Comparison of computational codes for direct numerical simulations of turbulent Rayleigh-B\'enard convection

Computational codes for direct numerical simulations of Rayleigh-B\'enard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at $Ra=10^8$ and $Pr=1$ in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBflow) and a specialized fourth-order finite-volume code (Goldfish) are compared with a general purpose finite-volume approach (OpenFOAM) and a general purpose spectral-element code (Nek5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBflow and Goldfish are found to excel in efficiency, outperforming the general purpose flow solvers Nek5000 and OpenFOAM by an order of magnitude with an error on the Nusselt number $Nu$ below $5\%$. However, we find that $Nu$ alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using Nek5000 clearly indicate inadequate flow resolution even when $Nu$ is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.Comment: 12 pages, 5 figure

Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution

Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-B\'enard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as $0.588Pr^{-1/2}$ for $Pr\ll Pr^*$ and as $0.982 Pr^{-1/3}$ for $Pr^*\ll\Pr$, with $Pr^*= 0.046$. Comparison of the Prandtl--Blasius velocity boundary layer thickness with that evaluated in the direct numerical simulations by Stevens, Verzicco, and Lohse (J. Fluid Mech. 643, 495 (2010)) gives very good agreement. Based on the Prandtl--Blasius type considerations, we derive a lower-bound estimate for the minimum number of the computational mesh nodes, required to conduct accurate numerical simulations of moderately high (boundary layer dominated) turbulent Rayleigh-B\'enard convection, in the thermal and kinetic boundary layers close to bottom and top plates. It is shown that the number of required nodes within each boundary layer depends on Nu and Pr and grows with the Rayleigh number Ra not slower than \sim\Ra^{0.15}. This estimate agrees excellently with empirical results, which were based on the convergence of the Nusselt number in numerical simulations

QED radiative corrections to impact factors

We consider the radiative corrections to the impact factors of electron and photon. According to a generalized eikonal representation the e\bar e scattering amplitude at high energies and fixed momentum transfers is proportional to the electron form factor. But we show that this representation is violated due to the presence of non-planar diagrams. One loop correction to the photon impact factor for small virtualities of the exchanged photon is obtained using the known results for the cross section of the e\bar e production at photon-nuclei interactions.Comment: 8 pp, plain LaTe

Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point

As shown in earlier work (Ahlers et al., J. Fluid Mech. 569, p.409 (2006)), non-Oberbeck Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Benard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric thermal expansion coefficient \beta(T). More precisely, it is the nonlinear T-dependence of the density \rho(T) in the buoyancy force which causes another type of NOB effect. We demonstrate that through a combination of experimental, numerical, and theoretical work, the latter in the framework of the extended Prandtl-Blasius boundary layer theory developed in Ahlers et al., J. Fluid Mech. 569, p.409 (2006). The latter comes to its limits, if the temperature dependence of the thermal expension coefficient \beta(T) is significant.Comment: 18 pages, 15 figures, 3 table

On generalized Bessel potentials and perfect functional completions

In this paper, perfect completions are constructed using the norm associated with the kernel of the generalized Bessel potentia

On the theory of spaces of generalized Bessel potentials

Potential theory originates from the theory of electrostatic and gravitational potentials and the study of the Laplace, wave, Helmholtz, and Poisson equations. The celebrated Riesz potentials are the realizations of the real negative powers of the Laplace and wave operators. In the meantime, much attention in potential theory is paid to the Bessel potential generating the spaces of fractional smoothnes

Solving the Euler-Poisson-Darboux equation of fractional order

We consider the Cauchy problem for the one-dimensional, homogeneous Euler-Poisson-Darboux equation with a differential operator of fractional order in time being the left-sided fractional Bessel operator. At the same time, we use the ordinary differential operator in the space variable of the second orde

THE INFLUENCE OF PHYSICAL METHODS OF VEGETABLES PROCESSING ON THE QUALITY OF FROZEN PRODUCTS

The indicators of quality and microbiological safety of frozen vegetable semi-finished products (for example, cut beet) processed in a microwave field for subsequent storage for 18 months at a temperature of minus 20±2 °C. Treatment with a microwave field was carried out under the following conditions: power 600 W and duration 5 min (180 kJ); power 1000 W and duration 4 min (240 kJ). It is established that treatment in the microwave field contributes to the preservation of consumer qualities of the product (total solids content, mass fraction of soluble solids, bound moisture content, microbiological effect) in the process of long-term low-temperature storage.The indicators of quality and microbiological safety of frozen vegetable semi-finished products (for example, cut beet) processed in a microwave field for subsequent storage for 18 months at a temperature of minus 20±2 °C. Treatment with a microwave field was carried out under the following conditions: power 600 W and duration 5 min (180 kJ); power 1000 W and duration 4 min (240 kJ). It is established that treatment in the microwave field contributes to the preservation of consumer qualities of the product (total solids content, mass fraction of soluble solids, bound moisture content, microbiological effect) in the process of long-term low-temperature storage

Comparison of simulation and analytical models for the distribution of a group of agents moving in random directions

This study focused on simulation and analytical models of a multidimensional random walk of many agent

Narratives in School History Textbooks: An East African Perspective

This study conducts an in-depth analysis of the content in school history textbooks from East Africa, assessing the potential impact on socio-political stability in the region. Citing relevant research on multiperspectivity, the authors recognize the significant role power dynamics play in shaping historical narratives, suggesting that colonial structures and the dominance of majority groups continue to influence these dynamics. The paper provides a historical context by reviewing the educational systems in Kenya, Uganda, and Tanzania. It particularly notes the challenges faced by the Muslim minority group, who often study in public schools managed by Christian entities or the government. These institutions predominantly use secular history textbooks, potentially biasing the learning experience. The researchers employed content analysis to qualitatively evaluate the data in these textbooks. The results indicated a biased presentation of historical events, overemphasizing the colonial and post-colonial periods while underrepresenting the influence of Islam in East Africa. The authors argue that this selective approach to history education might intensify existing socio-political tensions in the region. They advocate for a more balanced and inclusive representation of historical events in educational materials to foster a more equitable learning environment