104 Cullen: Trees and Wind—Drag Equation Velocity Exponent Even if data for variable A are readily available, however, there is a procedural reason not to vary A. It is conventional in aerodynamic analysis to determine A for V = 0 and treat A as a constant “reference area” as V increases (Vogel 1994, pp. 90–91; Benson 2001b, 2001d). Vogel (1994, p. 91) suggests that initial reference area should never be varied. The Drag Coefficient ( ) The notion that the V exponent determines the shape and slope of a curve of FWIND values, found with Equation 1, Figure 1. Curves of FWIND 1 using the conventional V2 (the lower curve) with = 10 m2 ), and constant CD = 1.2 kg/m3 values found with Equation (the upper curve) and V , constant A (here (here = 1.0). estimated with the conventional form of drag equation for individual velocities), or to actually suggest that the conven- tional form of drag equation using V2 the drag curve (composed of a number of FWIND their own analyses even while explicitly questioning the exponent. This raises practical questions: • If the curve of actual FWIND . values over a range of V varies at some rate other than as the square of V, what is the best way to calculate FWIND •Would a curve of calculated FWIND of V, found with Equation 1 using V2 1? The alternative choices are to vary the V exponent, A or CD values over a range , be expected to vary purely as the square of V as shown in Figure 1? •Does the curve of actual FWIND values over V vary Area ( ) If, as acknowledged in the explanations above, the actual drag curve varies more “linearly” than with the square of V because A actually decreases as V increases and the tree crown reconfigures, then it might seem most straightforward and most descriptive of the facts to vary A with V. Hedden et al. (1995) suggest this approach. Peltola et al. (1999), Gardiner et al. (2000), and Gaffrey and Kniemeyer (2002) acually account for changes in area with velocity in their analyses of forest conifers. These are, however, exceptions. Measuring or estimating these actual changes in A on individual, urban trees would be a difficult practical exercise (Sinn and Wessolly 1989). © International Society of Arboriculture Pinus spp.) over the range of V tested. Mayhead assumed the dashed vertical line would be the limit of crown reconfiguration and that CD Figure 2. Mayhead’s (1973) CD would be constant beyond. Grace (1977, p. 89) reported a similar pattern found by Raymer (1962). Kouwen and Fathi-Moghadam (2000) similarly found the friction factor (a dimensionless param- eter used in hydrology and similar to CD ) of trees tested in water and air to decrease with increasing V. Sauer (1951), curves (here for various “linearly,” as some sources suggest, rather than with the square of V? values with Equation values should be discarded in favor of a form using V. The confusion is compounded because some of these sources employ the conventional form, V2 , in must be constant. Bonser and Ennos (1998) note that the “hypothesis is based on the naïve assumption that trees do not deform and, hence, their drag coefficient remains constant.” Niklas (2003) observes that this may be a common assumption. Mattheck and Breloer (1994, p. 81) explicitly consider CD over a range of velocity (see Figure 1) seems to assume that A and CD expected to decrease as V increases. Mayhead (1973) in fact found CD with V (Sinn and Wessolly 1989; Niklas 1992, p. 438; Vogel 1994, p. 90). Mayhead (1973), the classically cited source for the “linear” argument, reported that actual A is expected to decrease as V increases. If reference A (for V = 0) remains constant in Equation 1, then CD a constant. In fact, CD would also be and Kniemeyer (2002) described a “parabolic decrease” in Mayhead’s CD Figure 2. Ezquerra and Gil (2001) noted a “non-uniform” decrease with increasing V in Mayhead’s CD to decrease for all tested species, as shown in data. Gaffrey data. is not constant CD ρ A
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