Convective heat transport in a fluid layer of infinite Prandtl number: upper bounds for the case of rigid lower boundary and stress-free upper boundary

We present the theory of the multi--solutions of the variational problem for the upper bounds on the convective heat transport in a heated from below horizontal fluid layer with rigid lower boundary and stress-free upper boundary. A sequence of upper bounds on the convective heat transport is obtained. The highest bound is between the bounds for the case of a fluid layer with two rigid boundaries and for the case of a fluid layer with two stress-free boundaries. As an additional result of the presented theory we obtain small corrections of the boundary layer thicknesses of the optimum fields for the case of fluid layer with two rigid boundaries. These corrections lead to systematically lower upper bounds on the convective heat transport in comparison to the bounds obtained in [5]. Received 29 September 1999     

Thermal Performance of Laser Diode Array under Constant Convective Heat Transfer Boundary Condition

Three-dimensional heat transfer model of laser diode array under constant convective heat transfer coefficient boundary condition is established and analytical temperature profiles within its heat sink are obtained by separation of variables. The influences on thermal resistance and maximum temperature variation among emitters from heat sink structure parameters and convective heat transfer coefficient are brought forward. The derived formula enables the thermal optimization of laser diode array.     

Third-order dispersion for generating optical rogue solitons

We theoretically and numerically evidence that optical rare and strong temporal events generated in fiber supercontinua originate from convective modulational instabilities. This convective nature is induced by higher-order terms (odd-order dispersion and stimulated Raman scattering) that break the time reversal symmetry of the nonlinear Schrödinger equation. We demonstrate (i) analytically that the third-order dispersion term alone turns the system to be convectively unstable and (ii) numerically that the sign of the curvature of the tail of the probability density function changes (in logarithmic scale) when the third-order dispersion term is added. This latter feature results in more powerful rare events. If, in addition, stimulated Raman scattering is taken into account, both the convective instabilities and the power of extreme events are further enhanced giving rise to a probability density function with a more pronounced curvature.     

On numerical realizability of thermal convection

Astounded at the regularity of convective structures observed in simulations of mesoscale flow past realistic topography, we investigate the computational aspects of a classical problem of flow over a heated plane. We find that the numerical solutions are sensitive to viscosity, either incorporated a priori or effectively realized in computational models. In particular, anisotropic viscosity can lead to regular convective structures that mimic naturally realizable Rayleigh–Bénard cells, which are unphysical for the specified external parameter range. Details of the viscosity appear to play a secondary role; that is, similar structures can occur for prescribed constant viscosities, explicit subgrid-scale turbulence models, ad-hoc numerical filters, or implicit dissipation of numerical schemes. This implies the need for a careful selection of numerical tools suitable for convection-resolving simulations of atmospheric circulations. The implicit large-eddy-simulation (ILES) approach using non-oscillatory schemes is especially attractive, as for under-resolved calculations it reproduces well the coarsened results of finely-resolved boundary layer convection.     

Upper bounds on the convective heat transport in a rotating fluid layer of infinite Prandtl number: case of large Taylor numbers

By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a horizontal fluid layer heated from below and rotating about a vertical axis. We consider the interval of large Taylor numbers where the intermediate layers of the optimum fields expand in the direction of the corresponding internal layers. We consider the 1 - α-solution of the arising variational problem for the cases of rigid-stress-free, stress-free, and rigid boundary conditions. For each kind of boundary condition we discuss four cases: two cases where the boundary layers are thinner than the Ekman layers of the optimum field and two cases where the boundary layers are thicker than the Ekman layers. In most cases we use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate layers of the optimum fields. This solution leads to corrections of the thicknesses of the boundary layers of the optimum fields and to lower upper bounds on the convective heat transport in comparison to the bounds obtained by Chan [J. Fluid Mech. 64, 477 (1974)] and Hunter and Riahi [J. Fluid Mech. 72, 433 (1975)]. Compared to the existing experimental data for the case of a fluid layer with rigid boundaries the corresponding upper bounds on the convective heat transport is less than two times larger than the experimental results, the corresponding upper bound on the convective heat transport, obtained by Hunter and Riahi is about 10% higher than the bound obtained in this article. When Rayleigh number and Taylor number are high enough the upper bound on the convective heat transport ceases to depend on the boundary conditions. Received 30 January 2001 and Received in final form 28 May 2001     

Experimental Investigation of Influence of Interfacial Tension on Convection of Two-Layer Immiscible Liquid

Bdnard-Marangoni convections of two-layer fluids heated from the bottom are investigated experimentally with a particle imagine velocimetry. The flows are visualized from the side, and various velocity fields near the onset of convection, such as three-layer vortex convective patterns, are observed when the depth ratio varies in a wide range. A new classification of the convective patterns is proposed with more detail than in previous studies. The analysis of the results indicates that the interface tension greatly influences the motion intensities of the bottom and top layers. The dimensionless wave number increases with the Bond number when the motion in the top layer is not more intense than that in the bottom layer, which agrees with the theoretical prediction.     

Eddy diffusivity in convective hydromagnetic systems

An eigenvalue equation, for linear instability modes involving large scales in a convective hydromagnetic system, is derived in the framework of multiscale analysis. We consider a horizontal layer with electrically conducting boundaries, kept at fixed temperatures and with free surface boundary conditions for the velocity field; periodicity in horizontal directions is assumed. The steady states must be stable to short (fast) scale perturbations and possess symmetry about the vertical axis, allowing instabilities involving large (slow) scales to develop. We expand the modes and their growth rates in power series in the scale separation parameter and obtain a hierarchy of equations, which are solved numerically. Second order solvability condition yields a closed equation for the leading terms of the asymptotic expansions and respective growth rate, whose origin is in the (combined) eddy diffusivity phenomenon. For about 10% of randomly generated steady convective hydromagnetic regimes, negative eddy diffusivity is found.     

Heat and mass transport in the photoacoustic effect on liquids

The temperature oscillation accompanying the photoacoustic effect generates a periodic variation of the vapor pressure of a liquid. The propagation of the oscillating concentration of the vapor in the inert cell gas (air) is described by a mass diffusion wave on which a convective motion of the gas is superposed. The diffusion wave characterized by the diffusion coefficient of the cell gas alone can be measured by the Mirage effect, whereas a microphone detects the total mass flux including the convective flux, which increases with temperature. On approaching the boiling temperature, the convective flow will govern the oscillating transport of mass. The photoacoustic signal is determined directly from the flux of heat and mass at the boundary between liquid and gas using the Gauss' divergence theorem. We have found that the temperature behaviour of the amplitude and phase angle of the photoacoustic signal depends on the length of the gas column in the cell. The contribution of thermal expansion to the photoacoustic signal is considered using the composite piston model. The results of the calculations agree fairly well with the experimental data.     

Nonlinear Evolution of Driven Electron Plasma Oscillations in Inhomogeneous Plasmas

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Study of Heat Transfer Control with Magnetic Field Using Higher Order Finite Difference Scheme

The control of convective heat transfer from a heated circular cylinder immersed in an electrically conducting fluid is achieved using an externally imposed magnetic field. A Higher Order Compact Scheme (HOCS) is used to solve the governing energy equation in cylindrical polar coordinates. The HOCS gives fourth order accurate results for the temperature field. The behavior of local Nusselt number, mean Nusselt number and temperature field due to variation in the aligned magnetic field is evaluated for the parameters 5≤$Re$≤40, 0≤$N$≤20 and 0.065≤$Pr$≤7. It is found that the convective heat transfer is suppressed by increasing the strength of the imposed magnetic field until a critical value of $N$, the interaction parameter, beyond which the heat transfer increases with further increase in $N$. The results are found to be in good agreement with recent experimental studies.     

Magnetic field generation by convective flows in a plane layer

Hydrodynamic and magnetohydrodynamic convective attractors in a plane horizontal layer 0≤z≤1 are investigated numerically. We consider Rayleigh-Bénard convection in Boussinesq approximation assuming stress-free boundary conditions on horizontal boundaries and periodicity with the same period L in the x and y directions. Computations have been performed for the Prandtl number P=1 for and Rayleigh numbers 0<R≤4000, and for L=4, 0<R≤2000. Fifteen different types of hydrodynamic attractors are found, including two types of steady states distinct from rolls, travelling waves, periodic and quasiperiodic flows, and chaotic attractors of heteroclinic nature. Kinematic dynamo problem has been solved for the computed convective attractors. Out of the 15 types of the observed attractors only 6 can act as kinematic dynamos. Nonlinear magnetohydrodynamic regimes have been explored assuming as initial conditions convective attractors capable of magnetic field generation, and a small seed magnetic field. After initial exponential growth, in the saturated regime magnetic energy remains much smaller than the flow kinetic energy. The final magnetohydrodynamic attractors are either quasiperiodic or chaotic.     

Possibility of surface plasmon-polaritons amplification by direct current in two-interface systems with 2D Bragg structure on the surface

The method for calculation of the local field factor that describes electrodynamical properties of three-layered systems with 2D photonic crystal is proposed. Analytical calculations are performed in the frame of Green function method on the base of the effective susceptibility concept. The possibility of surface waves amplification by direct electrical current in the systems under consideration is studied. Numerical analysis clearly proves the presence of convective instability. It means that three-layered systems can be considered as a surface waves’ amplifier. Because of periodical structure the phase synchronism domain lies in the range of entirely accessible in practice values of wave vector and frequency. The experimental setup and real systems parameters for observation of surface waves amplification effect are discussed.     

Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction

The laminar flow problem of convective heat transfer for a second grade fluid over a semi-infinite plate in the presence of species concentration and chemical reaction is investigated. The governing equations are transformed into a dimensionless system of three non-linear coupled partial differential equations. These equations have been solved analytically subject to the relevant boundary conditions by employing a homotopy analysis method (HAM). It is noted that for the arising system, the HAM performs extremely well in terms of efficiency and simplicity. The influence of dimensionless pertinent parameters on the velocity, temperature and concentration fields has been examined carefully.     

Transient oscillations in Soret-driven convection in a colloidal suspension

We present measurements of the transient stage of Soret-driven convective instability. The sample is a diluted colloidal suspension of silica spheres in water with an unusually large negative Soret coefficient S_T. A large temperature gradient (heating from above) is rapidly applied over the sample, while a shadowgraph imaging technique provides images of the convective flow. From the processing of the variance of the intensity of the images we are able to recover the time evolution of the overall intensity of the convective flow. A typical evolution of such signal exhibits, after a latency time, a peak followed by some damped oscillations leading to a steady-state value. Both the onset time _p (the temporal position of the first peak), and the oscillation period _osc show a power law dependence as a function of the solutal Rayleigh number R_s. The exponents found are compared with predictions from existing models.     

Soliton-like solutions of certain types of nonlinear diffusion-reaction equations with variable coefficient

With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion-reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.     

Self-organization analysis for a nonlocal convective Fisher equation

Using both an analytical method and a numerical approach we have investigated pattern formation for a nonlocal convective Fisher equation with constant and spatial velocity fields. We analyze the limits of the influence function due to nonlocal interaction and we obtain the phase diagram of critical velocities vc as function of the width μ of the influence function, which characterize the self-organization of a finite system.     

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.     

Homoclinic Breather-Wave with Convective Effect for the （1＋1）-Dimensional Boussinesq Equation

A new type of two-wave solution, i.e. a homoclinic breather-wave solution with convective effect, for the （1＋1）- dimensional Boussinesq equation is obtained using the extended homoelinic test method. Moreover, the mechanical feature of the wave solution is investigated and the phenomenon of homoelinic convection of the two-wave is exhibited on both sides of the equilibrium. These results enrich the dynamical behavior of （1＋1）-dimensional nonlinear wave fields.     

混合层流场中涡结构对流速度的特性

基于大涡模拟和光线追踪方法, 对光线穿越流场后的光程分布与混合层流场中涡结构之间的关系进行了分析, 提出了一种基于涡核位置提取的涡结构瞬时对流速度定量计算方法, 并使用直接几何测量数据进行了验证. 通过对不同尺寸的涡结构、涡-涡配对及融合过程中的涡结构和强压缩性流场中涡结构瞬时对流速度的定量数值计算, 揭示了混合层流场中涡结构对流速度的特性: 对单个涡结构而言, 其瞬时对流速度具有脉动特性, 且脉动幅度随涡结构尺寸和流场压缩性而变化; 在涡-涡配对及融合过程中, 涡对中各个涡结构的瞬时对流速度都表现出类似正弦波动的特点. 针对混合层流场中涡结构对流速度的特性, 给出了其背后的物理原因.