Numerical-Value Perturbation Method with Application of Solving Convective-Diffusion Integral Equation

The convective--diffusion equation is of primary importance in such fields as fluid dynamics and heat transfer hi the numerical methods solving the convective-diffusion equation, the finite volume method can use conveniently diversified grids (structured and unstructured grids) and is suitable for very complex geometry The disadvantage of FV methods compared to the finite difference method is that FV-methods of order higher than second are more difficult to develop in three-dimensional cases. The second-order central scheme (2cs) offers a good compromise among accuracy, simplicity and efficiency, however, it will produce oscillatory solutions when the grid Reynolds numbers are large and then very fine grids are required to obtain accurate solution. The simplest first-order upwind (IUW) scheme satisfies the convective boundedness criteria, however. Its numerical diffusion is large. The power-law scheme, QMCK and second-order upwind (2UW) schemes are also often used in some commercial codes. Their numerical accurate are roughly consistent with that of ZCS. Therefore, it is meaningful to offer higher-accurate three point FV scheme. In this paper, the numerical-value perturbational method suggested by Zhi Gao is used to develop an upwind and mixed FV scheme using any higher-order interpolation and second-order integration approximations, which is called perturbational finite volume (PFV) scheme. The PFV scheme uses the least nodes similar to the standard three-point schemes, namely, the number of the nodes needed equals to unity plus the face-number of the control volume. For instanc6, in the two-dimensional (2-D) case, only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized, respectively. The PFV scheme is applied on a number of 1-D problems, 2~Dand 3-D flow model equations. Comparing with other standard three-point schemes, The PFV scheme has much smaller numerical diffusion than the first-order upwind (IUW) scheme, its numerical accuracy are also higher than the second-order central scheme (2CS), the power-law scheme (PLS), the QUICK scheme and the second-order upwind(ZUW) scheme.

The thermo-visco-plastic instability analysis of saturated soil

A theoretical analysis of instability of saturated soil is presented considering the simple shearing of a heat conducting thermo-visco-plastic material. It is shown that the instability is mainly the consequence of thermal softening which overcomes the strain hardening and the other type of instability is controlled by strain softening. The effects of other factors such as permeability to the instability are discussed in this paper.

On the shear instability of saturated soil

In this paper. the dynamic instability of simple shear of saturated soil is discussed. The governing equations are obtained based on mixture theory in which the inertia effect and the compressibility of grains are considered. Perturbation method is used to analyze and it is shown that two types of instability may exist. One of them is dominated by pore-pressure-softening, while the other by strain-softening.

Effect of liquid bridge volume on the instability in small-Prandtl-number half zones

A perturbation method is used to examine the linear instability of thermocapillary convection in a liquid bridge of floating half-zone filled with a small Prandtl number fluid. The influence of liquid bridge volume on critical Marangoni number and flow features is analyzed. The neutral modes show that the instability is mainly caused by the bulk flow that is driven by the nonuniform thermocapillary forces acting on the free surface. The hydrodynamic instability is dominant in the case of small Prandtl number fluid and the first instability mode is a stationary bifurcation. The azimuthal wave number for the most dangerous mode depends on the liquid bridge volume, and is not always two as in the case of a cylindrical liquid bridge with aspect ratio near 0.6. Its value may be equal to unity when the liquid bridge is relatively slender.

The Pressure Transient Analysis Of Deformation Of Fractal Medium

The assumption of constant rock properties in pressure-transient analysis of stress-sensitive reservoirs can cause significant errors in the estimation of temporal and spatial variation of pressure. In this article, the pressure transient response of the fractal medium in stress-sensitive reservoirs was studied by using the self-similarity solution method and the regular perturbation method. The dependence of permeability on pore pressure makes the flow equation strongly nonlinear. The nonlinearities associated with the governing equation become weaker by using the logarithm transformation. The perturbation solutions for a constant pressure production and a constant rate production of a linear-source well were obtained by using the self-similarity solution method and the regular perturbation method in an infinitely large system, and inquire into the changing rule of pressure when the fractal and deformation parameters change. The plots of typical pressure curves were given in a few cases, and the results can be applied to well test analysis.

Rossby Waves With Linear Topography In Barotropic Fluids

Rossby waves are the most important waves in the atmosphere and ocean, and are parts of a large-scale system in fluid. The theory and observation show that, they satisfy quasi-geostrophic and quasi-static equilibrium approximations. In this paper, solitary Rossby waves induced by linear topography in barotropic fluids with a shear flow are studied. In order to simplify the problem, the topography is taken as a linear function of latitude variable y, then employing a weakly nonlinear method and a perturbation method, a KdV (Korteweg-de Vries) equation describing evolution of the amplitude of solitary Rossby waves induced by linear topography is derived. The results show that the variation of linear topography can induce the solitary Rossby waves in barotropic fluids with a shear flow, and extend the classical geophysical theory of fluid dynamics.

Crack propagation in the power-law nonlinear viscoelastic material

An analysis on crack creep propagation problem of power-law nonlinear viscoelastic materials is presented. The creep incompressilility assumption is used. To simulate fracture behavior of craze region, it is assumed that in the fracture process zone near the crack tip, the cohesive stress sigma(f) acts upon the crack surfaces and resists crack opening. Through a perturbation method, i. e., by superposing the Mode-I applied force onto a referential uniform stress state, which has a trivial solution and gives no effect on the solution of the original problem, the nonlinear viscoelastic problem is reduced to linear problem. For weak nonlinear materials, for which the power-law index n similar or equal to 1, the expressions of stress and crack surface displacement are derived. Then, the fracture process zone local energy criterion is proposed and based on which the formulas of cracking incubation time t

Transient effect on ideally plastic stationary crack-tip fields

This paper analyses the transient effect on ideally plastic stationary crack-tip fields under mode I plane strain conditions, when the inertial forces are not negligible. It is shown that the governing equation for such a problem can be expressed in formal simplicity when referred to a system of moving curvilinear coordinates, which is a generalization of the system defined by the slip-line field in quasi-static plasticity. A perturbation method of solving the equations is described and illustrated by application to problems of ideally plastic stationary crack-tip fields when the inertia forces are not negligible.

Transient effect on ideally plastic stationary crack-tip fields

This paper analyses the transient effect on ideally plastic stationary crack tip fields under mode I plane strain conditions, when the inertial forces are not negligible. It is shown that the governing equation for such a problem can be expressed in formal simplicity when referred to a system of moving curvilinear coordinates, which is a generalization of the system defined by the slip-line field in quasi-static plasticity. A perturbation method of solving the equations is described and illustrated by application to problems of ideally plastic stationary crack tip fields when the inertial forces are not negligible.

A model of an isolated magnetic-flux tube in a stratified atmosphere

A two-dimensional model of a magnetic flux tube confined in a gravitational stratified atmosphere is discussed. The magnetic field in the flux tube is assumed to be force-free. By using the approximation of large scale height, the problem of a free boundary with nonlinear conditions may be reduced to one involving a fixed boundary. The two-dimensional features are obtained by applying the perturbation method and adopting the Luest-Schlueter model as the basic state. The results show that the configuration of a flux tube confined in a gravitational stratified atmosphere is divergent, and the more twisted the magnetic field, the more divergent is the flux tube.

A passive scalar field convected by turbulence

Classical statistical mechanics is applied to the study of a passive scalar field convected by isotropic turbulence. A complete set of independent real parameters and dynamic equations are worked out to describe the dynamic state of the passive scalar field. The corresponding Liouville equation is solved by a perturbation method based upon a Langevin–Fokker–Planck model. The closure problem is treated by a variational approach reported in earlier papers. Two integral equations are obtained for two unknown functions: the scalar variance spectrum F(k) and the effective damping coefficient (k). The appearance of the energy spectrum of the velocity field in the two integral equations represents the coupling of the scalar field with the velocity field. As an application of the theory, the two integral equations are solved to derive the inertial-convective-range spectrum, obtaining F(k)=0.61 −1/3 k−5/3. Here is the dissipation rate of the scalar variance and is the dissipation rate of the energy of the velocity field. This theoretical value of the scalar Kolmogorov constant, 0.61, is in good agreement with experiments.

The interaction of two pairs of solitary waves in a two-fluid system

This paper deals with the interaction of solitary waves in a two-fluid system which consistsof two superimposed incompressible inviscid fluids with a free surface and a horizontal rigidbottom. Under the assumption of shallow water wave, we first derive the basic equationssuitable for the model considered, a generalized form of the Boussinesq equations, then usingthe PLK method and the reductive perturbation method, obtain the second-order approximatesolution for the head-on collision between two pairs of interface and surface solitary waves,and give their maximum amplitudes during the collision and the nonuniform phase shiftsafter the collision which lead to the distortion of the wave profiles.

Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin-Fokker-Planck (LFP) model. The dynamic damping coefficient eta of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for eta to replace the divergent response equation of Kraichnan's direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko = 1.2, which is compatible with the experimental data given by Gibson and Schwartz, (1963).

Dimensionless Analysis of the Simulation of Bucket Foundations under Dynamic Load

Firstly, the main factors are obtained by use of dimensionless analysis. Secondly, the time scaling factors in centrifuge modeling of bucket foundations under dynamic load are analyzed based on dimensionless analysis and control- ling equation. A simplified method for dealing with the conflict of scaling factors of the inertial and the percolation in sand foundation is presented. The presented method is that the material for experiments is not changed while the effects are modified by perturbation method. Thirdly, the characteristic time of liquefaction state and the characteristic scale of affected zone are analyzed.

Corrections to the paraxial approximation of an arbitrary free-propagation beam

A relatively simple transform from an arbitrary solution of the paraxial wave equation to the corresponding exact solution of the Helmholtz wave equation is derived in the condition that the evanescent waves are ignored and is used to study the corrections to the paraxial approximation of an arbitrary free-propagation beam. Specifically, the general lowest-order correction field is given in a very simple form and is proved to be exactly consistent with the perturbation method developed by Lax et nl. [Phys. Rev. A 11, 1365 (1975)]. Some special examples, such as the lowest-order correction to the paraxial approximation of a fundamental Gaussian beam whose waist plane has a parallel shin from the z = 0 plane, are presented. (C) 1998 Optical Society of America.