106 Cullen: Trees and Wind—Drag Equation Velocity Exponent C = D 2()(VA) FWIND 2 Understanding that Equations 1 and 6 are “definitional” and merely allow conversion of CD should now be clear that one of the terms must be known to solve for a value of the other. To solve Equation 1 for FWIND to FWIND must have been found by experiment, for example in a wind tunnel (e.g., Mayhead 1973), on a moving vehicle (e.g., Sauer et al. 1951; Lai 1955; Hoag et al. 1971; Kouwen and Fathi-Moghadam 2000; Smiley 2000), or by direct field measurement (e.g., Roodbaraky et al. 1994; Grant and Nickling 1998). similar characteristics. To solve Equation 6 for CD FWIND on a tree, an actual CD The Velocity ( ) Exponent As shown in Figure 1, reducing the velocity exponent to 1 from the conventional 2 and treating A and CD over velocity than would be expected using a velocity exponent of 2 and treating A and CD results in a more “linear” curve of calculated FWIND as constants. It is clear from the discussion of A that actual A may vary with V. While some analysts reflect variations in A in the drag equation, such data may be problematic and, in any case, it is conventional to treat A as a constant reference value. It is clear from the discussion of CD is constant and actual A decreases over V, then CD decreases over V (see Figure 2). CD constant CD than the curve expected using V2 Even if it is convenient or otherwise appealing to vary the exponent—say the analyst prefers a constant CD to reflect a drag curve which is “more linear” and a constant CD and a variable CD or shows that once reconfiguration ceases and both actual A and CD wants to project a drag curve without having to solve Equation 1 iteratively for values of FWIND become constant, the drag curve becomes “less linear.” Recall Tirén’s (1926; 1928) conclusion “that the exponent for the velocity is not constant with crown drag.” Baker (1995, citing Roodbaraky 1994) similarly observed “that the form of [exponential] relationship might vary depending upon whether or not the tree is in leaf.” If the exponent must be varied over the range of V or seasonally or with type of tree, any perceived advantage as compared to varying CD likely to be concerned with relatively high, “storm” veloci- ties that are above the “linear” range of the drag curve in FWIND © International Society of Arboriculture —Figure 3 clearly changes in drag over V, so on this basis alone it seems more appropriate to use V2 is intended to reflect rather than V and a . that if reference A also as constants values or FWIND to CD , it must be known for a tree with , an actual Figure 4. The curve of FWIND Equation 1 with the conventional V2 constant (here = 10 m2 beyond the dashed vertical line “W” (~25–28 m/s). Brudi and van Wassenaer (2002, Figure 3) suggest range of crown reconfiguration but constant beyond the dashed vertical line “M” as assumed by Mayhead (1973). Wessolly (1995) suggests CD ), and CD is constant there is little crown reconfiguration or decrease in CD beyond the dashed vertical line “B” (~17–21 m/s). The “hurricane” standard as applied by the SAG- Baumstatik group is the dotted vertical line. The ASCE (U.S.) standard as applied by Cullen (2002a) is the solid vertical line. The Australian standard as applied by James (2003b) is the dashed-dotted vertical line. (Also see Figure 3.) (Mayhead 1973). As shown in Figure 4, engineering standards which may be applied to tree risk assessment will be concerned with this higher range of V (Standards Australia 1989; Wessolly 1995; ASCE 1999; Mehta and Perry 2001; Cullen 2002a; ASCE 2003; James 2003a). The “SAG-Baumstatik” group of consultants (referred to by Brudi and van Wassenaer, 2002) similarly consider stability at higher wind speeds. Niklas (2002) describes such “a priori specifications for tree safety” which will be in this higher range of V. There are also compelling procedural reasons to use the conventional form, V2 : • First, the practioner is unlikely to develop CD CD data experimentally and will therefore look to the catalog of CD for those characteristics is minimized. In addition, while some applications may be interested across a wide range of V, practical risk assessment is reviewed sources describing some form of Equation 6 did so with the conventional form V2 , even if they noted the “linear” form of drag curve or questioned the V exponent. (The exception was Roodbaraky et al. 1994. They tested both V2 and V forms of Equations 1 and 6. Their “experiments did not lend support to the data available from the literature. It is now clear that is derived using Equation 6. Almost all of the values found using , (6) = 1.2 kg/m3 , A decreasing over the ρ ρ V
May 2005
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