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The 13-speed thermal lattice Bhatnagar–Gross–Krook model on hexagonal lattice is a single relaxation time model with an adjustable parameter λ which makes the Prandtl number tunable. This model maintains the simplicity of the lattice Boltzmann method (LBM) and is also suitable for various thermal fluids. In this paper, it is applied to simulations of the lid-driven flow in a square cavity at a wide range of Reynolds numbers. Numerical experiments show that this model can give the same accurate results as those by the conventional numerical methods.

The entropic lattice Boltzmann (LB) method has recently been extended to include energy conservation in order to simulate weakly compressible flows. One of the limitations of this method when using the BGK collision model is the fixed Prandtl number. In this paper a new simple method is proposed and validated in order to simulate fluids of arbitrary Prandtl number.

A lattice Boltzmann model is developed for viscous compressible flows with flexible specific-heat ratio and Prandtl number. Unlike the Maxwellian distribution function or circle function used in the existing lattice Boltzmann models, a polynomial kernel function in the phase space is introduced to recover the Navier–Stokes–Fourier equations. A discrete equilibrium density distribution function and a discrete equilibrium total energy distribution function are obtained from the discretization of the polynomial kernel function with Lagrangian interpolation. The equilibrium distribution functions are then coupled via the equation of state. In this framework, a model for viscous compressible flows is proposed. Several numerical tests from subsonic to supersonic flows, including the Sod shock tube, the double Mach reflection and the thermal Couette flow, are simulated to validate the present model. In particular, the discrete Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite-difference method. Numerical results agree well with the exact or analytic solutions. The present model has potential application in the study of complex fluid systems such as thermal compressible flows.

In this paper, we demonstrate a set of fundamental conditions required for the formulation of a thermohydrodynamic lattice Boltzmann model at an arbitrary Prandtl number. A specific collision operator form is then proposed that is in compliance with these conditions. It admits two independent relaxation times, one for viscosity and another for thermal conductivity. But more importantly, the resulting thermohydrodynamic equations based on such a collision operator form is theoretically shown to remove the well-known non-Galilean invariant artifact at nonunity Prandtl numbers in previous thermal lattice Boltzmann models with multiple relaxation times.

We evaluate the universal turbulent Prandtl numbers in the energy and enstrophy régimes of the Kraichnan-Batchelor spectra of two-dimensional turbulence using a self-consistent mode-coupling formulation coming from a renormalized perturbation expansion coupled with dynamic scaling ideas. The turbulent Prandtl number is found to be exactly unity in the (logarithmic) enstrophy régime, where the theory is infrared marginal. In the energy régime, the theory being finite, we extract singularities coming from both ultraviolet and infrared ends by means of Laurent expansions about these poles. This yields the turbulent Prandtl number σ ≈ 0.9 in the energy régime.

Rayleigh–Bénard convection in a horizontal layer of nanofluid in the presence of uniform vertical magnetic field is investigated by using Galerkin weighted residuals method. The model used for the nanofluid describes the effects of Brownian motion and thermophoresis. Linear stability theory based upon normal mode analysis is employed to find expressions for Rayleigh number and critical Rayleigh number. The boundaries are considered to be free–free, rigid–rigid and rigid–free. The influence of magnetic field on the stability is investigated and it is found that magnetic field stabilizes the fluid layer. It is also observed that the system is more stable in the case of rigid–rigid boundaries and least stable in case of free–free boundaries. The expression for Rayleigh number for oscillatory convection has also been derived for free–free boundaries.

The atmospheric boundary layer (ABL) is the lowest part of the atmosphere that is continuously under the influence of the underlying surfaces through mechanical (roughness and shear) and thermal effects (cooling and warming), and the overlying, more free layers. Such boundary layers and the related geophysical turbulence exist also in oceans, seas, lakes and rivers. Here we focus on those in the atmosphere; however, similar reasoning as presented here also applies to the other geophysical flows mentioned. Since most of human activities and overall life take place in the ABL, it is easy to grasp the need for an ever better understanding of the ABL: its nature, state and future evolution. In order to provide a reasonable and reliable short- or medium-range weather forecast, a decent climate scenario, or an applied micrometeorological study (for e.g. agriculture, road construction, forestry, traffic), etc., the state of the ABL and its turbulence should be properly characterized and marched forward in time in concert with the other prognostic fields. This is one of many tasks of numerical weather prediction and climate models. Many of these models have problems in handling rapid surface cooling under weak or without synoptic forcing (e.g. calm nighttime mountainous or even hilly conditions).

Overall research during the last ∼ 10 years or so, strongly suggests that the evolution of the stable ABL is still poorly understood today. There we make a contribution by assessing some recent advances in the understanding of nature, theory and modeling of the stable ABL (SABL). In particular, we address inclined very (or strongly) stratified SABL in more details. We show that a relatively thin and very SABL, as recently modeled using an improved “z-less” mixing length scale, can be successfully treated nowadays; the result is quietly extended to other types of the SABL. Finally, a new generalized “z-less” mixing length-scale is proposed. At the same time, no major improvements in modeling weak-wind strongly-stable ABL is reported yet.