Adomian decomposition method and Padé approximants for solving the Blaszak--Marciniak lattice

The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.     

Three-dimensional MHD flow over a shrinking sheet: Analytical solution and stability analysis

The magnetohydrodynamic(MHD) steady and unsteady axisymmetric flows of a viscous fluid over a two-dimensional shrinking sheet are addressed. The mathematical analysis is carried out in the presence of a large magnetic field. The steady state problem results in a singular perturbation problem having an infinite domain singularity. The secular term appearing in the solution is removed and a two-term uniformly valid solution is derived using the Lindstedt–Poincaré technique. This asymptotic solution is validated by comparing it with the numerical solution. The solution for the unsteady problem is also presented analytically in the asymptotic limit of large magnetic field. The results of velocity profile and skin friction are shown graphically to explore the physical features of the flow field. The stability analysis of the unsteady flow is made to validate the asymptotic solution.     

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.     

Topology optimization of unsteady incompressible Navier–Stokes flows

This paper discusses the topology optimization of unsteady incompressible Navier–Stokes flows. An optimization problem is formulated by adding the artificial Darcy frictional force into the incompressible Navier–Stokes equations. The optimization procedure is implemented using the continuous adjoint method and the finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier–Stokes flows are presented. Numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady Navier–Stokes flows.     

Computation of three-dimensional Brinkman flows using regularized methods

The Brinkman equations of fluid motion are a model of flows in a porous medium. We develop the exact solution of the Brinkman equations for three-dimensional incompressible flow driven by regularized forces. Two different approaches to the regularization are discussed and compared on test problems. The regularized Brinkman model is also applied to the unsteady Stokes equation for oscillatory flows since the latter leads to the Brinkman equations with complex permeability parameter. We provide validation studies of the method based on the flow and drag of a solid sphere translating in a Brinkman medium and the flow inside a cylindrical channel of circular cross-section. We present a numerical example of a swimming organism in a Brinkman flow which shows that the maximum swimming speed is obtained with a small but non-zero value of the porosity. We also demonstrate that unsteady Stokes flows with oscillatory forcing fall within the same framework and are computed with the same method by applying it to the motion of the oscillating feeding appendage of a copepod.     

Cytotoxic effects and in vitro reversal of multidrug resistance by therapeutic ultrasound in human hepatocarcinoma cell line (HepG2)

Multidrug resistance (MDR) is one of the major obstacles to successful chemotherapy of human malignancies. Although many strategies have been explored to overcome MDR, none of them have been proven to be clinically useful until now. The aim of this study was to investigate whether a novel therapeutic ultrasound (US) approach would have useful effects on the reversal of MDR in cancer cells. Wild-type and MDR phenotype (HepG2/ADM) cells of the human hepatocarcinoma cell line HepG2 were exposed to 0.8 MHz US at an intensity of 0.43 W/cm² for a 9 s exposure (total energy density: 3.87 J/cm²). After US exposure, cell number and viability were counted immediately, and flow cytometry was performed to measure retention of rhodamine 123 and adriamycin in HepG2 and HepG2/MDR cells. Both cell lines were then incubated in suspension with adriamycin, vincristine, etoposide, cisplatin and 5-fluorouracil, respectively, and the MTT assay was used to determine cytotoxicity. The results showed that US exposure could significantly increase the uptake of Rh123 and ADM by HepG2/ADM tumor cells. The resistant index for the chemotherapeutic drugs was significantly lower in the US-exposed HepG2/ADM cells than in those not exposed to US. It was therefore concluded that US exposure could enhance the sensitivity of HepG2/ADM tumor cells to these chemotherapeutic agents, and the functional and structural changes induced by previous US exposure in MDR tumor cells may be responsible for it. However, further study is needed to investigate the mechanism behind US-mediated reversal of MDR.     

任意马赫数非定常流动数值模拟的统一算法

发展适用于从低速到高速任意马赫数非定常流动数值模拟的统一算法.通过引入一个伪时间导数项和一个新的预处理矩阵,得到双时间非定常预处理可压缩Navier-Stokes方程.方程的对流项采用三阶Roe通量近似差分格式离散,粘性项采用二阶中心差分格式离散.基于数值通量的线性化技术,实现伪时间步的隐式ADI-LU格式迭代,进而获得物理时间步的二阶推进精度.重点以低马赫数流动为例,求解了圆柱绕流和NACA0015翼型等速上仰动态失速问题.计算结果表明该统一算法能够较好地模拟低马赫数乃至任意马赫数非定常流动.     

Series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field

In this Letter, the boundary-layer equation of flow over a nonlinearly stretching sheet in the presence of a chemical reaction and a magnetic field is investigated by employing the Adomian decomposition method (ADM). The series solution of the governing nonlinear problem is developed. The present solution is shown to agree very well with the existing solution.     

高阶精度CE/SE算法及其应用

对时-空守恒元解元算法(CE/SE)的网格设置做较大改进,提出一种新的六面体解元和元定义;同时在解元中对物理量进行高阶Taylor展开,给出一种在时间和空间上均具有高阶精度CE/SE算法.在此基础上,把新型的高阶精度CE/SE算法推广应用于高速流动捕捉激波间断、气相化学反应流动、计及固体动态效应的流体-弹塑性流动和非稳态多相不可压缩粘性流动中.数值实践表明,提出的新型网格结构上的高阶精度CE/SE算法具有算法简单、计算精度高、计算效率和计算效果好的优点,并大大改进和拓展了CE/SE算法的应用范围.     

含双时间步法的化学非平衡流解耦算法

发展基于隐式双时间步法的化学非平衡流解耦型计算方法.采用算子分裂法对流动和反应进行解耦处理,流动方程组通过双时间步方法求解;源项方程组采用二阶梯形公式迭代求解;提出"源项消去"法,以消除化学反应源项对流动求解引入的误差,从而保证流动方程组求解的时间精度.理论分析和计算结果表明,方法既可以保证双时间步法的求解效率,又可以获得比较精确的非定常计算结果.     

Dark and kink soliton solutions of the generalized ZK–BBM equation by iterative scheme

The generalized ZK–BBM equation is solved using iterative scheme of the Adomian decomposition method (ADM) and variational iteration method (VIM). A dark and a kink soliton solutions of the generalized ZK–BBM equation are obtained under initial conditions. The convergence analysis of the ADM and VIM solution shows that these solutions are convergent. The comparison of the ADM and VIM solutions with the exact solution shows that the solutions of the generalized ZK–BBM equation by the iterative methods are almost exact. The absolute errors show that the accuracy and efficiency of the ADM and VIM depend on the problem and its domain. It is found that the iterative scheme of Adomian decomposition method and variational iteration method are quite efficient for the soliton solution of the generalized ZK–BBM equation.     

Fast unsteady flow computations with a Jacobian-free Newton–Krylov algorithm

Despite the advances in computer power and numerical algorithms over the last decades, solutions to unsteady flow problems remain computing time intensive. Especially for large Reynolds number flows, nonlinear multigrid, which is commonly used to solve the nonlinear systems of equations, converges slowly. The stiffness induced by the large aspect ratio cells and turbulence is not tackled well by this solution method.     

Vortical and acoustical mode coupling inside a porous tube with uniform wall suction

This paper considers the oscillatory motion of gases inside a long porous tube of the closed-open type. In particular, the focus is placed on describing an analytical solution for the internal acoustico-vortical coupling that arises in the presence of appreciable wall suction. This unsteady field is driven by longitudinal oscillatory waves that are triggered by small unavoidable fluctuations in the wall suction speed. Under the assumption of small amplitude oscillations, the time-dependent governing equations are linearized through a regular perturbation of the dependent variables. Further application of the Helmholtz vector decomposition theorem enables us to discriminate between acoustical and vortical equations. After solving the wave equation for the acoustical contribution, the boundary-driven vortical field is considered. The method of matched-asymptotic expansions is then used to obtain a closed-form solution for the unsteady momentum equation developing from flow decomposition. An exact series expansion is also derived and shown to coincide with the numerical solution for the problem. The numerically verified end results suggest that the asymptotic scheme is capable of providing a sufficiently accurate solution. This is due to the error associated with the matched-asymptotic expansion being smaller than the error introduced in the Navier-Stokes linearization. A basis for comparison is established by examining the evolution of the oscillatory field in both space and time. The corresponding boundary-layer behavior is also characterized over a range of oscillation frequencies and wall suction velocities. In general, the current solution is found to exhibit features that are consistent with the laminar theory of periodic flows. By comparison to the Sexl profile in nonporous tubes, the critically damped solution obtained here exhibits a slightly smaller overshoot and depth of penetration. These features may be attributed to the suction effect that tends to attract the shear layers closer the wall.     

The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet

The flow and heat transfer problem of a second grade fluid film over an unsteady stretching sheet is considered. The fluid is incompressible and electrically conducting in the presence of a uniform applied magnetic field. The series solutions of the governing boundary value problems are obtained by employing homotopy analysis method (HAM). The convergence of the developed solutions is discussed explicitly. The dependence of velocity and temperature profiles on various parameters is shown and discussed through graphs. The values of skin-friction coefficient, Nusselt number and free surface temperature are given in tabular form for various emerging parameters.     

Nested Cartesian grid method in incompressible viscous fluid flow

In this work, the local grid refinement procedure is focused by using a nested Cartesian grid formulation. The method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries. A finite-volume formulation based on globally second-order accurate central-difference schemes is adopted here in conjunction with a two-step fractional-step procedure. The key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested-block boundaries, and accurate discretization of the governing equations in cells that are with block-interface as a control-surface. The interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second-order spatial accuracy of the underlying solver, and as a result high efficiency/accuracy nested grid discretization method is developed. Herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows, thereby demonstrating its performance in the solution of various complex flow–structure interactions. The numerical examples include a lid-driven cavity flow and Pearson vortex problems, flow past a circular cylinder symmetrically installed in a channel, flow past an elliptic cylinder at an angle of attack, and flow past two tandem circular cylinders of unequal diameters. For the numerical simulations of flows past bluff bodies an immersed boundary (IB) method has been implemented in which the solid object is represented by a distributed body force in the Navier–Stokes equations. The main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular Cartesian grid and applied to multiple nested-block (Cartesian) structured grids without any difficulty. Through the numerical experiments the strength of the solver in effectively/accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated, in which the nested Cartesian grid method was suitably combined together with the fractional-step algorithm to speed up the solution procedure.     

Augmented versions of the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow flow applications

Shallow water flows are found in a variety of engineering problems always dominated by the presence of bed friction and irregular bathymetry. These source terms determine completely the possible evolution of the flooded area in time. It is well known that appropriate numerical schemes for this type of flows must be well-balanced. Well-balanced numerical schemes are based on the preservation of cases of quiescent equilibrium over variable bed elevation. Commonly they are formulated as an adaptation of numerical solvers defined for cases without source terms. This procedure is insufficient when applied to real situations. Then, it is possible to argue that appropriate numerical schemes cannot arise directly from those derived from the simplest homogeneous case without source terms. New solutions are presented in this work by defining weak solutions that include the presence of source terms. To do that, the solvers presented in this work extend the number of waves in the well known HLL and HLLC solvers involving a stationary jump in the solution. This is done without modifying the original solution vector of conserved quantities. The resulting approximate Riemann solvers include variable bed level surface and friction. Solvers are systematically assessed via a series of test problems with exact solutions for one and two dimensions, including steady and unsteady flow configurations, variation of the flooded area in time and comparisons with experimental data. The obtained results point out that the new method is able to predict faithfully the overall behavior of the solution and of any type of waves.     

Simulation of Vortex Convection in a Compressible Viscous Flow with Dynamic Mesh Adaptation

In this work, vortex convection is simulated using a dynamic mesh adaptation procedure. In each adaptation period, the mesh is refined in the regions where the phenomena evolve and is coarsened in the regions where the phenomena deviate since the last adaptation. A simple indicator of mesh adaptation that accounts for the solution progression is defined. The generation of dynamic adaptive meshes is based on multilevel refinement/coarsening. The efficiency and accuracy of the present procedure are validated by simulating vortex convection in a uniform flow. Two unsteady compressible turbulent flows involving blade-vortex interactions are investigated to demonstrate further the applicability of the procedure. Computed results agree well with the published experimental data or numerical results.     

叶栅内定常及非定常粘性流动数值模拟

本文给出了一个模拟叶栅内准三维定常和非定常粘性流动的数值方法。对于定常流动,采用ＴＶＤ Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式和代数湍流模型求解雷诺平均Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程,使用当地时间步长和多网格技术使计算加速收敛到定常状态;对于非定常流动,使用双时间步长和全隐式离散,采用与求解定常流动相似的多网格方法求解隐式离散方程。文中给出了ＶＫＩ透平叶栅内的定常流结果和１．５级透平叶栅内的非定常数值结果。     

Numerical Simulations of Unsteady Flows from Rarefied Transition to Continuum Using Gas-Kinetic Unified Algorithm

Numerical simulations of unsteady gas flows are studied on the basis of Gas-Kinetic Unified Algorithm (GKUA) from rarefied transition to continuum flow regimes. Several typical examples are adopted. An unsteady flow solver is developed by solving the Boltzmann model equations, including the Shakhov model and the Rykov model etc. The Rykov kinetic equation involving the effect of rotational energy can be transformed into two kinetic governing equations with inelastic and elastic collisions by integrating the molecular velocity distribution function with the weight factor on the energy of rotational motion. Then, the reduced velocity distribution functions are devised to further simplify the governing equation for one- and two-dimensional flows. The simultaneous equations are numerically solved by the discrete velocity ordinate (DVO) method in velocity space and the finite-difference schemes in physical space. The time-explicit operator-splitting scheme is constructed, and numerical stability conditions to ascertain the time step are discussed. As the application of the newly developed GKUA, several unsteady varying processes of one- and two-dimensional flows with different Knudsen number are simulated, and the unsteady transport phenomena and rarefied effects are revealed and analyzed. It is validated that the GKUA solver is competent for simulations of unsteady gas dynamics covering various flow regimes.     

Digital speckle photography and speckle tomography in heat transfer studies

The line-of-sight speckle photography of transparent media is used for quantitative measurements of the instantaneous temperature fields in 3D unsteady flows. Both electronic and photographic methods are employed for specklegram recording. The subsequent specklegram treatment uses the Young's fringes method as well as cross-correlation analysis of small interrogation areas of the recordings. Experimental data for three different heat transfer configurations are obtained and discussed. The first one is natural convection over extended vertical heated plates with forward facing steps, the second is unsteady 3D convective flow around a suddenly heated vertical thin wire, and the third one is a convective plume above a multi-jet flame. Both local and global Nusselt numbers are determined via measuring local surface temperature gradients for these convective flows. The results are compared with Ostrach's theory for a single vertical plate and with the data obtained by Mach–Zehnder interferometry. The 3D temperature fields are reconstructed for axisymmetric convective flows around a suddenly heated vertical wire using quasi-double projection measurement and the Radon inversion. 3D temperature distributions above the combustion zone are reconstructed using multi-projection speckle photography measurements and computerised tomography.