A Modified Homotopy Analysis Method for Solving Boundary Layer Equations

A new modification of the Homotopy Analysis Method (HAM) is presented for highly nonlinear ODEs on a semi-infinite domain. The main advantage of the modified HAM is that the number of terms in the series solution can be greatly reduced;meanwhile the accuracy of the solution can be well retained. In this way, much less CPU is needed. Two typical examples are used to illustrate the efficiency of the proposed approach.

Optimal Homotopy Analysis of MHD Natural Convection Flow of Thixotropic Fluid under Subjection of Thermal Stratification: Boundary Layer Analysis

In this article, the model of a non-Newtonian fluid (Thixotropic) flow past a vertical surface in the presence of exponential space and temperature dependent heat source in a thermally stratified medium is studied. It is assumed that free convection is induced by buoyancy and exponentially decaying internal heat source across the space. The dynamic viscosity is taken to be constant and thermal conductivity of this particular fluid model is assumed to vary linearly with temperature. Thermal stratification has been properly incorporated into the governing equation so that its effect can be revealed and properly reported. The governing partial differential equations describing the model are transformed and parameterized to a system of non-linear ordinary differential equation using similarity transformations. Approximate analytic solutions were obtained by adopting Optimal Homotopy Analysis Method (OHAM). The results show that for both cases of non-Newtonian parameters (Thixotropic) (K1=K2=0?& K1=K2=1.0), increasing stratification parameters, relate to decreasing in the heat energy entering into the fluid region and thus reducing the temperature of the Thixotropic fluid as it flows.

An Iteration Method to Solve the Boundary Layer Flow past a Flat Plate

An iteration method similar to the thin-wing-expansion method for the compressible flow has been proposed to solve the boundary layer flow past a flat plate. Using such an iteration, the first step of which is Oseen’s approximation, the boundary layer past a flat plate is studied. As proceeding from the first approximation to the second and third approximations, it is realized that our solution approaches to a well known Howarth’s bench mark one gradually. Hence, it is concluded that the usefulness of the present method has been confirmed.

Retrieval of Boundary Layer Height and Its Influence on PM_(2.5) Concentration Based on Lidar Observation over Guangzhou

Wavelet analysis was applied to lidar observations to retrieve the planetary boundary layer height(PBLH)over Guangzhou from September 2013 to November 2014 over Guangzhou.Impact of the boundary effect and the wavelet scale factor on the accuracy of the retrieved PBLH has been explored thoroughly.In addition,the PBLH diurnal variations and the relationship between PM_(2.5) concentration and PBLH during polluted and clean episodes were studied.Results indicate that the most steady retrieved PBLH can be obtained when scale factor is chosen between 300-390 m.The retrieved maximum and minimum PBLH in the annual mean diurnal cycle were~1100 m and~650 m,respectively.The PBLH was significantly lower in the dry season than in the wet season,with the average highest PBLH in the dry season and the wet season being~1050 m and~1200 m respectively.Compared to the wet season,the development of PBLH in the dry season was delayed by at least one hour due to the seasonal cycle of solar radiation.Episode analysis indicated that the PBLH was~50%higher during clean episodes than during haze episodes.The average highest PBLH in the haze episodes and clean episodes were~800 m and~1300 m,respectively.A significant negative correlation between PBLH and PM_(2.5) concentration(r=-0.55**)is discovered.According to China"Ambient Air Quality Standard",the PBLH values in good and slightly polluted conditions were 1/6-1/3 lower than that in excellent conditions,while the corresponding PM_(2.5) concentration were~2-2.5 times higher.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

Wavelet-Galerkin Method for the Singular Perturbation Problem with Boundary Layers

A Wavelet-Galerkin method is proposed to solve the singular perturbation problem with boundary layers numerically. Because there are boundary layers in the solution of the singular perturbation problem, the approximation spaces with different scale wavelets and boundary bases are chosen. In addition, the computation of the inner integrals is transformed to an eigenvalue problem. Therefore, a high accuracy method with reasonable computation is obtained. On the other hand, there is an explicit diagonal preconditioning which makes the condition number of the stiff matrix become bounded by a constant. The error estimate of the Wavelet-Galerkin method and the analysis of the computation complexity are given. The numerical examples show that the method is feasible and effective for solving the singular perturbation problem with boundary layers numerically.

A sixth-order wavelet integral collocation method for solving nonlinear boundary value problems in three dimensions

A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.

RATIONAL SPECTRAL COLLOCATION METHOD FOR A COUPLED SYSTEM OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS

A, novel collocation method for a coupled system of singularly perturbed linear equations is presented. This method is based on rational spectral collocation method in barycentric form with sinh transform. By sinh transform, the original Chebyshev points are mapped into the transformed ones clustered near the singular points of the solution. The results from asymptotic analysis about the singularity solution are employed to determine the parameters in this sinh transform. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.

Wavelet Numerical Detection for Singularly Perturbed Elliptic Boundary Value Problems

This paper is concerned with the numerical solution, and with the detection of singularity for a singularly perturbed elliptic boundary value problems in two space dimensions. Specifically, a wavelet-collocation method is presented for a 2-D linear reaction-diffusion model. By using two-dimensional B-splinewavelet, we test the efficiency of the method.

Time-Domain Analysis of Underground Station-Layered Soil Interaction Based on High-Order Doubly Asymptotic Transmitting Boundary

Based on the modified scale boundary finite element method and continued fraction solution,a high-order doubly asymptotic transmitting boundary(DATB)is derived and extended to the simulation of vector wave propagation in complex layered soils.The high-order DATB converges rapidly to the exact solution throughout the entire frequency range and its formulation is local in the time domain,possessing high accuracy and good efficiency.Combining with finite element method,a coupled model is constructed for time-domain analysis of underground station-layered soil interaction.The coupled model is divided into the near and far field by the truncated boundary,of which the near field is modelled by FEM while the far field is modelled by the high-order DATB.The coupled model is implemented in an open source finite element software,OpenSees,in which the DATB is employed as a super element.Numerical examples demonstrate that results of the coupled model are stable,accurate and efficient compared with those of the extended mesh model and the viscous-spring boundary model.Besides,it has also shown the fitness for long-time seismic response analysis of underground station-layered soil interaction.Therefore,it is believed that the coupled model could provide a new approach for seismic analysis of underground station-layered soil interaction and could be further developed for engineering.

Chebyshev collocation approach for vibration analysis of functionally graded porous beams based on third-order shear deformation theory

In this paper,vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory.The beams have uniform and non-uniform porosity distributions across their thickness and both ends are supported by rotational and translational springs.The material properties of the beams such as elastic moduli and mass density can be related to the porosity and mass coefficient utilizing the typical mechanical features of open-cell metal foams.The Chebyshev collocation method is applied to solve the governing equations derived from Hamilton's principle,which is used in order to obtain the accurate natural frequencies for the vibration problem of beams with various general and elastic boundary conditions.Based on the numerical experiments,it is revealed that the natural frequencies of the beams with asymmetric and non-uniform porosity distributions are higher than those of other beams with uniform and symmetric porosity distributions.

Haar Wavelet Quasilinearization Approach for Solving Nonlinear Boundary Value Problems

Objective of our paper is to present the Haar wavelet based solutions of boundary value problems by Haar collocation method and utilizing Quasilinearization technique to resolve quadratic nonlinearity in y. More accurate solutions are obtained by wavelet decomposition in the form of a multiresolution analysis of the function which represents solution of boundary value problems. Through this analysis, solutions are found on the coarse grid points and refined towards higher accuracy by increasing the level of the Haar wavelets. A distinctive feature of the proposed method is its simplicity and applicability for a variety of boundary conditions. Numerical tests are performed to check the applicability and efficiency. C++ program is developed to find the wavelet solution.

Series Solution of Non-Similarity Boundary-Layer Flow in Porous Medium

This paper aims to present complete series solution of non-similarity boundary-layer flow of an incompressible viscous fluid over a porous wedge. The corresponding nonlinear partial differential equations are solved analytically by means of the homotopy analysis method (HAM). An auxiliary parameter is introduced to ensure the convergence of solution series. As a result, series solutions valid for all physical parameters in the whole domain are given. Then, the effects of physical parameters γ and Prandtl number Pr on the local Nusselt number and momentum thickness are investigated. To the best of our knowledge, it is the first time that the series solutions of this kind of non-similarity boundary-layer flows are reported.

A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations.Then,a Legendre-based spectral collocation method is developed for solving the transformed system.Therefore,we can make good use of the advantages of the Gauss quadrature rule.We present the construction and analysis of the collocation method.These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations.Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

Numerical Studies on the Generation and Propagation of Tsunami Waves Based on the High-Order Spectral Method

An effective numerical model for wave propagation over three-dimensional(3D)bathymetry was developed based on the High-Order Spectral(HOS)method and combined with a moving bottom boundary.Based on this model,tsunami waves caused by various mechanisms were simulated and analyzed.Two-dimensional bed upthrust and the effect of the uplift velocity of the bathymetry on the wave profiles of tsunami waves were studied.Next,tsunami waves caused by 3D submarine slides were generated and the effects of the slide velocity,slide dimension and water depth on the tsunami waves were analyzed.Based on wavelet analysis,the properties of the tsunami wave propagation were investigated.The results show that the bottom movement can significantly affect the generation and propagation of tsunami waves and the studies could help understand the mechanisms of tsunamis caused by a moving bottom boundary.

基于小波多尺度分析大气边界层高度提取方法研究

大气边界层物理演变特征及复杂边界层结构对大气污染过程的影响机制研究中,迫切需要边界层精细化结构的探测手段。通常使用梯度法和小波协方差变换法的方式进行边界层高度提取,但由于该方式容易受到噪声和气溶胶层结构的影响,实验误差较大。提出使用小波多尺度分析方法细化边界层特征结构,从而筛选出有效的细节信息,提高对探索边界层高度的准确性。此外,基于GBQ L-01激光雷达设备实测合肥地区全天气溶胶分布情况,使用小波多尺度分析方法分析对应的平行分量距离平方矫正信号、垂直分量距离平方矫正信号和退偏振比全天分布情况。实验结果表明,该方法准确性高,与梯度法、小波协方差法相比具有更高的稳定性与连续性。对特殊时间点进行数据分析,明确主要影响边界层高度的主要因素,并确定各个时间段的大气边界层高度值。

The Effects of an Inclined Magnetic Field, Brownian Motion, and Thermophoresis on the Flow of Electrically Conducting and Chemically Reacting Casson Nanofluids Using Soret-Dufour Mechanisms

This research explored the effects of an angled magnetic field, Brownian motion, and thermophoresis on the flow of an electrically conducting and chemically reacting Casson nanofluid under the influence of the Soret-Dufour mechanism. A set of partial differential equations is generated by the flow mode. The governing partial differential equations are solved numerically using the spectral collocation method after being transformed to self-similar forms. The effect of various fluid parameters on the velocity profile, temperature profile, and nanoparticle concentration is addressed. A quantitative agreement is observed when previous findings are compared to the current results. The skin friction, local Nusselt number, and local Sherwood number are also examined, and the results are presented in the table. This study discovered that the inclined magnetic field has a significant impact on the flow of the electrically conducting fluid by delaying its mobility within the boundary layer. The plastic dynamic viscosity, which acts as a barrier to fluid flow, is shown to degenerate the fluid velocity when the Casson parameter is increased. As a consequence, the findings may be used to improve thermal science instruments and increase industrial output.

沟槽壁面减阻机理实验研究

利用IFA300型热线风速仪,测量了光滑壁面和沟槽减阻壁面湍流边界层内的瞬时速度,利用自行设计的阻力天平仪测量了壁面摩擦力。得到了边界层无量纲速度分布和平均湍动能分布。对测得的脉动速度信号,利用离散正交小波变换按时间和尺度分解,得到各尺度分量的湍动能,并且发现其分布在湍流惯性区具有极大值。分析表明,当沟槽有减阻效果时,边界层内的平均湍动能减小,湍流惯性区各分量的湍动能极大值亦减小。

激光雷达反演边界层高度方法评估及其在北京的应用

边界层高度是影响大气边界层发展和空气污染程度的重要因子,是环境和气候研究的重要参数。本文利用激光雷达对北京地区2011年5月至2012年4月的边界层高度进行探测分析,采用小波协方差方法反演边界层高度,评估了该方法的适用性。得到基于小波协方差方法自动判断边界层高度的最优参数组合,激光雷达与飞机探测结果对比一致性较好;与探空结果相关系数0.88,激光雷达反演的边界层高度略偏高。当激光雷达的垂直分辨率为30 m时,更加适合北京地区的步长和阈值分别为210 m和0.05;当激光雷达的垂直分辨率为15 m时,步长和阈值分别为135 m和0.05。分析期间,不同季节边界层高度日变化有明显的不同,夏季14:00(北京时)左右达到最高,较高的边界层高度可维持3~4 h,平均可达1.30 km;冬季较高边界层高度只能维持2 h左右,平均为1.08 km。有云与无云天气边界层日变化特征以及边界层高度存在显著的差异,云的存在减少了到达地面的直接辐射,抑制了湍流的发展,进一步抑制了边界层的发展;本文也将激光雷达反演边界层高度结果应用于观测时期边界层高度与地面污染的关系研究中,统计得到边界层高度与PM2.5浓度的相关系数为-0.340。

测量大气边界层高度的激光雷达数据反演方法研究

大气边界层与人类关系最为密切,它的高度分布直接反映了近地面的大气状况。而今激光雷达已成为探测大气边界层时空演变特征的最有效手段,但如何从大量的测量数据中精确提取大气边界层高度则成为限制其应用的主要问题。介绍了四种常用的大气边界层高度提取方法,即梯度法、标准偏差法、曲线拟合法和小波协方差变换法,并结合自行研制的偏振拉曼-米散射激光雷达的实测数据,分别对四种方法的提取结果进行分析。结果表明:四种方法各有优缺点,梯度法、标准偏差法和小波协方差变换法比较相近,准确性高但不稳定;而曲线拟合法的稳定性好,但提取结果相对折中。总体而言,曲线拟合法更适用于大量数据的批处理运算。