may provide the necessary information on the shelter area provided by roughness elements, and thus adjustments to some of the parameters in the derivation of the Raupach [1992] model.

Figure 12.

Graph of measured and predicted values of R plotted as a function of log (λ). Lines represent values of R predicted by the Raupach et al. [1993] model with measured values of β for each freestream velocity. Points represent R-values measured by the tiered force balance for each configuration and freestream velocity.

[37] Despite these underestimated values, the equation works equally well for staggered roughness arrangements and more randomly arranged surface features. Li and Shao [2003] performed a numerical simulation on two small random arrays of roughness elements with the same value of λ and elements of varying dimensions and found similar results. They concluded that the application of the Raupach [1992] model to irregular arrays of nonuniform cylinders to predict drag partitions produces only a relatively small error when compared to the prediction for staggered arrays. Raupach et al. [2006] also concluded from their wind tunnel study that clustered roughness elements increase the local horizontal heterogeneity in the wind field compared to equivalent staggered arrays, but any difference in the momentum absorption between the two types of roughness distributions could not be distinguished.

[38] Predictions of R″ are complicated by the inclusion of the m parameter in the Raupach et al. [1993] model, which is necessary to adjust the average drag partition to that of a maximum. Values of m suggested by previous researchers [Wolfe and Nickling, 1996; Wyatt and Nickling, 1997; Crawley and Nickling, 2003] show that no consistent value is suitable for most surfaces [King et al., 2005]. Results from the Irwin sensor measurements in this study show a relatively consistent relationship between τS and τS and was therefore used to estimate m; therefore, a simple approach similar to that of Luttmer [2002] was used for the calculation of this parameter. Assuming that m is represented by the difference between τS and τS and that the Irwin sensors adequately represent the spatial distribution of surface shear stress, its value can be estimated by:

equation image

An inverse linear relationship arose between m and λ due to the decreasing difference between τS and τS with λ. ANOVA tests confirmed that no significant differences existed for the m values between the configurations. Therefore, values of m at each Uf and λ were averaged over the configurations and the values are presented with the β parameter in Table 2.

[39] Values of R″ measured in this study were compared to the drag partition predicted by the Raupach et al. [1993] model using each value of β and the appropriate value of m from Table 2 to evaluate the performance of equation (6) for nonstaggered arrays. Figure 13 displays the comparison between the measured and predicted values of R″ for a representative Uf. A good level of agreement was observed between the measured and predicted drag partitions, as the changes in m and β with Uf and λ relate closely to the response of R″ to those variables. Any minor differences between measured and predicted values were insignificant for all configurations. This comparison is exceptional considering that both the m and β parameters were solved for independently. Previous researchers have found strong relationships between measured and predicted values only when one parameter is solved for residually [e.g., Musick et al., 1996; Wyatt and Nickling, 1997; Crawley and Nickling, 2003]. Results from this study indicate that a good correlation between measured and predicted values of R″ can be achieved using m values derived from the relationship between τS and τS. However, additional experimental data are required to assess the relationship between these values and the m parameter, as the calculated m values are specific to this study.

Figure 13.

Values of R″ predicted by the Raupach et al. [1993] model plot against measured values for a representative freestream velocity (Uf = 12.92 m s−1). The β value for this Uf and individual m values for each λ were used in the model prediction of R″.

6. Conclusions

[40] The measurement of the drag on the surface and the roughness elements, and point measurements of surface shear stress for arrays of roughness elements with the same roughness density and different configurations provided the information required to evaluate the effect of roughness configuration on drag partitioning. Drag partition decreased with increasing λ for these arrays, thus confirming the increase in surface roughness enhanced the sheltering of the surface, regardless of roughness configuration. Irregular arrays displaying this degree of spacing and roughness density can therefore be considered analogous to staggered configurations in terms of drag partitioning, as no major differences were found between the R and R″ values of each configuration. This is advantageous as staggered arrays generally reduce the number of surface-characterizing variables and is reasonably described by the λ parameter. This configuration is also logistically simple to set up in wind tunnel experiments [Marshall, 1971]. However, results from previous studies indicate that the roughness arrangement has an effect on sediment flux [Gillette and Pitchford, 2004; Okin and Gillette, 2004]; therefore, the relationship between roughness arrangement and actual sediment entrainment, transportation, and deposition requires evaluation.

[41] Results from this study also provided a means to evaluate the parameters of the Raupach et al. [1993] drag partitioning model, and assess the model's applicability to heterogeneous surfaces. The Raupach et al. [1993] model is capable of predicting the average drag partition for irregular arrays of roughness elements, although there is a slight degree of underprediction. Additional studies on the properties of wakes and sheltering areas for several different types of roughness elements would aid to improve the model predictions. The evaluation of the Raupach et al. [1993] model emphasizes the necessity of proper parameters for accurate predictions of R. The dependency of β on Uf complicates the convenience of the model, yet the range of measured β values provided a better prediction of R than a single value. However, the scaling of β in this experiment is a result of the very smooth surface that the roughness elements were placed on, which resulted in dynamically smooth flow in which CS scales with u*. Using a variable CS created a variable β, but this is less likely to occur over natural surfaces where wind erosion occurs.

[42] The nature of element drag coefficients within a roughness array needs to be examined in more detail in order to provide a collection of β values for different types of roughness elements, particularly for complex obstacles such as vegetation. The high degree of agreement between the Raupach et al. [1993] model prediction and measured R″'s implies that this simple relationship between τS and τS can provide a satisfactory m parameter. Establishing a data set of m values for a range of λ and Uf values from the relationship between τS and τS will aid to improve the practicality of the model.

Notation
b

element breadth, m.

CR

drag coefficient of an individual roughness element, dimensionless.

CS

drag coefficient of the unobstructed surface, dimensionless.

F

total drag force imparted upon a surface with roughness elements, N.

FR

drag force on the roughness elements, N.

FS

drag force exerted on the intervening surface, N.

h

element height, m.

m

empirical constant between 0 and 1.

n

number of roughness elements occupying the ground area of the roughness array.

r2

correlation coefficient.

R

friction velocity ratio, dimensionless.

R

maximum friction velocity ratio, dimensionless.

ReR

roughness Reynolds number, u*h/ν, dimensionless.

Rt

threshold friction velocity ratio, dimensionless.

S

surface area, m2.

Uf

freestream wind speed, m s−1.

u*

wind friction velocity, m s−1.

u*t

threshold wind friction velocity, m s−1.

u*tR

threshold wind friction velocity with roughness elements, m s−1.

u*tS

threshold shear friction of bare surface, m s−1.

zo

aerodynamic roughness length, m.

β

ratio of element to surface drag coefficients, dimensionless.

λ

roughness density, dimensionless.

ν

kinematic viscosity, m2 s−1.

σ

roughness element basal area to frontal area ratio, dimensionless.

τ

total surface shear stress, N m−2.

τS

surface shear stress, N m−2.

τS

surface shear stress on the area not covered by the roughness elements, N m−2.

τS

maximum surface shear stress on the area not covered by the roughness elements, N m−2.

Ancillary