应用数学和力学

2016 Vol. 37, No. 6

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The Integral Form Fluid Momentum Theorem on Time-Varying Systems and Its Application to Aerodynamic Force Coefficient Analysis

LIN Xian-wu, LAN Wei-yao, LI Zhi-bin, LI He

2016, 37(6): 551-566. doi: 10.3879/j.issn.1000-0887.2016.06.001

Abstract(1589) PDF(1008)

Abstract:
To tackle the calculation problem on steady and unsteady hydrodynamic force coefficients of a moving body in viscous incompressible flow, a method for calculating hydrodynamic force coefficients in viscous flow was proposed based on the quasi-equilibrium hypothesis and the vorticity aerodynamics. Firstly, the concept of time-varying flow systems was defined, and its relationship with the space volume was clarified. Then, the momentum transport equation and the fluid momentum theorem for time-varying flow systems were developed respectively, so as to provide a basis for the further discussion. Secondly, the fluid momentum theorem was applied to a flow system enclosed in the boundary composed of the body surface and the outer fixed surface with an infinite radius, and the fluid dynamic force was related to the change of the total fluid momentum. Thirdly, the quasi-equilibrium hypothesis was proposed and the total fluid momentum was expressed as a function of the body velocity and angular velocity. At last, this function was determined with the CFD technology and the method for calculating the fluid dynamic force coefficients in viscous flow was established. The study also show that the variation of the flow system should be considered during the derivation of the fluid momentum, and consequently an additional steady fluid dynamic force would come forth. This additional steady force can be proved to be zero for the body in linear uniform motion in the ideal flow, which is in accordance with d’Alembert’s paradox and Lamb’s result. However, in the case of viscous flow, this additional steady force is not necessarily to be zero, which is in accordance with the experimental results.

A Modified Numerical Method for Arbitrary Mach Number Flows Based on the Preconditioning Technique

WANG Qiang, JIANG Peng

2016, 37(6): 567-573. doi: 10.3879/j.issn.1000-0887.2016.06.002

Abstract(1251) PDF(673)

Abstract:
The preconditioning technique was applied to modify the existent LU-SGS solver for compressible flows, which was unsuitable for the prediction of low-speed incompressible flows. The preconditioning modification included treatment of the eigen system of the governing equations, improvement of the implicit solving method and discretization of the convection terms with the low-diffusion difference scheme and the AUSM+-up scheme. The modified solver was applied to numerical simulations of inviscid bump flows, lid-driven square cavity viscous flows and Laval nozzle flows. The comparison between the present results and those in the previous literatures proves the feasibility of the preconditioning-modified numerical method in the simulation of arbitrary Mach number flows, including the low-speed, the subsonic, the transonic and the supersonic invicid or viscous flows.

Numerical Simulation of the Micro-PullingDown Method for YAG Crystal Growth

QU Jing-jing, ZENG Zhong, QIAO Long, DING Yu-chong, FU Chang-lu

2016, 37(6): 574-583. doi: 10.3879/j.issn.1000-0887.2016.06.003

Abstract(1370) PDF(599)

Abstract:
The global numerical simulation was performed for the YAG crystal growth with the micro-pulling-down method. The induction heating, the convection of both gas and melt and the heat transfer of solid/melt/gas were solved simultaneously. In the melt zone, buoyancy convection and thermocapillary flow were considered. In order to uniformly discretize the controlling equations with the finite volume method, the electromagnetic field was modelled with the complex function method, and the computation of the electromagnetic field was verified with the results from the stream function method. Both the temperature and flow fields in the global furnace (including gas and melt) were investigated. As for the low temperature gradient at the solid-liquid interface, the effects of the afterheater was parametrically investigated. This work is useful for the optimal design of crystal growth furnaces.

Study on the Super Viscoelastic Constitutive Theory for Saturated Porous Media

HU Ya-yuan

2016, 37(6): 584-598. doi: 10.3879/j.issn.1000-0887.2016.06.004

Abstract(1128) PDF(713)

Abstract:
In order to establish the super viscoelastic constitutive framework for saturated porous media in view of the reversible and irreversible deformations of solids, porous solids and fluids, an energy balance equation of which all terms were in the thermodynamically power-conjugated form, was built for saturated porous media according to the principle of homogeneous mixture response, with the porous solid selected as the reference configuration and the effective stress tensor, the material’s real hydrostatic stress and the fluid’s real pore pressure chosen as the state variables. The entropy flux and entropy production of the saturated porous medium were derived based on the decomposing principle of entropy in the non-equilibrium thermodynamics. The work shows that the super elastoplastic constitutive theory is only a special case of the proposed theory. The deformation rate of a porous solid is composed of 2 parts: the solid-phase interstice and the material deformation, of which the former is power-conjugated with the Terzaghi effective stress tensor and the latter with the material’s real hydrostatic stress. The free energy of a saturated porous medium consists of 2 parts: the porous solid-phase part and the fluid-phase part. If the solid-phase interstice is decoupled from the material deformation, the free energy of the solid can be further divided into 2 parts: the material strain and the interstitial change. The Skempton-type effective stress is proved not to be a basic state variable for saturated porous media.

A Fractional Structural Derivative Model for Ultra-Slow Diffusion

CHEN Wen, HEI Xin-dong, LIANG Ying-jie

2016, 37(6): 599-608. doi: 10.3879/j.issn.1000-0887.2016.06.005

Abstract(1472) PDF(1048)

Abstract:
The ultra-slow diffusion is even more slow than the power-law sub-diffusion and is widely observed in a variety of natural and engineering fields. The ultra-slow diffusion cannot be well described with the traditional anomalous diffusion models. The Sinai’s law of diffusion depicts a special type of ultra-slow diffusion which is characterized by a logarithmic stochastic relationship. In this study, the Sinai diffusion was extended to a general ultra-slow diffusion. In addition, in the proposed model the initial parameters were introduced to remedy the perplexing issue that the Sinai diffusion was not feasible around the initial period of the ultra-slow diffusion. As a generalized fractional-order derivative, the concept of the fractional structural derivative was also presented to establish the partial differential equation governing the ultra-slow diffusion.

Reliability Analysis Based on Bregman Distances

DU Xiu-yun, XUE Qi-wen, LIU Xu-dong

2016, 37(6): 609-616. doi: 10.3879/j.issn.1000-0887.2016.06.006

Abstract(989) PDF(740)

Abstract:
A reliability analysis model based on the homotopy algorithm was established to address probabilistic reliability problems of uncertain structures through introduction of Bregman distances. By means of the limit state equations, solution of the reliability index was transformed to a nonlinear constrained optimization problem. According to the homotopy theory and the Bregman distances, the system of homotopy equations was constructed and solved with the path-tracking algorithm. The reliability calculations for different types of functions and different degrees of nonlinear problems were discussed via numerical examples, and the results were compared with those out of previous methods. The results show that the proposed analysis model solves the probabilistic reliability problems of uncertain structures with high efficiency and good accuracy.

The Radial Basis Function Approximation Method for Solving Bratu-Type Equations

HONG Wen-qiang, XU Ji-qing, XU Xi-bin, ZHANG Chun, ZHOU Shi-liang

2016, 37(6): 617-625. doi: 10.3879/j.issn.1000-0887.2016.06.007

Abstract(1170) PDF(820)

Abstract:
Based on the powerful approximation capability of the radial basis function for almost all kinds of functions, and with reference to the interpolation method for elasto-plastic mechanics, the radial basis function expression of the interpolation combining displacement, velocity and acceleration was put forward. Then the MATLAB software was used for computer programming to successfully solve the strongly nonlinear Bratu-type equation, with the corresponding relative errors given and discussed. The analysis of several typical examples was conducted, where the present calculated results were compared with some of the existing numerical results as well as the exact solutions. The comparison shows the feasibility and high accuracy of the present method, which makes a new way of solving the strongly nonlinear Bratu-type equations.

The 2D Base Force Element Method Based on the Potential Energy Principle

SHAN Yan-yan, REN Cong, HUANG Si-bai, BAI Ya-qiong

2016, 37(6): 626-632. doi: 10.3879/j.issn.1000-0887.2016.06.008

Abstract(1066) PDF(537)

Abstract:
Based on the base force concept proposed by Gao Y C, the quadrangular element model of the base force element method (BFEM) for 2D linear elastic problems was proposed. Several typical examples of numerical calculation were given in view of the effect of the element’s length-to-width ratio. The BFEM results were compared with the analytic solution and the conventional triangular finite element solution as well as the isoperimetric quadrangular element (Q4 model) solution; in turn the correctness and the computing performance of the BFEM based on the potential energy principle were verified. It is shown that the BFEM gives results in good agreement with the analytic solution and has higher accuracy than the conventional finite element methods, but is not sensitive to the element’s length-to-width ratio. So the BFEM is expected to have wide application prospects.

Dynamic Responses of Nonlinear Vibro-Impact Systems Under Narrow-Band Random Parametric Excitation

HUANG Dong-mei, XU Wei

2016, 37(6): 633-643. doi: 10.3879/j.issn.1000-0887.2016.06.009

Abstract(1068) PDF(658)

Abstract:
The stochastic responses of nonlinear vibro-impact systems under random parametric excitation were investigated. Based on the Krylov-Bogoliubov averaging method, the largest Lyapunov exponent deciding the almost sure stability of the trivial solution was derived. Results show that the characteristics of the largest Lyapunov exponent of the vibro-impact system was different from that of the system without impact. Meanwhile, the backbone curve and the critical equation for the unstable region were also derived in the deterministic case. Then, the 1st- and 2nd-order non-trivial steady-state moments of the system were discussed, and the frequency island phenomenon was also found. Finally, the phenomenon of stochastic jump was analyzed via the finite difference method. The basic jump phenomena indicate that, under the conditions of system parameters within a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas within a larger bandwidth, the most probable motion is around the trivial one of the amplitude response curve.

Generalized Solution to the Thermoelastic Problem of Finite-Thickness Hollow Cylinders Subjected to Thermal Shock

WANG Ying-ze, WANG Qian, LIU Dong, GU Li-ping

2016, 37(6): 644-654. doi: 10.3879/j.issn.1000-0887.2016.06.010

Abstract(1355) PDF(587)

Abstract:
The thermoelastic problem of the bounded axisymmetric structures subjected to transient thermal shock was studied. The thermoelastic model for a thick hollow cylinder was built in the context of the generalized Lord-Shulman theory (L-S theory) for thermoelasticity. The Laplace transform technique and the asymptotic properties of the Bessel functions were introduced to derive the analytical solutions of displacement, temperature and stress, which were induced by a sudden temperature rise. The propagation, reflection and superposition of 2 waves, namely the thermoelastic wave and the thermal wave, respectively, were revealed through these solutions, and the jumps generated in the positions of wavefronts were also obtained clearly.

The Adaptive Gauss Pseudospectral Method for the Optimal Control of Spacecraft Solar Array Deployment

DONG Xue-yang, GE Xin-sheng

2016, 37(6): 655-664. doi: 10.3879/j.issn.1000-0887.2016.06.011

Abstract(1423) PDF(666)

Abstract:
The adaptive Gauss pseudospectral method (GMP) for the optimal control of spacecraft solar array deployment was formulated. In a practical solar array deploying process, under the constraint conditions of limited control, limited states and limited initial and final attitudes, etc., the spacecraft attitude control problem was treated as an optimal control problem where the above constraints and boundary conditions shall be met and the optimal performance index function be realized at the same time. With the adaptive GMP, the solution to the solar array deployment control problem was obtained with satisfactory accuracy and highly promoted efficiency, through the judgement on the needs of time interval subdivision and time node number increase. Finally, the solar array deploying process was numerically simulated and the optimal attitude motion trajectory as well as the control rule was acquired. The results demonstrate the effectiveness of this method in the attitude control problems.