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

ON THE USE OF THE CENTRAL DIFFERENCE SCHEME FOR SOLVING THE PROBLEM OF GAS DYNAMICS

The paper proposes a method for solving the problem of gas dynamics, implemented on the basis of a central difference scheme, the stability of which is achieved by performing a correction of the calcu-lated flows. It is shown that when solving the problem of discontinuity decay, the proposed method is stable, comparable in accuracy with the McCormack and Lax – Wendroff methods and surpasses them in performance.

FAST NUMERICAL IMPLEMENTATION OF THE MDR TRANSFORMATIONS

In the present paper a numerical implementation technique for the transformations of the Method of Dimensionality Reduction (MDR) is described. The MDR has become, in the past few years, a standard tool in contact mechanics for solving axially-symmetric contacts. The numerical implementation of the integral transformations of the MDR can be performed in several different ways. In this study, the focus is on a simple and robust algorithm on the uniform grid using integration by parts, a central difference scheme to obtain the derivatives, and a trapezoidal rule to perform the summation. The results are compared to the analytical solutions for the contact of a cone and the Hertzian contact. For the tested examples, the proposed method gives more accurate results with the same number of discretization points than other tested numerical techniques. The implementation method is further tested in a wear simulation of a heterogeneous cylinder composed of rings of different material having the same elastic properties but different wear coefficients. These discontinuous transitions in the material properties are handled well with the proposed method.

The distribution of pollutants in the ground layer of the atmosphere in the presence of forest plantations

The aim of the work is to study the influence of forest plantations on the distribution of pollutants in the ground layer of the atmosphere. The model that takes into account a variety of factors: the presence of forest plantations, the variability of pressure, density and temperature, the presence of a multicomponent impurity, etc., was proposed for the numerical modeling of the process of transferring air pollutants to air. The scheme obtained as a result of a linear combination of the central difference scheme and the «CABARET» scheme was constructed to approximate the convection operator in this paper. An analysis of the results of numerical experiments allows to conclude that the distribution of pollutants in a multicomponent air environment is most significantly affected by the density of vegetation, and insignificantly influenced by the width of the forest plantations area.

Numerical comparison of nonstandard schemes for the Airy equation

This paper considers the Airy ordinary differential equation (ODE) and different ways it can be discretized. We first consider a standard discretization using the central difference scheme. We then consider two difference schemes which were created using a nonstandard methodology. Finally, we compare the different schemes and how well they approximate solutions to the Airy ODE.