Statistics in turbulence analysis - Turbulence in the Atmosphere - page 385

v f ( x h ,z,t) = w × v( x h − r ,z,t) + w × v( x h ,z,t) + w × v( x h + r ,z,t).

For this simple case what is the relation between the spectrum of v f in the
plane and that of v , assuming the field is homogeneous in the plane?What cri-
terion should w satisfy to give unity transfer function at small wavenumbers?
Choose w that satisfies that criterion and sketch the transfer function.
16.6 To leading order the budget of vorticity variance is (Problem 5.6)

∂
∂t

ω i ω i
2

= 0 = u i,j ω i ω j − νω i,j ω i,j .

What is the physical meaning of each of these terms? Use the definition
of vorticity and the assumption of local isotropy to evaluate the production
term. Write the destruction term as an integral of the velocity spectral den-
sity tensor. Simplify and express the skewness S of the streamwise velocity
derivative ∂u 1 /∂x 1 as an integral of the spectrum of u 1 .
16.7 Show that the integral in (16.63) yields the result (16.64) .
16.8 Show that 2 ν 0 κ 2 E(κ) dκ.

16.9 Explain why it is (Subsection 16.1.1.1) that isotropy requires that the m ean
part of the scalar, C( x ,t) , not depend on x , and that the turbulent flux cu j
vanish.
16.10 Show that if the separation vector r is in the streamwise direction, Eq. (16.70)
can be reduced to a transfer function times the one-dimensional spectrum.
16.11 Sketch the spectral variance budget of Eq. (16.39) .
16.12 Explain why a 13 vanishes on the centerplane ahead of a circular cylinder,
Figure 16.6 .
16.13 Derive Eq. (16.93) .
16.14 Derive Eq. (16.94) .
16.15 Derive Eq. (16.33) .

=

References
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Press.
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Gal-Chen, T., and J. C. Wyngaard, 1982: Effects of volume averaging on the line spectra
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1881-1899.
Hogstrom, U., 1982: A critical evaluation of the aerodynamical error of a turbulence
instrument. J. Appl. Meteor. , 21 , 1838-1844.
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