INTEGRAL SCALE

There is an intuitive notion of the scale of turbulence. Clearly there are significant differences of “size” between the turbulence in the wing boundary layer, in the wake of the airplane, and in the atmosphere itself. These differ­ences are quantified by a definition of integral scale derived from the corre­lation function. Thus let

lu = -2

и і Jo

be a line integral on the axis. It might be called “the j scale of the і velocity component.” There are in general nine such scales, e. g. for ux measured along the xx axis, or «3 measured along the x2 axis, etc.

A second notion of scale derives from the spectral representation of turbulence. The wavelength at which the energy density peaks (see Fig. 13.6) is also a scale parameter, and for any given spectrum shape is uniquely related to L (defined below).

In isotropic turbulence, only two different scales are found, associated with the basic correlations / and g, and these are of course simply the areas

under the / and g curves. Because the maximum ordinate is unity, Li} is equal to the width of a rectangle that contains the same area as the corre­lation curve—i. e. it is a measure of the spatial extent of significant correlation. The two scales are

L = Lijt і = j = area under /(£) = longitudinal scale L’ = Li}, і Ф j = area under g(£) = lateral scale

The continuity condition (13.2,7) yields L — 2L’.

The situation with respect to scale is unfortunately more complicated in the ground boundary layer where isotropy does not hold.