Journal of Arboriculture 31(3): May 2005 105 Lai (1955), and Bell et al. (1990) all similarly observed that a “linear” drag curve associated with a decreasing A would alternatively be represented by a constant reference A and a decreasing CD . velocities at which crown reconfiguration ceased and actual A became constant. Smiley (2000) acknowledged that the “linear” relationship of drag (FWIND Mayhead (1973) assumed CD Equation 1 with V2 2000) was likely to cease beyond the range of crown reconfiguration. The curve of FWIND ) and V (Smiley et al. values found using over the range of crown reconfiguration is shown in Figure 3 and can be compared to the curves in Figures 1 and 6. It is clear in Figure 3 that a curve of FWIND values found would not be expected to vary purely as the sqaure of V as , and constant using Equation 1 with V2 decreasing CD shown in Figure 1. Stated another way, a curve of FWIND values found using Equation 1 with V2 Some sources acknowledge that actual A and, hence, CD will be variable with V but find this a confusing or prob- reference A, will describe a curve of actual FWIND tree that reconfigures if CD lematic rather than an expected and useful phenomenon (e.g., Heisler and DeWalle 1988; Hedden et al. 1995). This is often related to limited data or particular goals, but it may be that biological researchers have an incomplete understanding of fluid mechanics tools like the drag equation (Vogel 1994, p. 399) or that the engineering reference literature offers little guidance on the complex geometry of biological shapes (Loudon 1999). values for a is appropriately varied over V. , constant reference A, and over the range of crown reconfiguration , constant reference A and decreasing CD to be constant only above and reference A—and an object or body with unique characteristics such as shape, porosity, flexibility or rigidity, texture, or orientation to the wind. CD It is useful at this point to consider exactly what CD CD represents the relationship of FWIND —at any given V, simplifies the modeling of the complex interdependencies of these various characteristics (Vogel 1994, pp. 89–90; Benson 2001b). Vogel (1994, pp. 89–91) explains that CD is “a dimen- sionless form of drag, the drag per unit area divided by the dynamic pressure.” Recalling Equations 2 and 3, above, CD is thus defined by Equation 4. CD = 2 2 () () V V 2 (4) 2 Sauer et al. (1951) call this relationship of drag and dynamic pressure a proportionality factor, and Bell et al. (1990) call it a ratio. Vogel further describes CD as the than of unit area and represented by Equation 4a. . In practice, CD CD = 2 2 ()( ) ()( ) VA VA 2 (4a) 2 (1990) explain that CD to CD Equations 4 and 4a at first seem confusing, as if a term is divided by itself. Vogel (1994, p. 89) explains that Equation 1 is “definitional” and merely allows conversion of CD FWIND or FWIND to . Grace (1977, p. 14) and Bell et al. can be understood as a ratio of “actual” force and the force predicted by Equation 1. Merging these explanations, CD C =D FWIND DEFINITIONAL WIND ACTUAL F − Table 2. Drag coefficient (CD If the definitional force is greater Figure 3. The lower curve shows FWIND with Equation 1 using the conventional V2 1.2 kg/m3 , A constant (here = 10 m2 values used in Equation 1. values found , with ing over the range of crown reconfiguration but constant beyond the dashed vertical line as assumed by Mayhead (1973). The upper, dotted curve shows CD ), and CD = decreas- − ) ranges. If the actual and definitional forces are equal If the actual force is greater CD CD CD = 1.0 > 1.0 < 1.0 In other words, actual and definitional values for FWIND for many reasons, and such variations are reflected by CD vary . Equation 6 is a generalized form of the equation used in practice (e.g., Mayhead 1973; Grace 1977, p. 14; Sinn and Wessolly 1989; Wood 1995; Grant and Nickling 1998; Kerzenmacher and Gardiner 1998). ©2005 International Society of Arboriculture can be understood as the ratio of actual and definitional forces as shown by Equation 5 and Table 2. (5) quality of “dragginess” as contrasted to the quantity of drag or FWIND is likely to be a function of A rather is. , ρ ρ ρ ρ ρ ρ
May 2005
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