A wind-tunnel study of windbreak drag

Agricultural and Forest Meteorology 118 (2003) 75–84

A wind-tunnel study of windbreak drag Dexin Guan a,∗ , Yushu Zhang b , Tingyao Zhu a a

b

Institute of Applied Ecology, Chinese Academy of Sciences, Shenyang 110016, PR China Institute of Atmospheric Environment, China Meteorological Administration, Shenyang 110016, PR China Received 22 February 2002; received in revised form 14 March 2003; accepted 17 March 2003

Abstract Drag coefficient is an important index of the wind protection effect of a windbreak. Although there have been many studies on this, there are very few with direct measurements, especially those on realistic windbreaks. We conducted wind-tunnel experiments on simulated models of narrow and realistic windbreaks with different porosities. The surface boundary layer on the wind-tunnel floor was simulated with a thickness of 600 mm. The model, with height (H) of 100 mm, was immersed in the simulated surface boundary layer during measurements. The ratio of the model height to roughness length of the surface (H/z0 ) was 774. Two indices, optical porosity (β) and aerodynamic porosity (α), were used. The bleed wind speed of the model was measured to calculate aerodynamic porosity. The relative bleed speed was generally lower at the canopy crown and higher at the bottom surface. Dense model canopy had smaller standard deviation (S.D.) in aerodynamic porosity along the crosswind direction. The relationship between α and β resulted in α = β0.4 for realistic windbreak models. The divergence of α and β was largest near 0.5 and 0.2, respectively and then decreased as the windbreak became denser or looser. The drag (D) and drag coefficient (Cd ) of the models decreased with increasing porosity. The model Cd remained nearly constant within the range of Reynolds number measured in this study. The resultant Cd of our realistic model was lower than that of other studies using two-dimensional windbreaks with the same optical porosity. Our empirical relationship between Cd and α (i.e. Cd = 1.08(1 − α1.8 )) agreed well with that of two-dimensional windbreaks reported in the literature, suggesting that α was a better measure as porosity index than β. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Realistic windbreak; Two-dimensional windbreak; Optical porosity; Aerodynamic porosity; Drag coefficient

1. Introduction Windbreaks are used throughout the world to improve surface climate and soil conditions for human and animal life and crop growth. Recently, large-scale agro-ecosystems consisting of vegetative windbreaks have been established for environmental sustainability (e.g. Wang and Takle, 1997a). The main effect of ∗ Corresponding author. Tel.: +86-24-23916272; fax: +86-24-23843313. E-mail address: [email protected] (D. Guan).

a windbreak is its aerodynamic influence on surface airflow, resulting in changes in microclimate, soil climate and crop growth. In principle, the windbreak exerts a drag force on the wind field, causing a net loss of momentum in the incompressible airflow and thus a sheltering effect (Rain and Stevenson, 1977). A common physical way to express the aerodynamic effect of a windbreak is in terms of its resistance to the flow, or in terms of dimensionless form such as a drag coefficient, Cd (Jacobs, 1985). Thus, there have been many studies of windbreak drag and drag coefficient.

0168-1923/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-1923(03)00069-8

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The windbreak drag or drag coefficient depends on the structure of a windbreak, which may be divided into two categories: artificial windbreaks (e.g. thin screens, fences) and vegetative windbreaks. The former is always constructed with optical porosities and is considered as “non-thickness” in aerodynamic analysis. Hence, these windbreaks are generally called “two-dimensional windbreaks.” In contrast, the latter has width and internal structure. Most studies on windbreak drag are designed for two-dimensional windbreaks. In earlier studies, the drag of windbreaks was determined indirectly. Woodruff et al. (1963) developed a method for calculating the drag of a shelterbelt based on a study of Betz (Schlichting, 1960). Hagen and Skidmore (1971) utilized this method to evaluate the drag of slat fence windbreaks. Seginer and Sagi (1972) and Seginer (1972) reformulated the momentum equation and proposed an improved method. Miller et al. (1975) and Schwartz et al. (1995) used this method to estimate the Cd of a very young tree windbreak and a forage sorghum windbreak. Guyot (1978) obtained expressions for the Cd of thin artificial windbreaks. Wilson (1985) and Wang and Takle (1997b) modeled Cd using closure schemes. The only direct measurements known to the authors are those of Seginer (1975) and Jacobs (1985), both using the two-dimensional windbreak. The former was on a fence with optical porosity of 0.5 and the latter was on a closed fence. Natural windbreaks are more common in extensive field studies. Their structures are different from those of so-called two-dimensional windbreaks. Direct measurement of their drag is critical to understanding the mechanism of their shelter effect. This has been difficult to carry out because natural trees or plants are often too big to measure. The alternative approach is to measure the drag of realistic windbreak models in a wind tunnel. Woodruff et al. (1963) attempted such an experiment but the models were widely designed and the porosities were not given. Windbreaks or shelterbelts on farmland are typically narrow in order to minimize their area. This paper aims to measure the drag of narrow and realistic windbreak models in a wind tunnel; to analyze the influence of the porosity of a realistic windbreak on its drag; and to compare the results with those of two-dimensional windbreaks in the literature.

2. Wind-tunnel experiment on windbreak models 2.1. Simulation of surface boundary layer with neutral stratification The experiments were carried out in the environmental wind tunnel of Beijing University’s Environmental Science Center. The test section of the wind tunnel was 32 m long, 3 m wide and 2 m high. The wind speed in the tunnel can be adjusted from 0.2 to 25 m s−1 , and was measured by a hot wire anemometer (DISA-55M). The anemometer was installed on a frame in the wind tunnel that can be displaced precisely (within 0.1 mm) in three dimensions by a digital operation system outside the tunnel. In a neutrally stratified flow, the wind profile in the surface boundary layer fits the well-known logarithmic function: u∗ z − d u¯ 0 (z) = (1) ln κ z0 where u0 (z) is the wind speed at height z, u∗ the friction velocity, κ (=0.4) the von Karman constant, z0 the roughness length of the surface and d the zero-plane displacement. To simulate the surface boundary layer, a layer of gravel (3–4 mm) was uniformly spread on the floor to increase the surface friction. We assumed d = 3 mm for the rough surface. The drag measuring position was 26 m downstream from the entrance of the test section of the wind tunnel. Wind profiles at 23, 24, 25, 26, 27, and 28 m were measured. The surface boundary layer was well formed, with a thickness of 600 mm at every fan rotation speed. Friction velocity and roughness length were estimated from the measurements according to Eq. (1). Average roughness length for these positions was 0.131 mm with a standard deviation (S.D.) of 0.036 mm. As an example, Table 1 summarizes the results of the measurements and computations in this experiment. 2.2. Models of trees and windbreaks A plastic tree was used to simulate a “crown” and a nail was tightly inserted into the stem sheath of the crown to create a “trunk.” In this way a tree model was constructed. The height of the model tree was

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Table 1 Wind speed (m s−1 ) profiles, u∗ , z0 , correlation R2 and H/z0 (where H is the height of windbreak models and is 100 mm) at drag measuring position at different rotation speeds of the fan Rotation speed of fan (min−1 ) 100

Mean

150

300

400

440

500

550

z − d (mm) 20 40 60 80 100 150 200 250 300 400 500 600

1.19 1.32 1.45 1.52 1.59 1.65 1.73 1.80 1.83 1.89 1.95 1.99

1.33 1.51 1.63 1.70 1.72 1.86 1.95 1.97 2.02 2.11 2.18 2.25

2.52 2.88 3.05 3.21 3.32 3.49 3.66 3.76 3.87 4.02 4.12 4.21

3.01 3.43 3.68 3.85 3.97 4.23 4.39 4.51 4.61 4.78 5.01 5.05

3.37 3.84 4.06 4.28 4.41 4.69 4.89 5.05 5.17 5.32 5.50 5.63

3.69 4.18 4.45 4.68 4.85 5.11 5.34 5.46 5.63 5.86 6.00 6.19

4.07 4.66 4.92 5.18 5.31 5.68 5.94 6.09 6.25 6.42 6.64 6.81

u∗ (m s−1 ) z0 (mm) R2 H/z0

0.095 0.140 0.997 714.3

0.105 0.128 0.995 780.8

0.199 0.127 0.999 788.5

0.240 0.133 0.998 751.9

0.266 0.128 0.998 781.9

0.291 0.128 0.999 781.3

0.359 0.122 0.998 819.7

100 mm and the diameter of the crown was 50 mm. The model trees were nailed on plywood in rows to make windbreak models. To eliminate the opening or gap at trunk height, simulated (plastic) conifer needle bunches were inserted to some trunks to simulate “shrubs.” Seven windbreak models with various porosities were constructed by varying row spacing, tree spacing and shrub spacing. The width of the model windbreaks was designed to range from 100 to 140 mm to resemble natural narrow windbreaks. The structural characteristics of the models are listed in Table 2. Photographic silhouettes of models 1, 4, and 7 are shown in Fig. 1.

0.129 0.998 774

2.3. Optical porosity (β) and aerodynamic porosity (α) Two indices, optical porosity (β) and aerodynamic porosity (α) were used to describe the porosity of the windbreak. Optical porosity was defined as the ratio between the open surface and the total surface of the windbreak. It was measured using digitized photographs, following Kenney (1987). Optical porosities ranged from 0.016 to 0.389, depending on the model structure (Table 2). The aerodynamic porosity was defined as the ratio of mean wind speed immediately leeward of the

Table 2 Structural characteristics of the windbreak models Model number

Number of rows Row spacing (mm) Tree spacing (mm) Rows with “shrubs” (downwind direction) Shrub spacing (mm) Optical porosity Aerodynamic porosity

1

2

3

4

5

6

7

4 30 30 1, 3 30

4 30 30 1, 3 60

3 25 40 2 40

3 25 50 2 50

3 25 60 2 60

2 50 40 1 40

2 50 50 1 50

0.016 0.133

0.058 0.303

0.120 0.401

0.142 0.450

0.185 0.503

0.295 0.605

0.389 0.685

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Fig. 1. Photographs of windbreak models with optical porosity: (a) 0.016, (b) 0.142, and (c) 0.389.

windbreak under the windbreak height to that of an open field provided wind direction was perpendicular to the windbreak. Values of α varied in windbreak extending direction because β was generally not uniform in that direction. We considered an infinitely long windbreak on flat ground with the z-axis upward, x-axis horizontally perpendicular, and y-axis parallel to the windbreak. Hence, aerodynamic porosity at any position on y-axis is expressed as
H (1/H) z0 u(y, ¯ z) dz α(y) =
H (1/H) z0 u¯ 0 (z) dz
H ¯ z) dz z0 u(y, (2) = (u∗ /κ)[(H/z0 )(ln(H/z0 ) − 1) + 1] where u(y, z) was the leeward edge wind speed (bleed wind speed) and u0 (z) was the approaching wind speed as expressed in Eq. (1).

In the experiment, we measured bleed wind speed in the y–z plane and approaching wind speed profile (upwind 1.5 m of the model windbreak) to calculate α(y). The measurement position of bleed wind speed is shown in Fig. 2. The windbreak model was fixed on the wind-tunnel floor perpendicularly to the wind direction, spanning the whole width of the tunnel. The x-axis (on the floor) passed through the base point P1 of a tree (in the leeward row) in the central part of the model. The origin of the coordinates (point O) was placed 28 mm downstream from P1 . The adjacent tree’s base point was P2 and projection of P2 on y-axis was L. Ten equidistant points in [0, L] were selected as the measurement positions of bleed speed. With a tree spacing P1 P2 = L, the designated positions on y-axis were at 0, L/9, 2L/9, 3L/9„ 8L/9, and L. At each position, wind speeds were measured at five heights (i.e. z − d = 20, 40, 60, 80, 100 mm, respectively). Also,

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Fig. 2. Sketch map of coordinate positions of bleed wind speed measurement.

α(y) was estimated from Eq. (2) and the integration was calculated using a trapezoidal formula. The relative bleed speed, u(z)/u0 (z), varied along the z- and y-axis as shown in Table 3. It was generally lowest at crown height (60 and 80 mm) but higher at

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the levels near the trunk (20 and 40 mm) and canopy top (100 mm). In the y direction, u(z)/u0 (z) varied obviously at the “trunk.” For porous or medium models, the results of u(z)/u0 (z) > 1 appeared occasionally at the trunk level when the probe was positioned facing a gap. For dense models, however, the distribution of u(z)/u0 (z) was relatively uniform along the z- and y-axis. The aerodynamic porosity of the models was relatively uniform with smaller S.D. along the y-axis for denser models (No. 1 and No. 2). For other models, S.D. increased with greater porosity (Table 4). More porous model had more inhomogeneous gaps, resulting in greater differences in α at different positions (on y-axis). The calculated mean aerodynamic porosities ranged from 0.133 to 0.685 (Table 2). Previous studies have used β more frequently than α because the former is easier to measure. However, α has more physical meaning in terms of aerodynamics. For practical purposes, it is useful to quantify the relationship between α and β. Based on the results from this experiment, an empirical relationship between α

Table 3 Relative bleed speed u(z)/u0 (z) of model 3 (β = 0.120) z − d (mm)

Y

Mean

0

L/9

2L/9

3L/9

4L/9

5L/9

6L/9

7L/9

8L/9

L

20 40 60 80 100

0.44 0.45 0.32 0.28 0.45

0.32 0.54 0.31 0.22 0.51

0.37 0.46 0.35 0.33 0.49

0.41 0.48 0.38 0.31 0.53

0.41 0.52 0.41 0.33 0.49

0.51 0.49 0.36 0.28 0.53

0.48 0.52 0.35 0.31 0.49

0.46 0.49 0.33 0.31 0.46

0.41 0.48 0.31 0.29 0.46

0.42 0.46 0.31 0.26 0.49

0.42 0.49 0.34 0.29 0.49

Mean

0.39

0.38

0.40

0.42

0.43

0.43

0.43

0.41

0.39

0.39

0.41

Mean

S.D.

0.133 0.303 0.401 0.450 0.503 0.605 0.685

0.011 0.012 0.019 0.023 0.028 0.034 0.032

Table 4 Aerodynamic porosity (α) of the models and their standard deviation (S.D.) Model No.

1 2 3 4 5 6 7

Y 0

L/9

2L/9

3L/9

4L/9

5L/9

6L/9

7L/9

8L/9

L

0.115 0.285 0.382 0.402 0.462 0.551 0.612

0.121 0.294 0.378 0.422 0.496 0.612 0.687

0.135 0.312 0.402 0.461 0.518 0.653 0.731

0.136 0.306 0.421 0.458 0.566 0.608 0.721

0.151 0.306 0.426 0.466 0.502 0.654 0.698

0.124 0.318 0.421 0.479 0.478 0.632 0.666

0.131 0.321 0.418 0.465 0.523 0.586 0.654

0.142 0.304 0.401 0.459 0.491 0.604 0.716

0.138 0.292 0.386 0.451 0.514 0.586 0.723

0.138 0.292 0.376 0.436 0.492 0.566 0.643

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Fig. 3. Correlations between aerodynamic porosity α and optical porosity β in realistic models (䊉) of this experiment and two-dimensional models (䊊) of Zhu et al. (2001).

Fig. 4. Correlation between β and α − β for realistic models: measurements (䊉) and empirical relation α = β0.4 (—).

2.4. Drag measurements of model windbreaks and β was obtained as α = β0.4

(3)

The related data and the empirical curve are shown in Fig. 3. Their agreement was very good. The representative measurement of α is difficult. The wind speed at the leeward edge varied at different points. The bleed speed was higher at gaps or holes and lower at slats or bars of the windbreak. Previous studies of α–β relationship are seldom found and the only report known to the authors is that of Zhu et al. (2001). Their data were obtained from the wind-tunnel experiment of perforated plywood windbreaks. They pointed out that the calculated values of α for their models in the range of low β (<0.2) had larger errors than those in the higher range of β (Fig. 3). This was because low β models caused jet flow at leeward edges and it was difficult to measure the mean bleed velocity (Zhu et al., 2001). Comparing our dataset against those of Zhu et al. (2001), α–β relationships from the two were obviously different with larger differences between α and β in realistic windbreaks (Fig. 3). The data from two-dimensional windbreaks were close to the line α = β but those from realistic windbreaks were relatively far off. We plotted the measured data of α − β versus β for realistic windbreak models in Fig. 4 along with the empirical curve of α − β = β0.4 − β. The largest divergence of α from β occurred at α = 0.5, β = 0.2 with the measured data and at α = 0.543, β = 0.217 with the empirical Eq. (3).

A shearing stress meter was installed under the wind-tunnel floor to measure the drag of windbreak models. Fig. 5 illustrates a schematic diagram of the drag meter. The meter was constructed in a fixed frame and a free platform was suspended from it by four 0.5 mm piano wires. Eight vertical posts fixed to the platform extended above the suspension points and supported sample trays of varying size. Two force sensors were placed at the end of the installations next to the two windward heads of the platform. The installations could be manually adjusted with accuracy. The loading capacity of each sensor was 0–60 N, with a maximum displacement of 30 ␮m. The displacement was very small, so the errors caused by the weight of platform, tray and sample could be ignored. The shearing stress on the sample surface measured by the sensors was converted to electric signals and displayed on the screen. The model windbreak for our experiment was fixed on the tunnel floor in the direction perpendicular to the wind flow, spanning the whole width of the tunnel. In the center of the windbreak, a section (504 mm long) was removed and replaced by the drag measurement model (with the same structure as the original model). The measurement model was placed on a 500 mm × 100 mm plywood base. This base was then kept on the same level as the floor. For every experiment the signals were read 50 times for every 5 min to obtain an average. The drag of the bare base plywood was also measured before and after each measurement. During

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Fig. 5. Schematic diagram of drag meter: (1) fixed frame; (2) free platform; (3) piano wires; (4) vertical posts; (5) suspension points; (6) sample tray; (7) force sensors; (8) adjustable installations.

the measurements the air temperature and atmospheric pressure were recorded using an Assmann ventilated psychrometer and a mercury barometer, respectively. The drag coefficient of the windbreak was defined as: D Cd = (4) (1/2)ρu2H S where ρ is the air density, uH the approaching wind speed (at windbreak height, H, upwind 1.5 m of the model windbreak), S the frontal area of the measured windbreak model (S = 500 ×100 mm2 in this case), ρ varies with air temperature T and atmospheric pressure P and it was calculated using the state equation of dry air: P = ρRd T

cult to measure and much smaller than D1 . The bare base plywood drag, D2 , was bigger than D2 but still much smaller than D1 (D2 /D1 < 0.018). Hence, we assigned D1 − D2 as the drag of the model windbreak in our study. The measurement results of D versus the square of approaching wind speed, uH , are shown in Fig. 6. It was obvious that D decreased as optical porosity increased. 3.2. Dependence of drag coefficients on Reynolds number The calculated Cd (Eq. (4)) for different optical porosities at different Reynolds numbers

(5)

where Rd is a constant (287.05 J kg−1 K−1 ) and T the absolute temperature (in K). 3. Results and discussion 3.1. Drag of the windbreak models If the drag of the windbreak model together with the base plywood was D1 (as measured by the meter) and the drag of the base plywood was D2 , the drag of the windbreak model, D, is D1 − D2 . D2 was diffi-

Fig. 6. D/u2H vs. β at different wind speed uH .

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Fig. 7. Drag coefficient Cd as a function of the Reynolds number Re for different optical porosities.

(Re = uH H/ν, where ν is the viscosity of air) are shown in Fig. 7. Although Cd slightly increased as the Reynolds number increased, the increment was smaller than the magnitude of random errors in measurements. Thus, the model Cd was relatively constant over the measured range of the Reynolds number. This is in accordance with the report of Jacobs (1985) for a solid fence. 3.3. Relationship between drag coefficient and porosity Fig. 8 shows the relationship between drag coefficient and optical porosity. For comparison, results of other studies on two-dimensional windbreaks (see Table 5) were also included in Fig. 8. In this analysis, the wind speed, uH , was used as a reference and all the data were converted into uH . Fig. 8 clearly shows that our results are different from those from two-dimensional windbreaks. For the

Fig. 8. Drag coefficient Cd as a function of the optical porosity β: measurements for realistic windbreak models (䊊) of present; measurements of Jacobs’ () and Seginer’s (䊐); calculations of Hagen’s (䊉) and Hoerner’s (×); simulations of Wilson’s (1985) (䉫) and Wang’s (—) and empirical formula of Guyot’s (- - - -) for two-dimensional windbreak.

same values of β, the Cd of a realistic windbreak was much lower than that of two-dimensional windbreaks. The largest difference between them was found near β = 0.2. Cd was 0.74 for the realistic windbreak whereas it was as high as 1.07–1.16 in Hagen and Skidmore (1971) and Wang and Takle (1997b). In other words, the Cd determined by two-dimensional windbreaks corresponded to much lower porosity of realistic windbreaks. The difference between these two different windbreaks became smaller as β deviated from 0.2. The discrepancy was generated mainly from their structural differences. Realistic windbreaks have width and internal structure that may be porous to wind flow even though they are optically dense. Therefore, in order to attain the same protection efficiency, realistic windbreak should be designed to be optically denser than the two-dimensional windbreaks.

Table 5 Authors, windbreak characteristics and approaches to estimate Cd cited in Fig. 8 Authors

Windbreak characteristics

Approaches

Hoerner (1965) Hagen and Skidmore (1971) Seginer (1975) Guyot (1978) Jacobs (1985) Wilson (1985) Wang and Takle (1997b)

Solid fence with width equal to height Fence with optical porosity 0, 0.2, 0.4, and 0.6 Fence with optical porosity 0.5 Artificial windbreak Solid fence Two-dimensional windbreak Two-dimensional windbreak

Calculated Calculated Field measurement Empirical formula Field measurement Numerical simulations Numerical simulations

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be different (Hoerner, 1965; Wang and Takle, 1996, 1997b).

4. Summary and conclusions

Fig. 9. Comparison of empirical formula (6) prediction (—) and other studies of drag coefficients Cd as a function of aerodynamic porosity α (symbols are same as Fig. 8).

Based on the above results, we may conclude that α may be more useful than β for comparing two-dimensional and narrow realistic windbreaks. In our study, an empirical relationship between Cd and α was obtained as: Cd = Cd0 (1 − α1.8 )

(6)

where Cd0 is constant (1.08 in our study). Unfortunately, all the literature cited for two-dimensional windbreaks in Fig. 8 do not provide values of α. In order to compare these cited data to Eq. (6), we approximated their α values using the data of Zhu et al. (2001) in Fig. 3 (i.e. α = β when β ≥ 0.2, α = 0 when β = 0). All the drag coefficient data were plotted against the aerodynamic porosity in Fig. 9. All the cited data agreed well with the predictions from Eq. (6), suggesting that aerodynamic porosity would be a more appropriate index for the two kinds of windbreaks. Eq. (6) provides not only better prediction of the drag coefficient of windbreak, but also better physical meaning. A solid or absolutely dense windbreak has the biggest drag coefficient Cd0 (=1.0–1.2). When the aerodynamic porosity increases, drag coefficient of the windbreak decreases approximately as a power function, down to zero for a fully open surface (i.e. α = β = 1). It should be noted that our conclusion applies only to two-dimensional or narrow, realistic windbreaks with nearly rectangular sectional shape. When windbreak becomes wide or changes its sectional shape, the drag coefficient and shelter effect will

Drag coefficient is the most often used, key index in analysis of the aerodynamic effect of windbreaks. Although there have been many studies on this, some issues have not been clearly described. The first issue is the lack of direct measurements of the drag coefficient of windbreaks. In particular, studies of narrow realistic windbreaks have not been seen in literature. The second issue is how realistic windbreaks differ from two-dimensional windbreaks in the relation between porosity and drag coefficient. Furthermore, how the two kinds of windbreaks are different in the relation between the two most often used porosities, optical porosity and aerodynamic porosity. The third issue is what is the analytical relation between drag coefficient and porosity, both for two-dimensional and narrow realistic windbreaks. We tried to answer these questions by measurement results and cited literature in this paper. First, narrow, realistic windbreak models were proposed and their drag in simulated surface boundary layer was measured in the wind tunnel. Optical porosity, β, and aerodynamic porosity, α, of the models were used to express the porosity of the windbreak models. β and α were calculated from digitized photographs and bleed wind speed, respectively. They were fitted to an empirical equation (α = β0.4 ) and the divergence of the two indices was largest near α and β of 0.5 and 0.2, respectively (Figs. 3 and 4). But for two-dimensional windbreaks, α was approximately equal to β as β ≥ 0.2 according to Zhu et al. (2001). Further analysis of the wind-tunnel experiments and cited studies showed that the drag coefficient of narrow, realistic windbreak models was much lower than that of two-dimensional windbreaks with the same optical porosity. The largest difference between them was found near β = 0.2 (Fig. 8). The conclusion means that the relation between drag coefficient and optical porosity for two-dimensional windbreaks is not fitted for realistic windbreaks. This argument was not stressed numerically in related studies before. To find the analytical relation between porosity and drag coefficient, fitting both two-dimensional and

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narrow realistic windbreaks, aerodynamic porosity α was used as the porosity index. We constructed an empirical equation Cd = Cd0 (1 − α1.8 ) estimated from our realistic windbreak models. Cited data agree well with this equation. If optical porosity β was used as the porosity index, we can extrapolate according to Zhu et al. (2001) so that the equation Cd = Cd0 (1 − β1.8 ) is fitted for two-dimensional windbreaks as β ≥ 0.2. But for narrow realistic windbreaks the relation was Cd = Cd0 (1 − β0.72 ) based on Eq. (3). The divergence between them was obvious and should be considered in aerodynamic studies of windbreaks and in their field application. So it is certain that aerodynamic porosity is a more appropriate index than β when we compare two-dimensional and narrow realistic windbreaks. The resultant empirical formula for the Cd –α relationship not only provided better prediction of the drag coefficient, but also expresses the physically reasonable correlation between the drag coefficient and porosity of windbreaks. We hope this study can provide a clearer conception of two-dimensional windbreaks and realistic windbreaks, by means of their differences in Cd –β, Cd –α and α–β relationships, and more importantly, can provide a numerical connection between the two kinds of windbreaks and their aerodynamic effect. This could broaden the application of studies on two-dimensional windbreaks to vegetative shelterbelts that are more widely established in farmland.

Acknowledgements The authors thank the anonymous reviewers for their helpful and constructive comments and suggestions. Our thanks also go to Drs. Joon Kim and Tao Dali, and Theresa Krebs for their help in revising the manuscript. Appreciation is extended to Professor Yin Jiefen and Wang Xinying of Environmental Science Center of Beijing University. This research was partly supported by Chinese Academy of Sciences and Doctoral Start-up Fund of Liaoning Province.

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