Optimal Control for an Obstruction Problem

A question of flow around an obstacle leads to an optimal control problem. If an optimum path exists, then it is calculable from the Pontryagin principle. The optimum is verified to be reached, using a discretization of the problem.     

Numerical Approaches to Optimal Control of a Model Equation for Shear Flow Instabilities

We investigate two different discretization approaches of a model optimal-control problem, chosen to be relevant for control of instabilities in shear flows. In the first method, a fully discrete approach has been used, together with a finite-element spatial discretization, to obtain the objective function gradient in terms of a discretely-derived adjoint equation. In the second method, Chebyshev collocation is used for spatial discretization, and the gradient is approximated by discretizing the continuously-derived adjoint equation. The discrete approach always results in a faster convergence of the conjugate-gradient optimization algorithm. Due to the shear in the convective velocity, a low diffusivity in the problem complicates the structure of the computed optimal control, resulting in particularly noticeable differences in convergence rate between the methods. When the diffusivity is higher, the control becomes less complicated, and the difference in convergence rate reduces. The use of approximate gradients results in a higher sensitivity to the degrees of freedom in time. When the system contains a strong instability, it only takes a few iteration to obtain an effective control for both methods,even if there are differences in the formal convergence rate. This indicates that it is possible to use the approximative gradients of the objective function in cases where the control problem mainly consists of controlling strong instabilities. This revised version was published online in July 2006 with corrections to the Cover Date.     

更新过程理论在交通流上的一个应用

一个更新过程的两个随机变量的分布:间隔分布、计数分布是1—1对应的,但由间隔分布求对应的计数分布的问题尚未很好地解决。在道路断面观测交通流可得到一更新过程,车头时距和车辆到达分别是其间隔和计数。时距分布容易观测得到,而到达分布的观测却较难。因此上述数学问题的解决对交通流理论是非常有意义的,本文将研究之。     

Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0＋ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω1 and Ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.     

Investigation and application of point implicit Runge–Kutta methods to inviscid flow problems

We derive and investigate point implicit Runge–Kutta methods to significantly improve the convergence rate to approximate steady‐state solutions of inviscid flows. It turns out that the point implicit Runge–Kutta can be interpreted as a preconditioned explicit Runge–Kutta method, where the preconditioner arises naturally as local derivative of the residual function. Moreover, many preconditioners suggested in the literature so far are identified as special case of our general ansatz. Conditions will be formulated such that explicit Runge–Kutta methods with local time stepping are equivalent to point implicit methods. In numerical examples, we will demonstrate the improved convergence rates. Copyright © 2011 John Wiley & Sons, Ltd.     

Accuracies and conservation errors of various ghost fluid methods for multi-medium Riemann problem

Since the (original) ghost fluid method (OGFM) was proposed by Fedkiw et al. in 1999 [5], a series of other GFM-based methods such as the gas–water version GFM (GWGFM), the modified GFM (MGFM) and the real GFM (RGFM) have been developed subsequently. Systematic analysis, however, has yet to be carried out for the various GFMs on their accuracies and conservation errors. In this paper, we develop a technique to rigorously analyze the accuracies and conservation errors of these different GFMs when applied to the multi-medium Riemann problem with a general equation of state (EOS). By analyzing and comparing the interfacial state provided by each GFM to the exact one of the original multi-medium Riemann problem, we show that the accuracy of interfacial treatment can achieve “third-order accuracy” in the sense of comparing to the exact solution of the original mutli-medium Riemann problem for the MGFM and the RGFM, while it is of at most “first-order accuracy” for the OGFM and the GWGFM when the interface approach is actually near in balance. Similar conclusions are also obtained in association with the local conservation errors. A special test method is exploited to validate these theoretical conclusions from the numerical viewpoint.     

Free surface Stokes flows obstructed by multiple obstacles

Gravity‐driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far‐field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in‐line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solutions of peristaltic flow of Williamson fluid with radially varying MHD in an endoscope

In this article we have studied the peristaltic motion of an incompressible Williamson fluid with constant and radially varying magnetohydrodynamics (MHD) in an endoscope. Using the low Reynolds and long wavelength assumptions, the equations of Williamson fluid model in simplified form are solved using (i) the HAM method and (ii) the Shooting method. The comparisons of both solutions have been found a very good agreement. Graphical results have been presented for various emerging parameters. © British Crown Copyright 2010/MOD. Reproduced with permission. Published by John Wiley & Sons, Ltd.     

A hybrid FVM–LBM method for single and multi‐fluid compressible flow problems

The lattice Boltzmann method (LBM) has established itself as an alternative approach to solve the fluid flow equations. In this work we combine LBM with the conventional finite volume method (FVM), and propose a non‐iterative hybrid method for the simulation of compressible flows. LBM is used to calculate the inter‐cell face fluxes and FVM is used to calculate the node parameters. The hybrid method is benchmarked for several one‐dimensional and two‐dimensional test cases. The results obtained by the hybrid method show a steeper and more accurate shock profile as compared with the results obtained by the widely used Godunov scheme or by a representative flux vector splitting scheme. Additional features of the proposed scheme are that it can be implemented on a non‐uniform grid, study of multi‐fluid problems is possible, and it is easily extendable to multi‐dimensions. These features have been demonstrated in this work. The proposed method is therefore robust and can possibly be applied to a variety of compressible flow situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Hydraulic analysis of free-surface flows with solidification

Summary The paper is concerned with a one-dimensional analysis of plane open-channel flow with continuous solidification. The process is of relevance for recent developments in the casting of steel and other metals. The bottom of the channel consists of a rotating casting roll and a horizontal cooling table, where the solidified material is withdrawn with given velocity. The study is restricted to the region downstream of the top of the casting roll. Surface tension is neglected. In the main part of the analysis inviscid fluid flow is considered since the Reynolds number is very large in the applications. It is found that the steady-state solutions are nonunique in a certain parameter range. In addition to a continuous solution, there are two solutions including hydraulic jumps, with one hydraulic jump being located on the casting roll, the other one on the cooling table. Regarding the stability of the non unique solutions, the evolution of disturbances is investigated numerically as an initial-value problem. It is concluded that the hydraulic jump on the cooling table is unstable, while the other discontinuous solution as well as the continuous solution are stable for sufficiently small disturbances. Which stable solution is attained in the steady state, depends on the history of the process. Friction at the liquid/solid interface is taken into account in the last part of the analysis. A constant friction coefficient is assumed. It is found that the history of the process determines the steady-state solution if, and only if, the friction coefficient is sufficiently small. For larger values of the friction coefficient, the steady-state solution is unique and independent of the history of the transient process. Furthermore, for sufficiently large friction coefficients, stable hydraulic jumps are found, in contrast to the inviscid case, also on the cooling table. Received 19 March 1999; accepted for publication 3 May 1999