102 Cullen: Trees and Wind—Drag Equation Velocity Exponent qV 2 The Velocity Exponent This conventional form of equation, using V2 , suggests that tree varies more “linearly” with V and that the velocity exponent approximates 1. Cullen (2003) addressed the question of the velocity exponent in the drag equation in a previous paper intended to provide feedback to researchers. The current paper, by contrast, is expanded and is intended to provide guidance to practitioners, particularly arborists and urban foresters. FWIND varies as the square of V. The wider literature review found, however, that a number of sources suggest FWIND Need and Purpose The drag equation is a potentially useful tool in urban tree risk management. Arborists and urban foresters are con- cerned with public safety. If the convential form (V2 ) is, as suggested by some of the literature, less appropriate for trees than a “linear” form (V), then the conventional form is likely to overstate FWIND in terms of the drag equation’s potential in urban tree risk management, if it is used as a management tool and a managed tree fails, litigation can result. In this setting, the conventional form might be attacked simply because the literature seems to suggest the “linear” form. This is especially true under rules of evidence in the United States focusing on “scientific reliability” under the Daubert doctrine (see, e.g., Babitsky 2004). Questioning the credibility or reliability of the form of analysis may expose the practitioner and the client or employer to liability regardless of the actual quality or reliability of analysis. This prospect could discourage practical use of the drag equation in urban tree risk managment. The purpose of this paper is twofold: • first, to consider whether the conventional form of drag equation using V2 Conversely, if a “linear” form is not in fact valid, it is likely to understate FWIND , leading to unnecessary tree removal. and overstate tree safety. More importantly, is appropriate for trees and to resolve the apparent cloud around the velocity exponent; • second, to enable the practitioner to explain and support selection of the velocity exponent used in the drag equation. Much of the paper addresses the first purpose. This technical material and analysis will primarily be interesting to the researcher or to the advanced pratitioner, particularly if a scientifically detailed defense of the conventional form, V2 , is required. The second purpose is fulfilled, more simply, by the conclusion and the summary explanations. The typical practitioner may be interested only in them. © International Society of Arboriculture on a = ()= 2 FWIND A CD (3) Comprehensive explanation of the drag equation and application guidance are beyond the scope of this paper. MATERIALS AND METHODS This paper reports questions or suggestions about whether the velocity exponent should be 1 rather than the conven- tional 2 and associated explanations as found in the literature that applies the drag equation (Equation 1) to trees. Basic explanations of the drag equation are also reviewed. No field or laboratory tests of actual trees were conducted. The charting facility of Microsoft® Excel 97 was used to model and compare curves of drag (FWIND ) arbitrary area (A). Model curves of FWIND Equation 1 with V and V2 compared to curves of actual FWIND in the literature for a constant, actual A. values found using Equation 1 with various velocity (V) exponent and drag coefficient (CD and various CD and CD ) values and a constant, values found using values were also values reported RESULTS Literature Sources for the “Linear” Case This subsection briefly reviews the sources that have observed or commented on a “linear” increase in tree–wind drag with velocity that may be associated with a velocity exponent of 1. For the sake of clarity, their explanations are provided separately in the following subsection. •Mayhead (1973) is perhaps the classically cited source. Working with conifer data originally developed by Fraser (1962) and Raymer (1962) in wind tunnel tests, he reported that “drag is found to vary linearly with windspeed (U), and not with U2 .” In fact, much earlier sources observed the same phenomenon. • Sauer et al. (1951) measured the drag on small conifers in a wind tunnel and on larger conifers mounted on a truck. They reported that, for at least one tested tree, “drag is linear with velocity in the range shown.” They found this result in agreement with even earlier work by Tirén (1926). In a related study, Lai (1955) measured the drag on broadleaved trees mounted on a truck. Lai cites Tirén’s (1926, 1928) conclusion that “the exponent for the velocity is not constant with crown drag.” •Grace (1977, p. 90), citing the wind tunnel work of Fraser (1962) and Raymer (1962) on conifers, noted that “it might be expected … that the force would increase with the square of velocity, but this was not the case. … [T]he force is linearly related to wind- speed (up to ~25 m/s [56 mph]).” • Bell et al. (1990) observed that “drag for trees becomes more nearly linearly proportional to V.” • Roodbaraky et al. (1994), citing Fraser (1962) and Mayhead (1973), observed that “there is some evidence to suggest that the drag of trees in winds is actually linearly proportional to velocity rather than velocity squared …” ρ
May 2005
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