In this paper, the fluid flow and heat transfer around a circular cylinder are studied under various conditions (Reynolds number 10 < Re < 200; Prandtl number, 0.1 ≤ Pr ≤ 2). To solve the governing equations, we use the simplified thermal lattice Boltzmann model based on double-distribution function approach, and present a corresponding boundary treatment for both velocity and temperature fields. Extensive numerical results have been obtained to the flow and heat transfer behaviors. The vortices and temperature evolution processes indicate that the flow and temperature fields change synchronously, and the vortex shedding plays a determinant role in the heat transfer. Furthermore, the effects of Reynolds and Prandtl number on the flow and isothermal patterns and local and averaged Nusselt numbers are discussed in detail. Our simulations show that the local and averaged Nusselt numbers increase with the Reynolds and Prandtl numbers, irrespective of the flow regime. However, the minimum value of the local Nusselt number can shift from the rear point at the back of the cylinder with higher Prandtl number even in the steady flow regime, and the distribution of the local Nusselt number is almost monotonous from front stagnation point to rear stagnation point with lower Prandtl number in the unsteady flow regime.

We present a promising approach to reduce the difficulties associated with meshingcomplex curved domain boundaries for higher-order finite elements. In this work,higher-order XFEM analyses for strong discontinuity in the case of linear elasticityproblems are presented. Curved implicit boundaries are approximated inside an unstructuredcoarse mesh by using parametric information extracted from the parametric representation(the most common in Computer Aided Design CAD). This approximation provides local gradedsub-mesh (GSM) inside boundary elements (i.e. an element split by the curved boundary)which will be used for integration purpose. Sample geometries and numerical experimentsillustrate the accuracy and robustness of the proposed approach.

The problem of finding the probability distribution of the first hitting time of a double integral process (DIP) such as the integrated Wiener process (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise-cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a Monte Carlo method.

We study the (two-sided) exit time and position of a random walk outside boundaries which are regularly varying functions of smaller order at infinity than the square root. A natural domain of interest is those random walks which are attracted without centring to a normal law, or are relatively stable. These are shown to have ‘stable’ exit positions, in that the overshoot of the curved boundary is of smaller order of magnitude (in probability) than the boundary, as the boundary expands. Surprisingly, this remains true regardless of the shape of the boundary. Furthermore, within the same natural domain of interest, norming of the exit position by, for example, the square root of the exit time (in the finite-variance case), produces limiting distributions which are computable from corresponding functionals of Brownian motion. We give a functional limit theorem for attraction of normed sums to general infinitely divisible random variables, as a means of making this, and more general, computations. These kinds of theorems have applications in sequential analysis, for example.

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t1(1 + t2), .

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres.

A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.

Daniels (1974) reduced the problem of approximating the distribution of the maximum size of a closed epidemic to that of finding the distribution of max0≦t≦2 {W(t) – N1/2c(t)}, where c is a smooth function with a unique minimum of 0 at t = 1, and he derived an approximation to this distribution which he showed to be accurate to order N–1/4. In this paper, his approximation is shown to be accurate to order N–1/3, and a refined approximation is given which is accurate to order N–1/2 log N. The new approximation is still normal, and its accuracy is similar to that of the original approximation of a discrete process by the Wiener process.

An approximation is found to the distribution of the maximum number of infectives present at any time during the course of a closed epidemic. The technique used is applicable to a commonly occurring type of random walk problem where there is a curved absorbing boundary which is far from the mean path except over a narrow range.