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A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with boundary layers

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Communications in Computational Physics 2011年第6期10卷 161-182页

作者： Po-Wen Hsieh Yintzer Shih Suh-Yuh Yang Department of Mathematics National Central UniversityJhongli CityTaoyuan County 32001Taiwan Department of Applied Mathematics National Chung Hsing UniversityTaichung 40227Taiwan.

In this paper we propose a development of the finite difference method,called the tailored finite point method,for solving steady magnetohydrodynamic(MHD)duct flow problems with a high Hartmann number.When the Hartmann number is large,the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer.Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy.However,the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh.Firstly,we devise the tailored finite point method for the scalar inhomogeneous convectiondiffusion problem,and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations.For each interior grid point of a given rectangular mesh,we construct a finite-point difference operator at that point with some nearby grid points,where the coefficients of the difference operator are tailored to some particular properties of the problem.Numerical examples are provided to show the high performance of the proposed method.

关键词： Magnetohydrodynamic equations Hartmann numbers convection-dominated problems boundary layers tailored finite point methods finite difference methods

Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with boundary and Interface layers

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Communications in Computational Physics 2014年第3期15卷 797-826页

作者： Houde Han Min Tang Wenjun Ying Department of Mathematical Sciences Tsinghua UniversityHaidianBeijing 100084P.R.China Department of Mathematics Institute of Natural Sciences and MOE-LSCShanghai Jiao Tong UniversityMinhangShanghai 200240P.R.China

This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.

关键词： Neutron transport equation discrete ordinates method tailored finite point method boundary layers interface layers

Analytic Regularity for a Singularly Perturbed Reaction-Convection-Diffusion boundary Value Problem with Two Small Parameters

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Communications in Mathematical Research 2024年第2期40卷 125-153页

作者： Irene Sykopetritou Christos Xenophontos Department of Mathematics and Statistics University of CyprusPo Box 20537Nicosia 1678Cyprus.

We consider a second order,two-point,singularly perturbed boundary value problem,of reaction-convection-diffusion type with two small parameters,and we obtain analytic regularity results for its solution,under the assumption of analytic input data.First,we establish classical differentiability bounds that are explicit in the order of differentiation and the singular perturbation parameters.Next,for small values of these parameters we show that the solution can be decomposed into a smooth part,boundary layers at the two endpoints,and a negligible remainder.Derivative estimates are obtained for each component of the solution,which again are explicit in the differentiation order and the singular perturbation parameters.

关键词： Singularly perturbed problem reaction-convection-diffusion boundary layers analytic regularity.

boundary layers Associated with a Coupled Navier-Stokes/Allem-Cahn System： The Non-Characteristic boundary Case

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Journal of Partial Differential Equations 2012年第1期25卷 66-78页

作者： XIE Xiaoqiang School of Science Shanghai Second Polytechnic UniversityShanghai 201209China

The goal of this article is to study the boundary layer ofNavier-Stokes/AllenCahn system in a channel at small viscosity. We prove that there exists a boundary layer at the outlet (down-wind) of thickness n, where n is the kinematic viscosity. The convergence in L-2 of the solutions of the Navier-Stokes/Allen-Cahn equations to that of the Euler/Allen-Cahn equations at the vanishing viscosity was established. In two dimensional case we are able to derive the physically relevant uniform in space and time estimates, which is derived by the idea of better control on the tangential derivative and the use of an anisotropic Sobolve imbedding.

关键词： boundary layers Navier-Stokes Euler equations Allen-Cahn vanishing viscosity limit

Decomposition of mean skin friction in incident shock wave/turbulent boundary layer interaction flows at Mach 2.25

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中国航空学报:英文版 2023年第9期36卷 178-194页

作者： Junyi DUAN Fulin TONG Xinliang LI Hongwei LIU LHD Institute of MechanicsChinese Academy of SciencesBeijing 100190China School of Engineering Science University of Chinese Academy of SciencesBeijing 100049China State Key Laboratory of Aerodynamics China Aerodynamics Research and Development CenterMianyang 621000China

The evolution characteristics of the mean skin friction beneath the supersonic turbulent boundary layer that interacts with incident shock waves at Mach 2.25 are analyzed using Direct Numerical Simulation(DNS). The separated and attached boundary layers in the interaction region that respectively correspond to 33.2° and 28° incident shock angles are considered. The mean skin friction recovery rate for the separated boundary layer is much gentler and distinctly less than that for the attached case where the skin friction completes its recovery within one boundary layer thickness. The novel mean skin friction decomposition method for compressible flows proposed by the recent research is applied in the interaction region to investigate the internal evolution characteristics quantitatively. The results reveal that the three decomposition components are distinctly unequal between the two cases. The contributions of the turbulent motions at different scales to the associated term are focused on using empirical mode decomposition technology. It indicates that the outer large-scale structures dominate separation and reattachment regions, while contributions from inner small-scale structures are limited. In contrast, contributions from the outer largescale structures are dramatically reduced in the attached case, which results in the outer large-scale and inner small-scale motions being of equal importance.

关键词： Compressible flow boundary layers Direct numerical simulation Shock waves Skin friction Turbulent flow

Experimental investigation of expansion effect on shock wave boundary layer interaction near a compression ramp

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Chinese Journal of Aeronautics 2022年第12期35卷 89-101页

作者： Junjie HUO Shihe YI Wenpeng ZHENG Haibo NIU Xiaoge LU Dundian GANG College of Aerospace Science and Engineering National University of Defense TechnologyChangsha 410073China

The experiment is conducted to investigate the effect of expansion on the shock wave boundary layer interaction near a compression ramp. The small-angle expansion with an angle degree of 5° occurs at different positions in front of the compression ramp. The particle image velocimetry and flow visualization technology show the flow structures, velocity field, and velocity fluctuation near the compression ramp. The mean pressure distribution, pressure fluctuation, and power spectral density are measured by high-frequency response pressure transducers. The experimental results indicate that the expansion before the compression ramp position affects the shock wave boundary layer interaction to induce a large-scale separation. But the velocity fluctuation and pressure fluctuation are attenuated near the large-scale flow separation region. When the expansion occurs closer to the compression ramp, the expansion has a more significant impact on the flow. The fluctuation of velocity and pressure is significantly attenuated, and the wall pressure rise of the separation point is reduced obviously. And the characteristic low-frequency spectrum signal related to the unsteadiness of the shock wave boundary layer interaction is significantly suppressed. In addition, variation of the separation region scale at different compression angle degrees is distinctive with the effect of expansion.

关键词： boundary layers Flow visualization Fluid dynamic Pressure measurement Shock waves Supersonic aerodynamics Velocity measurement

Bayesian parameter estimation of SST model for shock wave-boundary layer interaction flows with different strengths

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Chinese Journal of Aeronautics 2023年第4期36卷 217-236页

作者： Denggao TANG Jinping LI Fanzhi ZENG Yao LI Chao YAN School of Aeronautic Science and Engineering Beihang UniversityBeijing 100191China

The Shock Wave-boundary Layer Interaction(SWBLI)flow generated by compression corner widely occurs in engineering.As one of the primary methods in engineering,the Reynolds Averaged Navier-Stokes(RANS)methods usually cannot correctly predict strong SWBLI flows.In addition to the defects of the eddy viscosity assumption,the uncertainty of the closure coeffi-cients in RANS models often significantly impacts the simulation results.This study performs para-metric sensitivity analysis and Bayesian calibration on the closure coefficients of the Menter k-x Shear-Stress Transport(SST)model based on the SWBLI with different strengths.Firstly,the para-metric sensitivity on prediction results is analyzed using the Sobol index.The results indicate that the Sobol indices of wall pressure and skin friction exhibited opposite fluctuation trends with the increase of SWBLI strength.Then,the Bayesian uncertainty quantification method is adopted to obtain the posterior probability distributions and Maximum A Posteriori(MAP)estimates of the closure coefficients and the posterior uncertainty of the Quantities of Interests(QoIs).The results indicate that the prediction ability for strong SWBLI of the SST model is significantly improved by using the MAP estimates,and the relative errors of QoIs are reduced dramatically.

关键词： Bayesian calibration boundary layers Compression corner Sensitivity analysis Shear-stress transport turbu-lence model Shock waves

A MONOTONE DOMAIN DECOMPOSITION ALGORITHM FOR SOLVING WEIGHTED AVERAGE APPROXIMATIONS TO NONLINEAR SINGULARLY PERTURBED PARABOLIC PROBLEMS

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Journal of Computational Mathematics 2008年第1期26卷 76-97页

作者： Igor Boglaev Matthew Hardy Institute of Fundamental Sciences Massey UniversityPrivate Bag 11-222Palmerston NorthNew Zealand Mathematical Sciences Institute Australian National UniversityCanberraAustralia 0200

This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite difference scheme for the partial differential equation, we construct a monotone domain decomposition algorithm based on a Schwarz alternating method and a box-domain decomposition. This algorithm needs only to solve linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear discrete problem. domain decomposition algorithm is estimated The rate of convergence of the monotone Numerical experiments are presented.

关键词： Parabolic reaction-diffusion problem boundary layers O-method Monotone domain decomposition algorithm Uniform convergence

The effect of variable viscosity on the flow and heat transfer of a viscous Agwater and Cu-water nanofluids

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Journal of Hydrodynamics 2013年第1期25卷 1-9页

作者： VAJRAVELU Kuppalapalle PRASAD Kerehalli Vinayaka NG Chiu-On Department of Mathematics University of Central Florida Department of Mathematics VSK University Department of Mechanical Engineering The University of Hong Kong

A numerical study is carried out to study the effects of the temperature dependent viscosity on the flow and heat transfer of a nanofluid over a flat surface in the presence of viscous dissipation. The governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equations, and are solved numerically by the Keller-box method. The numerical results indicate that the effect of nanoparticle volume fraction is to increase the heat transfer and hence enhance the thermal boundary layer thickness. This is true even in the presence of variable viscosity and the viscous dissipation. Furthermore, the results obtained for heat transfer characteristics with nanoparticles reveal many interesting behaviors that warrant further study on the effects of the ＂nano-solid-particles＂.

关键词： nanofluid particles variable viscosity boundary layers viscous dissipation Keller-box method