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.     

高瑞利数下水平自然热对流的幂律关系

采用数值模拟方法，研究了高度和宽度比为1∶10的狭长矩形腔内的水平自然热对流. 根据对瑞利数(Rayleigh数)Ra在10⁴ <Ra<10¹¹内情形的计算结果，将流动分为三个不同的区间：线性区、连续过渡区、1/5次幂律区. 虽然流量和努塞尔数(Nusselt数)Nu随瑞利数的变化都包括了三个参数演化区间，但从一个区间到另外一个参数区间的转变时并不是同步的，其中努塞尔数的转变总是超前流量的转变. 对比前人的研究发现，流量1/3次幂律的结果是由于瑞利数不够高所致. 此外，模拟结果也表明Siggers等的理论分析过高估计了热通量强度，实际的温度边界层内努塞尔数和瑞利数为1/5次幂律关系.     

Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection

Direct numerical simulation (DNS) has shown that Rayleigh–Bénard convection in a fluid-saturated porous medium self-organizes into narrowly spaced plumes at (ostensibly) asymptotically high values of the Rayleigh number Ra. In this Letter a combination of DNS and upper bound theory is used to investigate the dependence of the Nusselt number Nu on the domain aspect ratio L at large Ra  . A novel algorithm is introduced to solve the optimization problems arising from the upper bound analysis, allowing for the best available bounds to be extended up to Ra≈2.65×10⁴$\mathit{Ra}\approx 2.65\times {10}^4$. The dependence of the bounds on L(Ra)$L(\mathit{Ra})$ is explored and a “minimal flow unit” is identified.     

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.     

ATLAS discovery potential for a heavy charged Higgs boson

The sensitivity of the ATLAS detector to the discovery of a heavy charged Higgs boson is presented. Assuming a heavy SUSY spectrum, the most promising channels above the top quark mass are H^± → tb and H^± → τ^±ν_τ which provide coverage in the low and high tanβ regions up to ∼ 600 GeV. The achievable precisions on the charged Higgs mass and tanβ determination are also discussed. The H^± → W^± h⁰ channel, though restricted to a small MSSM parameter space, shows a viable signal in NMSSM where the parameter space is less constrained. The observation of the channel H⁻ → τ⁻_L ν_τ + c.c. may constitute a distinctive evidence for models with singlet neutrinos in large extra dimensions.     

Laminar natural convection between vertical isothermal heated plates with different temperatures

Numerical investigation of laminar free convection heat transfer in the vertical parallel plate channel with asymmetric heating is presented. Both inlet and exit effects are included into the analysis. A numerical solution is obtained for a Prandtl number of 0.71 and for modified Rayleigh number [`(Ra)]\overline {Ra} = 10⁻¹ ÷ 10⁵, and varying heating ratio T_R = 0 ÷ 1 and aspect ratio A = 10. Fully elliptic Navier-Stokes and energy equations are solved using the finite volume techniques with staggered grid arrangements. The obtained results show a strong influence of the temperature ratio on local and average heat transfer coefficient on the hot and cold plates. With reduction of T_R the heat transfer parameter on the hot wall grows, and on the cold one, on the contrary, it decreases. As a result, the total heat exchange from two plates depends poorly on the parameter T_R.     

Beta decay of 103Sn

The β decay of ¹⁰³Sn, a three-neutron-particle nucleus with respect to the ¹⁰⁰Sn core, was investigated at the GSI on-line mass separator using an array of 17 germanium crystals and a total absorption spectrometer. A total of 31 β-delayed γ-rays (29 new) of the ¹⁰³Sn →¹⁰³ In decay were observed and, on the basis of β-γ-γ coincidences, the ¹⁰³Sn decay scheme was established for the first time. By means of total absorption spectroscopy, β intensities, the Gamow-Teller strength distribution and the summed Gamow-Teller strength value of 3.5±0.5 were determined for this decay. Its half-life and Q_EC value were found to be 7.0±0.2 s and 7.64±0.7 MeV, respectively. The β-delayed proton branching ratio was measured to be 1.2±0.1%. The results are discussed in comparison with shell-model predictions based on realistic and empirical interactions.     

Dynamical decay process of <Superscript>219, 220</Superscript>Ra<Superscript>*</Superscript> formed in <Superscript>10, 11</Superscript>B + <Superscript>209</Superscript>Bi reactions

The excitation functions for both the evaporation residue and fission have been calculated for ¹⁰B + ²⁰⁹Bi and ¹¹B + ²⁰⁹Bi reactions forming compound systems ^{219, 220}Ra^* , using the dynamical cluster-decay model (DCM) with effects of deformations and orientations of the nuclei included in it. In addition to this, the excitation functions for complete fusion (CF) are obtained by summing the fission cross-sections, neutron evaporation and charged particle evaporation residue cross-sections produced through the axn\ensuremath \alpha xn and pxn\ensuremath pxn (x = 2, 3, 4) emission channels for the ²¹⁹Ra system at various incident centre-of-mass energies. Experimentally the CF cross-sections are suppressed and the observed suppression is attributed to the low binding energy of ^{10, 11}B which breaks up into charged fragments. The reported complete fusion (CF) and incomplete fusion (ICF) excitation functions for the ²¹⁹Ra system are found to be nicely fitted by the calculations performed in the framework of DCM, without invoking a significant contribution from quasi-fission. Although DCM has been applied for a number of compound nucleus decay studies in the recent past, the same is being used here in reference to ICF and subsequent decay processes along with the CF process. Interestingly the main contribution to complete fusion cross-section comes from the fission cross-section at higher incident energies, which in DCM is found to consist of an asymmetric fission window, shown to arise due to the deformation and orientation effects of formation and decay fragments.     

Brownian Motion with Dry Friction

A small object (Solid or droplet) is placed on a horizontally vibrating plate, imposing an acceleration γ(t) in the form of a white noise. The object experiences dry friction (due to soild/solid interaction, or to contact angle hysteresis in the case of a droplet). The object is driven by a force γ(t) – Δσ(t) where σ(t), =±1, depending on the sign of the velocity. We discuss the motion at two levels: (i) in terms of simple scaling laws, (ii) by a propagator technique. (a) When Δ is below a certain crossover value Δ*, we expect an unperturbed (Langevin) Brownian motion. (b) When Δ > Δ*, we expect a reduced diffusion coefficient proportional to Δ⁻⁴ for small Δ.     

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     

Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection

A numerical study was performed on natural convection for water–CuO nanofluid filled enclosure where the top surface was partially exposed to convection. The cavity has a square cross-section and differentially heated. Except exposed convection part on the top, all sides are adiabatic on horizontal walls. Effects of Rayleigh number (10³ ? Ra ? 10⁵), Biot number (0 ? Bi ? ∞), length of partial convection (0.0 ? L ? 1.0) and volume fraction of nanoparticles (0.0 ? φ ? 0.1) on heat and fluid flow were investigated. The results showed that for the case of high Biot number that heat transfer along the heated was enhanced by increasing the Rayleigh number mainly at the upper portion of the heated wall. When the top wall was totally exposed to convection, the results prevail that the heat transfer was more effective at high Biot number especially at the upper portion of the heated wall. For the case of high Biot number, the results prevailed that the heat transfer at the upper portion of the heated wall increases considerably at high exposed length to convection (L); however, for L ? 0.75 the effect of L was less pronounced. Contour maps for percentage of heat transfer enhancement were presented and it was shown that the location of maximum enhancement in heat transfer was sensitive to Ra, φ and L.     

Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center

The problem of free convection fluid flow and heat transfer of Cu–water nanofluid inside a square cavity having adiabatic square bodies at its center has been investigated numerically. The governing equations have been discretized using the finite volume method. The SIMPLER algorithm was employed to couple velocity and pressure fields. Using the developed code, a parametric study was conducted and the effects of pertinent parameters such as Rayleigh number, size of the adiabatic square body, and volume fraction of the Cu nanoparticles on the fluid flow and thermal fields and heat transfer inside the cavity were investigated. The obtained results show that for all Rayleigh numbers with the exception of Ra = 10⁴ the average Nusselt number increases with increase in the volume fraction of the nanoparticles. At Ra = 10⁴ the average Nusselt number is a decreasing function of the nanoparticles volume fraction. Moreover at low Rayleigh numbers (10³ and 10⁴) the rate of heat transfer decreases when the size of the adiabatic square body increases while at high Rayleigh numbers (10⁵ and 10⁶) it increases.     

Scaling and excitation of combined convection in a rapidly rotating plane layer

The optimum (to my mind) scaling of the combined thermal and compositional convection in a rapidly rotating plane layer is proposed.This scaling follows from self-consistent estimates of typical physical quantities. Similarity coefficients are introduced for the ratio convection dissipation/convection generation (s) and the ratio thermal convection/compositional convection (r). The third new and most important coefficient δ is the ratio of the characteristic size normal to the axis of rotation to the layer thickness. The faster the rotation, the lower δ. In the case of the liquid Earth core, δ ~ 10^–3 substitutes for the generally accepted Ekman number (E ~ 10^–15) and s ~ 10^–6 substitutes for the inverse Rayleigh number 1/Ra ~ 10^–30. It is found that, at turbulent transport coefficients, number s and the Prandtl number are on the order of unity for any objects and δ is independent of transport coefficients. As a result of expansion in powers of δ, an initially 3D system of six variables is simplified to an almost 2D system of four variables without δ. The problem of convection excitation in the main volume is algebraically solved and this problem for critical values is analytically solved. Dispersion relations and general expressions for critical wavenumbers, numbers s (which determine Rayleigh numbers), other critical parameters, and asymptotic solutions are derived. Numerical estimates are made for the liquid cores in the planets that resemble the Earth. Further possible applications of the results obtained are proposed for the interior of planets, moons, their oceans, stars, and experimental objects.     

Long relaxation times and tilt sensitivity in Rayleigh Bénard turbulence

We study the influence of a small tilt angle ( rd) on the Nusselt number in a 1/2 aspect ratio Rayleigh-Bénard cell, at high Rayleigh number (5 x 10¹¹ < Ra < 4 x 10¹²). The small decrease observed is interpreted as revealing a two rolls structure of the flow. Transitions between different global flows are also observed, on very long times, comparable to the diffusion time on the whole cell. The consequence is that the Nusselt number observed in most high Ra experiments should significantly depend on initial conditions.Received: 19 May 2004, Published online: 12 August 2004PACS: 92.60.Ek Convection, turbulence, and diffusion - 47.27.Te Convection and heat transfer - 44.25. + f Natural convection     

Volume and grain boundary diffusion of implanted 113Sn in aluminium

Volume and grain boundary diffusion of ¹¹³Sn in aluminium was investigated with the radiotracer method. The implantation technique was used for tracer deposition to avoid problems of tracer hold-up caused by the oxide layer always present on aluminium. The diffusion penetration was chosen large enough to permit serial sectioning of samples with the aid of a microtome.The temperature dependence of the volume diffusivity was determined as D(T)=4.54×10^–5×exp[–(114.5±1.2)kJmol^–1/RT] m ² s ^–1. This confirms previous measurements from our group which already showed that Sn is the fastest foreign metal diffusor so far investigated in aluminium.Grain boundary diffusion of ¹¹³Sn in Al polycrystals was measured in the type-B kinetic regime. The grain boundary diffusion product P=sD _gb (s=segregation factor, =grain boundary width, D _gb=grain boundary diffusivity) was found to be strongly affected by the impurity content of aluminium. For Al polycrystals of 99.9992% nominal purity we obtained P _5N(T)=1.08×10^–8exp [–(96.9±7.5) kJ mol^–1/RT] m³ s^–1 and for less pure Al polycrystals of 99.99% nominal purity P _4N(T)=3.0×10^–10 exp [–(90.1±4.2) kJ mol^–1/RT] m³ s^–1 was determined. The grain boundary diffusion product in the purer material is more than one order of magnitude higher than in the less pure material. Very likely this is an effect of co-segregation of non-diffusant impurities into the grain boundaries.     

Experimental Study of Natural Convection from Electrically Heated Vertical Cylinders Immersed in Air

Abstract

An experimental study of laminar steady-state natural convection heat transfer from electrically heated vertical cylinders immersed in air has been undertaken. Three stainless steel (316 SS) test sections of different slenderness ratios were employed. Surface temperature profiles along the vertical cylinders were obtained using miniature thermocouples when the cylinders were heated with different power levels resulting in different total wall heat fluxes. After the mandatory corrections for the radiation heat fluxes were made, three correlation equations relating the local Nusselt number Nu_y with the local modified Rayleigh number Ra^* _y and the position-to-cylinder diameter y/d were developed. The correlation equations are valid for Ra^* _y ≤ 2 × 10¹².     

Angular distribution measurements of Li(p,α)He reaction at 140 keV proton energy using nuclear track detectors

Angular distributions of a ⁶Li(p,α) ³He reaction were measured at six angles for 140 keV proton energy using nuclear track detectors (NTDs). The measurements were carried out over 60°–160° lab. angles in 20° increments using a scattering chamber of 80° beam line of the 350 kV accelerator. A semiconductor silicon surface barrier (SSB) detector was placed at +160° and was used as a monitor. The results have shown that the CR-39 detector has excellent capabilities to distinguish 1.4–2.7 MeV α+ ³He particles from the ⁶Li(p,α) ³He reaction and 8–9.4 MeV α-particles from the ⁷Li(p,α) ⁴He reaction through their track diameters. However, it was not possible to distinguish between the 2.3 MeV ³He ions and the 1.7 MeV ⁴He ions from the ⁶Li(p,α) ³He reaction from their track diameter measurements, but it was possible to differentiate between the two, from the darker contrast of the ³He particles caused by its deeper tracks as compared to those of ⁴He.     

Apparent diffusion coefficient of line scan diffusion image in normal prostate and prostate cancer—comparison with single-shot echo planner image

Purpose

This retrospective study was designed to evaluate the apparent diffusion coefficient (ADC) of line scan diffusion images (LSDI) in normal prostate and prostate cancer. Single-shot echo planner images (SS-EPI) were used for comparison.

Materials and Methods

Twenty prostate tumors were examined by conventional MRI in 14 patients prior to radical prostatectomy. All patients were examined with a 1.5-T MR imager (Signa CV/i ver. 9.1 GE Medical System Milwaukee, WI, USA). Diffusion-weighted MR imaging (DWI) using LSDI was performed with a pelvic phased-array coil, with b values of 5 and 800 s/mm². DWI using SS-EPI was performed with a body coil, with b values of 0 and 800 s/mm². The ADCs of each sequence for 14 normal prostate and 20 prostate cancers were histopathologically assessed. Signal-to-noise ratio (SNR) on DWI was estimated and compared for each sequence.

Results

The mean ADCs (±S.D.) of normal peripheral zones (PZ), transition zones (TZ) and cancer (in 10⁻³ mm²/s) that used LSDI were 1.42±0.12, 1.23±0.10 and 0.79±0.19, respectively. Those that used SS-EPI were 1.76±0.26, 1.38±0.20 and 1.05±0.27, respectively. Using unpaired t test (P<.05), we found a significant difference in each sequence between normal tissue (both PZ and TZ) and the cancer. Paired t test (P<.05) also registered a significant difference between LSDI and SS-EPI. Mean SNR for DWI using LSDI was 16.49±5.03, while the DWI using SS-EPI was 18.85±9.26. The difference between the SNR of each sequence was not statistically significant by paired t test.

Conclusion

We found that ADCs using LSDI and SS-EPI showed similar tendencies in the same patients. However, in all regions, LSDI ADCs had smaller standard deviations than SS-EPI ADCs.     

Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

Single and multiple shot near-infrared femtosecond laser pulse ablation thresholds of copper

The single-shot ablation threshold and incubation coefficient of copper were investigated using an amplified near-infrared, femtosecond Ti:sapphire laser. To date, the near-infrared femtosecond ablation threshold of copper has been reported in the range of several hundred millijoules per cm² based primarily on multiple shot ablation studies. A careful study of the single shot ablation threshold for copper was carried out yielding an incident single-shot ablation threshold of (1.06±0.12) J/cm² for a clean copper foil surface. This was determined by measuring the diameters of the ablation spots as a function of the laser pulse energy using scanning electron microscopy for spatially Gaussian laser spots. When multiple shots were taken on the same spot, a reduction in ablation threshold was observed, consistent with a multiple shot incubation coefficient of 0.76±0.02. Similar experiments on 250 nm and 500 nm copper thin films sputtered on a silicon substrate demonstrated that scaling the threshold values with the absorbance of energy at the surface yields a consistent absorbed fluence threshold for copper of (59±10) mJ/cm². This absorbed threshold value is consistent with the expected value from a two-temperature model for the heating of copper with an electron-lattice coupling constant of g=10¹⁷ Wm^-3 K^-1. Single-shot rippling of the surface in the threshold ablation intensity regime was also observed for the foil target but not for the smooth thin film target. PACS 61.80.Ba; 61.82.Bg