Application of the multi distribution function lattice Boltzmann approach to thermal flows

Numerical methods able to model high Rayleigh (Ra) and high Prandtl (Pr) number thermal convection are important to study large-scale geophysical phenomena occuring in very viscous fluids such as magma chamber dynamics (10⁴ < Pr < 10⁷ and 10⁷ < Ra < 10¹¹). The important variable to quantify the thermal state of a convective fluid is a generalized dimensionless heat transfer coefficient (the Nusselt number) whose measure indicates the relative efficiency of the thermal convection. In this paper we test the ability of Multi-distribution Function approach (MDF) Thermal Lattice Boltzmann method to study the well-established scaling result for the Nusselt number (Nu ∝ Ra ^1/3) in Rayleigh Bénard convection for 10⁴ ≤ Ra ≤ 10⁹ and 10¹ ≤ Pr ≤ 10⁴. We explore its main drawbacks in the range of Pr and Ra number under investigation: (1) high computational time N _c required for the algorithm to converge and (2) high spatial accuracy needed to resolve the thickness of thermal plumes and both thermal and velocity boundary layer. We try to decrease the computational demands of the method using a multiscale approach based on the implicit dependence of the Pr number on the relaxation time, the spatial and temporal resolution characteristic of the MDF thermal model.     

Temperature gradient spectra and temperature dissipation rate in a turbulent convective flow

Determining mixing coefficients in oceanographic flows relies on the form of temperature gradient spectra in turbulent water flows at large wavenumbers. Several recent investigations concluded that these spectra are best described by the functional form proposed by Kraichnan rather than by the Batchelor form, more commonly used in oceanography. In this study, we provide additional support for this conclusion using laboratory measurements of the temperature gradient spectra in a Rayleigh–Bénard convective cell, in order to avoid difficulties inherent in oceanographic field measurements. The range of Rayleigh numbers in experiments is between Ra = 3×10⁷ and Ra = 5×10⁹. In addition to a traditional method of traversing thermistors, a novel optical technique recently introduced for oceanic measurements was used to obtain the spectra; comparison between these two methods serves as a validation test for the new optical technique. The temperature measurements were also augmented by 2D particle image velocimetry (PIV) observations. The measured dependence of the Nusselt number on the Rayleigh number followed Nu ∝ Ra^0.29 at Pr = 6 and was consistent with the literature data. We observed the temperature dissipation rate to vary by an order of magnitude over a horizontal transect at Ra > 10⁹. The temperature dissipation spectra obtained by both methods were in agreement over the Ra interval considered. The location of the temperature dissipation peaks was also consistent with PIV measured energy dissipation rates. Our data suggest increasing importance of top/bottom boundaries for the momentum and the temperature dissipation with increasing Ra number. Applied to oceanic upper ocean convection, our results imply that most of the dissipation occurs close to the air–sea boundary. Thus, attempts to parameterise or measure air–sea turbulent convective fluxes have to reflect the dominant role of near boundary dissipation at large Ra.     

Low-Prandtl-number Rayleigh-Bénard convection with stress-free boundaries

Results of direct numerical simulations on Rayleigh-Bénard convection in low-Prandtl-number convection with stress-free horizontal boundaries are presented. Simulations are done in a three dimensional rectangular simulation box of dimensions L _x × L _y × 1. Instabilities and the corresponding fluid patterns near onset of convection are investigated by varying the horizontal aspect ratio η = L _y /L _x in a range 1 ≤ η ≤ 4. Fluid patterns are complex and unsteady at the instability onset for η ≥ 2. They consist of wavy rolls, rhombic patterns and oblique wavy rolls. The patterns near onset are time periodic for η < 2. We observe periodic wavy rolls for η = 4 / 3. Homoclinic bifurcations are observed for η = 1 for 0 ≤ Pr ≤ 0.03. We observe spontaneous breaking of a single limit cycle in two and again spontaneous merging of two limit cycles into one in a simulation box with η = 1, as the reduced Rayleigh number r = Ra/R a _c is raised at a fixed value of Pr. The effect of Prandtl number on the homoclinic bifurcations is also investigated. We also present a low-dimensional model, which captures the instability sequence quite accurately for η = 1.     

Stability conditions for fermionic Ising spin-glass models in the presence of a transverse field

The stability of a spin-glass (SG) phase is analyzed in detail for a fermionic Ising SG (FISG) model in the presence of a magnetic transverse field Γ. The fermionic path integral formalism, replica method and static approach have been used to obtain the thermodynamic potential within one step replica symmetry breaking ansatz. The replica symmetry (RS) results show that the SG phase is always unstable against the replicon. Moreover, the two other eigenvalues λ_± of the Hessian matrix (related to the diagonal elements of the replica matrix) can indicate an additional instability to the SG phase, which enhances when Γ is increased. Therefore, this result suggests that the study of the replicon cannot be enough to guarantee the RS stability in the present quantum FISG model, especially near the quantum critical point. In particular, the FISG model allows changing the occupation number of sites, so one can get a first order transition when the chemical potential exceeds a certain value. In this region, the replicon and the λ_± indicate instability problems for the SG solution close to all ranges of a first order boundary.     

The zone boundary mode in periodic nonlinear electrical lattices

We study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of N inductors and N voltage-dependent capacitors with periodic boundary conditions. The inductances are allowed to alternate, while the capacitors are identical and each have a quadratic dependence on voltage. By block-diagonalizing a 2N×2N Floquet problem, we reduce the question of the stability of the mode to a single Hill’s equation that is analyzed using methods of perturbation theory and averaging. We show that periodicity of the lattice inductances degrades stability, and also show that the instability threshold is proportional to N⁻². Numerical computations validate the perturbative results.     

Scaling in large Prandtl number turbulent thermal convection

We study the scaling properties of heat transfer Nu in turbulent thermal convection at large Prandtl number Pr using a quasi-linear theory. We show that two regimes arise, depending on the Reynolds number Re. At low Reynolds number, NuPr ^-1/2 and Re are a function of RaPr ^-3/2. At large Reynolds number NuPr ^1/3 and RePr are function only of RaPr ^2/3 (within logarithmic corrections). In practice, since Nu is always close to Ra ^1/3, this corresponds to a much weaker dependence of the heat transfer in the Prandtl number at low Reynolds number than at large Reynolds number. This difference may solve an existing controversy between measurements in SF₆ (large Re) and in alcohol/water (lower Re). We link these regimes with a possible global bifurcation in the turbulent mean flow. We further show how a scaling theory could be used to describe these two regimes through a single universal function. This function presents a bimodal character for intermediate range of Reynolds number. We explain this bimodality in term of two dissipation regimes, one in which fluctuation dominate, and one in which mean flow dominates. Altogether, our results provide a six parameters fit of the curve Nu(Ra, Pr) which may be used to describe all measurements at Pr≥0.7. Received 27 February 2002 / Received in final form 29 May 2002 Published online 31 July 2002     

Study of heat transfer in a horizontal cylinder with fins

The purpose of this paper is to investigate, numerically, the effect of internal fins on the flow pattern, temperature distribution and heat transfer between concentric horizontal cylinders. A Galerkin finite element method is adopted for the discretization of the governing equations. The numerical procedure consists in solving series of transient problems of increasing Ra. Results are presented using air (Pr = 0.7) with Rayleigh numbers ranging from 10³ to 10⁶ for different fin configurations (1 and 2), geometries (sharp, round and divergent tip) and lengths (l = 0.25, 0.5 and 0.75). They are illustrated in terms of isotherms, velocity fields, Nusselt numbers and fin efficiencies. Configuration 2 presents a heat transfer rate 10% above that of configuration 1, at Ra = 10⁶. The heat transfer is about the same for the three geometries, but the best fin efficiency is associated with the fin with a round tip.     

Stabilities of one-dimensional stationary states of Bose-Einstein condensates

We explore the dynamical stabilities of a quasi-one-dimensional (1D) Bose-Einstein condensate (BEC) consisting of fixed N atoms with time-independent external potential. For the stationary states with zero flow density the general solution of the perturbed time evolution equation is constructed, and the stability criterions concerning the initial conditions and system parameters are established. Taking the lattice potential case as an example, the stability and instability regions on the parameter space are found. The results suggest a method for selecting experimental parameters and adjusting initial conditions to suppress the instabilities.     

Nature of the quantum spin correlations through the superconducting-normal phase transition in electron-doped superconducting Pr0.88LaCe0.12CuO4

We use neutron scattering and specific heat measurements to relate the response of the spin fluctuations and static antiferromagnetic (AF) order to the superconductivity in the electron-doped high-transition-temperature superconductor, Pr_.88LaCe_.12CuO_4−δ (PLCCO) (T_c=24 K), as the system is tuned via a magnetic field applied beyond the upper critical field (H_c2) and driven into the normal state. The strength of the collective magnetic excitation commonly termed “resonance” decreases smoothly with increasing field and vanishes in the normal state, paralleling the behavior of the superconducting condensation energy. The suppression of superconductivity is accompanied by a smooth reduction in the very low energy spin fluctuations, and the concomitant emergence of static AF order. Our results suggest an intimate connection between the resonance and the superconducting condensation energy.     

Magnetic stabilization,transition and energy analysis in the Marangoni driven Full-Zone at low Prandtl numbers

The linear stability of the Marangoni-driven Full-Zone is investigated for low Prandtl number fluids. A constant and uniform magnetic field is applied along the axial axis of the liquid bridge to stabilize the axisymmetric base state. Dramatic contraction of the flow circulation in both radial and azimuthal directions is observed with moderate magnetic fields. The numerical solution utilizes a vorticity transport formulation and high resolution spectral collocation scheme with Chebyshev polynomial basis functions. Critical transitions to three-dimensional, stationary flows are observed up to Ha = 300 for Pr = 0.02 and Ha = 500 for Pr = 0.001. A hydrodynamically driven instability is suggested by the perturbation flows and confirmed through an energy analysis.     

Effect of interface states associated with transitional nanophase precipitates in the enhancement of red emission from SrAl12O19:Pr by Ti incorporation

SrAl₁₂O₁₉:Pr³⁺, Ti⁴⁺ phosphor suitable for field emission displays is prepared by the wet chemical gel-carbonate method and the mechanism of enhancement in red photoluminescence (PL) intensity with Ti⁴⁺ therein has been investigated. The PL spectra of Pr³⁺ show both ¹D₂-³H₄ and ³P₀-³H₆ emission in the red region with very weak intensity when excited at 355 nm. The emission intensity has increased by about 100 times at room temperature in the compositional range SrAl_12−xTi_xO_19+x/2:Pr³⁺, with 0.1≤x≤0.3 in comparison to Ti-free SrAl₁₂O₁₉:Pr³⁺. TEM investigations show the presence of exsolved nanophase of SrAl₈Ti₃O₁₉, the precipitation of which is preceded by the presence of defect centers at the interfacial regions between the semicoherent transient phase and the parent SrAl₁₂O₁₉ matrix. The presence of transitional nanophase and the associated defects modify the excitation-emission process by way of formation of electronic sub-levels at lower energy (3.5 eV) than the band gap of SrAl₁₂O₁₉ (∼7 eV) followed by non-resonance energy transfer to Pr³⁺ level, leading to magnetic-dipole related red emission with enhanced intensity. The PL intensity of Pr³⁺ decreases at high Ti⁴⁺ concentrations (x>0.3) due to higher extent of segregation of non-emissive SrAl₈Ti₃O₁₉:Pr³⁺ phase.     

Scaling in convective evaporation and sidewall boundary layer

Turbulent convection at aspect ratios from 0.06 to 2 is investigated in the laboratory with evaporation experiments from vertical cylinders having different diameters and liquid levels. With alcohol, only diffusive evaporation takes place. With water, for small diameters, evaporation proceeds by diffusion whereas convective evaporation develops when the diameter is increased. This onset can be effectively interpreted in terms of a viscous sidewall boundary layer, whose thickness δ varies with respect to the available height h according to δ/h = 3.4 Ra^-0.28±0.01 versus Rayleigh number Ra. The Sherwood number Sh, analog of the Nusselt number, exhibits a power law variation Sh = 0.6 Ra^0.27±0.02 for Ra varying from 10⁴ to 3 ×10⁸. The scaling observed in this case of an open boundary is thus similar to the scaling measured in confined Rayleigh-Bénard convection.     

Influence of the electron-electron coupling parameters on the temperature of the superconducting transition in compounds with the A-15 structure

The electronic structure of the A-15 compounds is investigated with allowance for the Hubbard energy as the largest energy parameter. The conditions for the onset of Cooper instability are obtained. The dependence of the superconducting transition temperature on the degree of filling of the d and p shells of the transition elements and nontransition elements is determined. The relation between the transition temperature and the energy of a hop along the chains is established, permitting a comparison with experiment. A qualitative explanation is given for the empirical results of Matthias.     

Importance of direction of vibration on the onset of Soret-driven convection under gravity or weightlessness

This paper considers the influence of the direction of vibration on the stability threshold of two-dimensional Soret-driven convection. The configuration is an infinite layer filled with a binary mixture, which can be heated from below or from above. The limiting case of high-frequency and small-amplitude vibration is considered for which the time-averaged formulation has been adopted. The linear stability analysis of the quasi-mechanical equilibrium shows that the problem depends on five non-dimensional parameters. These include the thermal Rayleigh number ( Ra_T), the vibrational parameter (R), the Prandtl number ( Pr), the Lewis number (Le), the separation ratio (S) and the orientation of vibration with respect to the horizontal heated plate (). For different sets of parameters, the bifurcation diagrams are plotted Ra_c = f (S) and k_c = g(S), which are the critical thermal Rayleigh and the critical wave numbers, respectively. Our results indicate that, relative to the classical case of static gravity, vibration may affect all regions in Ra_c-S stability diagram. In the case of mono-cellular convection, by using a regular perturbation method, a closed-form relation for the critical Rayleigh number is found. Several physical situations in the presence or in the absence of gravity (micro-gravity) are discussed.     

Onset of Unsteady Horizontal Convection in Rectangular Tank at Pr=1

The horizontal convection within a rectangular tank is numerically simulated. The flow is found to be unsteady at high Rayleigh numbers. There is a Hopf bifurcation of Ra from steady solutions to periodic solutions, and the critical Rayleigh number Rac is obtained to be Rac = 5.5377×10^8 for the middle plume forcing at Pr = 1, which is much larger than the value previously obtained. In addition, the unstable perturbations are always generated from the central jet, which implies that the onset of instability is due to velocity shear （shear instability） other than thermally dynamics （thermal instability）. Finally, Paparella and Young＇s first hypotheses [J. Fluid Mech. 466 （2002） 205] about the destabilization of the flow is numerically proven, i.e. the middle plume forcing can lead to a destabilization of the flow.     

Investigation on stability of directionally solidified CBr4-C2Cl6 lamellar eutectic by using multiphase field simulation

With the multiphase field method, the stability of lamellar basic state is investigated during the directional solidification of eutectic alloy CBr4-C2Cl6. A great number of lamellar patterns observed in experiments are simulated, and a stability diagram for lamellar pattern selections is presented. The simulated growth behaviours of these patterns are found to be qualitatively consistent with Karma et al＇s numerical calculations and experimental results. The formation of the primary instability is attributed to the destabilization of solute boundary layer.     

Ce and Pr5d-energy levels in the (pseudo) perovskites KMgF3 and NaMgF3

The location of the 5d-energy levels of Ce³⁺ and Pr³⁺ in the cubic perovskite KMgF₃ and in the distorted perovskite NaMgF₃ was determined from spectroscopic studies in the vacuum ultraviolet. It is established that Ce³⁺ and Pr³⁺ ions both occupy the same site in each host: K⁺ sites for KMgF₃ and Na⁺ sites for NaMgF₃. The small crystal field splitting and the small value of the centroid shift of the 4fⁿ⁻¹5d-configuration yield a relatively high energy for the lowest 5d state of both Ce³⁺ and Pr³⁺. The lowest 5d state of Pr³⁺ in both hosts is found at energy higher than the 4f² state, enabling the photon cascade emission to occur.     

A nonlinear stability analysis of a double-diffusive magnetized ferrofluid with magnetic field-dependent viscosity

A rigorous nonlinear stability result is derived by introducing a suitable generalized energy functional for a magnetized ferrofluid layer heated and soluted from below with magnetic field-dependent (MFD) viscosity, for stress-free boundaries. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. For ferrofluids, we find that there is possibility of existence of subcritical instabilities, however, it is noted that in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effects of magnetic parameter, M₃, solute gradient, S₁ and MFD viscosity parameter, δ, on the subcritical instability region have also been analyzed.     

The onset of convection in a couple stress fluid saturated porous layer using a thermal non-equilibrium model

The stability of a couple stress fluid saturated horizontal porous layer heated from below and cooled from above when the fluid and solid phases are not in local thermal equilibrium is investigated. The Darcy model is used for the momentum equation and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is employed to obtain the condition for the onset of convection. The effect of thermal non-equilibrium on the onset of convection is discussed. It is shown that the results of the thermal non-equilibrium Darcy model for the Newtonian fluid case can be recovered in the limit as couple stress parameter C→0. We also present asymptotic analysis for both small and large values of the inter phase heat transfer coefficient H. We found an excellent agreement between the exact solutions and asymptotic solutions when H is very small.     

Heat transfer over a stretching surface with variable heat flux in micropolar fluids

Heat transfer over a stretching surface with uniform or variable heat flux in micropolar fluids is investigated in this Letter. The boundary layer equations are transformed into ordinary differential equations, and then they are solved numerically by a finite-difference method. The effects of the material parameter K, Prandtl number Pr, velocity exponent parameter m, and heat flux exponent parameter n on the heat transfer characteristics are studied. It is found that the local Nusselt number is higher for micropolar fluids compared to Newtonian fluids.