Destruction of the family of steady states in the planar problem of Darcy convection

We consider natural convection of an incompressible fluid in a porous medium described by the planar Darcy equation. For some boundary conditions, Darcy problem may have non-unique solutions in form of a continuous family of steady states. We are interested in the situation when these boundary conditions are violated. The resulting destruction of the family of steady states is studied via computer experiments based on a mimetic finite-difference approach. Convection in a rectangular enclosure is considered under different perturbations of boundary conditions (heat sources, infiltration). Two scenario of the family of equilibria are found: the transformation to a limit cycle and the formation of isolated convective patterns.     

Studying anomalous scaling and heat transport of turbulent thermal convection using a dynamical model

Turbulent thermal convection is a well-studied problem with various issues of interest. In this paper, we review our work which shows the nature and origin of anomalous scaling and heat transport in the limit of very strong thermal forcing, can be gained by studying a dynamical model, known as shell model, of homogeneous turbulent thermal convection in which buoyancy acts directly at most scales. Specifically, we have obtained two results. The first result is that when buoyancy acts directly at most scales such that the dynamics are governed by a cascade of entropy, the scaling behavior is described by Bolgiano and Obukhov scaling plus corrections that are due to the variations of the local entropy transfer rate. This result indicates the validity of the extension of refined similarity hypothesis to turbulent thermal convection. The second result is that when buoyancy is acting directly at most scales, a damping term acting on the largest scale, which has to be added for the system to achieve stationarity, plays a crucial role in heat transport, and that the heat transport depends on the strength of thermal forcing in the same manner as that predicted for the ultimate state of very strong thermal forcing. With our interpretation of the damping term representing the effect of the boundaries, this result indicates that in the ultimate state of turbulent thermal convection, when buoyancy is acting at most scales, boundaries would play a significant role in heat transport.     

Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection

A three-dimensional set of ordinary differential equations that constitutes a simple abstract model of Darcy convection is investigated. The model reproduces a number of effects that are typical for dynamic systems with nontrivial cosymmetry. Nontrivial cosymmetry can give rise to a continuous family of equilibria where, in this case, the equilibrium stability spectrum varies along the family. The family of equilibria and its stability are examined analytically, and special bifurcations that occur in the system are investigated. It is shown that discrete and continual symmetries, called "flash symmetries," can be present in the system for certain parameter values. Computer experiments on the selection of equilibria in the symmetric and cosymmetric cases have been carried out. They showed that, for initial points that are far enough from a cycle of equilibria, the neighborhood of a single equilibrium is established in the case of cosymmetry, but all the equilibria are equivalent in the case of symmetry. The authors hope that these results, as well as the formulation of the problems and the approach to their solution, will serve as a sample in the investigation of more complex systems in mathematical physics. (c) 1999 American Institute of Physics.     

Determination of forced convection parameters by interferometric imaging of the concentration field during growth of KDP crystals

Growth of a potassium dihydrogen phosphate (KDP) crystal from its aqueous solution has been considered under forced convection conditions. The KDP crystal is grown in a conventional top hanging geometry. Forced convection conditions are created by rotating the crystal about a vertical axis. The rotational RPM is varied in a cycle, creating an accelerated rotation (AR) paradigm. The effect of varying the rotational RPM on the concentration field around the crystal was investigated. Mach-Zehnder interferometry was adopted as an optical technique to image the evolving concentration fields. Six different experiments were performed to obtain the specific set of time periods and rotation rates of the acceleration cycle that result in a uniform concentration field around the growing crystal. The Reynolds number, an index of the strength of forced convection, was optimized through the experiments. The optimized parameters of the accelerated rotation cycle were found to be as follows: maximum rotation rate of 32 RPM, spin up period=40 s, spin down period=40 s, steady period=40 s, and stationary period=40 s. The parametric study further revealed that concentration was highly sensitive to the maximum rotation rate adopted during the AR cycle. It did not depend crucially on the time periods that could be varied by as much as ±25% around the respective average values. Finally, a KDP crystal was grown using the optimized forced convection parameters and the crystal quality was found to be good.     

Bifurcations in autonomous mechanical systems under the influence of joint damping

This contribution deals with the impact of joint damping on two classes of stability problems which are often found in engineering problems. In a first part, the principle structure of the equations of motion is derived when joints are modeled using Masing-, Prandtl- and Coulomb-elements. For these general formulations, some fundamental statements concerning stability and attractiveness of steady-state solutions may be given for large amplitudes and configurations which are not too close to the linear stability threshold. The second part focuses on analyzing the behavior at small amplitudes and in the vicinity of the linear stability threshold in more detail: to this end, a static stability problem (buckling) and two oscillatory self-excitation mechanisms (negative damping, non-conservative coupling) are discussed. For all considered problems, adding joint damping transforms the equilibrium points into sets of equilibria and bifurcations of the non-smooth problems occur near the linear stability threshold. Concerning the buckling problem adding joint damping does not alter the behavior fundamentally: still a local bifurcation occurs and attractiveness or instability of equilibrium solutions is preserved. In contrast, the oscillatory instability examples are strongly influenced by joint damping: here, global discontinuous bifurcations may occur. Besides the joint friction also the joint-stiffness may play a crucial role, since it determines whether attractive solutions in or about the equilibrium set exist. It is found that only in some cases a linear stability analysis of the corresponding system without joints may give correct indications on the behavior: consequently, neglecting joint-damping in stability analyses may lead to wrong results concerning self-excitation.     

Semi-analytical modeling of tokamak density evolution

Tokamak plasma density evolution is generally modeled by a diffusion--convection equation in cylindrical geometry. By using a semi-analytical approach, we solve such an equation for a given diffusion coefficient and inward convection velocity as an arbitrary function of the radial position. Through variable separation, a Sturm--Liouville-type eigenvalue problem is solved, thereby constructing a complete set of orthogonal eigenfunctions. Based on the decomposition of the solution, the initial function, and the source function in these eigenfunctions, several problems of practical interest about the density evolution are analyzed. They include the density evolution, with boundary density not being zero; the density profile with internal transport barrier; the damping profile during particle source being shut-down. Results are found to be qualitatively consistent with the tokamak experiments.     

Selection of steady states in planar Darcy convection

The planar natural convection of an incompressible fluid in a porous medium is considered. We study the selection of steady states under temperature perturbations on the boundary. A selection map is introduced in order to analyze the selection of a steady state from a continuous family of equilibria which exists under zero boundary conditions. The results of finite-difference modeling for a rectangular enclosure are presented.     

Poiseuille-Rayleigh-Bénard流动中对流斑图的分区和成长

采用SIMPLE算法对二维流体力学基本方程组进行了数值模拟,研究了Poiseuille-Rayleigh-Bénard流动中对流斑图的分区、成长及水平流动对不同斑图特征物理量的影响.结果表明,上下临界雷诺数Re_u,Re_l将流动分成三个区域,即行波区、局部行波区、水平流区.Re_u和Re_l随着相对瑞利数r的增大而增大.在对流斑图的成长阶段,三种斑图随时间的成长过程是不同的,但对流圈都是从下游区开始成长;特征物理量随着时间的变化也是不同的,行波对流和局部行波对流的最大垂直流速wmax和努塞尔数Nu经过指数增长阶段后进入周期变化的稳定阶段;水平流斑图的w_(max)和Nu经过缓慢增长后又缓慢降到稳定值.三种斑图的w_(max)和Nu随雷诺数Re增大而减小,不同斑图区域有不同的变化规律.本文给出了Re_u和Re_l随r的变化关系式及不同斑图的w_(max)和Nu随着Re的变化关系式.     

Theoretical and numerical study of a thermal convection problem with temperature-dependent viscosity in an infinite layer

A convection problem with temperature-dependent viscosity in an infinite layer is presented. This problem has important applications in mantle convection. The existence of a stationary bifurcation is proven together with a condition to obtain the critical parameters at which the bifurcation takes place. A numerical strategy has been developed to calculate the critical bifurcation curves and the most unstable modes for a general dependence of viscosity on temperature. An exponential dependence of viscosity on temperature has been considered in the numerical calculations. Comparisons with the classic Rayleigh-Bénard problem with constant viscosity indicate that the critical temperature difference threshold decreases as the exponential rate parameter increases. The vertical velocity of the marginal mode exhibits motion concentrated in the region where viscosity is smaller.     

Bifurcation of the branching of a cycle in n-parameter family of dynamic systems with cosymmetry

A study is reported of the bifurcation of the branching of a cycle (Poincare-Andronov-Hopf bifurcation) from a smooth one-dimensional submanifold of equilibria of a dynamical system that depends on a vector parameter and admits cosymmetry. The paper reports a topological classification of local phase portraits near a known equilibrium, when the system parameter is close to its critical value that corresponds to an oscillatory instability. New phenomena that are not observed in the classical case of an isolated equilibrium include a delay of cycle creation with respect to the system parameter, loss of stability by the family of equilibria without loss of attraction, and the possibility of unstable supercritical self-oscillations. (c) 1997 American Institute of Physics.     

Bifurcations accompanying monotonic instability of an equilibrium of a cosymmetric dynamical system

It is well known that equilibrium in a cosymmetric system in the general position is a member of a one-parameter family. In the present paper the Lyapunov-Schmidt method and the method of the central manifold are used to analyze bifurcations of such a family of equilibria as well as internal bifurcations: transitions of the type focus-node, node-saddle, and so on during motion along the family. A series of scenarios of branching of families of equilibria and the change in the structure of their arcs, consisting of equilibria of the same type, is described. Bifurcations of stable and unstable arcs, coalescence and decomposition of families of equilibria, bifurcation of the loss of smoothness by the family of equilibria, and branching of a small equilibrium cycle from a corner point of the family of equilibria are investigated in detail. The variability of the spectrum along a family gives rise to a variety of new phenomena that are not encountered in the classical case of an isolated equilibrium or in bifurcations of families of equilibria of a system with symmetry. They include protraction with respect to the branching parameter of the family of equilibria, Lyapunov instability of a family of equilibria with the attraction properties being preserved, and the appearance and disappearance of new stable and unstable arcs on the family of equilibria. (c) 2000 American Institute of Physics.     

Relating PAC damping to EFG fluctuation rates through the PAC relaxation peak

A perturbed angular correlation (PAC) experiment that measures dynamic damping also needs information about the fundamental quadrupole frequency to relate the damping as a function of temperature to the EFG fluctuation rate. When the experiment is unable to access slow electric field gradient (EFG) fluctuations that show the fundamental quadrupole frequency directly, one needs additional information to determine the hyperfine field parameters and thereby the connection between observed damping and EFG fluctuation rates. One way to solve this problem is to estimate the hyperfine parameters from the fluctuation rate for maximum damping (i.e. at the relaxation peak) or from the rate of maximum damping. This work relates both the maximum damping rate and the fluctuation rate at the relaxation peak to EFG magnitudes (or quadrupole frequencies) for five dynamic N-state symmetric models of fluctuating EFGs.     

Natural convection with damped thermal flux in a vertical circular cylinder

Natural convection flows of an incompressible Newtonian fluid inside a circular cylinder are studied. The heat transfer process is described by a generalized fractional constitutive equation for the thermal flux-temperature gradient. Caputo time-fractional derivative operator, which provides the damping of thermal flux, is considered into the studied model.Analytical solutions to the fluid temperature, thermal flux, fluid velocity and volume flow rate are obtained with the integral transforms method (Laplace transform and finite Hankel transform).Temperature behaviors for small and large values of the time t, as well as the post-transient and transient velocity components are determined. The influence of the memory parameter (the order of the time-fractional derivative) on the temperature, thermal flux, velocity and the volume flow rate is numerically and graphically studied.     

Classifying relative equilibria. III

We announce several theorems on the evolution of relative equilibria classes in the planar n-body problem. In an earlier paper [1] we announced a partial classification of relative equilibria of four equal masses. In [2] we described these new relative equilibria classes and showed the way in which a degeneracy arose in the four body problem. These results point the way toward classifying relative equilibria for any n > 4.Research supported in part by NSF grant MPS-73-03735 A04.     

Dependence of heat transport on the strength and shear rate of prescribed circulating flows

We study numerically the dependence of heat transport on the maximum velocity and shear rate of physical circulating flows, which are prescribed to have the key characteristics of the large-scale mean flow observed in turbulent convection. When the side-boundary thermal layer is thinner than the viscous boundary layer, the Nusselt number (Nu), which measures the heat transport, scales with the normalized shear rate to an exponent 1/3. On the other hand, when the side-boundary thermal layer is thicker, the dependence of Nu on the Peclet number, which measures the maximum velocity, or the normalized shear rate when the viscous boundary layer thickness is fixed, is generally not a power law. Scaling behavior is obtained only in an asymptotic regime. The relevance of our results to the problem of heat transport in turbulent convection is also discussed. Received 28 November 2001 Published online 25 June 2002     

Finite-amplitude waves on the surface of a viscous deep liquid

For the first time a rigorous solution to the problem on time evolution of the periodic wave shape on the surface of a viscous infinitely deep liquid is found in the quadratic approximation with respect to the wave amplitude. It is found, in particular, that the damping rate of the quadratic component with respect to the wave amplitude is twice as high as the damping rate of the linear term. It is shown that inclusion of viscosity leads to asymmetry of the wave profile.     

On the weakly damped vibrations of a vertical beam with a tip-mass

In this paper the wind-induced, horizontal vibrations of a weakly damped vertical Euler-Bernoulli beam with and without a tip-mass will be studied. The damping is assumed to be boundary damping and global Kelvin-Voigt damping. The boundary damping is assumed to be proportional to the velocity of the beam at the top. The horizontal vibrations of the beam can be described by an initial-boundary value problem. In this paper, the multiple-timescales perturbation method will be applied to construct approximations of the solutions of the problem. Also it will be shown that a combination of boundary damping and Kelvin-Voigt damping can be used to damp the wind-induced vibrations of a vertical beam with tip-mass uniformly.     

Modeling of heat explosion with convection

The work is devoted to numerical simulations of the interaction of heat explosion with natural convection. The model consists of the heat equation with a nonlinear source term describing heat production due to an exothermic chemical reaction coupled with the Navier-Stokes equations under the Boussinesq approximation. We show how complex regimes appear through successive bifurcations leading from a stable stationary temperature distribution without convection to a stationary symmetric convective solution, stationary asymmetric convection, periodic in time oscillations, and finally aperiodic oscillations. A simplified model problem is suggested. It describes the main features of solutions of the complete problem.     

数值模拟水环境污染的一种L∞稳定的分步杂交方法

本文提出了一种数值求解对流扩散方程的分步杂交方法。在不规则的三角形网格上,采用迎风离散格式或改型特征线方法处理对流算子;采用集中质量的有限元方法处理扩散算子。详细分析了这种算法的稳定性同题,在数学上严格证明了在满足①Δt≤min((2d)/v,(d²)/(3K)),其中d是三角形网格中最短垂线的长度,V和K分别为流场中的最大速度和扩散系数。②所有三角形的内角θ≤π/2的条件下,整个计算格式是L_∞稳定的,从而保证了在海洋环境和水质的数值模拟中海水的盐度、污染物的浓度和核电站冷却水系统中的超温不会出现负值。应用非线性的对流扩散方程对此方法的精度和收敛性进行了检验。通过数值解与精确解的比较,表明本方法的数值耗散很小,用改型特征线方法处理对流算子较迎风离散格式有更高的精度;两种处理对流算子的方法都没有伪振荡现象发生。本方法由于具有算法简单、L_∞稳定、计算网格灵活等优点,可推广使用于实际的海洋环境(潮波、海流、海洋污染)、港口和海湾的数值模拟以及不可压粘性流和对流传热同题的数值计算。     

Analysis of Frequency Spectrum of Laser-Induced Vibration of Microbeam Resonators

The vibration phenomenon during pulsed laser heating of micro-beams is investigated. The beam is made of silicon and is heated by a laser pulse with a non-Gaussian temporal profile and with an ultrashort pulse duration of 2ps, which incites vibration due to the thermoelastic damping effect. This coupled thermoelastic problem is solved using an analytical-numerical technique based on the Laplace transformation. The damping ratio and resonant frequency shift ratio of beams due to the air damping effect and the thermoelastic damping effect are also examined and discussed.