In this paper, we study the sum rate performance of zero-forcing (ZF) and regularized ZF (RZF) precoding in large MISO broadcast systems under the assumptions of imperfect channel state information at the transmitter and per-user channel transmit correlation. Our analysis assumes that the number of transmit antennas M and the number of single-antenna users K are large while their ratio remains bounded. We derive deterministic approximations of the empirical signal-tointerference plus noise ratio (SINR) at the receivers, which are tight as M, K → ∞. In the course of this derivation, the per-user channel correlation model requires the development of a novel deterministic equivalent of the empirical Stieltjes transform of large dimensional random matrices with generalized variance profile. The deterministic SINR approximations enable us to solve various practical optimization problems. Under sum rate maximization, we derive (i) for RZF the optimal regularization parameter, (ii) for ZF the optimal number of users, (iii) for ZF and RZF the optimal power allocation scheme and (iv) the optimal amount of feedback in large FDD/TDD multi-user systems. Numerical simulations suggest that the deterministic approximations are accurate even for small M, K.

The recent implementation of new environmental legislations led to a change in the manufacturing of composites that has repercussions on printed wiring boards (PWB). This in turn led to alternate processing methods (e.g., lead-free soldering), which affected the required physical and chemical properties of the additives used to impart flame retardancy. This review will discuss the latest advancements in phosphorus containing flame retardants for electrical and electronic (EE) applications and compare them with commercially available ones. The mechanism of degradation and flame retardancy of phosphorus flame retardants in epoxy resins will also be discussed.

We analyse the wind and boundary layer properties of turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio one for Prandtl number $\mathit{Pr}= 0. 786$ and Rayleigh numbers ($\mathit{Ra}$) up to $1{0}^{9} $ by means of highly resolved direct numerical simulations. We identify time periods in which the orientation of the large-scale circulation (LSC) is nearly constant in order to perform a statistical analysis of the LSC. The analysis is then reduced to two dimensions by considering only the plane of the LSC. Within this plane the LSC is treated as a wind with thermal and viscous boundary layers developing close to the horizontal plates. Special focus is on the spatial development of the wind magnitude and the boundary layer thicknesses along the bottom plate. A method for the local analysis of the instantaneous boundary layer thicknesses is introduced which shows a dramatically changing wind magnitude along the wind path. Furthermore a linear increase of the viscous and thermal boundary layer thickness along the wind direction is observed for all $\mathit{Ra}$ considered while their ratio is spatially constant but depends weakly on $\mathit{Ra}$. A possible explanation is a strong spatial variation of the wind magnitude and fluctuations in the boundary layer region.

We report a new thermal boundary layer equation for turbulent Rayleigh-Bénard convection for Prandtl number Pr > 1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr 1 and (b) Pr ≫ 1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr = 4.38 (water) and Pr = 2547.9 (glycerol) respectively.PACS numbers: 44.20.+b, 44.25.+f, 47.27.ek, 47.27.te Turbulent Rayleigh-Bénard convection (RBC) [1][2][3][4][5], consisting of a fluid confined between two horizontal plates, heated from below and cooled from above, is a system of great research interest. It is a paradigm system for studying turbulent thermal convection, which is ubiquitous in nature, occurring in the atmosphere and the mantle of the Earth as well as in stars like our Sun. Convective heat transfer is also an important problem in engineering and technological applications. The state of fluid motion in RBC is determined by the Rayleigh number Ra = αg∆H 3 /(κν) and Prandtl number Pr = ν/κ. Here α denotes the isobaric thermal expansion coefficient, ν the kinematic viscosity and κ the thermal diffusivity of the fluid, g the acceleration due to gravity, ∆ the temperature difference between the bottom and top plates, and H the distance between the plates.In turbulent RBC, there are viscous boundary layers (BLs) near all rigid walls and two thermal BLs, one above the bottom plate and one below the top plate. We denote the thicknesses of the viscous and thermal BLs by l and λ respectively. Both viscous and thermal BLs play a critical role in the turbulent heat transfer of the system and in particular λ is inversely proportional to the heat transport. Grossmann and Lohse (GL) [6], [7] developed a scaling theory of how the Reynolds number Re, determined by the mean large-scale circulation velocity U 0 above the viscous BL, and the dimensionless Nusselt number Nu, measuring the heat transport, depend on Ra and Pr for moderate Ra. The theory makes explicit use of the result l/H ∝ Re −1/2 with the proportionality constant depending only on Pr. This result follows from the assumptions that the BLs are laminar and their mean profiles, averaged over time, are described by the Prandtl-Blasius-Pohlhausen (PBP) theory [8-10] for steady-state forced convection above an infinite weakly-heated plate. Although the GL theory gives perfect agreement with the heat transport measurements, the assumption that the BLs are described by PBP theory is not fulfilled. Systematic deviations of the mean velocity and temperature profiles from the PBP predictions have been reported both in experiments and in direct numerical simulations (DNS) [11][12][13][14][15]. These deviations remain even after a dynamical rescaling procedure [16] that takes into account of the time variations of λ is used, and increase ...

We report on a numerical study of the aspect-ratio dependency of Rayleigh-Bénard convection, using direct numerical simulations. The investigated domains have equal height and width while the aspect ratio Γ of depth per height is varied between 1/10 and 1. The Rayleigh numbers \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra for this study variate between 105 and 109, while the Prandtl number is \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Pr}} = 0.786$\end{document}Pr=0.786. The main focus of the study concerns the dependency of the Nusselt number \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Nu}}$\end{document}Nu and the Reynolds number \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Re}}$\end{document}Re on \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra and Γ. It turns out that due to Γ, differences to the cubic case (i.e., Γ = 1) in \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Nu}}$\end{document}Nu of up to 55% and in \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Re}}$\end{document}Re of up to 97% occur, which decrease for increasing \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra. In particular for small Γ sudden drops in the \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra-scaling of \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Nu}}$\end{document}Nu and \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Re}}$\end{document}Re appear for \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}\approx 10^6$\end{document}Ra≈106. Further analysis reveals that these correspond to the onset of unsteady motion accompanied by changes in the global flow structure. The latter is investigated by statistical analysis of the heat flux distribution on the bottom and top plates and a decomposition of the instantaneous flow fields into two-dimensional modes. For \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra slightly above the onset of unsteady motion (i.e., \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}\approx 10^6$\end{document}Ra≈106) for all considered Γ ⩽ 1/3 a four-roll structure is present, which corresponds to thermal plumes moving vertically through the domain's center. For \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}\ge 10^7$\end{document}Ra≥107, also for small Γ, a single-roll structure is dominant, in agreement with two-dimensional simulations and experiments at larger \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra and \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Pr}}$\end{document}Pr.