Probability Distribution of Vertical Longitudinal Shear Fluctuations

Abstract

In order to properly design aerospace systems (conventional airplanes, V/STOL vehicles, space vehicles, etc.) the engineer must consider the vertical structure of the horizontal wind during the launch and landing phases of flight. One way do this is with vertical two-point wind differences (wind shear). In turbulent flows like those found near the ground, wind shear is composed a steady-state part associated with the mean wind profile and a fluctuating part produced by atmospheric turbulence. Mean wind profile theory can be used to specify the steady-state wind shear; however, the fluctuating part is a stochastic process which can only be specified statistically. This paper discusses some recent measurements of third and fourth moments of vertical differences (shears) of longitudinal velocity fluctuations obtained in unstable air at the NASA 150 m meteorological lower site at Cape Kennedy, Fla. Each set of measurements consisted of longitudinal velocity fluctuation time histories obtained at the 18, 30, 60, 90, 120 and 150 m levels, so that 15 wind-shear time histories were obtained from each set of measurements. It appears that standardized third and forth moments S and K of wind shear are universal functions of ?z/z? and z?/L0, where ?z? is the vertical distance between the two points over which the wind difference is calculated, z? the height of the mid-point of ?z above natural grade, and L0 the surface Monin-Obukhov stability length. As ?z/z? ? 2, K ? 3, S ? 0, and S > 0, K > 3 for ?z/z? < 2. Thus, it appears that the joint distribution function of the longitudinal wind fluctuations at two levels is not bivariate Gaussian and that it can only be approximated with a Gaussian distribution for sufficiently large value of ?z/z?. The kurtosis K appears to be independent of z?/L0. However, the skewness S seems to experience a rather abrupt transition at z?/L0?O(?1). The implications of these and other results relative to the design and operation of aerospace vehicles are discussed.