On the determination of the dissipation rate of turbulence kinetic energy

The paper presents a comparison of the dissipation rate obtained from numerical differentiation of the time-resolved velocity, analog differentiation of the hot-wire signal, integration of the velocity derivative spectra obtained from the velocity spectra, and the application of a power decay law. Hot-wire measurements downstream of an active-grid provide the time-resolved velocity with a Taylor Reynolds number in the range of 200–470, turbulence intensities in the range of 5.8–11%, and nominal mean velocities of 4, 6, and 8 m s$$^{-1}$$ - 1 . The dissipation rate calculated using a ninth-order central-difference scheme differs at most by $${\pm }$$ ± 4% from the value obtained by analog differentiation. For comparison, a 23rd-order central-difference scheme offers negligible (0.02%) difference relative to the ninth-order scheme. Correction for an apparent uncertainty in the calibration of the analog differentiator reduces the difference to $${\pm }$$ ± 2.5%. In contrast, integration of the velocity derivative spectra obtained from the velocity spectra leads to a dissipation rate 14–45% larger than the corresponding values obtained using analog differentiation. Results obtained from the application of a power decay law of turbulence kinetic energy with a nonzero virtual origin to determine the dissipation rate deviate by 1.7%, 1.6%, and 3.6% relative to the corresponding values obtained from the analog differentiator based on the ensemble average of downstream locations with a $${\pm }$$ ± 5.6% scatter about the ensemble average. Graphic abstract

Frequency derivative of Rayleigh wave phase velocity for fundamental mode dispersion inversion: parametric study and experimental application

SUMMARY Monitoring the small variations of a medium is increasingly important in subsurface geophysics due to climate change. Classical seismic surface wave dispersion methods are limited to quantitative estimations of these small variations when the variation ratio is smaller than 10 per cent, especially in the case of variations in deep media. Based on these findings, we propose to study the contributions of the Rayleigh wave phase velocity derivative with respect to frequency. More precisely, in the first step of assessing its feasibility, we analyse the effects of the phase velocity derivative on the inversion of the fundamental mode in the simple case of a two-layer model. The behaviour of the phase velocity derivative is first analysed qualitatively: the dispersion curves of phase velocity, group velocity and the phase velocity derivative are calculated theoretically for several series of media with small variations. It is shown that the phase velocity derivatives are more sensitive to variations of a medium. The sensitivity curves are then calculated for the phase velocity, the group velocity and the phase velocity derivative to perform quantitative analyses. Compared to the phase and group velocities, the phase velocity derivative is sensitive to variations of the shallow layer and the deep layer shear wave velocity in the same wavelength (frequency) range. Numerical data are used and processed to obtain dispersion curves to test the feasibility of the phase velocity derivative in the inversion. The inversion results of the phase velocity derivative are compared with those of phase and group velocities and show improved estimations for small variations (variation ratio less than 5 per cent) of deep layer shear wave velocities. The study is focused on laboratory experiments using two reduced-scale resin-epoxy models. The differences of these two-layer models are in the deep layer in which the variation ratio is estimated as 16.4 ± 1.1 per cent for the phase velocity inversion and 17.1 ± 0.3 per cent for the phase velocity derivative. The latter is closer to the reference value 17 per cent, with a smaller error.

Comparison of Different Techniques to Calculate Properties of Atmospheric Turbulence from Low-Resolution Data

In this work we study different techniques to estimate basic properties of turbulence, that is its characteristic velocity and length scale from low-resolution data. The methods are based on statistics of the signals like the velocity spectra, second-order structure function, number of signal’s zero-crossings and the variance of velocity derivative. First, in depth analysis of estimates from artificial velocity time series is performed. Errors due to finite averaging window, finite cut-off frequencies and different fitting ranges are discussed. Next, real atmospheric measurement data are studied. It is demonstrated that differences between results of the methods can indicate deviations from the Kolmogorov’s theory or the presence of external intermittency, that is the existence of alternating laminar/turbulent flow patches.

Adaptive-Uniform-Experimental-Design-Based Fractional-Order Particle Swarm Optimizer with Non-Linear Time-Varying Evolution

An adaptive-uniform-experimental-design-based fractional particle swarm optimizer (AUFPSO) with non-linear time-varying evolution (NTE) is proposed. A particle swarm optimizer (PSO) is an excellent evolutionary algorithm due to its simple structure and rapid convergence. Nevertheless, PSO has notable drawbacks. Although many proposed methods and strategies have enhanced its effectiveness and performance, PSO is limited by its tendency to fall into local optima and its tendency to obtain different solutions in each search (i.e., its weak robustness). Introducing fractional-order calculus in PSO (FPSO) can correct the order of the velocity derivative for each particle, which enhances the diversity and algorithmic effectiveness. This study used NTE of the order of the velocity derivative, inertia weight, cognitive parameter, and social parameter in an FPSO used to search for a global optimal solution. To obtain the best combination of FPSO and NTE, an adaptive uniform experimental design (AUED) method was used to deal with this essential issue. The AUED method integrates a uniform layout with the best combination phase and a stepwise ratio to assist in selecting the best combination for FPSO-NTE. Experimental applications in 15 global numerical optimization problems confirmed that the AUFPSO-NTE had a better performance and robustness than existing PSO-related algorithms.