On the Stochastic Closure Theory of Homogeneous Turbulence
We compare the predictions of the stochastic closure theory (SCT)  with experimental
data obtained in the Variable Density Turbulence Tunnel (VDTT) , at the Max Planck Institute
for Dynamics and Self-Organization in G¨ottingen. The mean flow in the homogeneous
turbulence experiment reduces the number of parameters in SCT to just three, one characterizing
the variance of the mean field noise and another characterizing the rate in the large
deviations of the mean. The third parameter is the decay exponent of the Fourier variables in
the Fourier expansion of the noise. This characterizes the smoothness of the turbulent velocity.
We compare the data for the even-order longitudinal structure functions ranging from the
the second to the eighth structure function as well as the third-order structure function, to
the SCT theory with generic noise, depending on the above three parameters, at five Taylor-
Reynolds (Rl) numbers ranging from 110 to 1450. The theory gives excellent comparisons
with data for all the structure functions and for all the Taylor-Reynolds numbers. This highlights
the advantage of the SCT theory, where the structure functions can be computed explicitly
and their dependence on the Rl number computed. These results are robust with respect
to the size of dissipation range filters applied to the data, and comparisons to the fits without
the Rl number corrections show a clear improvement when the corrections are present. This
improvement is significant for the lower Rl number and disappears as the Rl number becomes
large, as expected. Very surprisingly the comparison of SCT and the data also gives information about the
smoothness of the turbulent velocity as Rl becomes very large.
We then compare the SCT to the Townsend-Perry constants generated in the flow physics
facility (FPF) at the University of New Hampshire. The Reynolds correction terms in the SCT
changed the initial derivation of this similarity, forcing a refinement of the theory. Once done,
we see good agreement between the data and the SCT.