Journal of Arboriculture 31(3): May 2005 103 •Vogel, citing Mayhead (1973), observed (1994, p. 121) that “the increase in drag [on a tree] was more nearly proportional to the first than to the second power of velocity” and (1996) that “drag [on a tree] increased with an exponent of less than 1 (0.72) rather than the expected 2.0 up to a speed [of] 38 m/s, or 85 mph.” • Baker (1995), citing Mayhead (1973), Johnson et al. (1982), and Roodbaraky et al. (1994), observed that “there is some doubt as to whether tree drag varies in proportion in the square of the velocity … or simply in proportion to velocity.” •Haritos and James (1996), citing Baker and Bell (1992), observe that the velocity exponent “may depart from the ‘classical’ value of 2 to a lower value closer to unity.” • Bonser and Ennos (1998), working with saplings in wind tunnel experiments, observed that “the strains measured in the stems of these plants show a non- linear relationship with the square of windspeed.” Such strains would be a function of wind force. •Smiley et al. (2000) mounted trees on a truck and measured wind loads on the trees at various truck speeds. They reported that “[measured] wind resistance showed a linear increase with vehicle speed.” A calcu- lated regression line extrapolating their data beyond the range of V tested was also “linear.” They did not explicitly employ or consider the drag equation. •Moore and Maguire (2002), citing Hoag et al. (1971), Mayhead et al. (1975), and Roodbaraky et al. (1994), observed that “there is some debate as to whether the relationship between drag force and wind speed is quadratic or linear.” Taken together, the independent observations (Tirén 1926, 1928; Sauer et al. 1951; Lai 1955; Fraser 1962; Raymer 1962; Mayhead 1973; Bonser and Ennos 1998; Smiley et al. 2000) and the repetitive citations of Mayhead (1973) and Mayhead et al. (1975) by others seem to have some weight. Explanations for the “Linear” Case The “linear” sources quite consistently explain that if the drag on a tree varies more “linearly” than would be ex- pected using V2 rigid bodies and effectively reduce A. Sauer et al. (1951) observed that “most variation in drag force acting on tree crowns is due to their deformation” and noted similar indications in Tirén (1926). Lai (1955) observed that “for a flexible, porous body such as a tree crown, the area and porosity change constantly with dynamic pressure.” As noted above, dynamic pressure is a function of velocity (see Equation 2). Mayhead (1973) noted that drag was found to vary linearly with windspeed because “as is to be expected with a tree, the projected frontal area varies with windspeed.” Grace (1977) explained the linear relationship was observed because “at higher wind-speeds, the trees became streamlined, exposing less area to the wind.” Bell et al. (1990) explained, “For a building, the drag … is directly proportional to the square of the wind velocity. … The ability of trees to streamline reduces the cross-sectional area of the tree … and corre- spondingly the wind interception. Thus, the drag for trees becomes more nearly linearly proportional to V.” Roodbaraky et al. (1994) noted the linear phenomenon “would be expected, due to streamlining of the trees and branches.” Vogel (1989; 1994, pp. 121–124; 1996) cited Mayhead (1973) in this regard and also detailed his own research showing that individual broadleaves and broadleaf clusters change their shapes under wind load. Bonser and Ennos (1998) observed that “since sapling trees are relatively flexible, they deform easily in airflows; the stem bends and the needles fold up.” Moore and Maguire (2002) observed that “departure from the quadratic relationship [toward a linear one] can be explained by streamlining, … which acts to reduce the crown frontal area.” Many sources, in addition to the “linear” ones, similarly acknowledge the reduction in drag resulting from flexibility and reduction in effective area (e.g., Heisler and DeWalle 1988; Sinn and Wessolly 1989; Hedden et al. 1995; Gardiner et al. 2000; Mattheck and Bethge 2000; Spatz and Bruechert 2000). Vogel (1994, p. 115) has suggested the term “recon- figuration” to describe this reversible reduction in crown area to distinguish it from permanent deformation, which is a different result of tree–wind interaction (see Robertson 1987; Cullen 2002c). “Reconfiguration” is used throughout this paper to include actual reduction in A as well as actual streamlining, which technically is an increase in the propor- tion of friction or skin drag relative to the proportion of form or profile drag (Grace 1977, p. 13; Niklas 1992, pp. 437– 438; Vogel 1994, pp. 96–97). Modeled using V2 , it is because trees are flexible rather than Curves Figure 1 illustrates the apparent question raised in the literature: whether the expected curve of FWIND using V in Equation 1 more accurately describes the change in tree–wind drag over increasing V. The remaining modeled curves are presented in follow- ing sections to illustrate the discussion. in Equation 1 or the curve of FWIND DISCUSSION Practitioners’ Questions It is not entirely clear from the “linear” sources, particularly those merely citing the earlier independent studies, whether the “linear” observations are simply intended to describe the rate of change in drag over velocity, or the shape or slope of ©2005 International Society of Arboriculture values found values found FWIND
May 2005
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