应用数学和力学

2021 Vol. 42, No. 2

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Steady-State Solutions of Traffic Flow in a Simple Circled Road Network

ZHANG Peng, Lü Yupei, GUO Mingmin, LIN Zhiyang, FANG Rui, LI Xiaoyang, ZHANG Xiaoning

2021, 42(2): 123-132. doi: 10.21656/1000-0887.410100

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Abstract:
The steady-state solutions of traffic flow in a circled road network composed of 3 road sections and 2 junctions were studied under the assumption of the user equilibrium principle at the diverging junction. The results show that, the solution parameters and the dynamic behavior depend continuously on the total number of vehicles in the network. More precisely, the solution suggests a constant density in each road section when the total number of vehicles is not greater than the 1st critical density and not smaller than the 2nd critical density. When the total number of vehicles is between the 2 critical densities, the shock discontinuity or queuing appears upstream towards a bottleneck or a junction where the upstream capacity is greater than the downstream capacity. Complete analytical results were presented with the diverging and the merging junctions as bottlenecks, respectively.

Experimental Investigation of Natural Frequencies of Gas-Liquid Coupled Systems in Tanks

WEI Zhijun, ZHAI Gangjun, WU Chuijie

2021, 42(2): 133-141. doi: 10.21656/1000-0887.410207

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Abstract:
The wave impact phenomenon widely exists in nature, ocean and aerospace engineering. When the wave impacts on the large-scale structure, the violent free surface may break and the wave tip entraps the air. As a consequence, wave impact with gas-entrapment may cause localized and large load, which may lead to structural failure. During the slamming process, the influence of gas on the natural modes of the free surface has not been systematically reported. A series of experiments were designed and conducted to study the influences of 2 different ullage space pressures on the natural frequencies and damping ratios of gas-liquid coupled systems. High-speed cameras were employed to record the free-surface vibration. Furthermore, the surface wave height was extracted with a self-made image-processing software. The results show that, the sloshing energy mainly concentrates on the lowest natural frequency of the free surface under a low ullage space pressure; while the sloshing energy mainly concentrates on the 2nd natural frequency of the free surface under a high ullage space pressure. As the ullage space pressure of the sloshing tank increases, the dominant natural frequency of the free surface will increase, while the corresponding damping ratio will decrease. Therefore, the gas compressibility is an important factor for the sloshing issue in the tank.

A P_N×P_N-2 Spectral Element Method Based on the Picard Iteration for Steady Incompressible Navier-Stokes Equations

QIU Zhouhua, ZENG Zhong, LIU Hao

2021, 42(2): 142-150. doi: 10.21656/1000-0887.410289

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Abstract:
A P_N×P_N-2 spectral element method based on the Picard linearized iteration was presented for the solution of 2D steady incompressible Navier-Stokes equations. Through the Picard iteration, the Navier-Stokes equations were converted to a series of Stokes-type equations to be solved with the P_N×P_N-2 spectral element method on the non-staggered grid in each iteration step. In order to eliminate the pseudo pressure mode, the pressure discretization is 2 orders lower than the velocity discretization, and the application of non-staggered grids makes the discretization of the equation convenient and avoids the interpolation error. The Stokes flow, the Kovasznay flow and the lid-driven cavity flow were simulated with the present method. The numerical results show that, the error converges with the spectral accuracy. In addition, avoidance of the pressure oscillation phenomenon indicates the accuracy and reliability of the present method.

Numerical Study of Aqueous Humor Flow in Human Eyes

CAI Jiancheng, ZHANG Baoyun, CAO Yuehong, BRAZHENKO Volodymyr

2021, 42(2): 151-161. doi: 10.21656/1000-0887.410113

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Abstract:
The study of intraocular aqueous humor (AH) flow is helpful to understand the mechanism of some eye diseases such as glaucoma. A numerical study of AH flow inside human eyes with the computational fluid dynamics (CFD) were carried out to address the following flow processes: AH secreted in the posterior chamber by the ciliary body entering the anterior chamber through the iris-lens gap of 5 μm and 30 μm respectively, and discharging through the trabecular meshwork (TM). Detailed flow fields in different eye orientations were analyzed. Results show that, in general the intraocular pressure distributions are similar in all cases; the pressure in the posterior chamber is around 30 Pa higher than that in the anterior chamber with the 5 μm iris-lens gap, and they are almost equal with the 30 μm iris-lens gap. The pressure drop in the TM is noticeable. The pressure difference between the anterior and posterior chambers drives AH from the posterior chamber to the anterior chamber, while the temperature difference between cornea and iris surfaces causes the natural convection, which is the dominant flow in the anterior chamber. The pressure difference due to natural convection is in the magnitude of millipascal, and the higher-pressure region forms where warmer AH gathers. For the vertical orientation, warmer fluid rises along the iris surface and then turns downwards as it encounters the higher resistance in the upper TM regions. The flow then descends along the corneal surface toward the lower TM. For the upward-facing position, AH entering the pupil rises along the center line of the anterior chamber, and moves down along the cornea surface leading to 2 large symmetric recirculation zones. For the downward-facing position, the circulation route is opposite to that of the upward-facing position. With no buoyancy, the averaged velocity will be 1 to 2 orders of magnitude smaller than that with buoyancy.

A Thermal Protective Clothing-Air-Skin Heat Conduction Model and Its Analytical Solution

LI Changyu, FANG Yankui, LIU Fuxu, RUAN Yuhang

2021, 42(2): 162-169. doi: 10.21656/1000-0887.400290

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Abstract:
A thermal protective clothing-air-skin heat conduction model in high temperature environment was established. The analytical solution of heat conduction in a small time period was derived by means of the method of separation of variables under the conditions of equal temperature and continuous heat flux at the laminate interface during heat conduction, and then the analytical solution in the whole time domain was obtained through iteration. The variations of temperature and heat flux in different positions in the model at an ambient temperature of 80 ℃ were analyzed with the analytical solution. Then the surface temperature and thermal damage of skin at different ambient temperatures were calculated. The solution method applies to heat transfer problems of general laminate structures. The calculation results are of referential significance for the design and effect evaluation of thermal protective clothing.

A Hybrid Scheme of Rotational Flux for Solving 2D Euler Equations

JIA Dou, ZHENG Supei

2021, 42(2): 170-179. doi: 10.21656/1000-0887.410196

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Abstract:
To improve the resolution of the numerical results for the 2D Euler equations, a hybrid scheme of rotational flux was proposed. The algorithm was built under the quasi-1D idea of the rotational flux method, and the flux function was given in the form of a hybrid scheme coupling the entropy stable numerical flux satisfying the 2nd law of thermodynamics with the HLL numerical flux of good robustness. The time was advanced with the 3rd-order strongly stable Runge-Kutta method. The hybrid scheme of rotational flux has the advantages of simple structure and high resolution. Numerical results show good characteristics of the algorithm.

Global Solutions of the Asymptotically Periodic Curvature Flow Equations in Band Domains

LIU Qian, CHEN Ruiqi

2021, 42(2): 180-187. doi: 10.21656/1000-0887.410087

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Abstract:
The curvature flow equations were studied with Neumann boundary conditions and asymptotically periodic coefficients. First, a series of initial value problems and corresponding global solutions were considered. By uniform prior estimates, a subsequence converging to the global solution was obtained. Second, the uniqueness of the global solution was proved with the renormalization method in the direction of negative infinite time and the strong maximum principle. Finally, to study the ω-limit and α-limit of the global solution, the renormalization method was used again. Through the construction of the pullback function, the uniform prior estimation and the convergent subsequence with the Cantor diagonalization method, it is shown that, the ω-limit and α-limit of global solutions are the global solutions of the corresponding limit problems, i.e., they both are periodic traveling waves.

A Self-Adaptive Uzawa Block Relaxation Method for Stokes Problems With Slip Boundary Conditions

ZHANG Maolin, RAN Jing, ZHANG Shougui

2021, 42(2): 188-198. doi: 10.21656/1000-0887.410170

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Abstract:
A self-adaptive Uzawa block relaxation method was designed for Stokes problems under nonlinear slip boundary conditions. For the variational formulation of the problem, an auxiliary unknown was introduced to transform the problem into a saddle-point one based on an augmented Lagrangian function, which can be solved with the Uzawa block relaxation method. To improve the performance of the method, a self-adaptive rule was proposed with the proper penalty parameter chosen automatically. The main advantage of this method is that each iterative step consists of a linear problem while the auxiliary unknown can be computed explicitly. The convergence of the algorithm was analyzed. The numerical results show the feasibility and effectiveness of the proposed method.

Evaluation and Prediction of Prevention and Control Effects of the COVID-19 Epidemic Based on the SEIR Model

CHEN Xingzhi, TIAN Baodan, WANG Daiwen, HUANG Feixiang, FU Lingyan, XU Haoying

2021, 42(2): 199-211. doi: 10.21656/1000-0887.410139

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Abstract:
Based on the analysis of the spread of the COVID-19 epidemic in China, a SEIR epidemic model was established with the diagnosed population divided into 2 categories: the admitted population and the non-admitted population. Through theoretical analysis, the basic reproduction number, the disease-free equilibrium of the model and its stability were derived. Further, several numerical simulations and comparative analysis were conducted on the development trend of the epidemic situation in Wuhan before and after the city closure, as well as the influences of some important parameters in the model on the number of diagnosed cases. Finally, according to the results of above theoretical analysis and numerical simulations, some control strategies previously adopted were analyzed and evaluated, and predictions were made for the development of the epidemic.

A Fuzzy Portfolio Model With Background Risk and Liquidity Considered

SONG Huihui, LONG Xianjun, HE Guang, PENG Zaiyun

2021, 42(2): 212-220. doi: 10.21656/1000-0887.400298

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Abstract:
With the 2nd moment of the investment return as the risk measurement function, a fuzzy portfolio model with the background risk and liquidity considered was established. The 2nd moment of the investment return was minimized under the constraint conditions satisfying the preset return rate, the average level of turnover probability and the investment proportion of risk assets. Finally, the historical data of some stocks in the 100 indices were selected for numerical analysis, and the model was proved to conform to the law of ‘high return and high risk’. The results show that, the model is suitable for the actual financial market. The 2nd moment instead of the variance as the risk measurement function, overcomes the complexity of variance calculation and simplifies the problem of fuzzy portfolio solution.

Solitary Periodic Waves and Local Bifurcations of Critical Periods for a Class of Reaction-Diffusion Equations

GU Jieping, HUANG Wentao, CHEN Ting

2021, 42(2): 221-232. doi: 10.21656/1000-0887.410263

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Abstract:
The small-amplitude solitary periodic wave solutions and the local critical periodic bifurcations of the traveling wave equations for a class of reaction-diffusion equations with quintic nonlinear reaction terms and constant diffusion terms were studied. First, the reaction-diffusion equation was transformed into the corresponding traveling wave system through traveling wave transformation. The first 8 singular point quantities of the system were calculated with the singular point value method and the computer algebra software MATHEMATICA. Then, 2 center conditions for the singular point of the system were obtained, 8 limit cycles were proved to bifurcate at the origin of the traveling wave system, and 8 small-amplitude solitary periodic wave solutions were found to exist in the corresponding nonlinear reaction-diffusion equation. Furthermore, through computation of the period constants, the weak center order for the origin of the traveling wave system was derived. Then, the system was proved to have at most 3 local critical periodic bifurcations and be able to reach the 3 bifurcations. Moreover, the analysis of the critical periodic bifurcations of the traveling wave system reveals that the reaction-diffusion equation has 3 critical periodic wavelengths.