Mixed convection over a horizontal plate with streamwise non-uniform surface temperature distribution

Numerical investigation on mixed convection of a two-dimensional incompressible laminar flow over a horizontal flat plate with streamwise sinusoidal distribution of surface temperature has been performed for different values of Rayleigh number, Reynolds number and frequency of periodic temperature for constant Prandtl number and amplitude of periodic temperature. Finite element method adapted to rectangular non-uniform mesh elements by a non-linear parametric solution algorithm basis numerical scheme has been employed. The investigating parameters are the Rayleigh number, the Reynolds number and frequency of periodic temperature. The effect of variation of individual investigating parameters on mixed convection flow characteristics has been studied to observe the hydrodynamic and thermal behavior for while keeping the other parameters constant. The fluid considered in this study is air with Prandtl number 0.72. The results are obtained for the Rayleigh number range of 102 to 104, Reynolds number ranging from 1 to 100 and the frequency of periodic temperature from 1 to 5. Isotherms, streamlines, average and local Nusselt numbers are presented to show the effect of the different values of aforementioned investigating parameters on fluid flow and heat transfer.

The role of randomly mixed-layered chlorite/smectite in the transformation of smectite to chlorite

Vesicular and groundmass phyllosilicates in a hydrothermally altered basalt from the Point Sal ophiolite, California, have been studied using transmission electron microscopy (TEM). Pore-filling phyllosilicates are texturally characterized as having coherent, relatively thick and defect-free crystals of chlorite (14 Å) with occasional 24-Å periodicities. Groundmass phyllosilicates are texturally characterized as 1) randomly oriented crystals up to 200 Å in width and 2) larger, more coherent crystals up to 1000 Å in width. Small crystallites contain predominantly 14-Å layers with some 24-Å units. Large crystals show randomly interlayered chlorite/smectite (C/S), with approximately 50% chlorite on average. Adjacent smectite-like layers are not uncommon in the groundmass phyllosilicates. Electron microprobe analyses show that Fe/Mg ratios of both groundmass and vesicular phyllosilicates are fairly constant. Termination of brucite-like interlayers has been identified in some of the TEM images. The transformation mechanisms represented by these layer terminations are 1) growth of a brucite-like interlayer within smectite interlayer regions and 2) the dissolution and reprecipitation of elements to form chlorite layers. Both mechanisms require an increase in volume as smectite transforms to chlorite. The data, combined with that from previously published reports, suggest that randomly interlayered C/S is a metastable phase formed in microenvironments with low water/rock ratios. Chlorite forms in microenvironments in the same sample dominated by higher water/rock ratios. The relatively constant number of Mg's in the structure (Mg#) of both structures indicates that in both microenvironments the bulk rock composition has influence over the composition of phyllosilicates.

Integrated TEM, XRD and electron-microprobe investigation of mixed-layer chlorite smectite from the Point-Sal ophiolite, California

Five basalt samples from the Point Sal ophiolite, California, were examined using HRTEM and AEM in order to compare observations with interpretations of XRD patterns and microprobe analyses. XRD data from ethylene-glycol-saturated samples indicate the following percentages of chlorite in mixed-layer chlorite-smectite identified for each specimen: (i) L2036 almost-equal-to 50%, (ii) L2035 almost-equal-to 70 and 20%, (iii) 1A-13 almost-equal-to 70%, (iv) 1B-42 almost-equal-to 70%, and (v) 1B-55 = 100%. Detailed electron microprobe analyses show that 'chlorite' analyses with high Si, K, Na and Ca contents are the result of interlayering with smectite-like layers. The Fe/(Fe + Mg) ratios of mixed-layer phyllosilicates from Point Sal samples are influenced by the bulk rock composition, not by the percentage of chlorite nor the structure of the phyllosilicate. Measurements of lattice-fringe images indicate that both smectite and chlorite layers are present in the Point Sal samples in abundances similar to those predicted with XRD techniques and that regular alternation of chlorite and smectite occurs at the unit-cell scale. Both 10- and 14-angstrom layers were recorded with HRTEM and interpreted to be smectite and chlorite, respectively. Regular alternation of chlorite and smectite (24-angstrom periodicity) occurs in upper lava samples L2036 and 1A-13, and lower lava sample 1B-42 for as many as seven alternations per crystallite with local layer mistakes. Sample L2035 shows disordered alternation of chlorite and smectite, with juxtaposition of smectite-like layers, suggesting that randomly interlayered chlorite (< 0.5)-smectite exists. Images of lower lava sample 1B-55 show predominantly 14-angstrom layers. Units of 24 angstrom tend to cluster in what may otherwise appear to be disordered mixtures, suggesting the existence of a corrensite end-member having thermodynamic significance.

High-resolution imaging of ordered mixed-layer clays

HRTEM has been used to examine illite/smectite from the Mancos shale, rectorite from Garland County, Arkansas; illite from Silver Hill, Montana; Na-smectite from Crook County, Wyoming; corrensite from Packwood, Washington; and diagenetic chlorite from the Tuscaloosa formation. Thin specimens were prepared by ion milling, ultra-microtome sectioning and/or grain dispersal on a porous carbon substrate. Some smectite-bearing clays were also examined after intercalation with dodecylamine hydrochloride (DH). Intercalation of smectite with DH proved to be a reliable method of HRTEM imaging of expanded smectite, d(001) 16 A which could then be distinguished from unexpanded illite, d(001) 10 A. Lattice fringes of basal spacings of DH-intercalated rectorite and illite/smectite showed 26 A periodicity. These data support XRD studies which suggest that these samples are ordered, interstratified varieties of illite and smectite. The ion-thinned, unexpanded corrensite sample showed discrete crystallites containing 10 A and 14 A basal spacings corresponding with collapsed smectite and chlorite, respectively. Regions containing disordered layers of chlorite and smectite were also noted. Crystallites containing regular alternations of smectite and chlorite were not common. These HRTEM observations of corrensite did not corroborate XRD data. Particle sizes parallel to the c axis ranged widely for each sample studied, and many particles showed basal dimensions equivalent to > five layers. -J.M.H.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focused recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the "best" choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.

Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

Order conditions of stochastic Runge-Kutta methods by B-series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Numerical solutions of stochastic differential equations – implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

Cement-in-cement revision for selected Vancouver Type B1 femoral periprosthetic fractures : A biomechanical analysis

The aim of this study was to perform a biomechanical analysis of the cement-in-cement (c-in-c) technique for fixation of selected Vancouver Type B1 femoral periprosthetic fractures and to assess the degree of cement interposition at the fracture site. Six embalmed cadaveric femora were implanted with a cemented femoral stem. Vancouver Type B1 fractures were created by applying a combined axial and rotational load to failure. The femora were repaired using the c-in-c technique and reloaded to failure. The mean primary fracture torque was 117 Nm (SD 16.6, range 89–133). The mean revision fracture torque was 50 Nm (SD 16.6, range 29–74), which is above the torque previously observed for activities of daily living. Cement interposition at the fracture site was found to be minimal.

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.

Visualization of thermal characteristics around the absorber tube of a standard parabolic trough thermal collector by 3D simulation

Three dimensional conjugate heat transfer simulation of a standard parabolic trough thermal collector receiver is performed numerically in order to visualize and analyze the surface thermal characteristics. The computational model is developed in Ansys Fluent environment based on some simplified assumptions. Three test conditions are selected from the existing literature to verify the numerical model directly, and reasonably good agreement between the model and the test results confirms the reliability of the simulation. Solar radiation flux profile around the tube is also approximated from the literature. An in house macro is written to read the input solar flux as a heat flux wall boundary condition for the tube wall. The numerical results show that there is an abrupt variation in the resultant heat flux along the circumference of the receiver. Consequently, the temperature varies throughout the tube surface. The lower half of the horizontal receiver enjoys the maximum solar flux, and therefore, experiences the maximum temperature rise compared to the upper part with almost leveled temperature. Reasonable attributions and suggestions are made on this particular type of conjugate thermal system. The knowledge that gained so far from this study will be used to further the analysis and to design an efficient concentrator photovoltaic collector in near future.