应用数学和力学

Modelling and Dynamics Analysis of Optimal Dynamical Systems of Fluctuation Velocity Equations for Incompressible Navier-Stokes Equations

WANG Jincheng, QI Jin, WU Chuijie

2020, 41(3): 235-249. doi: 10.21656/1000-0887.400277

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Abstract:
The modelling theory of the optimal low-dimensional dynamical system of the fluctuation velocity equations of the Navier-Stokes equations was studied. The optimal target functional is the incompressibility and orthogonality of the fluctuation velocity basis function. The fully developed side-by-side two-column flow problem was numerically simulated. Based on the two-scale global optimization method, the optimal dynamical system model for its fluctuation velocity was established. The dynamics properties of the phase portraits, the Poincaré section, the bifurcation characteristics, the power spectrum and the Lyapunov exponent set were analyzed. With the increase of the Reynolds number, the optimal dynamical systems of the fluctuation velocity equations of the flow around the two columns exhibit complex quasi-periodic bifurcation behaviors.

Multi-Roll Type Convection Patterns in Cavities Heated Laterally

NING Lizhong, ZHANG Ke, NING Bibo, WU Hao, TIAN Weili

2020, 41(3): 250-259. doi: 10.21656/1000-0887.400165

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Abstract:
The numerical simulation was carried out based on the hydrodynamic equations, and the convection patterns in cavities with large aspect ratios heated laterally at inclined angle θ=90° were studied. For the fluid with Prandtl number Pr=6.99 in the range of reduced Rayleigh number 2≤Ra_r≤ 25, a singleroll type convection pattern occurs in the cavity. For the fluid with Pr=0.027 2 and Ra_r=13.9, with the development of calculation time, the convection pattern in the cavity transforms from the original singleroll type to the multi-roll type, which is a new type of convection pattern pertinent to large-aspect-ratio cavities heated laterally. The calculation results for different Ra_r values show that,Ra_r has a significant effect on the formation of convection patterns. A single-roll type convection pattern occurs for Ra_r≤4.4; for Ra_r=8.9~11.1, the system is in a transitional state; and a multi-roll type convection pattern appears for Ra_r≥13.9. The maximum convection amplitude and Nusselt number Nu increase with Ra_r.The comparison with the convection pattern for Pr=6.99 shows that the formation of the convection pattern depends on Pr.

An Integer Nonlinear Programming Method for Underwater Training Spacesuit Buoyancy Balancing

PENG Yanglin, MA Aijun, LIU Hongying, DONG Rui, SHI Meng, LIU Lei, ZHANG Lei, LU Laijie

2020, 41(3): 260-267. doi: 10.21656/1000-0887.400083

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Abstract:
Aimed at neutral balancing and specific-angle stable balancing of the underwater training spacesuit, through the mathematical description of the buoyancy balancing process, a new method of optimizing calculation for the buoyancy balancing scheme was proposed based on integer nonlinear programming of the branch and bound algorithm. According to given initial values of the underwater training spacesuit and under conditions of neutral balancing and different-angle stable balancing, the optimal buoyancy balancing results were calculated. The results demonstrate the effectiveness and efficiency of the method in calculation of the balancing scheme. This method makes a good guidance for the buoyancy balancing process of different balancing objects and balancing tasks.

Numerical Simulation of Head-on Binary Collision Between Seawater Droplets

YIN Qiang, QI Xiaoni, LIANG Wei

2020, 41(3): 268-279. doi: 10.21656/1000-0887.400196

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Abstract:
Numerical simulations of head-on binary collisions between equal-size seawater droplets were conducted with the volume of fluid (VOF) method and the adaptive grid technique, to investigate the collision physics and mechanics of seawater droplets in shower cooling towers. Trial simulations of head-on collisions of tetradecane droplets in nitrogen medium were performed firstly to give results in good agreement with those of the previous experiments. The binary collisions of equal-size seawater droplets were simulated under room temperature and normal pressure conditions. The stream field and flow mechanism of seawater droplets collision were analyzed, and the effects of droplet diameters and concentrations on the collision process were studied. Two different types of collision outcomes were identified: the coalescence and the reflexive separation, for which the critical Weber numbers were given. The schematic diagrams for various head-on collision regimes of seawater droplets at various Ohnesorge numbers were also obtained.

Dynamic Characteristics Prediction of Launch Vehicles Based on the Equivalent Beam Model

PIAO Siyang, ZHU Chunyan, CHU Fuyun, ZHANG Yahui

2020, 41(3): 280-291. doi: 10.21656/1000-0887.400256

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Abstract:
An efficient and accurate method was proposed for dynamic characteristics prediction of launch vehicles based on the equivalent beam model, with which the influence of liquid propellants can be considered. The equivalent beam model for the launch vehicle was established based on the cross-sectional area equivalence principle, and improved through model updating against the natural frequencies and mode shapes of the fine finite element model. The present method was validated through computation of a certain type of launch vehicles. A highly representative equivalent beam model for the launch vehicle was established. The dynamic characteristics of the launch vehicle were predicted based on the equivalent beam model under the influence of liquid propellants in 2 ways of the lumped mass method and the coupled mass method, and the results were compared.

Fracture Strength Analysis of the Plane-Stress State by the Ellipse Criterion

DU Jingfeng, WU Qiangjie

2020, 41(3): 292-301. doi: 10.21656/1000-0887.400155

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Abstract:
The features of the ellipse criterion for fracture mechanics was briefly analyzed, and the general equation of the criterion was derived in the principal stress coordinate system. According to this general equation, a complete description of the theoretical fracture loci was proposed for the fracture under the plane stress state, then a discussion by this criterion was presented about the relationship between the failure plane direction, the fracture pattern and the material intrinsic parameters of mechanical properties. The comparison with previous theoretical results and experimental phenomena explains the limitation of the ellipse criterion in the determination of material parameters. With material characteristic parameters associated with the stress state as 2 constants in the tension and the compression zones, the theoretical fracture loci were obtained for the cast iron and the concrete. The theoretical results coincide well with the related experimental data in the tension zone, but vastly differ in the compression zone. The work illustrates the necessity to reveal the relation between the material intrinsic parameters and the stress state for the development of the ellipse criterion.

Analysis on Deformation and Stress of Bending Stratum Based on the Elastic Theory for Curved Beams

BU Wankui, XU Hui, ZHAO Yucheng

2020, 41(3): 302-318. doi: 10.21656/1000-0887.400081

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Abstract:
A displacement function suitable for plane curved beams in polar coordinates was introduced, and the partial differential governing equation for plane curved beams was obtained through theoretical analysis. Then, the displacement components and stress components were formulated with the displacement function. On this basis, the finite difference schemes of the partial differential governing equation, the displacement components and the stress components for the curved beam in polar coordinates were presented. Finally, these theoretical formulas were applied to analyze the displacement and stress distributions of the bending stratum. The results indicate that: 1) The bending stratum sinks down after excavation of the coal seam, and there are both tension and compression in the circumferential direction. 2) The radial stress reaches a peak value not far from the openoff cut and increases gradually from the inner surface to the outer surface along the radial direction; the circumferential stress reaches the peak value not far behind the working face and may cause circumferential compressive fracture in the bending stratum; the shear stress reaches the peak value not far from the openoff cut and increases from the inner surface to the outer surface along the radial direction for smallangle sections. The work provides a scientific basis and reference for coal mining engineering.

Reliability Analysis of Perilous Toppling Rock Considering Seismic Force Directions

TANG Hongmei, SHU Qingjiang, WANG Linfeng

2020, 41(3): 319-328. doi: 10.21656/1000-0887.400057

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Abstract:
Lateral unloading leads to development of many perilous rock masses on high and steep slopes. It is significant to judge the probability of instability under the action of rainfall and earthquake. For the perilous toppling rock, the physical and mechanical model with the most dangerous seismic force directions was established. Based on the extreme value theory for functions, the expression of the most dangerous seismic force direction was given. Combined with the reliability theory, the reliability index, the expression of the probability of instability and the judgment criterion for the perilous toppling rock were built. The proposed method was applied to analyze the stability of perilous toppling rock masses in the Jinfoshan district in Chongqing. The calculation results show that, the most dangerous seismic force direction in case 1 is within 5°, and in case 2 is about 10°; the most dangerous direction angle is not a fixed value, but related to the shape, the fracture water pressure and the depth of rock cavity, etc. When the control crack length is small, the most dangerous direction angle is small under other conditions but will increase significantly with the crack length. The probability of instability increases with the control crack length in case 2 more than in case 1. The research work is applicable to disaster prevention and mitigation of perilous rocks.

A New Nonlinear Scalarization Function and Its Applications in Vector Optimization With Variable Ordering Structures

LI Fei

2020, 41(3): 329-338. doi: 10.21656/1000-0887.400262

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Abstract:
In a topological vector space with variable ordering structures, a new nonlinear scalarization function was defined and its main properties were discussed. Meanwhile a family of semi-norms and a class of related normed linear spaces were constructed with this nonlinear scalarization function. Also the conclusions about upper, lower semi-continuity of this nonlinear scalarization function and the semi-norm function was established.

Convergence Results on Heat Source for 2D Viscous Primitive Equations of Ocean Dynamics

LI Yuanfei

2020, 41(3): 339-352. doi: 10.21656/1000-0887.400176

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Abstract:
The convergence of solutions to 2D viscous primitive equations of ocean dynamics in a cylindrical region was considered. A key parameter in this model is heat source, which is known to cause resonance between the inner layers of fluid and in turn trigger instability. Therefore, through derivation of the priori bounds of the equations, the convergence of solutions to the equations on the heat source itself was obtained.

2020 Vol. 41, No. 3