综合国力的数学建模

本文考虑社会文明与社会导恶现象的制约机制，建立一个综合国力的非线性数学模型，从数学上讨论了相应的二次微分系统的Ｈｏｐｆ分叉，中心与细焦点的判定，极限环的存在唯一性等问题，并对数学结论予以合理解释，把社会相平面划成社发展区域，社会动荡区域和社会崩溃区域，研讨社会走向，确定纳入社会发展区域的途径。     

Aerodynamic noise from wave-turbulence interaction

In order to estimate the acoustic energy scattered when a unit volume of free turbulence, such as in free jets, interacts with a plane steady sound wave, theoretical expressions are derived for two simple models of turbulence: eddy model and isotropic model. The effect of convection by mean motion of the energy-bearing eddies on the incident sound wave and on the sound generated from wave-turbulence interaction is taken into account. Finally, by means of a representative calculation, the directionality pattern and Mach number dependence of the noise so generated is discussed.     

A large-eddy-based lattice Boltzmann model for turbulent flow simulation

In this paper a novel and simple large-eddy-based lattice Boltzmann model is proposed to simulate two-dimensional turbulence. Unlike existing lattice Boltzmann models for turbulent flow simulation, which were based on primitive-variables Navier–Stokes equations, the target macroscopic equations of the present model are vorticity-streamfunction equations. Thanks to the intrinsic features of vorticity-streamfunction equations, the present model is efficient, stable and simple for two-dimensional turbulence simulation. The advantages of the present model are validated by numerical experiments.     

解线性约束优化问题的新锥模型信赖域法

本文提出了一个解线性等式约束优化问题的新锥模型信赖域方法.论文采用零空间技术消除了新锥模型子问题中的线性等式约束,用折线法求解转换后的子问题,并给出了解线性等式约束优化问题的信赖域方法.论文提出并证明了该方法的全局收敛性,并给出了该方法解线性等式约束优化问题的数值实验.理论和数值实验结果表明新锥模型信赖域方法是有效的,这给出了用新锥模型进一步研究非线性优化的基础.     

双尺度二阶矩颗粒相湍流模型经验系数的影响

基于将颗粒脉动分成湍流引起的大尺度脉动和颗粒间碰撞产生的小尺度脉动的概念,建立了双尺度二阶矩两相湍流模型．用该模型对下行床内两相流动进行了数值模拟,颗粒体积浓度、平均速度的计算结果和实验数据吻合较好．分析了双尺度二阶矩两相湍流模型经验系数变化对预报结果的影响:在经验系数的一定变化范围内,预报结果并无明显的影响,但是变化范围增大,预报结果会产生较大变化．     

Numerical modelling of subcritical open channel flow using the K-ε turbulence model and the penalty function finite element technique

A numerical model has been developed that employs the penalty function finite element technique to solve the vertically averaged hydrodynamic and turbulence model equations for a water body using isoparametric elements. The full elliptic forms of the equations are solved, thereby allowing recirculating flows to be calculated. Alternative momentum dispersion and turbulence closure models are proposed and evaluated by comparing model predictions with experimental data for strongly curved subcritical open channel flow. The results of these simulations indicate that the depth-averaged two-equation k-ε turbulence model yields excellent agreement with experimental observations. In addition, it appears that neither the streamline curvature modification of the depth-averaged k-ε model, nor the momentum dispersion models based on the assumption of helicoidal flow in a curved channel, yield significant improvement in the present model predictions. Overall model predictions are found to be as good as those of a more complex and restricted three-dimensional model.     

三维非定常/定常不可压缩流动N-S方程基于人工压缩性方法的数值模拟

基于人工压缩性方法提出—中心与迎风混合的算法,以数值模拟N-S方程的定常/非定常解．对半离散方程的左端采用中心差分, 方程右端数值流量采用迎风Roe近似算法,其精度可达三阶．湍流模式利用Baldwin-Lomax代数模式．计算例子包括二维平板、机翼剖面、扁椭球、颅动脉瘤等．计算结果表明,压力和摩擦系数与实验符合,在分离涡旋区计算值与实验有差别,这或许是由于湍流模式不够精确的缘故．     

Characterization of the Reynolds-stress and dissipation-rate decay and anisotropy from DNS of grid-generated turbulence

Grid–generated turbulence is a classical but still controversial topic, one open issue being the spatial decay rate of turbulent energy. We study the influence of grid geometry on the Reynolds–stress and dissipation–rate tensors, including the range and exponent of their self–similar spatial decay. DNS using a validated lattice Boltzmann code at mean–flow Reynolds numbers up to 1400 are performed, comparing square grids with blockage ratios from 0.05 to 0.49. A clear picture of spatial distribution and self–similarity emerges for the statistics of interest: Axisymmetry is excellently confirmed. A consistent power law decay is found in the self–similar decay region beyond 10 grid stride lengths downstream. Its exponent of –5/3 can be obtained, for weak turbulence, from a spatial flux balance reminiscent of the constant transport through the inertial range of isotropic turbulence. In the near–grid region, on the other hand, differences in Reynolds stress components are pronounced while those between dissipation tensor components are only recognizable very close to the grid, where a strong dependence on grid porosity is found. A normalization with respect to porosity is proposed that collapses the data from all runs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

宾汉流体稠密两相湍流流动中的二阶矩模型

建立的Bingham流体稠密两相流动的二阶矩-颗粒动力论湍流模型（USM-theta模型）既体现了两相的作用,又体现了屈服应力所引起的附加项,并提出了USM-theta模型下考虑浓度修正值影响的两相湍流流动的算法．利用该模型对圆管内Bingham流体的单相湍流流动、稠密液固两相的湍流流动进行了计算,并和五方程湍流模型进行了比较,结果表明该模型的预测效果更好．利用USM-theta模型对含颗粒的Bingham流体的两相湍流流动进行了模拟,随着屈服应力的增加,Bingham流体相与颗粒相在管道中心附近的主流速度减小．液固两相湍流和Bingham流体两相湍流的计算结果表明屈服应力引起的附加项对流动有很重要的影响．     

Assessment of an improved turbulence model in simulating the unsteady flows around a D-shaped cylinder and an open cavity

Most engineering flows are still predicted by the conventional Reynolds-averaged Navier-Stokes method because of the low requirements of the computational quantities. However, the resolution capability of Reynolds-averaged Navier-Stokes models is still open to deliberation, especially in the recirculation and wake regions, where the vortical flows dominate. In the present work, an improved turbulence model derived from the original shear stress transport k-ω model is proposed and its superiority is assessed by our modeling the unsteady flows around a D-shaped cylinder and an open cavity, corresponding to two different Reynolds numbers. The results are compared with results from experiments and other turbulence models in terms of the flow morphology and mean velocity profiles. This shows that the predictive accuracy of the modified turbulence model is increased significantly in the bluff body wake flows and in the shear layer and separation flows of the cavity. Some special vortex structures can be captured in the open cavity, in which the secondary vortex emerging from the shear layer and the separation vortex near the trailing edge can induce large flow instability, and this phenomenon should be eliminated in engineering applications. It is believed that this improved turbulence model can be used for the more complex turbomachinery flows with better prediction of the hydrodynamic/aerodynamic performance and the unsteady vortical flows, which can provide some guidelines to design or optimize rotating machines.     

Direct numerical simulation of turbulent non-Newtonian flow using a spectral element method

A spectral element—Fourier method (SEM) for Direct Numerical Simulation (DNS) of the turbulent flow of non-Newtonian fluids is described and the particular requirements for non-Newtonian rheology are discussed. The method is implemented in parallel using the MPI message passing kernel, and execution times scale somewhat less than linearly with the number of CPUs, however this is more than compensated by the improved simulation turn around times. The method is applied to the case of turbulent pipe flow, where simulation results for a shear-thinning (power law) fluid are compared to those of a yield stress (Herschel–Bulkley) fluid at the same generalised Reynolds number. It is seen that the yield stress significantly dampens turbulence intensities in the core of the flow where the quasi-laminar flow region there co-exists with a transitional wall zone. An additional simulation of the flow of blood in a channel is undertaken using a Carreau–Yasuda rheology model, and results compared to those of the one-equation Spalart-Allmaras RANS (Reynolds-Averaged Navier–Stokes) model. Agreement between the mean flow velocity profile predictions is seen to be good. Use of a DNS technique to study turbulence in non-Newtonian fluids shows great promise in understanding transition and turbulence in shear thinning, non-Newtonian flows.     

Response of a tethered aerostat to simulated turbulence

Aerostats are lighter-than-air vehicles tethered to the ground by a cable and used for broadcasting, communications, surveillance, and drug interdiction. The dynamic response of tethered aerostats subject to extreme atmospheric turbulence often dictates survivability. This paper develops a theoretical model that predicts the planar response of a tethered aerostat subject to atmospheric turbulence and simulates the response to 1000 simulated hurricane scale turbulent time histories. The aerostat dynamic model assumes the aerostat hull to be a rigid body with non-linear fluid loading, instantaneous weathervaning for planar response, and a continuous tether. Galerkin’s method discretizes the coupled aerostat and tether partial differential equations to produce a non-linear initial value problem that is integrated numerically given initial conditions and wind inputs. The proper orthogonal decomposition theorem generates, based on Hurricane Georges wind data, turbulent time histories that possess the sequential behavior of actual turbulence, are spectrally accurate, and have non-Gaussian density functions. The generated turbulent time histories are simulated to predict the aerostat response to severe turbulence. The resulting probability distributions for the aerostat position, pitch angle, and confluence point tension predict the aerostat behavior in high gust environments. The dynamic results can be up to twice as large as a static analysis indicating the importance of dynamics in aerostat modeling. The results uncover a worst case wind input consisting of a two-pulse vertical gust.     

On Transition to Turbulence and Synthesis of Alternative Approaches

In the study of complex systems, controversial debates often arise among advocates of different schools of thought. In this article, we examine how such controversies should be addressed, with the problem of transition to turbulence as a primary example. It is shown that, in many cases, these controversies may be resolved by first noting that the alternative theories proposed may not be mutually exclusive. Indeed, they may even be mutually complementary, if they were originally developed to address similar issues in different physical contexts. In any case, for the validity of the alternative theories proposed, each should be separately and fully supported from both the theoretical and empirical points of view. Each applies to a specific physical context, and each stands on its own merits and limitations. Synthesis into a broader theory may then be achieved, if commonality is identified among the different alternative theories proposed. To demonstrate this conciliatory approach, we begin with an examination of the move toward resolution of the well‐known controversy over the problem of transition to turbulence from the steady laminar flow in the boundary layer over a flat plate. Several other long‐standing controversies have been successfully addressed on the basis of this approach. In addition to the problem of transition to turbulence, we considered, in some detail, two additional examples: (1) the global structures of spiral galaxies; and (2) the theory of jet noise. In all three cases, it is shown that the model approach is meritorious despite the limitations. Synthesis, with a conciliatory approach to apparent conflicts, will be recommended in general as a new part of an extended paradigm in applied mathematics. It is an approach appropriate to situations where an ideal theory, with universal applicability, is elusive. Parallel development of several alternative theories is natural, and a final synthesis is needed. In contrast, it should be noted that the same perspective is generally not expected useful if the controversies concern the unique solution of well‐defined mathematical issues. The potential success of the application of this conciliatory perception and approach to other areas of science are discussed (see Section 5 ).     

Analysis of a Prototypical Multiscale Method Coupling Atomistic and Continuum Mechanics： the Convex Case

In order to describe a solid which deforms smoothly in some region,but non smoothly in someother region,many multiscale methods have been recently proposed that aim at coupling an atomistic model(discrete mechanics) with a macroscopic model (continuum mechanics).We provide here a theoretical basis forsuch a coupling in a one-dimensional setting,in the case of convex energy.     

An iterative langevin solution for turbulent dispersion in the atmosphere

In this work we present an alternative hybrid method to solve the Langevin equation and we apply it to simulate air pollution dispersion in inhomogeneous turbulence conditions. The method solves the Langevin equation, in semi-analytical manner, by the method of successive approximations or Picard's Iterative Method. Solutions for Gaussian and non-Gaussian turbulence conditions, considering Gaussian, bi-Gaussian and Gram–Charlier probability density functions are obtained. The models are applied to study the pollutant dispersion in all atmospheric stability and in low-wind speed condition. The proposed approach is evaluated through the comparison with experimental data and results from other different dispersion models. A statistical analysis reveals that the model simulates very well the experimental data and presents results comparable or even better than ones obtained by the other models.     

关于湍流理论中的不封闭性的讨论*

有一种观点认为湍流的不封闭性是由于N-S方程的非线性,这是一似是而非的错误说法。因为工程上要求的量并不仅仅限于湍流的平均速度和平均压力,即使方程式是线性的,也无法把工程上要求的这些量逐个求出来。本文论证了湍流的不封闭性来源于目前的湍流理论中缺少了一个与实际情况符合的有关分布函数,并且进一步说明了湍流模式理论的局限性和用N-S方程直接计算机模拟亦是相当困难的原因。     

Variation of drag coefficient with wind wave development

Turbulent air flows over developing wind waves in the air-sea boundary layer are numerically simulated without considering wave breaking. Influences of wind waves on air flows are considered using a model of significant wave and surface roughness, with a formula proposed for calculating the surface roughness. κ-ε model is adopted to simulate turbulent flows. The results of the drag coefficient and turbulence characteristics agree well with the observations. Project supported by the National Natural Science Foundation of China (Grant No. 19332010).     

Effects of plane strain on inhomogeneous turbulence

The influence of the mean plane strain on the turbulence transportation is investigated by large eddy simulation (LES) in the shearless turbulence mixing layer. It is found that the mean strains enhance the turbulent fluctuations in the mixing region. Compression in the inhomogeneous direction can greatly increase the transport of turbulent kinetic energy by triple correlation terms, while stretching in the inhomogeneous direction decreases the turbulence transportation. The gradient diffusion models for turbulent transportation are evaluated and it is found that the intermittency consideration can improve the prediction ability of the gradient-type models for the triple correlation terms. Project supported by the Sino-French Laboratory in Beijing and the National Natural Science Foundation of China (Grant No. 19572041).     

Assessment of an Eulerian CFD model for prediction of dilute droplet dispersion in a turbulent jet

The turbulent dispersion of non-evaporating droplets in an axisymmetric round jet issuing from a nozzle is investigated both experimentally and theoretically. The experimental data set has a well-defined inlet boundary with low turbulence intensity at the nozzle exit, so that droplet dispersion is not affected by the transport of nozzle-generated fluctuating motion into the jet, and is influenced solely by turbulence in the gas phase produced in the shear layer of the jet. This data set is thus ideal for testing algebraic models of droplet fluctuating motion that assume local equilibrium with the turbulence in the gas phase. Moreover, the droplet flux measurements are sufficiently accurate that conservation of the total volume flow of the droplet phase has been demonstrated. A two-fluid turbulence modelling approach is adopted, which uses the k–ε turbulence model and a simple algebraic model that assumes local equilibrium to predict the fluid and droplet turbulent correlations, respectively. We have shown that the k–ε turbulence model lacks generality for predicting the spread of momentum in jets with and without a potential core. However, in general, the model predicts the radial dispersion of droplets in the considered turbulent jet with reasonable accuracy over a broad range of droplet sizes, once deficiencies in the k–ε turbulence model are taken into account.     

Mathematical modeling of coupled turbulent flow and solidification in a single belt caster with electromagnetic brake

A mathematical model has been developed to simulate turbulent fluid flow and solidification in the presence of a DC magnetic field in an extended nozzle for metal delivery to a single belt caster. This paper reports on predicted effects of DC magnetic field conditions in modifying flows and solidification behavior in the metal delivery system. It is shown that the application of a DC magnetic brake to the proposed system can result in a reasonably uniform feeding of melt onto the cooled moving belt. This, in turn, optimises the rate of even shell growth along the chilled substrate. In order to account for the effects of turbulence, a revised low-Reynolds k–ε turbulent model was employed. A Darcy-porosity approach was used to simulate fluid flow within the mushy solidification region. Simulations were carried out for plain carbon steel strip casting. The fully coupled transport equations were numerically solved using the finite volume method. The computed flow patterns were compared with those reported in the literature. The performance of the magnetic flow control device proposed in this work is evaluated and compared with flow modifications obtained by inserting a ceramic filter within the reservoir.