Covariances, autocorrelations, and spectra - Turbulence in the Atmosphere - page 358

Do this by finding the spectrum of this approximation to the derivative.
15.8 (a) Find the spectral transfer function (the ratio of the spectra of the filtered
and unfiltered variables) of the filter defined by

τ du f (t)
dt

τ du u f (t) + =

dt ,

where u(t) is a time series and u f (t) is the filtered series. Sketch the transfer
function. What does this filter accomplish?
(b) Find the spectral transfer function of the filter defined by

τ du f (t)
dt

u f (t) + = u(t).

Sketch its transfer function. What does this filter accomplish?
15.9 In practice when dealing with observational data or the results of numerical
simulations of turbulence we estimate spectra through the coefficients of
finite Fourier series. Use a stationary, finite time series of a scalar variable
to show how this is done.
15.10 Spectra estimated from time series or numerical simulation results are typi-
cally “ragged” due to inadequate averaging. Showhow averaging the spectral
estimates over narrow frequency or wavenumber bands can smooth such
spectra. Develop a criterion that gives the maximum width of the averaging
band in the inertial subrange for a given distortion of the spectrum there.
15.11 Determine the time constant needed in a fine-wire resistance thermometer
in order to measure the finest scales in the turbulent temperature field in a
fluid of Pr

1.
15.12 Prove that if R αα is an even function of ξ ,then φ αα is purely real and an even
function of κ .
15.13 Relate the one-dimensional co- and quadrature spectra to the full ones in a
way similar to (15.49) .
15.14 Express λ/ and λ/η as functions of the large-eddy Reynolds number R t .
15.15 Prove Eq. (15.121) .
15.16 Prove Eq. (15.122) .
15.17 Show that in an isotropic field the integral scale determined from f is twice
that determined from g .(Hint:use (15.65) .)
15.18 Prove Eq. (15.75) by staying completely in wavenumber space.
15.19 Show that Eq. (15.46) implies that R αα ( ξ ,t) is an even function of ξ .
15.20 Derive Eq. (15.67) from Eq. (15.66) . Assume homogeneous turbulence.
15.21 If the three-dimensional pressure spectrumhas a κ − 7 / 3 inertial subrange, how
do its one- and two-dimensional spectra behave there? Relate the spectral
constants.

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