Large-eddy dynamics, the energy cascade, and large-eddy simulation - Turbulence in the Atmosphere - page 143

6.23 In Eq. (6.65) we assumed that ensemble means pass perfectly through the
filter. Is that a reasonable assumption? Explain.
6.24 Explain the intermediate steps in Eq. (6.39) .
6.25 Show that the transfer function of the Gaussian filter, Eq. (6.15) ,isalso
Gaussian.
6.26 Derive Eq. (6.52) .
6.27 Explain why we say that the filtered product ( u j ) r u i ˜ ˜ in the set (6.63) is

u j , then the set could be
solved numerically. Show that its kinetic-energy equation would lose its
sink term, however, and so could not give steady solutions.
6.28 Interpret the limit of Eq. (6.56) as the filter scale decreases within the inertial
subrange.
6.29 Discuss the nature of the gradient production term in the SFS flux conser-
vation equation (6.73) . Why does the SFS scalar flux it produces need not
be in the direction of the resolved gradient? Why could we call it a “tensor
diffusivity” term?

u i ˜ unknown. Show that if we approximate it as ˜

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