Spectral contributions of sections of power-law structure function of random processes with stationary increments. Part 2. The exponent is greater than one

2019 ◽  
Keyword(s):  
Structure Function ◽  
Power Law ◽  
Random Processes ◽  
Stationary Increments

Investigation of sub-Kolmogorov inertial particle pair dynamics in turbulence using novel satellite particle simulations

2013 ◽  
Vol 720 ◽  
pp. 192-211 ◽  
Author(s):  
Baidurja Ray ◽  
Lance R. Collins
Keyword(s):  
Structure Function ◽  
Power Law ◽  
Particle Motion ◽  
Relative Velocity ◽  
Velocity Structure ◽  
Inertial Particles ◽  
Particle Clustering ◽  
Particle Pair ◽  
Velocity Statistics ◽  
Particle Simulations

AbstractClustering (or preferential concentration) of weakly inertial particles suspended in a homogeneous isotropic turbulent flow is driven primarily by the smallest eddies at the so-called Kolmogorov scale. In particle-laden large-eddy simulations (LES), these small scales are not resolved by the grid and hence their effect on both the resolved flow scales and the particle motion have to be modelled. In order to predict clustering in a particle-laden LES, it is crucial that the subgrid model for the particles captures the mechanism by which the subgrid scales affect the particle motion (Ray & Collins, J. Fluid Mech., vol. 680, 2011, pp. 488–510). In this paper, we describe novel satellite particle simulations (SPS), in which we study the clustering and relative velocity statistics of inertial particles at separation distances well below the Kolmogorov length scale. SPS is designed to isolate pairwise interactions of particles, and is therefore well suited for developing two-particle models. We show that the power-law dependence of the radial distribution function (RDF), a statistical measure of clustering, is predicted by the SPS in excellent agreement with direct numerical simulations (DNS) for Stokes numbers up to 3, implying that no explicit information from the inertial range is required to accurately describe particle clustering. This result further explains our successful prediction of the RDF power using the drift-diffusion model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219–251) for $\mathit{St}\leq 0. 4$. We also consider the second-order longitudinal relative velocity structure function for the particles; we show that the SPS is able to capture its power-law exponent for $\mathit{St}\leq 0. 5$ and attribute the disagreement at larger $\mathit{St}$ to the effect of the larger scales of motion not captured by the SPS. Further, the SPS is able to capture the ‘caustic activation’ of the structure function at zero separation and predict the critical $\mathit{St}$ and rate of activation in agreement with the DNS (Salazar & Collins, J. Fluid. Mech., vol. 696, 2012, pp. 45–66). We show comparisons between filtered DNS and equivalently filtered SPS, and the findings are similar to the unfiltered case. Overall, SPS is an efficient and accurate computational tool for investigating particle pair dynamics at small separations, as well as an interesting platform for developing LES subgrid models designed to accurately reproduce particle clustering.

scholarly journals ON THE KOLMOGOROV-WIENER FILTER FOR RANDOM PROCESSES WITH A POWER-LAW STRUCTURE FUNCTION BASED ON THE WALSH FUNCTIONS

2021 ◽  
pp. 39-47
Author(s):  
V. N. Gorev ◽  
A. Yu. Gusev ◽  
V. I. Korniienko ◽  
A. A. Safarov
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Numerical Calculation ◽  
Weight Function ◽  
Structure Function ◽  
Random Process ◽  
Power Law ◽  
Wiener Filter ◽  
Walsh Function ◽  
Better Than

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of a continuous stationary random process with a power-law structure function. Objective. The aim of the work is to develop an algorithm of obtaining an approximate solution for the weight function without recourse to numerical calculation of integrals. Method. The weight function under consideration obeys the Wiener-Hopf integral equation. A search for an exact analytical solution for the corresponding integral equation meets difficulties, so an approximate solution for the weight function is sought in the framework of the Galerkin method on the basis of a truncated Walsh function series expansion. Results. An algorithm of the weight function obtaining is developed. All the integrals are calculated analytically rather than numerically. Moreover, it is shown that the accuracy of the Walsh function approximations is significantly better than the accuracy of polynomial approximations obtained in the authors’ previous papers. The Walsh function solutions are applicable in wider range of parameters than the polynomial ones. Conclusions. An algorithm of obtaining the Kolmogorov-Wiener filter weight function for the prediction of a stationary continuous random process with a power-law structure function is developed. A truncated Walsh function expansion is the basis of the developed algorithm. In opposite to the polynomial solutions investigated in the previous papers, the developed algorithm has the following advantages. First of all, all the integrals are calculated analytically, and any numerical calculation of the integrals is not needed. Secondly, the problem of the product of very small and very large numbers is absent in the framework of the developed algorithm. In our opinion, this is the reason why the accuracy of the Walsh function solutions is better than that of the polynomial solutions for many approximations and why the Walsh function solutions are applicable in a wider range of parameters than the polynomial ones. The results of the paper may be applied, for example, to practical traffic prediction in telecommunication systems with data packet transfer.

Study on the wave structure function and spatial coherence radius with a power-law spectrum of oceanic turbulence

Optik ◽  
2019 ◽  
Vol 187 ◽  
pp. 172-184
Author(s):  
Peng Yue ◽  
Dongling Xu ◽  
Xiang Yi ◽  
Zongmin Cui ◽  
Xiaohui Luan ◽  
...  
Keyword(s):  
Structure Function ◽  
Power Law ◽  
Spatial Coherence ◽  
Wave Structure ◽  
Oceanic Turbulence
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