应用数学和力学

2007 Vol. 28, No. 5

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Stability Analysis and Transition Prediction of Hypersonic Boundary Layer Over a Blunt Cone With Small Nose Bluntness at Zero Angle of Attack

SU Cai-hong, ZHOU Heng

2007, 28(5): 505-513.

Abstract(2866) PDF(921)

Abstract:
Stability and transition prediction of hypersonic boundary layer on a blunt cone with small nose bluntness at zero angle of attack had been investigated.The nose radius of the cone is 0.5 mm;the cone half-angle is 5 degree,and the Mach number of the oncoming flow is 6.The base flow of the blunt cone was obtained by direct numerical simulation.The linear stability theory was applied for the analysis of the first mode and the second mode unstable waves under both isothermal and adiabatic wall condition,and e-N method was used for the prediction of transition location.The N factor was tentatively taken as 10,as no experimentally confirmed value was available.It is found that the wall temperature condition has a great effect on the transition location.For adiabatic wall,transition would take place more rearward than those for isothermal wall.And despite that for high Mach number flows,the maximum amplification rate of the second mode wave is far bigger than the maximum amplification rate of the first mode wave.The transition location of the boundary layer with adiabatic wall is controlled by the growth of first mode unstable waves.The methods employed are expected to be also applicable to the transition prediction for the three dimensional boundary layers on cones with angle of attack.

Nonlinear Vibration of Corrugated Shallow Shells Under Uniform Load

YUAN Hong, LIU Ren-huai

2007, 28(5): 514-520.

Abstract(2347) PDF(704)

Abstract:
Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution,the nonlinear forced vibration of a corrugated shallow shell under uniform load had been investigated.The nonlinear partial differential equations of shallow shell were reduced to the nonlinear integral-differential equations by using the method of Green's function.To solve the integral-differential equations,expansion method was used to obtain Green's function.Then the integral-differential equations were reduced to the form with degenerate core by expanding Green's function as series of characteristic function.Therefore,the integral-differential equations became nonlinear ordinary differential equations with regard to time.The amplitude-frequency response under harmonic force was obtained by considering single mode vibration.As a numerical example,forced vibration phenomena of shallow spherical shells with sinusoidal corrugation were studied.The obtained solutions are available for reference to design of corrugated shells.

On the Two-Dimensional Large-Scale Primitive Equations in Oceanic Dynamics (Ⅰ)

HUANG Dai-wen, GUO Bo-ling

2007, 28(5): 521-531.

Abstract(2392) PDF(900)

Abstract:
The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered.It was assumed that the depth of the ocean is a positive constant.First,if the initial data are square integrable,then,by Fadeo-Galerkin method,the existence of the global weak sohctions for the problem was obtained.Second,if the initial data and their vertical derivatives are all square integrable,then by Faedo-Galerkin method and anisotropit inequahites,the existerce and uniqueness of the global weakly strong solution for the above initial boundary problem was obtained.

On the Two-Dimensional Large-Scale Primitive Equations in Oceanic Dynamics (Ⅱ)

HUANG Dai-wen, GUO Bo-ling

2007, 28(5): 532-538.

Abstract(1957) PDF(791)

Abstract:
The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially.Here the depth of the ocean is positive but not always a constant.By Faedo-Galerkin method and anisotropic inequalities,the existence,uniqueness of the global weakly strong solution and global strong solution for the probem were obtained.Moreover,by study the asymptotic behavior of solutions for the ablve problem,that the energy is exponential decay in time was proved.

Optimal Control of Nonholonomic Motion Planning for a Free-Falling Cat

GE Xin-Sheng, CHEN Li-qun

2007, 28(5): 539-545.

Abstract(3048) PDF(717)

Abstract:
The nonholonomic motion planning of a free-falling cat is investigated.Nonholonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws.When the total angular momentum is zero,the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum.The control of system can be converted to the problem of nonholonomic motion planning for a freefalling cat.Based on Ritz approximation theory,the Gauss-Newton method for motion planning by a falling cat is proposed.The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.

Generalized Synchronization of Discrete Systems

MA Zhong-jun, LIU Zeng-rong, ZHANG Gang

2007, 28(5): 546-550.

Abstract(2422) PDF(551)

Abstract:
Generalized synchr onization of two discrete systems is discussed.By constructing appr opriately nonlinear coupling terms,some sufficient conditions for determining the generalized synchronization between the drive and response systems were derived.In a positive invariant and bounded set,many chaotic maps satisfy the sufficient conditions.The effectiveness of the sufficient conditions is illustrated by three examples.

Dynamic Behavior of Two Collinear Interface Cracks Between Two Dissimilar Functionally Graded Piezoelectric/Piezomagnetic Material Strips

ZHANG Pei-wei, ZHOU Zhen-gong, WANG Biao

2007, 28(5): 551-560.

Abstract(2457) PDF(692)

Abstract:
The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated.By using the Fourier transform,the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is the jump of displacement across crack surfaces.These equations are solved using the Schmidt method.Numerical examples are provided to show the effect of the functionally graded parameter,the circular frequency of the incident waves and the thickness of the strip upon stress,electric displacement and magnetic flux intensity factors of cracks.

Propagation of Plane Waves in Thermoelastic Cubic Crystal Material With Two Relaxation Times

Rajneesh Kumar, Manjeet Singh

2007, 28(5): 561-574.

Abstract(2643) PDF(578)

Abstract:
A problem concerning with the reflection and refraction of thermoelastic plane waves at an imperfect interface between two generalized thermally conducting cubic crystal solid half-spaces of different elastic and thermal properties with two relaxation times has been investigated.The generalized thermoelastic theory with two relaxation times developed by Green and Lindsay has been used to study the problem in 1972.The expressions for the reflection and refraction coefficients which are the ratios of the amplitudes of reflected and refracted waves to the amplitude of incident waves were obtained for an imperfect boundary and deduced for normal stiffness,transverse stiffness,thermal contact conductance,slip and welded boundaries.Amplitude ratios of different reflected and refracted waves for different boundaries with angle of emergence were compared graphically for different incident waves.It is observed that the amplitude ratios of reflected and refracted waves are affected by the stiffness and thermal properties of the media.

Method to Calculate Fatigue Fracture Life of Control Fissure in Perilous Rock

CHEN Hong-kai, TANG Hong-mei

2007, 28(5): 575-580.

Abstract(2293) PDF(686)

Abstract:
Rupture and safety of perilous rock are dominated by control fissure behind perilous rock block.Based on model-Ⅰ and mode-Ⅱ stress strength factors of control fissure under acting of weight of perilous rock,water pressure in control fissure and earthquake forces,method to calculate critical length of control fissure is established.Water pressure in control fissure is taken as a variable periodic load,and abiding by P-M criterion,when control fissure is filled with water method is established to calculate fatigue fracture life of control fissure in critical status by contributing value of stress strength factor stemming from water pressure of control fissure in Paris's fatigue equation.Further,parameters C and m of sandstone with quartz and feldspar in the area of the Three Gorges Reservoir of China are obtained by fatigue fracture testing.

Some Theoretical Problems on Variational Data Assimilation

TENG Jia-jun, ZHANG Gui, HUANG Si-xun

2007, 28(5): 581-591.

Abstract(2555) PDF(773)

Abstract:
Theoretical aspects of variational data assimilation(VDA) for a simple model with both global and local observational data are discussed.For the VDA problems with global observational data,the initial conditions and parameters for the model are revisited and the model itself is modified.The estimates of both error and convergence rate are theoretically made and the validity of the method is proved.For VDA problem with local observation data,the conventional VDA method are out of use due to the ill-posedness of the problem.In order to overcome the difficulties caused by the illposedness,the initial conditions and parameters of the model are modified by using the improved VDA method,and the estimates of both error and convergence rate are also made.Finally,the validity of the improved VDA method is proved through theoretical analysis and illustrated with an example.And a theoretical criterion of the regularization parameters is proposed.

Research of Discrete Formulation of Mixed Finite Element Methods for the Vapor Deposition Chemical Reaction Equations

LUO Zhen-dong, ZHOU Yan-jie, ZHU Jiang

2007, 28(5): 592-602.

Abstract(2163) PDF(820)

Abstract:
The vapor deposition chemical reaction processes,which are of extremely extensive applications,can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector,temperature field,pressure field,and gas mass field.The mixed finite element(MFE) method is employed to study the system of equations for the vapor deposition chemical reaction processes.The semidiscrete and fully discrete MFE formulations are derived.And the existence and convergence(error estimate) of the semidiscrete and fully discrete MFE solutions are demonstrated.By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes,the numerical solutions of the velocity vector,the temperature field,the pressure field,and the gas mass field can be found out simultaneously.Thus,these researches are not only of important theoretical meaning,but also of extremely extensive applied vistas.

Global Solution for a Coupled Nonlinear Klein-Gordon System

GAN Zai-hui, ZHANG Jian

2007, 28(5): 603-613.

Abstract(2529) PDF(792)

Abstract:
The global solution for a coupled nonlinear Klein-Gordon system in two space dimensions is studied.First,a sharp threshold of blowup and global existence for the system is obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow.Then the result of how small the initial data are for which the solution of the system exists globally is proved by using the scaling argument.

Impact of Singularity of Navier-Sokes Equation Upon the Atmospheric Motion Equations

SHI Wei-hui, WANG Yue-peng

2007, 28(5): 614-618.

Abstract(2773) PDF(943)

Abstract:
Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized.On the basis of this,by taking the basic system of equations of atmospheric motion via Boussinesq approximation as an example to explain in detail that the instability about some simplified models of the basic system of equations for atmospheric motion is caused by the instability of Navie-rStokes equation,thereby,a principle to guarantee the stability of simplified equation is drawn in simplifying the basic system of equations.

Global Attractor for KGS Lattice System

YIN Fu-qi, ZHOU Sheng-fan, YIN Chang-ming, XIAO Cui-hui

2007, 28(5): 619-630.

Abstract(2841) PDF(868)

Abstract:
The longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon Schrêdinger equation (KGS lattice system) was considered.The existence of a global attractor for the system is proved here by introducing an equivalent norm and using "End Tails" of solutions.Then the upper bound of the Kolmogorov D-entropy of the global attractor is estimated by applying element decomposition and the covering property of a polyhedron by balls of radii D in the finite dimensional space.Finally,an approximation to the global attractor is presented by the global attractors of finitedimensional ordinary differential systems.