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2008 Vol. 40, No. 5

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Numerical investigation into the mechanism of under-expanded supersonic jet instability
The numerical investigation into the instability of theunder-expanded supersonic jets is presented in this paper by solving theaxisymmetric N-S equations with the NND finite difference scheme. Numericalresults demonstrated that there are three kinds of the instability for theunder-expanded supersonic jets according to their shock cell structurepatterns: (1) the regular reflection shock cell with a single shear layer,(2) the Mach reflection shock cell with two shear layers and (3) the curvedMach stem with a relatively high expansion ratio. Moreover, for theunder-expanded supersonic jets, the repetitively-generated quasi-steadyshock cells in the jets indicate their stable development and thedisappearance of these shock cells implies that the jets are about to fallinto instable states.
2008, 40(5): 577-584. doi: 10.6052/0459-1879-2008-5-2006-651
One dimensional shallow water equations with porosity and their numerical discretization scheme
Zhili Wang, Yongjun Lu, Yanfen Geng
In this paper, the introduction of porosity into theone-dimensional shallow water equations is investigated to lead to modifiedexpressions for the fluxes and source terms subjected to the effects ofbuildings and other structures. An extra porosity source term appears in themomentum equation. The numerical discretization scheme for shallow waterequations with porosity is presented using finite volume method and Roe-typeapproximate Riemann solver. The source terms of the bed slope and theporosity are both decomposed in the characteristic direction to make thescheme exactly satisfy the conservative property. Several cases have beenbrought, such as the dam break with variable porosity, the steady flowthrough continue porosity and the damming heights in different pier width,to show that the scheme can be used to simulate the influence of buildingsand other structures in the channel with harmonious, robust andhigh-resolution characteristics.
2008, 40(5): 585-592. doi: 10.6052/0459-1879-2008-5-2007-445
Gust response analysis using cfd-based reduced order models
Gust response analysis plays an important role in largeaircraft design. Most of existing aerodynamic models for gust responseanalysis are linear, not suitable for transonic nonlinear aeroelasticanalysis. In this paper, based on CFD technique and system identificationmethod, a CFD-based ROM (reduced order model) is firstly constructed instate space. Gust response analysis is then investigated by couplingstructural equations with CFD-based ROM. The case of a square-edged gustresponse for a typical wing is presented to obtain the responses ofgeneralized coordinates and root bending moment (RBM) and to validate thecorrectness and precision of the CFD-based ROM method, compared well withthe direct approach of a coupled CFD/CSD methodology
2008, 40(5): 593-598. doi: 10.6052/0459-1879-2008-5-2007-357
Numerical simulation and experimental investigation on steady flow foam in porous media
PANG Zhan-Xi
Porous media is firstly simplified as a bunch ofconstricted capillary tubes related to grain diameter, particle arrangement,pore-throat scaling ratio and saturation of immobile water. Based on thedeformation characteristics of a single bubble in throat and pore of porousmedia, pressure distribution of fluid phase is derived from the conservationof momentum in a single pore unit, to obtain the equation for pressuredistribution in porous media. Moreover, the experiments on flow foam inU-shape sandpack are investigated. The results demonstrate the lineardecreasing pressure distribution of steady foams in porous media and theinfluencing factors, i.e., pore-throat structure, flowing rate and qualityof foams, interfacial tension of gas and liquid, bubble size in pore media.Furthermore, the results show further that pore-throat structure and foamquality have the most influence on the blocking ability of foams in porousmedia.
2008, 40(5): 599-604. doi: 10.6052/0459-1879-2008-5-2007-141
Numerical study of ionized hypersonic air flow with radiation
A numerical method for an ionized hypersonic air flowwith radiation under thermo-chemical nonequilibrium condition has beendeveloped with multi-species N-S equations based on unstructured mesh. Inthe algorithm, there are 11 chemical species (O$_{2}$, N$_{2}$, O, N,NO, NO$^{+}$, N$^+$, O$^+$, N$_2^+$, O$_2^+$, e$^{ - }$), 20 chemicalreactions, and two temperature models subjectedto thermal nonequilibrium effect. Radiation transfer equation (RTE) wassolved by finite volume method. In particular, the Delta, Epsilon, Beta primeand Gamma prime bands of NO were introduced in this paper. The numericalresults of MUSES-C were shown and compared well with the reference data. Theinfluence of radiation was also discussed.
2008, 40(5): 605-610. doi: 10.6052/0459-1879-2008-5-2007-057
Inverse estimation of heat source term in three-dimensional transient heat conduction problem
Weiqi Qian
Inverse estimation of heat source term inthree-dimensional heat conduction problem from surface temperaturemeasurements is a typical Inverse Heat Conduction Problem (IHCP). In thispaper, based on the three-dimensional (3D) numerical simulation of transientheat conduction problem with the Finite Volume Method (FVM), the 3Dtransient IHCP is generally converted into an optimization problem toinversely estimate the heat source with Adjoint Equation Method (AEM) andConjugate Gradient Method (CGM). A typical case has been shown that the twoestimation methods are feasible and not sensitive to the measurement noise.Moreover, the selection of stopping criterion for the inverse estimationalgorithm is further investigated. The results demonstrate that the closeragreement between the estimated and exact heat sources hasn't made with thedecreasing value of the objective function due to the ill-posedness of theIHCP, which can be overcome with the proper stopping criterion to simulatethe regularization effect in the IHCP.
2008, 40(5): 611-618. doi: 10.6052/0459-1879-2008-5-2007-256
A holistic thermomechanical model of skin thermal pain
Skin thermal pain is one of the most common problems forhuman in everyday life and in thermal therapies. However, its underlyingphysical mechanisms are not clearly understood, and there are few attemptsto model it. In this paper, a holistic mathematical model for quantifyingskin thermal pain sensation is developed. The model is composed of threeinterconnected parts: peripheral modulation of noxious stimuli,which converts the energy from a noxious thermal stimulus intoelectrical energy via nerve impulses; transmission, which transports theseneural signals from the site of transduction in the skin to the spinal cordand brain; and modulation and perception in the spinal cord and brain. Withthe holistic model, a direct relationship is built between the level ofthermal pain sensation and the character of noxious stimuli.
2008, 40(5): 619-628. doi: 10.6052/0459-1879-2008-5-2007-240
Invariant tensor function representations of constitutive equations for the isotropic and rate independent materials
This paper combines the internal variable theory and thetensor function representation theory to establish the constitutiveequations of the deformation theory and the increment theory for theisotropic and rate independent materials. In the equations, there are threecomplete and irreducible base tensors, that is, the stress tensor of thezero order, the first order and the second order power, to show that theprincipal axes of plastic strain and its increment are coincident with thoseof the stresses. With the orthogonalization of the base tensors, thegeometrical explanation of the constitutive equations is obtained in theprincipal stress space. Furthermore, the coefficients in the constitutiveequations of deformation theory (or increment theory) can be derived withthree invariants of stresses and plastic strain ( or plastic strainincrements). Meanwhile, the present constitutive equations may reduce toclassical deformation theory (or plastic potential theory), and beconsistent to the singular yield surface theory.
2008, 40(5): 629-635. doi: 10.6052/0459-1879-2008-5-2007-155
Simplified theory and analytical solutions for functionally graded piezoelectric circular plate
Yaochen Li, Feng Qi, Zheng Zhong
Simplified theory and analytical solutions for afunctionally graded piezoelectric circular plate subjected to axisymmetricloading was presented with some assumptions, such as Kirchhoff assumption,Reissner-Mindlin assumption, and the assumption that material propertiesvary exponentially across the thickness of the plate. We derived thesolution of the rotation angle of the normal line of the neutral layer andthe solution of the electric potential in the plate expressed byFourier-Bessel series, in the cases that the edges of the plate was fixed orsimply supported and grounded. The presented solutions were obviously simplecompared with some exact analytical solutions and easy to perform thenumerical analysis. The numerical results for the edge-fixed/grounded platewere applied to validate the simplified theory presented in the paper.
2008, 40(5): 636-645. doi: 10.6052/0459-1879-2008-5-2007-145
Topology optimization of continuum structures with materials exhibiting different tensile and compressive properties
Hongwu Zhang
The effects of the difference between the tensileand compressive properties (TCPs) of materials on the final topologies ofstructures are investigated with floating reference intervals in theoptimization case of a bone growing process based on Wolff's law. There aretwo aspects in the difference on TCPs of basal materials to have equivalenteffects on the structural topology optimization: one is the differencebetween their tensile and compressive Young's moduli, , the other is thevariation of the intervals of their references strains in bone growingprocess under the tensile or compressive state . The numerical cases verifythis equivalence and further reveal a more reasonable structure topologysubjected to the differences on TCPs of the materials.
2008, 40(5): 646-653. doi: 10.6052/0459-1879-2008-5-2007-455
Reliability sensitivity analysis based on subset simulation and importance sampling
Shufang Song, Zhenzhou Lu
Reliability sensitivity algorithm is presented on thebasis of subset simulation and importance sampling due to the small failureprobability highly experienced in engineering. Firstly, a small failureprobability is expressed as a product of larger conditional failureprobabilities of some intermediate failure events. Secondly, the largerconditional failure probabilities can be estimated efficiently byconstructing the importance sampling density functions of the intermediatefailure events. Thirdly, the reliability sensitivity is transformed into thepartial derivatives of conditional failure probabilities with respect to thedistribution parameters of the basic variables in the paper. The estimationof the reliability sensitivity and its variance are then derived for thepresented algorithm. The results from several cases show that the presentmethod is efficient, precise and applicable to the structural system withsingle and multiple failure modes.
2008, 40(5): 654-662. doi: 10.6052/0459-1879-2008-5-2007-274
Wave components of acoustic logs in transversely isotropic porous formations
He Xiao, Hengshan HU
The acoutic field excited by a monopole point-source wasnumerically investigated with a model of a transversely isotropicfluid-saturated poroelastic formation. The wavefields inside and outside theborehole are formulated in the frequency-wavenumber domain. All the poles ofthe acoustic function are obtained to correspond to several modes of guidewaves: the normal modes (including Stoneley wave and pseudo-Rayleigh wave)and two leaky modes. The numerical results showed that the contribution ofthe lowest order leaky mode can not be ignored since its imaginary part hasthe same magnitude order as that of the normal mode. Moreover, the velocityof pseudo-Rayleigh wave at the low-frequency limit is not always equal tothe shear wave velocity in transversely isotropic formations. Meanwhile, theexcitation spectra and waveforms in the time-space domain are acquired forStoneley wave, pseudo-Rayleigh wave and two-leaky-modes wave, contributed bythe poles and two branchcuts (the quasi-shear and the quasi-fast-compressional branchcuts). The results showed that the contribution ofleaky-mode poles can not be omitted in the full waveforms.
2008, 40(5): 663-671. doi: 10.6052/0459-1879-2008-5-2007-311
A coupled elasto-plastic microplane damage model for jointed rock masses at great depth
Rockwall excavated at great depth often experiences largeductile deformation and may be failure due to high compressive in-situstresses. Such phenomena are closely related to anisotropic distribution ofjoints in rock masses and mechanical response of the rock block and thejoint. To take into account the shear resistance of joint surface andanisotropic inelastic deformation related to secondary joints formation, acoupled elasto-plastic microplane damage model for jointed rock masses isdeveloped in the framework of microplane model and damage mechanics. Eachmicroplane of a jointed rock mass is regarded as a binary medium, which iscomposed of rock and joint. Joint connectivity on the microplane isintroduced as the damage variable to characterize area damage of the rockmass on that orientation due to presence of the joints. By adoptingdifferent yield function and damage evolution law for the microplane undertensile and compressive normal stress respectively, the coupled mechanismsof inelastic deformation and damage evolution are modeled. Rate formconstitutive relation of stresses and strains on microplanes is developedbased on classical plastic theory. According to the kinematic constraintcondition of microplane model, macroscopic constitutive relation is obtainedthrough directional integration along all microplanes. The presentelasto-plastic microplane damage model is implemented to the commercial FEMsoftware MARC through a user subroutine. Simulations with respect to theuniaxial tension and compression of jointed rock masses and problem ofwellbore stability under inner mud pressure are presented respectively. Ithas demonstrated that the coupled effect between anisotropic inelasticdeformation and damage evolution of jointed rock masses can be characterizedefficiently with the model.
2008, 40(5): 672-683. doi: 10.6052/0459-1879-2008-5-2007-089
Recursive method for modeling flexible multibody systems with non-ideal constraints
Qi Zhaohui
With the recursive kinematics relationship between the twobodies interconnected with a joint, it is showed that the interactionbetween the outermost body and the inboard body in a tree-type multibodysystem can be measured by the transmission of the inertia mass and forces.The non-ideal constraint forces and the constraint forces at the cut-jointsin a closed-loop multibody system can be viewed as external applied forces,therefore, any mutibody system corresponds to an equivalent tree-typesystem. Moreover, the general accelerations of the system appearing in theconstraint equations can be systematically replaced by the Lagrangemultipliers which are related to the constraint forces. Based on the aboveideas, a recursive method is presented, by which, any closed-loop flexiblemultibody system with non-ideal constraints can be dynamically reduced to anequivalent system with only one body in a recursive manner. Consequently,the inverse of the mass matrix of multibody systems is not involved in thecalculations of the general accelerations, and the number of equations tosolve the non-ideal constraint forces can be reduced to minimum. An exampleis analyzed to illustrate and validate the presented method in details.
2008, 40(5): 684-694. doi: 10.6052/0459-1879-2008-5-2007-577
Dynamic propagation of asymmetrical mode III interface crack
Dynamic propagation of asymmetrical mode III interfacecrack were investigated with the theory of complex functions. It can betranslated into a Riemann-Hilbert problem by means of self-similarfunctions, and the universal representations of analytical solution wereobtained when there is $Pt / x$ or $Px^3 / t^2$ load at the origin of thecrack coordinates. The closed-form solutions can also be simply obtainedwith Muskhelishvili's theory.
2008, 40(5): 695-700. doi: 10.6052/0459-1879-2008-5-2007-104
Fracture analysis of pzt-4 compact tension specimen
Haijun Li, Feng Liu, T.C. Wang
Based on a complex potential theory of linearpiezoelectric materials and series expansion of the complex potential, ageneral method is proposed for solving the crack problem of a finitepiezoelectric plate. By this method, fracture parameters such as the stressintensity factor, the electric displacement intensity factor and energyrelease rate et al. can be easily evaluated under the impermeable boundarycondition. Numerical results of fracture parameters demonstrate that theproposed method gives satisfactory results compared with the existingsolutions given by Woo and Wang for a central crack in a finitepiezoelectric plate. This method is identified to be an accurate andefficient method. An experiment with a PZT-4 compact tension specimen hasbeen modeled under the impermeable boundary condition by the present method,and the critical values of fracture parameters are greatly varied and notthe material constants. Therefore, these fracture parameters are not used asthe single-parameter fracture criterion for the piezoelectric material.Then, in terms of the PZT-4 compact tension specimen, the influence ofpermeability of the crack has been studied based on the results of finiteelement calculations. The finite element model of the PZT-4 compact tensionspecimen with an actual crack profile is analyzed for two different electricpermeability of the medium in the crack cavity corresponding to animpermeable medium and silicone oil. The largest applied negative electricfield and the corresponding mechanical load for fracture are considered.Finally, the crack profile in the PZT-4 compact tension specimen can beregarded to be composed of a notch and an ideal crack (so-called microcrack)in front of the notch. The model of this assumed crack profile has beenanalyzed by the finite element method.
2008, 40(5): 701-706. doi: 10.6052/0459-1879-2008-5-2007-625
Damage failure analysis for composite bolted joints subjected to static loading
Danyong Wang
A progressive damage method for predicting the staticstrength and damage failure of mechanically fastened joints infibre-reinforced plastics is established in the paper based on the classicallaminate theory, the three-dimensional Hashin-type failure criteria , thefailure criteria of the structure and material property degradation rulesrelated to the correlation among four basic damage mechanisms. Damageaccumulation in composite bolted joints was investigated and compared withthe X-radiographs of fractured specimens under multistep loading. Numericalresults have been demonstrated that the method established in the paper iscapable of predicting the ultimate tensile strength and the failure mode ofjoints and can be applied to simulate the whole process of failureinitiation, propagation and catastrophic failure of the structure. We findan excellent agreement between the results from the present scheme and theexperimental data.
2008, 40(5): 707-715. doi: 10.6052/0459-1879-2008-5-2007-207
Frequency sensitivity analysis for dynamic structural systems reliability
A method of frequency reliability sensitivity for dynamicstructural systems is presented in the paper based on the frequencyreliability theory. With the criterion that the difference in value betweenthe natural frequency and driving frequency is controlled, the reliabilitymode and the safety probability of the random systems are then defined asseries mode to perform the frequency reliability sensitivity analysis. Acase of resonant problem in the dynamic structural systems is carried out byusing the random perturbation technique, the reliability theory andsensitivity approach. Numerical results have demonstrated that the proposedmethod is efficient and accurate. This scheme is helpful for the design andoptimization of dynamic structural systems.
2008, 40(5): 716-720. doi: 10.6052/0459-1879-2008-5-2008-060