Journal of Arboriculture 31(3): May 2005 101 TREES AND WIND: A PRACTICAL CONSIDERATION OF THE DRAG EQUATION VELOCITY EXPONENT FOR URBAN TREE RISK MANAGEMENT By Scott Cullen Abstract. Arborists and urban foresters are increasingly con- cerned with tree risk management. The aerodynamic drag equation is a potentially useful management tool. Some sources question the form of equation—specifically, the velocity expo- nent—that should be applied to trees. For the tree risk manager, concerned with public safety and legal liability, this is more than an academic curiosity. Uncertainty about the appropriate exponent questions the reliability of the conventional form. This paper reviews the literature, reports on modeling of both equation forms, and concludes that the conventional form—velocity squared—is appropriate for trees. Detailed analysis is presented for the researcher or advanced practitioner. A summary explana- tion is provided for the typical practitioner. Key Words. Aerodynamics; biomechanics; drag equation; trees and wind; tree risk management; velocity exponent; wind. literature from traditional forestry and other fields (Cullen 2002b). Cullen (2002a) reviewed these sources in order to support a practical application of the drag equation to an urban tree. There is heightened interest in North America (Brudi 2004) in methods that rely on the drag equation, and such methods are used routinely by European consult- ants (see, e.g., Brudi and van Wassenaer 2002). The Drag Equation Equation 1 is a generalized, conventional form of the drag equation, where FWIND is the horizontal wind force; INTRODUCTION Background Internationally, arborists and urban foresters are increas- ingly concerned with tree risk management (e.g., Matheny and Clark 1994; Coder 1996; Jim and Liu 1997; Wessolly and Erb 1998; Lonsdale 1999; Pokorny 2003). In an urban or landscape setting, the principal risk is personal injury or property damage caused by trees. In traditional forestry, by contrast, the principal risk is economic crop loss (e.g., Coutts and Grace 1995; Peltola et al. 2000). In either case, a key issue is the risk of structural failure of trees. Arborists and urban foresters have long recognized that risk of failure may be dependent on wind force, load, or drag. A number of sources in the arboriculture and urban forestry literature have applied biomechanics to tree failure (e.g., Sinn and Wessolly 1989; Mattheck and Breloer 1994; Coder 2000; James 2003a), and this growing awareness of biomechanical analysis has offered the possibility of actually quantifying the wind force on a tree crown. A relatively few sources in the arboriculture and urban forestry literature present the aerodynamic drag equation (e.g., Sinn and Wessolly 1989; Mattheck and Breloer 1994, p. 138; Alaoui et al. 1999). These are very brief treatments and do not enable informed application of the drag equation. Mattheck and Breloer (1994) even suggest that the drag equation cannot be reliably applied to trees. Fortunately, arborists and urban foresters can look to a much wider range of tree–wind cient. This conventional form is found widely in the scientific and engineering literature. It is explained in detail by Niklas (1992) and Vogel (1994). It is based, ultimately, on Newton’s laws of motion (Vogel 1994, p. 89; Benson 2001c). is the density of air; V is wind velocity; A is the area of the trunk and crown; and CD is a dimensionless drag coeffi- FV A CD 2 2 Table 1. Notation. area A FWIND q CD V drag coefficient, dimensionless wind force, load, or drag dynamic pressure rho, air density wind velocity or speed The particular arrangement of terms known as “dynamic pressure” is derived from Bernoulli’s equation for fluids (Niklas 1992, pp. 429–430, 438; Vogel 1994, pp. 52–62, 81; Benson 2001a) and is shown in Equation 2. Dynamic pressure = 2() V 2 (2) Dynamic pressure is simply a force per unit area, often designated q (Sinn and Wessolly 1989; Vogel 1994, pp. 59– 63; Brudi and van Wassenaer 2002; ASCE 2003), found by Equation 3. In practice, pressure (q) may be specified by building codes or design standards, which may be appli- cable to tree risk management (Cullen 2002a). ©2005 International Society of Arboriculture see Equations 4–6 see Equation 1 see Equations 2–3 taken as 1.2 kg/m3 WIND = ()( )( ) (1) (rho) ρ ρ ρ ρ
May 2005
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