On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.      

Closure method for spatially averaged dynamics of particle chains

We study the closure problem for continuum balance equations that model the mesoscale dynamics of large ODE systems. The underlying microscale model consists of classical Newton equations of particle dynamics. As a mesoscale model we use the balance equations for spatial averages obtained earlier by a number of authors: Murdoch and Bedeaux, Hardy, Noll and others. The momentum balance equation contains a flux (stress), which is given by an exact function of particle positions and velocities. We propose a method for approximating this function by a sequence of operators applied to the average density and momentum. The resulting approximate mesoscopic models are systems in closed form. The closed form property allows one to work directly with the mesoscale equations without the need to calculate the underlying particle trajectories, which is useful for the modeling and simulation of large particle systems. The proposed closure method utilizes the theory of ill-posed problems, in particular iterative regularization methods for solving first order linear integral equations. The closed form approximations are obtained in two steps. First, we use Landweber regularization to (approximately) reconstruct the interpolants of the relevant microscale quantities from the average density and momentum. Second, these reconstructions are substituted into the exact formulas for stress. The developed general theory is then applied to non-linear oscillator chains. We conduct a detailed study of the simplest zero-order approximation, and show numerically that it works well as long as the fluctuations of velocity are nearly constant.     

Thermal effects in the description of a multicomponent mixture using the density functional method

The use of a model, based on an expression for the total entropy in the form of a functional with the temperature and density gradients of the components, is proposed to describe a multicomponent, multiphase system using continuous hydrodynamics (that is, within the framework of the approach of the continuum mechanics without discontinuities in the hydrodynamic quantities). It is proved that this model is consistent with the zeroth law of thermodynamics. Expressions for the stress tensor, the diffusion fluxes and the heat flux are found from the condition that the entropy production is non-negative. Compared with the classical Newton, Fick and Fourier laws, these expressions contain third-order spatial derivatives, The problem of a mixture between two parallel and impermeable walls at different temperatures is analysed. In this case, the system of dynamic equations reduces to a system of ordinary differential equations. It is shown that the number of free parameters, on which the solution depends, corresponds to the number of boundary and general integral conditions.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Initial boundary value problem for the 3D quasilinear hyperbolic equations with nonlinear damping

We investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy’s-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy’s-type law in large time.     

The heat and moisture transfer balance theory of garment simulation

To establish the human body model to analyze the heat and moisture transfer on body surface, a new explicit definition of rational L-recursion surface is given and the L-recursion surfaces, in Grassmann spaces, are constructed by using blossom method of the homogeneous normal pyramid form. Based on our human body model, the balance theory of garment simulation, the heat and moisture transfer balance equations, called ICAD-balance equations are obtained. The balance theory of garment simulation integrally studies the complex system of human body-fabric-environment. At the same time, the method of obtaining the heat and moisture transfer balance equations is also based on the mass conservation law, the energy conservation law and the Fish law of capillarity. A finite volume method is employed to solve the ICAD-balance equations.     

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)     

The heat and moisture transfer balance theory of garment simulation

To establish the human body model to analyze the heat and moisture transfer on body surface, a new explicit definition of rational L-recursion surface is given and the L-recursion surfaces, in Grassmann spaces, are constructed by using blossom method of the homogeneous normal pyramid form. Based on our human body model, the balance theory of garment simulation, the heat and moisture transfer balance equations, called ICAD-balance equations are obtained. The balance theory of garment simulation integrally studies the complex system of human body–fabric–environment. At the same time, the method of obtaining the heat and moisture transfer balance equations is also based on the mass conservation law, the energy conservation law and the Fish law of capillarity. A finite volume method is employed to solve the ICAD-balance equations.     

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Fractal Fracture of an Elastic Plane Weakened by a Lunate Notch

The possibility of satisfying the energy balance conditions is justified for the problem of tension on a plane weakened by a notch in the shape of a symmetric lune. A macrocrack growing from a corner of an elastic domain is modeled by a fractional-dimensional fractal. The dependence of the dimension of a fractal crack on the angle of the lunate notch is determined such that the Griffith energy balance equation can be satisfied. The same kind of determination is impossible in the realm of classical fracture mechanics and is used as a basis for calculating the breaking load in the stated problem.     

Adaptation of the law of the wall boundary treatment to include volumetric heating effects

Computational Fluid Dynamics (CFD) simulations of liquid–metal spallation targets, such as MEGAPIE and ESS, which utilize the High Reynolds number k–ε turbulence model, invariably incorporate an implicit law of the wall treatment in which a linear or logarithmic fit to the velocity and temperature profiles is made next to heated, non-slip surfaces. The law is well-established, but has been derived from the assumptions that the wall shear stress and the normal heat flux are constant through the viscous sub-layer and buffer zone, which lie beneath the turbulent boundary layer. However, in the case of the heat flux, this condition will be violated for applications in which there is intense volumetric heating in the near-wall layers. This is just the case for the spallation reactions taking place in liquid–metal targets as a result of proton bombardment. In this article, a modified law of the wall is derived to be used under such conditions. Use of the law is illustrated by means of flow in a flat channel and one application to a spallation target. From the applications considered, it is found that the effect of the modification is small, provided the local mesh resolution is chosen appropriately. Specific recommendations regarding optimum mesh size for liquid–metal heat transfer problems are given, which will be of general interest, with or without volumetric heating.     

重建极性连续统理论的基本定律和原理(Ⅱ)——微态连续统理论和偶应力理论

重建微态连续统理论和偶应力理论的动量和动量矩均衡定律以及能量守恒定律,并由这些定律自然地推导出相应的局部和非局部均衡方程。这些结果可由耦合型微极连续统理论过渡和归结而得到。把推导出的结果和传统的质量和微惯性守恒定律以及熵不等式结合在一起就构成微态连续统理论和偶应力理论的基本均衡定律和方程体系。还弄清了以前的各种连续统理论的不完整性层次。最后,给出了几种特殊情形。     

半导体磁流体动力学模型经典解的整体适定性

研究一类半导体磁流体动力学模型,它是由关于电子的质量和速度的守恒律方程耦合Maxwell方程构成的流体动力学方程组.在小初值条件下,运用经典的双曲能量方法,得到了磁流体动力学模型Cauchy问题经典解的整体适定性.     

Weak solution of incompressible Euler equations with decreasing energy

Weak solution of incompressible Euler equations are L²-vector fields, satisfying integral relations, which express the mass and momentum balance. They are believed to describe the turbulent fluid motion at high Reynolds numbers. We justify this conjecture by constructing a weak solution with decreasing kinetic energy. The construction is based on Generalized Flows, introduced by Y. Brenier.     

Analytical solutions for heat transfer on fractal and pre-fractal domains

Fractals can be used to represent intricate self-similar geometries, but their application to the representation of physical systems is beset with difficulties which stem from an inability to define traditionally derived-physical quantities such as stress, pressure, strain, heat etc. This paper describes a method for the determination of analytical heat-transfer solutions on pre-fractal and fractal domains. The approach requires the construction of maps from pre-fractal domains to the continuum, which facilitate the application of traditional continuum solution methods. Solutions on fractal domains are achievable with this approach, and are defined to be the limit solution of analytical solutions obtained on the pre-fractals approximating the fractal of interest. This approach avoids many of the complications and technical difficulties arising from the use of measure theory and fractional derivatives, but also infers that the governing heat transfer equations are valid on all pre-fractals. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions presented are limited to one and two-dimensional domains and, in 1-D, are applied to an idealised composite material consisting of relatively small particles of infinitely low thermal conductivity embedded in a relatively large matrix of infinitely high thermal conductivity. The fractal composite system is thus not truly representative of a realistic physical system, but the methods presented do serve to demonstrate how analytical solutions can be attained on dust-like fractal domains. It is demonstrated that a measurable temperature is possible on a fractal structure along with finite measures of heat flux and energy. Transient and steady state thermal solutions are presented. The solutions on a selection of the pre-fractals are compared against finite element predictions to reinforce the validity of the approach.     

Effects of fractality on the accessible surface area values of zeolite adsorbents

The effects of the molecular sizes of adsorbates on the accessible surface area values of the zeolites 13X, 5A, silicalite and NaY as well as the modified forms of NaY are investigated by taking into account the concept of fractality. For this aim, a relationship developed by combining the Pfeifer–Avnir and the surface area equations, which relates the surface areas of adsorbents to the molar volumes and the cross-sectional areas of adsorbates is utilized. The expected relationship, signifying that the accessible surface areas of adsorbents having fractal dimensions above and below 2, increase and decrease, respectively, with decreasing size of the adsorbate, is quantified in this study for the above zeolite adsorbents. Modifying the properties of adsorbents by using various treatment methods is seen to be potentially useful for enhancing the performances of processes involving adsorption, e.g. adsorption heat pump applications. Since the treatments employed may change the fractal dimension of the original sample, adsorbates having proper sizes should be used with the modified forms in order to achieve a good result. The modified hydrogen form of NaY provides the opportunity to increase the efficiency of the adsorption heat pumps by almost 40% with respect to the utilization of the original sample when methanol is used as the adsorbate.     

Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation

For the quasi-gasdynamic system of equations, there holds the law of nondecreasing entropy. Difference methods based on this system have been successfully used in numerous applications and test gasdynamic computations. In theoretical terms, however, for standard spatial discretizations of this system, the nondecreasing entropy law does not hold exactly even in the one-dimensional case because of the mesh imbalance terms. For the quasi-gasdynamic equations, a new conservative spatial discretization is proposed for which the entropy balance equation has an appropriate form and the entropy production is guaranteed to be nonnegative (which also holds in the presence of body forces and heat sources). An important element of this discretization is that it makes use of nonstandard space-averaging techniques, including a nonlinear ??logarithmic?? averaging of the density and internal energy. The results hold on arbitrary nonuniform meshes.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

In this work, the effects of slip velocity on the flow and heat transfer for an electrically conducting micropolar fluid over a permeable stretching surface with variable heat flux in the presence of heat generation (absorption) and a transverse magnetic field are investigated. The governing partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformation, which is solved numerically using the Chebyshev spectral method. The effects of the slip parameter on the flow, micro-rotation and temperature profiles as well as on the local skin-friction coefficient, the wall couple stress and the local Nusselt number are presented graphically. The numerical results of the local skin-friction coefficient, the wall couple stress and the local Nusselt number are given in a tabular form and discussed.