By Richard O Fimmel, Lawrence Colin, Eric Burgess
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Additional info for Pioneering Venus: A Planet Unveiled
The selfafﬁne nature of particle displacement ﬁelds means that uniform strain in a granular ﬂow is accommodated via correlations at all scales. This behavior does not imply that velocity correlations extend to inﬁnity under arbitrary boundary conditions. The observed scaling characteristics are a consequence of uniform shear. An interesting consequence is that uniform strain is easily disturbed as a result of the details of conﬁning conditions or symmetry breaking agents such as a bulk force. In other words, disturbing the long-range correlations leads to non-uniform behavior.
29 for a small integration time τ = 10−3 and for a large integration time τ = 10−1 . 29. Probability density functions of the y components of ﬂuctuating velocities for two different time resolutions: 10−3 (broad curve) and 10−1 (narrow curve). 30. Averaged power spectrum of the x and y components of the ﬂuctuating velocity ﬁeld with τ = 10−7 for 1D cross-sections along the mean ﬂow tails, extending nearly to the center of the distribution, at small τ . This non-Gaussian broadening of the distributions as a function of τ is also observed for the component vx .
The Kronecker function δ is equal to 1 when its argument is zero, and 0 otherwise. When there are no correlations, we have K = 1. The deviations from K = 1 therefore reﬂect the positive and negative correlations. The peaks result from local ordering of the particles and we see that the correlations persist at distances as great as 10 times the mean particle diameter d, even although this is only an average over all directions. 37, can be larger. 39 shows the pdfs of normal forces fn for two isotropic samples simulated by molecular dynamics and contact dynamics methods [RIC 09].