Numerical Investigations on the Impact of Turbulent Prandtl Number and Schmidt Number on Supersonic Combustion

The flow field inside the combustor of a scramjet is highly complicated and the related turbulent Prandtl and Schmidt numbers have a significant impact on the effective numerical prediction of such dynamics.As in many cases researchers set these parameters on the basis of purely empirical laws,assessing their impact(via parametric numerical simulations)is a subject of great importance.In the present work,in particular,two test cases with different characteristics are selected for further evaluation of the role played by these non-dimensional numbers:Burrows-Kurkov case and DLR case.The numerical results indicate that these parameters influence ignition location.Moreover,the temperature distribution is more sensitive to them than to H2O mass fraction and velocity distributions.

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-Bénard 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 10^(7) to 10^(10).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 byδθ~Ra^(-0.30) approximately,but the local values vary a lot for different Pr,which become more uniform with Pr increasing.

Effect of particles on turbulent thermal field of channel flow with different Prandtl numbers

The direct numerical simulation （DNS） of heat transfer in a fully developed non-isothermal particle-laden turbulent channel flow is performed. The focus of this paper is on the modulation of the particles on turbulent thermal statistics in the particle-laden flow with three Prandtl numbers （Pτ = 0.71, 1.5, and 3.0） and a shear Reynolds number （Reτ = 180）. Some typical thermal statistics, including normalized mean temperature and their fluctuations, turbulent heat fluxes, Nusselt number and so on, are analyzed. The results show that the particles have less effects on turbulent thermal fields with the increase of Prandtl number. Two reasons can explain this. First, the correlation between fluid thermal field and velocity field decreases as the Prandtl number increases, and the modulation of turbulent velocity field induced by the particles has less influence on the turbulent thermal field. Second, the heat exchange between turbulence and particles decreases for the particle-laden flow with the larger Prandtl number, and the thermal feedback of the particles to turbulence becomes weak.

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

Benard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating 1 solution with expansions up to any order are given and the convergence rates O（εm＋1/2）and the optimal convergence rates O（εm＋1） are obtained respectively. This improves the result of J.G. SHI.

Flux Limiter Lattice Boltzmann Scheme Approach to Compressible Flows with Flexible Specific-Heat Ratio and Prandtl Number

We further develop the lattice Boltzmann （LB） model [Physica A 382 （2007） 502] for compressible flows from two aspects. Firstly, we modify the Bhatnagar Gross Krook （BGK） collision term in the LB equation, which makes the model suitable for simulating flows with different Prandtl numbers. Secondly, the flux limiter finite difference （FLFD） scheme is employed to calculate the convection term of the LB equation, which makes the unphysical oscillations at discontinuities be effectively suppressed and the numerical dissipations be significantly diminished. The proposed model is validated by recovering results of some well-known benchmarks, including （i） The thermal Couette flow; （ii） One- and two-dlmenslonal FLiemann problems. Good agreements are obtained between LB results and the exact ones or previously reported solutions. The flexibility, together with the high accuracy of the new model, endows the proposed model considerable potential for tracking some long-standing problems and for investigating nonlinear nonequilibrium complex systems.

Prandtl number effects in MRT lattice Boltzmann models for shocked and unshocked compressible fluids

This paper constructs a new multiple relaxation time lattice Boltzmann model which is not only for the shocked compressible fluids,but also for the unshocked compressible fluids.To make the model work for unshocked compressible fluids,a key step is to modify the collision operators of energy flux so that the viscous coefficient in momentum equation is consistent with that in energy equation even in the unshocked system.The unnecessity of the modification for systems under strong shock is analyzed.The model is validated by some well-known benchmark tests,including thermal Couette flow,Riemann problem.The first system is unshocked and the latter is shocked.In both systems,the Prandtl number effects are checked.Satisfying agreements are obtained between new model results and analytical ones.

Influence of Prandtl Number on Thin Film Condensation in Forced Convection in an Inclined Wall Covered with a Porous Material

The numerical study of thin film type condensation in forced convection of a saturated pure vapor in an inclined wall covered with a porous material is presented. The generalized Darcy-Brinkman-Forchheimer (DBF) model is used to describe the flow in the porous medium while the classical boundary layer equations have been exploited in the case of a pure liquid. The dimensionless equations are solved by an implicit finite difference method and the iterative Gauss-Seidel method. The objective of this study is to examine the influence of the Prandtl number on the hydrodynamic and thermal fields but also on the local Nusselt number and on the boundary layer thickness. For Pr ≤ 0.7 (low) the velocity and the longitudinal temperature increase with the Prandtl number. On the other hand, when Pr ≥ 2 (high) the Prandtl number no longer influences the velocity and the longitudinal temperature. The local Nusselt number increases as the Prandtl number increases and the thickness of the hydrodynamic boundary layer increases as the Prandtl number decreases.

Study of Variable Turbulent Prandtl Number Model for Heat Transfer to Supercritical Fluids in Vertical Tubes

In order to improve the prediction performance of the numerical simulations for heat transfer of supercritical pressure fluids, a variable turbulent Prandtl number(Pr_t) model for vertical upward flow at supercritical pressures was developed in this study. The effects of Pr_t on the numerical simulation were analyzed, especially for the heat transfer deterioration conditions. Based on the analyses, the turbulent Prandtl number was modeled as a function of the turbulent viscosity ratio and molecular Prandtl number. The model was evaluated using experimental heat transfer data of CO_2, water and Freon. The wall temperatures, including the heat transfer deterioration cases, were more accurately predicted by this model than by traditional numerical calculations with a constant Pr_t. By analyzing the predicted results with and without the variable Pr_t model, it was found that the predicted velocity distribution and turbulent mixing characteristics with the variable Pr_t model are quite different from that predicted by a constant Pr_t. When heat transfer deterioration occurs, the radial velocity profile deviates from the log-law profile and the restrained turbulent mixing then leads to the deteriorated heat transfer.

Local Convective Heat Transfer from Small Heaters to Impinging Submerged Axisymmetric Jets of Seven Coolants with Prandtl Number Ranging from 0.7 to 348

Using seven working fluids, a systematic experimental study was performed to investigate the local convective heat transfer from vertical heaters to impinging circular submerged jets in the range of Reynolds number between 1.17 ×102 and 3.69 × 104 with the emphasis placed on the examination of Prandtl number dependence. Heat transfer coefficiellts at the stagnation point were collected and correlated with the plate held within and beyond the potential core. Radial distribution of the local heat transfer coefficient was measured with five test liquids. Based on the measured profiles of the local heat transfer, a correlation was developed to cover the entire range of the radial distance. Besides the presellt data, the correlations developed in this work were also compared with a large quantity of available data of circular air jets. General agreement was observed between the air data and the correlat ions.

Numerical Prediction of Film Cooling Effectiveness over Flat Plate using Variable Turbulent Prandtl Number Closures

Numerical simulations were performed to predict the film cooling effectiveness on the fiat plate with a three- dimensienal discrete-hole film cooling arrangement. The effects of basic geometrical characteristics of the holes, i.e diameter D, length L and pitch S/D were studied. Different turbulent heat transfer models based on constant and variable turbulent Prandtl number approaches were considered. The variability of the turbulent Prandtl number Prt in the energy equation was assumed using an algebraic relation proposed by Kays and Crawford, or employing the Abe, Kondoh and Nagano eddy heat diffusivity closure with two differential transport equations for the temperature variance ko and its destruction rate εθ The obtained numerical results were directly compared with the data that came from an experiment based on Transient Liquid Crystal methodology. All implemented models for turbulent heat transfer performed sufficiently well for the considered case. It was confirmed, however, that the two- equation closure can give a detailed look into film cooling problems without using any time-consuming and inherently unsteady models.

Free Convention Heat Transfer from an Isothermal Horizontal Thin Strip:the Influence of the Prandtl Number

Simultaneous free convection above and below a uniformly heated horizontal plate has been widely investigated,both in the case of an isothermal surface,and of a uniformly heated surface,but always assuming only air as fluid(Pr=0.7).Nevertheless,there are works dealing with horizontal plates whose results show that the Nu dependence on Pr may not be simply expressed by a power law with the same exponent of the Gr one.So it was considered useful to study the Prandtl number influence in the case of the isothermal horizontal strip.Results show that,while for the lower surface of the strip the Nu dependence in Gr can be expressed by a power law with an exponent close to the Gr one,for the upper surface the exponent is sensibly different.Correlating equations related to the investigated situations are proposed.

Numerical Investigation on Flow and Heat Transfer Performance of Supercritical Carbon Dioxide Based on Variable Turbulent Prandtlnumber Model

Flow and heat transfer characteristic of supercritical carbon dioxide(SCO_(2))are numerically investigated in the horizontal and vertical tubes.TWL turbulent Prandtl number model could well describe the behavior of SCO_(2) affected by the buoyancy.Under the cooling condition,the heat transfer performance of SCO_(2) along the upward direction is best and that along the downward direction is worst when bulk fluid temperatures are below the pseudocritical temperature.Reducing the ratio of heat flux to mass flux could decrease the difference of convective heat transfer coefficient in three flow directions.Under the heating condition,heat transfer deterioration only occurs in vertical upward and horizontal flow directions.Heat transfer deterioration of SCO_(2) could be delayed by increasing the mass flux and the deterioration degree is weakened in the second half of tube along the vertical upward flow direction.Compared with the straight tube,the corrugated tube shows better comprehensive thermal performance.

Turbulent Prandtl Number

The method of determining eddy transport coefficients of turbulence is utilized to study how the turbulent Prandtl number Prturb depends upon the molecular Prandtl number Pr and turbulence conditions.A comparison between the theoretical prediction and the relevant experimental data is made.

底部加热的低Prandtl数流体自然对流换热的分岔

本文用具有QUICK方案的SIMPLE算法对底部加热的低Prandtl数流体自然对流换热进行了数值计算,研究了这种流动与换热问题数值解的分岔问题。计算发现,在2200<Ra<5000区间数值解不唯一。当Ra≤2200时,流动和换热是稳态的和唯一的,最终的流场是稳定的4涡型流场。当Ra≥5000时,流动和换热是非稳态的,流场分别有5涡型的和4涡型的两种, 5涡型流场能够长期稳定地存在,但4涡型流场不会长期存在,当有足够的时间积累,会突然转变为5涡型流场。

低Prandtl数水平流体层自然对流的振荡和分歧

本文用具有QUICK方案的有限差分法对底部加热的低Prandtl数水平流体层自然对流换热进行了数值计算，研究了这种问题中存在的振荡和分歧问题。结果显示，在Ra的一定取值区间，有4涡型流场和5涡型流场两个解的分支。但在这个区间以外，最终的结果没有出现分歧。在所发现的两个解的分支中，问题由稳态转变为非稳态的临界Racr是不同的。

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Pr>4.3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).

环形浅液池内低Prandtl数流体超临界热毛细对流演化

为研究低Prandtl数(Pr)流体热毛细对流演化过程,对环形浅液池内Pr=0.011的流体热毛细对流进行三维数值模拟.研究发现:当Marangoni数较小时,流动为轴对称稳态流动;当Marangoni数超过某一临界值后,流动失稳并转变为热流体波,其波数随Marangoni数增加而减小,而波动主频增大;随着Marangoni数增加,流动加强,沿周向运动的热流体波演变为沿径向运动的径向波,其波数大大减小;当Marangoni数继续增加时,波动频谱曲线噪声增加,呈广谱特性.因此,在计算范围内热毛细对流的演化过程为:轴对称稳态流动-热流体波-单周期径向波-多周期三维振荡流动.

高Prandtl数双层流体的热毛细对流数值模拟

不相溶混的双层流体系统的传热传质现象广泛地存在于自然界和工程技术领域中,尤其是芯片工业与微电子工业中半导体材料的液封提拉法生长制备工艺中,双层流体的流动稳定性直接影响着晶体材料的生长质量。微重力条件下,由于没有了重力的影响,可以增强双层流体流动的稳定性以获得更为优异的晶体生长环境,然而,两相界面处的热毛细力引起的热毛细对流却对双层流体系统中的传热传质现象影响甚大。目前,可采用实验研究的方法对水平温度梯度作用下的双层流体的热毛细对流进行可视化研究,以观测流动特性及失稳的耗散结构。在空间实验中,由于低Prandtl数流体不可透视的特性,故在实验中鲜少采用其作为观察工质,多采用高Prandtl数流体。关于双层流体的热毛细对流,前人的研究主要集中在矩形腔体内环形池内的低Prandtl数双层流体的流动特性,对环形液池中高Prandtl数双层流体的流动特性及失稳耗散结构的研究还鲜见报道。本工作针对微重力条件下,水平温度梯度作用下的环形液池内B_(2)O_(3)/蓝宝石熔体、5cSt硅油/HT-70、水/FC-75三组工质对的热毛细对流特性进行了研究,采用数值模拟的方法获得了R-Z截面的流函数、温度分布,以及监测点的速度和温度周期波动,揭示了流动失稳后的振荡流型。研究结果表明,运动粘度会影响流动流型,对于运动粘度较低的5cSt硅油/HT-70、水/FC-75的工质对,流动失稳为热毛细对流失稳,而高运动粘度的B_(2)O_(3)/蓝宝石熔体则出现了热毛细对流失稳与Marangoni失稳。对比三种工质对的流动失稳临界Marangoni数,发现上下液层Prandtl数比值越大,则临界Marangoni数越大,选择较大的上下液层Prandtl数比值,可增强环形双液层流体流动稳定性。

不同流体Prandtl数时颗粒有序填充床的流动传热性能

对小管径/粒径比颗粒有序填充床进行数值模拟,研究流体普朗特数(Pr)变化及不同管径/粒径比对填充床流动与传热性能的影响。结果表明,颗粒填充通道可分为入口阶段、充分发展阶段以及出口阶段,各个阶段的流动与传热特点不同。在相同雷诺数(Re)时,流动传热充分发展阶段传热系数和压降均随管径/粒径比的增大而增大,且后者增大的幅度大于前者。流动工质的 Pr 越大,填充床努塞尔数(Nu)越高,而 Pr 变化对阻力系数的影响可忽略。Pr 不同,Nu随管径/粒径比的变化趋势不同,而阻力系数随管径/粒径比的变化趋势相似。

湍流普朗特数在高超声速绕流中的修正

为精确模拟高超声速激波/湍流边界层干扰导致的复杂非平衡湍流流动,提出一种湍流普朗特数修正模型。应用数值模拟及理论分析方法,对高超声速来流马赫数为9.22的平板、压缩拐角等绕流进行数值分析,评估了所提出的湍流普朗特数修正模型。数值模拟结果与实验数据及Kays湍流普朗特数模型的对比表明:对高超声速复杂流动,湍流普朗特数应进行修正,提出的经湍流非平衡参数修正的湍流普朗特数修正模型与原模型及Kays模型相比,给出的壁面压强、壁面热流更精确,壁面最大热流相对误差小于6%。