Analytical and numerical analysis for the FGM thick sphere under combined pressure and temperature loading

Thermo-mechanical analysis of functionally graded hollow sphere subjected to mechanical loads and one-dimensional steady-state thermal stresses is carried out in this study. The material properties are assumed to vary non-linearly in the radial direction, and the Poisson’s ratio is assumed constant. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. In the analysis presented here, the effect of non-homogeneity in FGM thick sphere was implemented by choosing a dimensionless parameter, named β _i (i = 1, . . . , 3), which could be assigned an arbitrary value affecting the stresses in the sphere. It is observed that the solution process for β _i (i = 3) = −1 are different from those obtained for other values of β _i (i = 1, . . . , 3). Cases of pressure, temperature, and combined loadings were considered separately. It is concluded that by changing the value of β _i (i = 1 . . . 3), the properties of FGM can be so modified that the lowest stress levels are reached. The present results agree well with existing results. Using FEM simulations, the analytical findings for FGM spheres under the influence of internal pressure and temperature gradient were compared to finite element results.     

A form function method for form finding of tension structure

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Methodology to Assess the Effects of Rice Cultivation Under Flooded Conditions on van Genuchten’s Model Parameters and Pore Size Distribution

The effects of rice (Oryza sativa L.) cultivation under flooded conditions on soil’s physical-hydraulic properties were studied in this article, using a new methodology based on a combined analysis on soil water retention curve (WRC) and pore size distribution (PSD). WRC analysis was carried out through the changes of van Genuchten’s model parameters, the characteristics of WRC at the inflection point, and the specific water capacity curve. Analysis of PSD was performed on the volume changes of porosity fractions through a detailed pore size classification, while different pore size classifications based on their hydraulic and structural characteristics were also used. The methodology was applied using a small dataset obtained from fine-textured Entisol soils which were subjected to rice cultivation under flooded conditions in Axios River plain (Northern Greece). Measurements of WRC were obtained at four depths of the soil profile from two fields, before and after the growing season of rice. The analysis indicated that the van Genuchten’s model parameters (θ _s, a, and n) and the WRC characteristics at the inflection point (pressure head h _i , pore equivalent diameter D _i , and slope S _i ) significantly changed after the growing season following similar patterns, along the soil profile in both fields. The parameters θ _s, a, D _i , and S _i were decreased, while n and h _i were increased. The h _i and a were the most sensitive parameters, while the values of (h _i and 1/a) in each layer before and after the growing season for each field were linearly correlated and shifted to higher values because of compaction, indicating that it could be applied as a tool to evaluate the degree of soil compaction to similarly textured soils. The peaks of the specific water capacity curves were compressed (lower values of slope S _i ) and shifted to lower water potentials (h _i ) that corresponded to pores of equivalent diameter D _i between 2 and 6 μm. The soils had few structural pores (>9 μm) and low air-filled porosity (>30 μm) before the growing season, which presented accessory reduction after the growing season in both fields. Total porosity was reduced at the expense of structural porosity along the soil profile, while the pore size class of 5–3 μm was identified as the threshold where the smaller pores’ volume started to increase in all layers of both fields. The results indicated that the changes in the WRC and the PSD follow specific trends, which can be used in future studies to model temporal variability of soil’s physical-hydraulic properties.     

Slip-flow heat transfer in a microchannel with viscous dissipation

Forced convection flow in a microchannel with constant wall temperature is studied, including viscous dissipation effect. The slip-flow regime is considered by incorporating both the velocity-slip and the temperature-jump conditions at the surface. The energy equation is solved for the developing temperature field using finite integral transform. To increase β_v Kn is to increase the slip velocity at the wall surface, and hence to decrease the friction factor. Effects of the parameters β_v Kn, β, and Br on the heat transfer results are illustrated and discussed in detail. For a fixed Br, the Nusselt number may be either higher or lower than those of the continuum regime, depending on the competition between the effects of β_v Kn and β. At a given β_v Kn the variation of local Nusselt number becomes more even when β becomes larger, accompanied by a shorter thermal entrance length. The fully developed Nusselt number decreases with increasing β irrelevant to β_v Kn. The increase in Nusselt number due to viscous heating is found to be more pronounced at small β_v Kn.     

ROSSBY WAVES WITH THE CHANGE OF β

In this paper, the change of the Rossby parameter β with latitude is considered and the parameter γ≡-dβ/dy=2sinφ/a² is introduced and the β-plane approximation is extended into f=f₀+β₀y-γ₀y²/2 which includes the parameter γ. Such approximation closes further to practice especially in the high latitude regions.We give emphasis to the research of the effect of the parameter γ on the Rossby waves. It is seen that the effect of the parameter γ is remarkable in the high latitude regions. It can produce the Rossby waves caused by the pure parameter γ. And the phase speed formula of Rossby waves with the change of ft is generally given, which is degenerated into the well-known Rossby formula when γ₀=0. The researches also point out that when the change of β is regarded, even if the basic current u is a linear function of y the unstable modes can also take place. However,the parameter γ usually plays a stable part in the Rossby waves and it does affect the longitudinal scale and the structure of constant phase lines(trough-ridge lines)of Rossby waves and slow down the growing or decaying of Rossby waves.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 2: Numerical Simulations

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Some issues of turbulence statistics

The issue of dropping the random forcef _i and the arbitrariness of choosing the basic variable in the variational approach to turbulence closure problem, pointed out recently by the Russian scientists Bazdenkov and Kukharkin, are discussed. According to the mean-square estimation method, the random forcef _i should be dropped in the error expression of the LFP (Langevin-Fokker-Planck) model. However,f _i is not neglected, its effect has been taken into account by the variational approach. In order to optimize the perturbation solution of the Liouville equation, the LFP model requires that the basic variable is as near to Gaussian as possible. Hence, the velocity, instead of the vorticity, should be chosen as the basic variable in the three-dimensional turbulence. Although the LFP model and the zero-order Gaussian term of PDF (probability density function) imply whiteness assumption (zero correlation time off _i ), the higher-order non-Gaussian terms of PDF correspond to the nonwhiteness of turbulence dynamics, the variational approach does calculate the nonwhiteness effect properly. The work is supported by the National Basic Research Program “Non-linear Sciences” and the National Natural Science Foundation of China     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Range-pair-counting fatigue gage

In this paper, the surface roughening phenomenon of an aluminum foil during fatigue process is utilized for monitoring of a variable amplitude stress. It is found that constant mean stress does not affect surface roughness. When the mean stress varies, however, it causes surface roughening. In such a case, the equivalent stress defined by (Σσ _i ^α N _i /ΣN _i )^1/α dominates surface roughness of an aluminum foil, where σ _i andN _i are stress amplitude and the number of cycles counted by the rangepair method, respectively, and α is a gage factor which is peculiar to the foil used. It is concluded that the aluminum foil can be used as the range-pair counting fatigue gage with high accuracy.     

Slip coefficients and macroparameter jumps of a diatomic gas with rotational degrees of freedom on a weakly curved spherical surface

Using the half-space moment method, the problem of the slip of a diatomic gas along a rigid spherical surface is solved within the framework of a model kinetic equation previously proposed which takes into account the rotational degrees of freedom of the gas. Second-order slip coefficients (correctionsC _m ^′ , β _R ^′ , and β _R to the isothermal and thermal slip which are linear with respect to the Knudsen number Kn) are obtained. The gas macroparameter jump coefficientsC _v andC _q, which are of the second order in the Knudsen number and characterize the discontinuity of the normal mass and heat fluxes on the gas-rigid phase interface, are calculated. These coefficients are given as functions of the tangential momentum accommodation coefficient, the translational and rotational energy accommodation coefficients, and the Prandtl number. The coefficients are calculated for certain diatomic gases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–173, January–February, 2000.     

Prediction of CHF in concentric-tube open thermosiphon using artificial neural network and genetic algorithm

In this paper, an artificial neural network (ANN) for predicting critical heat flux (CHF) of concentric-tube open thermosiphon has been trained successfully based on the experimental data from the literature. The dimensionless input parameters of the ANN are density ratio, ρ _l/ρ _v; the ratio of the heated tube length to the inner diameter of the outer tube, L/D _i; the ratio of frictional area, d _i/(D _i + d _o); and the ratio of equivalent heated diameter to characteristic bubble size, D _he/[σ/g(ρ _l−ρ _v)]^0.5, the output is Kutateladze number, Ku. The predicted values of ANN are found to be in reasonable agreement with the actual values from the experiments with a mean relative error (MRE) of 8.46%. New correlations for predicting CHF were also proposed by using genetic algorithm (GA) and succeeded to correlate the existing CHF data with better accuracy than the existing empirical correlations.     

Development of an Analytical <Emphasis Type="Italic">β</Emphasis>-Function PDF Integration Algorithm for Simulation of Non-premixed Turbulent Combustion

Among many presumed-shape pdf approaches for modeling non-premixed turbulent combustion, the presumed β-function pdf is widely used in the literature. However, numerical integration of the β-function pdf may encounter singularity difficulties at mixture fraction values of Z = 0 or 1. To date, this issue has been addressed by few publications. The present study proposes the Piecewise Integration Method (PIM), an efficient, robust and accurate algorithm to overcome these numerical difficulties with the added benefit of improving computational efficiency. Comparison of this method to the existing numerical integration methods shows that the PIM exhibits better accuracy and greatly increases computational efficiency. The PIM treatment of the β-function pdf integration is first applied to the Burke–Schumann solution in conjunction with the k − ε turbulence model to simulate a CH₄/H₂ bluff-body turbulent flame. The proposed new method is then applied to the same flow using a more complex combustion model, the laminar flamelet model. Numerical predictions obtained by using the proposed β-function pdf integration method are compared to experimental values of the velocity field, temperature and species mass fractions to illustrate the efficiency and accuracy of the present method.     

On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder

Here the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. At time t=0 the fluid is at rest and the motion is produced by the rotation of the cylinder around its axis. The solutions that have been obtained are presented under series form in terms of the generalized G-functions. The similar solutions for ordinary second grade and Newtonian fluids are obtained from general solution for β→1, respectively, β→1 and α ₁→0. Finally, the influences of the pertinent parameters on the fluid motion, as well as a comparison between models, is underlined by graphical illustrations.     

Numerical simulation of fully developed turbulent flow and heat transfer in annular-sector ducts

 The mixing length theory is employed to simulate the fully developed turbulent heat transfer in annular-sector ducts with five apex angles (θ₀=18,20,24,30,40^∘) and four radius ratios (R _o/R _i=2,3,4,5). The Reynolds number range is 10⁴10⁵. The numerical results agree well with an available correlation which was obtained in following parameter range: θ₀=18,20,24,30,40^∘, R _o/R _i=4 and Re=10⁴5×10⁴. The present work demonstrates that the application range of the correlation can be much extended. Apart from the mixing length theory, the kɛ model with wall function and the Reynolds stress model are also employed. None of the friction factor results predicted by the three models agrees well with the test data. For the heat transfer prediction the mixing length theory seems the best for the cases studied. Received on 17 July 2000 / Published online: 29 November 2001     

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI^λD^μ controllers. It is based on plotting the global stability region in the (k _p, k _i)-plane for the PI^λ controller and in the (k _p , k _i , k _d)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.     

Further Comments on “Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work”

This note is concerned with the assertion of Barletta and Nield (2009a) that “a fluid with a thermal expansion coefficient greater than that of a perfect gas (β > β _perfect gas) is of marginal or no interest in the framework of convection in porous media”, and that for a remark of Magyari (Transp. Porous Media, 2009) about the forced convection eigenflow solutions, the circumstance β > β _perfect gas does not represent “a sound physical basis”. Here, it is shown, however, that these assertions are in contradiction with the experimentally measured values of β for important technical fluids as e.g., air, nitrogen, carbon dioxide, and ammonia where, in the temperature range between −20 and +100°C, just the inequality β > β _perfect gas holds.     

On the unsteady rotational flow of a generalized Maxwell fluid through a circular cylinder

In this paper, the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized Maxwell fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. Initially, the fluid is at rest, and the motion is produced by the rotation of the cylinder about its axis with a unsteady angular velocity. The solutions that have been obtained are presented under series form in terms of the generalized G _a,b,c (, t)-functions. The similar solutions for the ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases, when β → 1, respectively β → 1 and λ → 0, from general solutions. Finally, the solutions that have been obtained are compared by graphical illustrations, and the influence of the pertinent parameters on the fluid motion is also underlined by graphical illustrations.     

Effects of Large Scales Motion in Unstable Stratified Shear Flows

Homogeneous turbulence under unstable uniform stratification (N ² < 0) and vertical shear is investigated by using the linear theory (or the so-called rapid distortion theory, RDT) for an initial isotropic turbulence over a range −∞ ≤ R _i =N ²/S ² ≤ 0. The initial potential energy is zero and P _r =1 (i.e. the molecular Prandtl number).One-dimensional (streamwise) k ₁−spectra, especially Θ₃₃(k ₁) (i.e., that of the vertical kinetic energy, are investigated. In agreement with previous experiments, it is found that the unstable stratification affects the turbulence quantities at all scales. A significant increase of the vertical kinetic energy is observed at low wavenumbers k ₁ (i.e. large scales) due to an increase of the stratification . The effect of the shear (S) is appreciable only at high wavenumbers k ₁ (i.e. small scales).Based on the importance of the spectral components with k ₁ = 0, the asymptotic forms of Θ _ij (k ₁ = 0) or equivalently the so-called “two-dimensional” energy components (2DEC) are analyzed in detail. The asymptotic form for the ratio of 2DEC is compared to the long-time limit of the ratio of real energies. In the unstable stratified shearless case (S=0,N ² ≠ 0) the variances and the covariances of the velocity and the density are derived analytically in terms of the Weber functions, while when S ≠ 0 and N ² ≠ 0 they are obtained numerically (−100 ≤ R _i <0 and . The results are discussed in connection to previous experimental results in unstable stratified open channel flows cooled from above by Komori et al. Phy Fluids 25, 1539–1546 (1982).It is shown that the Richardson number dependence of the long-time limit of the ratios of real energies is well described by this “simple” model (i.e. the dependence of the long-time limit of 2DEC on R _i ). For example, the ratio of the potential energy to the kinetic energy (q ²/2), approaches −R _i /(1−R _i ), the ratio of turbulent energy production by buoyancy forces to production by shearing forces (i.e. the flux Richardson number, R _if ), approaches R _i . Also, the Richardson number dependence of the principal angle (β) of the Reynolds stress tensor and the angle (β_ρ) of the scalar flux vector is fairly predicted by this model .On the other hand, it is found that the above ratios are insensitive to viscosity, while the ratios ɛ /q ² and , depend on the viscosity and they evolve asymptotically like t ⁻¹. The turbulent Froude number, F _rt =(L _Oz /L _E )^2/3, where L _Oz and L _E are the Ozmidov length scale and the Ellison length scale, respectively, evolves asymptotically like t ^−1/3.     

泡沫铝材料动态本构参数的实验确定

基于泡沫材料的动态刚性-线性硬化塑性-刚性卸载(D-R-LHP-R)模型,结合连续性方程,动量守恒方程及刚体的运动方程,得到了激波在泡沫材料中的量纲一消失位置X_s/L₀和动态屈服应力Y_i、激波波速c_p、冲击初始应变ε_i之间的如下关系式: $\frac{X_{\mathrm{s}}}{L_{0}}=\exp \left(-\frac{\rho_{0} c_{\mathrm{p}} v_{\mathrm{i}}}{Y}\right)=\exp \left(1-\frac{\sigma_{\mathrm{i}}}{Y}\right)=\exp \left(-\frac{\rho_{0} c_{\mathrm{p}}^{2} \varepsilon_{\mathrm{i}}}{Y}\right)$ 采用Taylor-Hopkinson装置进行实验,当直接测得泡沫铝试样密度ρ₀、边界初始应力σ_i、初始打击速度v_i、泡沫铝杆原长L₀及激波在泡沫铝杆中消失长度X_s后,利用方程式(a)可反演求得D-R-LHP-R模型下的泡沫铝动态应力应变曲线。最后通过与泡沫铝准静态实验数据对比,表明该泡沫铝是应变率敏感性材料。     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.