应用数学和力学

Periodicity of Convection Under Lateral Local Heating

NING Lizhong, ZHANG Di, NING Bibo, LI Kaiji, TIAN Weili, TENG Sufen

2020, 41(2): 125-133. doi: 10.21656/1000-0887.400091

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Abstract:
The periodicity of convection under lateral local heating with Prandtl number Pr=0.027 2 was studied through numerical simulation of hydrodynamic equations. The results show that, convection develops in the order of steady-state convection, single-local-period convection, double-local-period convection and quasi-period convection with the increase of Grashof number Gr. Convection is steady for Gr＜3.6×10³. In the range of 3.6×10³＜Gr＜6.78×10⁴,convection has a single local period; in the range of 6.78×10⁴＜Gr＜3.5×10⁶, convection has double local periods; for Gr＞3.5×10⁶,convection has a quasi period. In the steady convection case, the position of the convection roll corresponding to the heating zone on the wall does not change with time. In the single-local-period convection case, the core of the convection roll corresponding to the upper heating zone on the wall moves periodically with time. In the double-local-period convection case, the cores of the convection rolls corresponding to the 2 heating zones on the wall move periodically with time. In the quasi-period convection case, there are small convection rolls with quasi-period variations in the upper and lower parts of the convection loop corresponding to the lower unheated zone on the wall. In the range of the convection periods mentioned above, the convection period decreases with the increase of Grashof number Gr for given Prandtl number Pr.

Derivation and Application of Productivity Equations for High-Pressure Gas Reservoirs With Gas Acceleration Effects

JIANG Hailong, ZHU Peiwang, XU Donghua

2020, 41(2): 134-142. doi: 10.21656/1000-0887.400030

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Abstract:
Non-Darcy flow in a gas well cannot be expressed with traditional binomial equations or trinomial equations for ultra-deep high pressure gas reservoirs, which leads to large errors of open flow capacities for the neglect of gas acceleration effects. A new method was presented with gas acceleration effects. Under the assumption of high-pressure radial gas flow, through combination of the continuity equation, the Darcy-Forchheimer equation with acceleration and the isothermal state equation, the productivity equation was derived. The proposed equation is also able to replace traditional binomial equations, but has no analytical solution. It can be simplified by analogy, and the related coefficients can be solved by trial and error. The application of the proposed method in data process of ultra-deep high pressure gas well Xi35-1 in Sichuan Basin is effective, and the comparison with the real production data proves the precision of the calculated productivity. The proposed method avoids the disadvantages of the relative error increase with the differential pressure. The work enriches the productivity prediction methods, with accurate calculation of open flow capacities and reasonable determination of high pressure gas well productivities.

Wall Effects on Floating Characteristics of Bubbles in Shear-Thinning Fluids

PANG Mingjun, NIU Ruipeng, LU Minjie

2020, 41(2): 143-155. doi: 10.21656/1000-0887.400194

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Abstract:
The wall effects on the floating motion of bubbles in shear-thinning fluids were investigated with the numerical method. The interface between the bubble and the liquid phase was tracked with the volume of fluid (VOF) method. The Carreau model and the continuous surface tension model were used to calculate the rheological properties of the shear-thinning fluid and to compute the surface tension between gas and liquid phases, respectively. For different rheological indexes, the wall effects on the bubble shape, the liquid-phase flow field and the bubble terminal velocity were studied in detail. The results show that, the strong wall effect or the weak shear-shining effect can hinder the bubble deformation and the formation of wake vortices, and can reduce the bubble terminal velocity. Among all physical parameters, the bubble terminal velocity is most liable to the wall effect. For the strong wall effect and the strong shear-shining effect, the high-shear-rate region will occur near the wall, which results in a significant decrease in the apparent viscosity of the liquid phase near the wall.

Isogeometric in-Plane Vibration Analysis of Functionally Graded Triangular Plates

CHEN Mingfei, LIU Kunpeng, JIN Guoyong, ZHANG Yantao, YE Tiangui, LIU Zhigang

2020, 41(2): 156-170. doi: 10.21656/1000-0887.400171

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Abstract:
The in-plane vibration of the triangular plates of in-plane functionally graded (IFG) materials based on the plane stress theory was investigated by means of isogeometric analysis (IGA). The material of the triangular plate is homogenous in the thickness direction, but functionally graded along the in-plane direction. The geometry and displacement field of the considered plate were constructed with the non-uniform rational B-splines (NURBS) basis functions, then a seamless integration of the geometric design and the vibration characteristic analysis of the triangular plate was realized. The arbitrary boundary conditions of the triangular plate were obtained through adjustment of the stiffness of artificial springs introduced into the boundary of the triangular plate. The flexibility, high accuracy and quick convergency of the proposed method were verified through different refinements and results comparison. Finally, effects of boundary conditions, material properties and geometry parameters were investigated systematically. The vibration solutions of many kinds of triangular plates of in-plane functionally graded materials with elastic boundary conditions were given. The work provides a good reference for engineering application.

Theoretical Study on Multi-Parameter Inversion Non-Uniqueness Based on Elastic Displacements of Concrete Gravity Dams

HUANG Yaoying, YIN Xiaohui, LI Chunguang

2020, 41(2): 171-181. doi: 10.21656/1000-0887.400164

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Abstract:
Inversion analysis is an important part of the closed-loop system of on-site monitoring, inversion analysis, engineering practice test, forward analysis and prediction, and the back analysis problem in engineering practice mainly involves the parameter back analysis. Aimed at the uniqueness of multi-parameter inversion analysis on concrete gravity dams, the objective functions were established based on the theoretical solution of gravity dam displacements on homogeneous foundation under water pressure, and a convex programming problem was constructed with the objective function and the non-empty convex set. Then the positive definiteness of the Hessian matrix of the objective function was analyzed to verify the strict convexity of the objective function, thereby to identify whether the constructed convex programming problem has a unique global minimum. The analysis on different combinations of elastic constants of dams and rock foundations shows that, when the l₁ norm of the difference between the theoretical value and the measured value is used as the objective function, the Hessian matrix of the objective function cannot be guaranteed to be a positive definite matrix, that is, the multi-parameter elastic displacement inversion analysis of concrete gravity dams does not have a unique global minimum point.

Study on a 2D Mesoscopic Modeling Method for Concrete With Voids

CHEN Qingqing, ZHANG Yuhang, ZHANG Jie, WANG Zhiyong, WANG Zhihua

2020, 41(2): 182-194. doi: 10.21656/1000-0887.400058

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Abstract:
According to the meso-structure of concrete, a high-efficiency stepwise overlap checking method was proposed based on traditional 2D modeling. Voids were introduced directly into the models and different concrete meso-structures were generated. The method involves various models containing circular, elliptical or polygonal aggregates and circular or elliptical voids. Meanwhile, several modeling examples based on different area fractions and different shapes of aggregates and voids were given to prove the computation efficiency of the proposed algorithm. At last, the mechanical behaviors of standard concrete specimens under uniaxial compression were simulated. The effects of internal voids on the main paths of crack propagation, concrete failure modes and macroscopic mechanical performances were analyzed.

Existence of Time-Dependent Global Attractors for Beam Equations

SU Xiaohu, JIANG Jinping

2020, 41(2): 195-203. doi: 10.21656/1000-0887.400088

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Abstract:
The existence of time-dependent global attractors for beam equations was studied. Based on the existence theorem for time-dependent global attractors, if nonlinear term f met the critical increase condition, the asymptotic compactness of process family {U(t,τ)} corresponding to the beam equations was proved with the prior estimate method and the operator decomposition method when the coefficient was related to time t.In turn, the existence and regularity of the time-dependent global attractors for beam equations were obtained.

Optimization of Multimodal Transport Paths for Refrigerated Containers Under Carbon Emission Restriction

LIU Song, SHAO Yiming, PENG Yong

2020, 41(2): 204-215. doi: 10.21656/1000-0887.400159

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Abstract:
To cut the multimodal transportation cost of refrigerated containers under carbon emission restriction, save energy and reduce carbon emission accordingly, efficient path selection is essential. From the perspective of carbon emission limit, based on the points that in a multimodal transport network, the railway and waterway transport modes are restricted by scheduling timetables and the refrigerated containers involve cooling costs, cargo damages and carbon emissions, a path optimization model was established with the lowest total cost. All the costs of transportation, transshipment, refrigeration and cargo damage, dynamically changing with the scheduling timetable, were considered. In addition, a genetic algorithm was designed and applied to example analyses. The results show that, the proposed model and algorithm can quickly select the transportation plan with the least cost according to the requirements of the decision maker, thus providing a useful decision support.

Existence and Uniqueness of Solutions to Boundary Value Problems of a Class of Nonlinear 3rd-Order Differential Equations

YANG Jingbao, MO Jiaqi

2020, 41(2): 216-222. doi: 10.21656/1000-0887.400158

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Abstract:
The existence and uniqueness of solutions to the boundary value problem of a class of nonlinear 3rd-order differential equations were studied. Firstly, the results of the research on the boundary values of 3rd-order differential equations at home and abroad in recent years were combed. Then the boundary value problem of nonlinear 3rd-order differential equations with nonlinear boundary value conditions was put forth, and the solution to the related linear problem was explored. Finally, the Banach fixed point theorem was used to prove that the proposed boundary value problem has a unique solution. An example illustrates the applicability of the main results.

Front-Like Entire Solutions to Lattice Periodic Dynamic Systems With Delays and Global Interaction

XUE Xue

2020, 41(2): 223-234. doi: 10.21656/1000-0887.400170

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Abstract:
The front-like entire solutions to lattice periodic dynamic systems with delays and global interaction were investigated. Through establishment of appropriate comparison theorems, some front-like entire solutions were constructed out of mixture of the traveling fronts with different directions of propagation and spatially periodic solutions connecting unstable equilibrium and stable equilibrium. Some properties of these entire solutions were also discussed. The front-like entire solutions, exhibiting new characteristic behaviors in the front dynamics, are different from the traveling fronts.

2020 Vol. 41, No. 2