Application of differential quadrature method to simulate natural convection in a concentric annulus

In this paper, the Fourier expansion‐based differential quadrature (FDQ) and the polynomial‐based differential quadrature (PDQ) methods are applied to simulate the natural convection in a concentric annulus with a horizontal axis. The comparison and grid independence of PDQ and FDQ results are studied in detail. It was found that both PDQ and FDQ can obtain accurate numerical solutions using just a few grid points and requiring very small computational resources. It was demonstrated in the paper that the FDQ method can be applied to a periodic problem or a non‐periodic problem. When FDQ is applied to a non‐periodic problem (half of annulus), it can achieve the same order of accuracy as the PDQ method. And when FDQ is applied to the periodic problem (whole annulus), it is very efficient for low Rayleigh numbers. However, its efficiency is greatly reduced for the high Rayleigh numbers. The benchmark solution for Ra=10², 10³, 3×10³, 6×10³, 10⁴, 5×10⁴ are also presented in the paper. Copyright © 1999 John Wiley & Sons, Ltd.     

高速小展弦比机翼颤振的微分求积法分析

引入微分求积法,分析高速小展弦比机翼的气动弹性问题。将小展弦比机翼等效为悬臂板,基于一阶活塞气动力理论建立机翼颤振偏微分方程,采用微分求积法将偏微分方程转化为常微分方程,根据频率重合理论对颤振问题进行求解。分析了机翼的固有频率及颤振速度,并与有限元软件计算结果进行比较,误差在2％以内,很好的验证了微分求积法求解小展弦比机翼颤振问题的有效性。分析了机翼面积、展弦比及厚度对颤振速度的影响,结果表明,小展弦比机翼的颤振速度受结构尺寸的影响较大,颤振速度随面积和展弦比的增大而减小,随机翼厚度的增大而增大。     

基于小波微分求积法的薄板弯曲分析

利用小波微分求积法(WDQM)对任意荷载作用下的薄板弯曲问题进行了求解分析。数值算例表明,小波微分求积法与一般的DQ法相比具有很好的适用性,特别是薄板受集中荷载或不连续分布荷载作用时,由于小波基函数的紧支撑特性与其对突变信号良好的描述能力,WDQ法的精度明显优于一般的DQ法,具有良好的应用前景。     

Experimental analysis of thermal fields in horizontally eccentric cylindrical annuli

This paper presents new experimental results on thermal field and heat transfer in a two-dimensional annulus between horizontally eccentric cylinders. The study is conducted by means of optical techniques, for 1.07×10⁴≤Ra _L≤8.27×10⁴ and a wide eccentricity range. The horizontal eccentricity of the inner cylinder substantially alters the thermal field and the geometry of the plume, but, in analogy to the behaviour for vertical eccentricity, the average Nu is slightly affected in the investigated range of eccentricity. The concentric geometry is also considered mainly to validate the experimental technique and evaluate the accuracy of the adopted methodology by comparison with available results. Both shearing interferometer and reference beam interferometer are obtained by means of Wollaston prisms with appropriate splitting angles, so that the temperature and local Nu distributions may be evaluated quantitatively from the original pictures via digital image processing.     

基于局部DQ-RBF不可压缩N-S方程的数值求解

以RBF作为DQ方法的基函数,将迎风机制引入DQ-RBF中,建立了二维不可压缩黏性N-S方程数值求解模型,采用Levenberg-Marquardt算法求解非线性方程组.求解时分析了形状参数对求解精度的影响,改进了边界速度的处理方法.对平板Couette流及有限宽台阶绕流流动问题进行了数值求解.比较了本文方法和FLUE...     

Hydrodynamic and thermal fields analysis in gas-solid two-phase flow

The present work aims to investigate numerically the flowfield and heat transfer process in gas-solid suspension in a vertical pneumatic conveying pipe. The Eulerian-Lagrangian model is used to simulate the flow of the two-phases. The gas phase is simulated based on Reynolds Average Navier-Stokes equations (RANS) with low Reynolds number k-ε model, while particle tracking procedure is used for the solid phase. An anisotropic model is used to calculate the Reynolds stresses and the turbulent Prandtl number is calculated as a function of the turbulent viscosity. The model takes into account the lift and drag forces and the effect of particle rotation as well as the particles dispersion by turbulence effect. The effects of inter-particles collisions and turbulence modulation by the solid particles, i.e. four-way coupling, are also included in the model. Comparisons between different models for turbulence modulation with experimental data are carried out to select the best model. The model is validated against published experimental data for velocities of the two phases, turbulence intensity, solids concentration, pressure drop, heat transfer rates and Nusselt number distribution. The comparisons indicate that the present model is able to predict the complex interaction between the two phases in non-isothermal gas-solid flow in the tested range. The results indicate that the particle-particle collision, turbulence dispersion and lift force play a key role in the concentration distribution. In addition, the heat transfer rate increases as the mass loading ratio increases and Nusselt number increases as the pipe diameter increases.     

Thermal bending analysis of laminated orthotropic plates by the generalized differential quadrature method

The interlaminar stresses in a thin laminated rectangular orthotropic plate with four sides simply supported edges under bending was determined by using the generalized differential quadrature (GDQ) method involving the effects of thermal expansion strain and transverse load. The approximate stress and displacement solutions are obtained under the effects of thermal expansion force and uniform pressure load for eight-layer unidirectional laminates, symmetric cross-ply laminates. Numerical results on the dominant interlaminar stresses and displacement of bending analysis are compared to the Navier solution. The thermal induced forces have significant effect on the bending of plates.     

最大偏心圆环空间自然对流传热的数值分析

采用正切圆坐标变换 ,对不同直径比以及上、下、侧面三种偏心位置 ,偏心率达到最大值± 1的变壁温水平圆柱环形封闭空间内空气的自然对流传热进行了数学模拟 ,求出的二维空间温度分布与实验拍摄相应的温度干涉条纹图片吻合良好。计算结果同时给出流线分布及内、外壁面的局部传热系数、热流量。并与现有的偏心率小于 1的有关资料作对比分析。数值计算的范围是 :2 .0× 1 0 2 ≤ Ra≤ 3 .0× 1 0 5,1 .3≤ Do/Di≤ 3 .8,Pr=0 .70 6,|ε|=1 .     

A boundary integral equation method for Navier-Stokes equations—application to flow in annulus of eccentric cylinders

A novel Navier-Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non-linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.     

In-plane vibration analysis of plates in curvilinear domains by a differential quadrature hierarchical finite element method

Free in-plane vibration analysis of plates is carried out by a differential quadrature hierarchical finite element method (DQHFEM). The NURBS (Non-Uniform Rational B-Splines) patches of geometries were first transformed into differential quadrature hierarchical (DQH) patches, and then the elastic field was discretized by the same DQH basis. The DQHFEM solved the compatibility problem caused by different parametrization of neighbouring patches of isogeometric analysis using NURBS. And mesh refinement in DQHFEM does not propagate from patch to patch. The DQHFEM matrices also have the embedding property as the hierarchical finite element method (HFEM). In-plane vibration analyses of plates of several planforms showed that the DQHFEM is similar as the fixed interface mode synthesis method that can analyse a structure using a few nodes on the boundary of substructure elements and only several clamped modes inside each substructure element, but the DQHFEM does not need modal analysis and is of high accuracy. The accuracy and convergence of the DQHFEM were validated through comparison with exact and approximate results in literatures and computed by the authors.     

Bending analysis of thin functionally graded plates using generalized differential quadrature method

In this paper, the differential quadrature (DQ) method is presented for easy and effective analysis of isotropic functionally graded (FG) and functionally graded coated (FGC) thin plates with constant Poisson’s ratio and varying Young’s modulus in the thickness direction. The bending of FG and FGC plates under transverse loading has been studied using the polynomial differential quadrature (PDQ) and the harmonic differential quadrature (HDQ) methods. A three-dimensional elasticity solution for a moderately thick FG plate with exponential Young’s modulus is used as the benchmark. Two examples, including a thin FG rectangular plate and a thin FGC rectangular plate with sigmoidal Young’s modulus, are investigated. The numerical results of PDQ and HDQ methods reveal good agreement with other solutions. Also, it is shown that the formulations for thin FG plates and homogeneous plates are similar, except that the plane strain components of the middle surface in FG plates are not zero.     

Thermoelastic analysis of functionally graded annulus with arbitrary gradient

A thermoelastic problem of a circular annulus made of functionally graded materials with an arbitrary gradient is investigated. Different from previous works, our analysis neither requires a special form of the gradient of material properties nor demands partitioning the entire structure into a multilayered homogeneous structure. Instead, we propose a new method for solving the thermoelastic problem of a functionally graded circular annulus by transforming it to a Fredholm integral equation. The distribution of thermal stresses and radial displacement can be obtained by solving the resulting equation. Illustrative examples are given to show the effects of varying gradients on the thermal stresses and radial displacement for given temperature changes at the inner and outer surfaces. The results indicate that the thermal stresses can be relaxed for specified gradients, which is beneficial to design an inhomogeneous annulus to maintain structural integrity.     

基于微分求积法求解非粘滞阻尼系统的时程响应

研究一种含有指数型非粘滞阻尼线性多自由度振动系统的时程分析问题。该非粘滞阻尼模型假设阻尼力与质点速度的时间历程相关,数学表述为质点速度与核函数的卷积。由于阻尼模型的改变,常用的数值积分方法（如Newmark-β法、Wilson-θ法）不能直接应用于这种非粘滞阻尼系统。基于一种无条件稳定的微分求积方法,给出了这种非粘滞阻...     

微分求积法的特性及其改进

微分求积法已在科学和工程计算中得到了广泛应用。然而,有关时域微分求积法的数值稳定性、计算精度即阶数等基本特性,仍缺乏系统性的分析结论。依据微分求积法的基本原理,推导证明了微分求积法的权系数矩阵满足V-变换这一重要特性;利用微分求积法和隐式Runge-Kutta法的等值性,证明了时域微分求积法是A-稳定、s级s阶的数值方法。在此基础上,为进一步提高传统微分求积法的计算精度,利用待定系数法和Padé逼近,推导出了一类新的s级2s阶的微分求积法。数值计算对比结果验证了所提出的新微分求积法比传统的微分求积法具有更高的计算精度。     

Numerical predictions of flow and thermal fields in an enclosure with two periodically moving walls

Numerical predictions of transient flow and thermal fields in a rectangular enclosure with two periodically moving vertical walls are presented. The combined influence of the movement of the walls and the buoyancy as well on the flow pattern and heat transfer performance is evaluated. The compressible‐flow model is adopted, and governing equations are expressed in integral form and discretized on the moving grids, which deform in resonance with the walls to accommodate the variation in the volume of the enclosure. A two‐stage pressure‐correction scheme is applied for simultaneously determining the distributions of absolute pressure, density, temperature, and velocity of the compressible flow field in the enclosure during the periodically stable periods. Effects of the frequency, stroke, and the phase angle of the wall oscillations on the flow are of major concerns in this study. The frequency is ranged between 5 and 25 Hz and the dimensionless strokes (l/H) of the wall are varied from 0.4 to 1.0. Results for Nusselt numbers on the walls as well as the dimensionless input work required to excite the wall oscillation are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of differential quadrature method to solve entry flow of viscoelastic second‐order fluid

The entry flow of viscoelastic second‐order fluid between two parallel plates is discussed. The governing equations of vorticity and the streamfunction are expanded with respect to a small parameter that characterizes the elasticity of the fluid by means of the standard perturbation method. By using the differential quadrature method with only a few grid points, high‐accurate numerical solutions are obtained. The numerical results show a lot of the features of a viscoelastic second‐order fluid. Copyright © 1999 John Wiley & Sons, Ltd.     

Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order

The present paper investigates the free vibration characteristics of Timoshenko beams whose cross-sectional profile and material properties vary along the beam axis with any arbitrary functions. Free vibration analysis of these beams is carried out through solving the governing differential equations of motion. Since the application of differential transformation method (DTM) does not necessarily converge to satisfactory results, an element-based differential transformation method, namely differential transformation element method (DTEM), is introduced which significantly enhances the accuracy of the results. Furthermore, differential quadrature element of the lowest order (DQEL) is introduced which is based on differential quadrature element method (DQEM). DQEL formulates the problem on the basis of the interpolation of the first differential of the functions; therefore, in contrast with DQEM higher differentials of functions are not employed in DQEL. The competency of DQEL and DTEM in free vibration analysis is verified through several numerical examples. The effects of taper ratio and material non-homogeneity on natural frequencies are investigated.     

A numerical investigation of developing flow in an eccentric curved annulus in the presence of gravity

In this article developing incompressible viscous flow in an eccentric curved annulus in the presence of gravity is numerically studied using a second order finite difference method based on the projection algorithm to solve the governing equations including the continuity and full Navier–Stokes equations. The equations written in a bipolar–toroidal coordinate system are discretized in a three dimensional staggered grid. The effects of governing non-dimensional parameters including the eccentricity, non-dimensional curvature ratio, Dean number, Froude number, aspect ratio, and the Reynolds number on the flow field in the entrance and fully developed region are investigated. The numerical results indicate that at the small Froude numbers, the flow field distorts from the symmetrical condition due to the larger body force effect and the axial velocity formation mostly takes place at the lower half of the annulus. In addition, at the constant Froude number, by decreasing the curvature radius, the peak axial velocity and its sharp gradient appear on the outer curvature region due to the larger centrifugal forces and by increasing the eccentricity the flow rate intensifies at the wider region and weakens at the narrower region due to the larger flow resistance. Furthermore, the friction factor increases by decreasing the Froude number and increasing the Dean number.     

Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment

This study reports the results of an investigation into the static analysis of microbeams based on nonlocal thermal elasticity theory by differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods. To show the accuracy of the method, the results of present work are compared with those of other works. Different parameters such as temperature, length-to-thickness ratio, length-to-width ratio and boundary conditions are studied, too. From the knowledge of author, it is the first time that results for static analysis of microbeams by nonlocal thermal elasticity theory in thermal environment using two types of differential quadrature method are investigated and the results may be used as benchmarks for the future works.     

Photoelastic-coating analysis in thermal fields

If photoelastic-coating materials exhibited thermal conductivity and thermal expansion equal to that of structural materials, and if strain-optical sensitivity did not vary with temperature, photoelasticcoating analyses could be conducted in thermal fields exactly as in room-temperature test. Methods for circumventing problems associated with these material properties are presented. Corrections are introduced as analytically and empirically derived factors to account for birefringence resulting from differential thermal expansion of coating and workpiece. Surface strains induced by external loading and by thermal stresses can be performed in the temperature range of ?60° F to +350° F for tests of extended duration and to +500° F for brief periods.