Piezo-driven thermoacoustic refrigerators with dynamic magnifiers

Thermoacoustic refrigeration is an emerging cooling technology which does not rely for in its operation on the use of any moving parts or harmful refrigerants. This technology uses acoustic waves to pump heat across a temperature gradient. The temperature gradient forms across the ends of a porous body, called the stack, enclosed in a resonator. The vast majority of thermoacoustic refrigerators to date have used electromagnetic loudspeakers to generate the acoustic input. In this paper, the design, construction, operation, and modeling of a piezo-driven thermoacoustic refrigerator are detailed. The performance of the refrigerator is significantly enhanced by coupling the acoustic driver with an elastic structure, referred to as a dynamic magnifier. Proper selection of the magnifier parameters can increase the magnitude of the pressure oscillations across the stack, and consequently the temperature difference. The magnified refrigerator demonstrates the effectiveness of piezoelectric actuation in moving 0.3 W of heat across a 10 °C temperature difference with an input power of 7 W. All the theoretical predictions are validated against data from experimental prototypes. The developed theoretical and experimental tools can serve as invaluable means for the design and testing of piezo-driven thermoacoustic refrigerator configurations.     

Optimal acoustic fields in compact thermoacoustic refrigerators

Thermoacoustic refrigerators have been developed during the last 15 years, employing quasi-standing resonant acoustic waves inside fluid-filled cavities to transfer heat along a stack region. Because higher efficiency can be reached when a significant travelling wave component exists in the resonator, specific resonant thermoacoustic devices have been designed allowing to adjust more or less the ratio of travelling and standing wave components. However, the acoustic pressure field and the particle velocity field do not appear to be the optimal ones, for the thermal quantities of interest. Thus, it is the aim of the paper to present a new kind of thermoacoustic standing wave-like device which allows to control easily and independently the pressure field and the particle velocity field, after investigating the optimal acoustic field, in the stack region, for the main parameters of interest, i.e. the temperature gradient, the thermoacoustic heat flow and the coefficient of performance.     

Use of electrodynamic drivers in thermoacoustic refrigerators

Some issues involved in matching electrodynamic drivers to thermoacoustic refrigerators are examined using an equivalent circuit model. Conclusions are that the driver should be chosen to have a large product (Bl)2/(ReRm); the suspension stiffness should be chosen to make the combined impedance of the mechanical and acoustical parts of the system entirely real at the operating frequency; and the piston area should be selected to maximize electroacoustic efficiency, or other desired parameter, by matching the acoustic load to the optimum mechanical load for the particular driver. Alternately, if the piston area is fixed, the operating frequency can be adjusted to make this same match.     

压电扬声器驱动的热声制冷机中阻抗匹配研究

为了减小热声制冷机的体积,在热声系统中,采用具有较高工作频率的压电扬声器作为驱动源,因此,压电扬声器与热声谐振管的匹配成为研究和改进此类热声制冷系统性能的重要因素。采用电力声等效电路的方法,推导出扬声器输出声功率和电声转换效率的表达式,并为获得高声功率和高制冷效率,对扬声器和谐振管结构以及工作频率进行优化。结果表明,谐振管的谐振频率即为系统的最佳工作频率,而当系统总的抗性为零时,扬声器输出较大的声功率。然而系统效率与输出声功率不能同时达到最大值,因此在实际应用中只能采取折中的效果。     

Critical magnetic Prandtl number for small-scale dynamo

We report a series of numerical simulations showing that the critical magnetic Reynolds number Rm(c) for the nonhelical small-scale dynamo depends on the Reynolds number Re. Namely, the dynamo is shut down if the magnetic Prandtl number Pr(m)=Rm/Re is less than some critical value Pr(m,c)< approximately 1 even for Rm for which dynamo exists at Pr(m)> or =1. We argue that, in the limit of Re-->infinity, a finite Pr(m,c) may exist. The second possibility is that Pr(m,c)-->0 as Re--> infinity, while Rm(c) tends to a very large constant value inaccessible at current resolutions. If there is a finite Pr(m,c), the dynamo is sustainable only if magnetic fields can exist at scales smaller than the flow scale, i.e., it is always effectively a large-Pr(m) dynamo. If there is a finite Rm(c), our results provide a lower bound: Rm(c) greater, similar 220 for Pr(m)< or =1/8. This is larger than Rm in many planets and in all liquid-metal experiments.     

Simulation of induction at low magnetic Prandtl number

We consider the induction of a magnetic field in flows of an electrically conducting fluid at low magnetic Prandtl number and large kinetic Reynolds number. Using the separation between the magnetic and kinetic diffusive length scales, we propose a new numerical approach. The coupled magnetic and fluid equations are solved using a mixed scheme, where the magnetic field fluctuations are fully resolved and the velocity fluctuations at small scale are modeled using a large eddy simulation (LES) scheme. We study the response of a forced Taylor-Green flow to an externally applied field: topology of the mean induction and time fluctuations at fixed locations. The results are in remarkable agreement with existing experimental data; a global 1/f behavior at long times is also evidenced.     

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     

Effects of Prandtl number in two-dimensional turbulent convection

We report a numerical study of the Prandtl-number(Pr) effects in two-dimensional turbulent Rayleigh-Benard convection.The simulations were conducted in a square box over the Pr range from 0.25 to 100 and over the Rayleigh number(Ra) range from 107 to 1010.We find that both the strength and the stability of the large-scale flow decrease with the increasing of Pr,and the flow pattern becomes plume-dominated at high Pr.The evolution in flow pattern is quantified by the Reynolds number(Re),with the Ra and the Pr scaling exponents varying from 0.54 to 0.67 and-0.87 to-0.93,respectively.It is further found that the non-dimensional heat flux at small Ra diverges strongly for different Pr,but their difference becomes marginal as Ra increases.For the thermal boundary layer,the spatially averaged thicknesses for all the Pr numbers can be described by δ_θ～Ra~(-0.30) approximately,but the local values vary a lot for different Pr,which become more uniform with Pr increasing.     

Bound on vertical heat transport at large Prandtl number

We prove a new upper bound on the vertical heat transport in Rayleigh-Bénard convection of the form under the assumption that the ratio of Prandtl number over Rayleigh number satisfies where the non-dimensional constant c₀ depends on the aspect ratio of the domain only. This new rigorous bound agrees with the (optimal) bound (modulo logarithmic correction) on vertical heat transport for the infinite Prandtl number model for convection due to Constantin and Doering [P. Constantin, C.R. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1) (1999) 159-172] and Doering, Otto and Reznikoff [C.R. Doering, F. Otto, M.G. Reznikoff, Bounds on vertical heat transport for infinite Prandtl number Rayleigh-Bénard convection, J. Fluid Mech. 560 (2006) 229-241]. It also improves a uniform (in Prandtl number) bound for the Nusselt number [P. Constantin, C.R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049-1060] in the case of large Prandtl number.     

Turbulent Prandtl number in the boundary layer on a plate: effect of the molecular Prandtl number,injection (suction), and longitudinal pressure gradient

Using the differential turbulence model supplemented by the transport equation for the turbulent heat flux, a numerical study is carried out of the dependence of the Prandtl turbulent number on the molecular Prandtl number, the intensity of the gas injection (suction) through the permeable wall, and the freestream acceleration (deceleration). The air and mixtures of helium with xenon and argon are considered as gas carriers, and mercury, water, and transformer oil are used as liquid carriers. The obtained results of calculations are consistent with the available experimental data for the turbulent Prandtl number and the quantities included in its definition.     

湍流热对流Prandtl数效应的数值研究

本文采用直接数值模拟的并行直接求解方法,计算了Ra=10~(10),0.05≤Pr≤20的系列Prandtl(Pr)数二维湍流热对流.通过流动显示技术,讨论了Pr数对羽流形态和大尺度环流结构的影响.在Ra=10~(10)时,随着Pr数减小,羽流的运动和分布表现出更强的湍流性质,较高Pr数的羽流则表现出较强的规律性,当Pr4.3时,流场中存在明显的大尺度环流和角涡结构.不同Pr数的温度边界层厚度差异不大,并随Pr数存在标度率变化关系.当Pr数较低时,系统的传热Nusselt(Nu)数随着Pr数增加而增加,当Pr数较高时,Nu数随Pr数的变化不敏感.靠近底板处速度脉动随Pr数有显著的变化,Pr数越低速度波动越剧烈.通过底板中心位置水平脉动速度和平均场水平速度最大值给出的雷诺数Re_((u))和Re_(U_(max)),两种Re数随Pr数的变化满足同一标度律,为Re～Pr~(-0.81).     

Convective pattern dynamics at low Prandtl number: Part II

Part II of the paper concerns the erratic time dependence of natural patterns of fluids, particularly those fluids with low Prandtl number. Possible explanations of the phenomena and experimental tests in simple situations are presented. Part I of the paper concerned the concept of pattern instabilities and major pattern defects.     

Similarities between 2D and 3D convection for large Prandtl number

Using direct numerical simulations of Rayleigh–Bénard convection (RBC), we perform a comparative study of the spectra and fluxes of energy and entropy, and the scaling of large-scale quantities for large and infinite Prandtl numbers in two (2D) and three (3D) dimensions. We observe close similarities between the 2D and 3D RBC, in particular, the kinetic energy spectrum E_u(k)～k^?13/3, and the entropy spectrum exhibits a dual branch with a dominant k^?2 spectrum. We showed that the dominant Fourier modes in 2D and 3D flows are very close. Consequently, the 3D RBC is quasi-two-dimensional, which is the reason for the similarities between the 2D and 3D RBC for large and infinite Prandtl numbers.     

A BGK model for small Prandtl number in the Navier-Stokes approximation

We present a BGK-type collision model which approximates, by a Chapman-Enskog expansion, the compressible Navier-Stokes equations with a Prandtl number that can be chosen arbitrarily between 0 and 1. This model has the basic properties of the Boltzmann equation, including theH-theorem, but contains an extra parameter in comparison with the standard BGK model. This parameter is introduced multiplying the collision operator by a nonlinear functional of the distribution function. It is adjusted to the Prandtl number.     

Low Mach number analysis of idealized thermoacoustic engines with numerical solution

A model of an idealized thermoacoustic engine is formulated, coupling nonlinear flow and heat exchange in the heat exchangers and stack with a simple linear acoustic model of the resonator and load. Correct coupling results in an asymptotically consistent global model, in the small Mach number approximation. A well-resolved numerical solution is obtained for two-dimensional heat exchangers and stack. The model assumes that the heat exchangers and stack are shorter than the overall length by a factor of the order of a representative Mach number. The model is well-suited for simulation of the entire startup process, whereby as a result of some excitation, an initially specified temperature profile in the stack evolves toward a near-steady profile, eventually reaching stationary operation. A validation analysis is presented, together with results showing the early amplitude growth and approach of a stationary regime. Two types of initial excitation are used: Random noise and a small periodic wave. The set of assumptions made leads to a heat-exchanger section that acts as a source of volume but is transparent to pressure and to a local heat-exchanger model characterized by a dynamically incompressible flow to which a locally spatially uniform acoustic pressure fluctuation is superimposed.     

Prandtl number effects in decaying homogeneous isotropic turbulence with a mean scalar gradient

Decaying homogeneous isotropic turbulence with an imposed mean scalar gradient is investigated numerically, thanks to a specific eddy-damped quasi-normal Markovian closure developed recently for passive scalar mixing in homogeneous anisotropic turbulence (BGC). The present modelling is compared successfully with recent direct numerical simulations and other models, for both very large and small Prandtl numbers. First, scalings for the cospectrum and scalar variance spectrum in the inertial range are recovered analytically and numerically. Then, at large Reynolds numbers, the decay and growth laws for the scalar variance and mixed velocity–scalar correlations, respectively, derived in BGC, are shown numerically to remain valid when the Prandtl number strongly departs from unity. Afterwards, the normalised correlation ρ_wθ is found to decrease in magnitude at a fixed Reynolds number when Pr either increases or decreases, in agreement with earlier predictions. Finally, the small scales return to isotropy of the scalar second-order moments is found to depend not only on the Reynolds number, but also on the Prandtl number.     

Viscosity,heat conductivity,and Prandtl number effects in the Rayleigh–Taylor Instability

The two-dimensional Rayleigh–Taylor instability problem is simulated with a multiple-relaxation-time discrete Boltzmann model with a gravity term. Viscosity, heat conductivity, and Prandtl number effects are probed from macroscopic and nonequilibrium viewpoints. In the macro sense, both viscosity and heat conduction show a significant inhibitory effect in the reacceleration stage, which is mainly achieved by inhibiting the development of the Kelvin–Helmholtz instability. Before this, the Prandtl number effect is not sensitive. Viscosity, heat conductivity, and Prandtl number effects on nonequilibrium manifestations and the degree of correlation between the nonuniformity and the nonequilibrium strength in the complex flow are systematically investigated.