108 Cullen: Trees and Wind—Drag Equation Velocity Exponent fit the curve of FWIND V1.5 actual drag curve is “linear,” meaning it should be described by using V rather than V2 simply by manipulating CD , or even what the various sources mean by “linear.” For summary comparison, the model drag curves from Figures 1, 3, and 5 are shown in Figure 6. values found using Equation 1 with . It is not at all clear that the described by Equation 1 using V, a constant reference A, and a constant CD curve for Raymer’s largest tree is closely approximated by FWIND values found using Equation 1 with V2 reference A, and Raymer’s actual CD CD = 1.0 . It is clear in Figure 7 that the , a constant using Equation 1 with V and a constant reference A can be forced to fit the curve of actual FWIND with increasing V. As noted at Figure 5, FWIND values by employing values with no relationship to conventional reference data. The curves of CD values used in Equation 1 with V and V2 actual R4 FWIND and a constant reference A to describe Raymer’s values are compared in Figure 8. Figure 7 suggests that “linear,” as used by Grace and Raymer at least, describes a “straight line” shape (constant rate of change) in FWIND with increasing V rather than a slope associated with a velocity exponent of 1 and fixed CD curves over V may vary and could not all be described by a single V exponent unless CD Figure 6. The drag curves from Figures 1, 3, and 5 are compared. which Raymer calculated from the experimental FWIND values using Equation 6. The CD tested trees are shown with manually superimposed trendlines in Figure 7. Raymer noted that the four sets vary because A varied among the four trees. Reported FWIND CD range of V. Raymer’s experimental FWIND and values for the largest Raymer tree (R4) were used with Equation 7 to solve for an approximate reference A (i.e., for V = 0) for that tree. A = 2()( ) VC WIND F 2 D This reference A was used first with V2 values and then with V and a constant CD mental values. These estimated FWIND calculate FWIND and Raymer’s CD in Equation 1 to values for comparison to Raymer’s experi- values and their trendlines are also shown in Figure 7. Grace agreed with Raymer that the curves of Raymer’s Figure 7 that neither the slope nor the amplitude of the curve of actual FWIND © International Society of Arboriculture FWIND values, shown in Figure 7, are “linear.” It is clear in values for Raymer’s largest tree are (7) experimentally by Raymer (1962) and noted that “the force on the trees is linearly related to wind-speed above 10 m/s [up to ~25 m/s].” Grace also presented curves of CD Grace (1977) presented curves of FWIND values, values declined over that values for four values found It is clear from lines R1–R4 in Figure 7 that the slopes of actual FWIND is varied for each. It is also clear in Figure 7 that this “linear” relation- ship was observed for a range of V below the limit of crown reconfiguration. The Drag Equation Does Not Describe a Curve Perhaps the most basic argument against using V rather than V2 in the drag equation in order to describe a “linear” drag curve over velocity is that the drag equation does not describe a curve at all. The drag equation solves for a single quantity or point (FWIND ) given a single value of V. Vogel (1994 p. 90), in fact, explains that the drag equation “is most definitely not the equation for drag,” and, as already noted above, “it’s just a definitional equation that converts drag to drag coefficient and vice versa.” It should now be clear from the preceding discussion that for a tree at any given value of V, actual A and, hence, CD are likely to vary. The equation can be solved iteratively for a number of points that will describe a curve, but either actual A or CD may vary in any interation as V varies. Recent research (Alben et al. 2002, 2004; Steinberg 2002) has proposed an equation (Equation 8) to calculate the drag on a simple, flexible body using the material characteristics of the body rather than an experimetally derived drag coefficient. Drag = ∫∫ 243 fibre []=+ dy /" / + − 1 23 2 23/ / / 2 2 3 K K dY dS dS (8) But even this complex form of equation solves for a single value of drag at a single value of V (Shelley 2003). It does not describe a curve of drag values over a range of V. In this light, it would seem difficult to suggest that simply manipulating the velocity exponent in Equation 1 will do so. . values, which decreased values found η η η ηρ ρ
May 2005
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