Poiseuille-Bénard流的出口边界条件及其谱元法计算

研究二维矩形管道中底部加热的不可压缩Ｐｏｉｓｅｕｉｌｌｅ－Ｂｅｎａｒｄ流的谱元法数值计算问题．讨论各种不同的出口边界条件的处理及其对谱元法数值模拟的影响．通过干扰区、干扰幅度和计算时间的比较,确定比较理想的出口边界条件．     

Résumé and remarks on the open boundary condition minisymposium

The incompressible Navier-Stokes equations—and their thermal convection and stratified flow analogue, the Boussinesq equations—possess solutions in bounded domains only when appropriate/legitimate boundary conditions (BCs) are appended at all points on the domain boundary. When the boundary—or, more commonly, a portion of it—is not endowed with a Dirichlet BC, we are faced with selecting what are called open boundary conditions (OBCs), because the fluid may presumably enter or leave the domain through such boundaries. The two minisymposia on OBCs that are summarized in this paper had the objective of finding the best OBCs for a small subset of two-dimensional test problems. This objective, which of course is not really well-defined, was not met (we believe), but the contributions obtained probably raised many more questions/issues than were resolved—notable among them being the advent of a new class of OBCs that we call FBCs (fuzzy boundary conditions).     

AUGMENTED LAGRANGIAN METHOD AND OPEN BOUNDARY CONDITIONS IN 2D SIMULATION OF POISEUILLE–BÉNARD CHANNEL FLOW

The main objective of this study is to compare the influence of different boundary conditions upon the incompressible Poiseuille –Bénard channel flow (PBCF) in a 2D rectangular duct heated from below. In a first technical part the algorithm used to carry out this work, based on the augmented Lagrangian method, is presented. The implementation details of the five different open boundary conditions (OBCs) and the periodic boundary conditions (PBCs) tested in the present paper are also given. The study is then carried out for 1800<Ra≤ 10,000, 0<Re≤10 and 0·67≤Pr≤6·4. The five selected OBCs, applied at the outlet of the computational domain, respectively express the following conditions: a square profile for the velocity (OBC1), mass conservation (OBC2), zero second derivative of the horizontal velocity component (OBC3), a mixed boundary condition combining Dirichlet and Neumann conditions (OBC4) and an Orlanski-type boundary condition (OBC5). A good estimation of the perturbation amplitude and of the length of the perturbed zone at the outlet boundary is proposed. It is shown that OBC5 causes very little perturbation in the recirculating flow compared with the other OBCs. © 1997 John Wiley & Sons, Ltd.     

High‐order boundary conditions for linearized shallow water equations with stratification,dispersion and advection

The two‐dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ??, and a high‐order open boundary condition (OBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd.     

Characteristic‐based inlet and outlet boundary conditions for incompressible flows

Determining boundary conditions (BCs) for incompressible flows is such a delicate matter that affects the accuracy of the results. In this research, a new characteristic‐based BC for incompressible Navier‐Stokes equations is introduced. Discretization of equations has been done via finite volume. Additionally, artificial compressibility correction has been employed to deal with equations. Ordinary extrapolation from inner cells of a domain was used as a traditional way to estimate pressure and velocities on solid wall and inlet/outlet boundaries. Here, this method was substituted by the newly proposed BCs based on the characteristics of artificial compressibility equations. To follow this purpose, a computer code has been developed to carry out series of numerical tests for a flow over a backward‐facing step and was applied to a wide range of Reynolds numbers and grid combinations. Calculation of convective and viscous fluxes was done using Jameson's averaging scheme. Employing the characteristic‐based method for determining BCs has shown an improved convergence rate and reduced calculation time comparing with those of traditional ones. Furthermore, with the reduction of domain and computational cells, a similar accuracy was achieved for the results in comparison with the ones obtained from the traditional extrapolation method, and these results were in good agreement with the ones in the literature.     

A three‐dimensional numerical internal tidal model involving adjoint method

A three‐dimensional internal tidal model involving the adjoint method is constructed based on the nonlinear, time‐dependent, free‐surface hydrodynamic equations in spherical coordinates horizontally, and isopycnic coordinates vertically, subject to the hydrostatic approximations. This model consists of two submodels: the forward model is used for the simulation of internal tides, while the adjoint model is used for optimization of modal parameters. Mode splitting technique is employed in both forward and adjoint models. In this model, the adjoint method is employed to estimate model parameters by assimilating the interior observations. As a preliminary feasibility study, a set of ideal experiments with the model‐generated pseudo‐observations of surface currents are performed to invert the open boundary conditions (OBCs). In the ideal experiments, 14 kinds of bottom topographies and six kinds of predetermined distributions of OBCs are considered to examine their influence on experiment results. The inversion obtained satisfying results and all the predetermined distributions were successfully inverted. Analysis of results suggests the following: in the case where the spatial variation of the OBC distribution is great or the open boundary is close to a rough topography, the results will be comparatively poor, but still satisfactory; both the tidal elevations and currents can be simulated very accurately with the surface currents at several observation points; the assimilation precision could be reliable and able to reflect both of the inversion and simulation results in the whole field. The performance and results of ideal experiments give a preliminary indication that the construction of this model is successful. Copyright © 2011 John Wiley & Sons, Ltd.     

Unsteady supersonic flow over an oscillatory aerofoil using a second-order TVD scheme

The unsteady flow over an oscillatory NACA0012 aerofoil has been simulated by the calculation with Euler equations. The equations are discretized by an implicit Euler in time, and a second-order space-accurate TVD scheme based on flux vector splitting with van Leer's limiter. Modified eigenvalues are proposed to overcome the slope discontinuities of split eigenvalues at Mach = 0·0 and ± 1·0, and to generate a bow shock in front of the aerofoil. A moving grid system around the aerofoil is generated by Sorenson's boundary fitted co-ordinates for each time step. The calculations have been done for two angles of attack θ = 5·0° sin (ωt) and θ = 3·0° + 3·0° sin (ωt) for the free-stream Mach numbers 2·0 and 3·0. The results show that pressure and Mach cells flow along characteristic lines. To examine unsteady effects, the responses of wall pressure and normal force coefficients are analysed by a Fourier series expansion.     

Consistent inlet and outlet boundary conditions for particle methods

In this paper, simple and consistent open boundary conditions are presented for the numerical simulation of viscous incompressible laminar flows. The present approach is based on an arbitrary Lagrangian-Eulerian particle method using upwind interpolation. Three kinds of inlet/outlet boundary conditions are proposed for particle methods, a pressure specified inlet/outlet condition, a velocity profile specified inlet/outlet condition, and a fully developed flow outlet condition. These inlet/outlet conditions are realized by using boundary particles and modification to the physical value such as velocity. Poiseuille flows, flows over a backward-facing step, and flows in a T-shape branch are calculated. The results are compared with those of mesh-based methods such as the finite volume method. The method presented herein exhibits accuracy and numerical stability.     

Simulation of dynamic fluid–solid interactions with an improved direct-forcing immersed boundary method

Dynamic fluid–solid interactions are widely found in chemical engineering, such as in particle-laden flows, which usually contain complex moving boundaries. The immersed boundary method (IBM) is a convenient approach to handle fluid–solid interactions with complex geometries. In this work, Uhlmann's direct-forcing IBM is improved and implemented on a supercomputer with CPU–GPU hybrid architecture. The direct-forcing IBM is modified as follows: the Poisson's equation for pressure is solved before evaluation of the body force, and the force is only distributed to the Cartesian grids inside the immersed boundary. A multidirect forcing scheme is used to evaluate the body force. These modifications result in a divergence-free flow field in the fluid domain and the no-slip boundary condition at the immersed boundary simultaneously. This method is implemented in an explicit finite-difference fractional-step scheme, and validated by 2D simulations of lid-driven cavity flow, Couette flow between two concentric cylinders and flow over a circular cylinder. Finally, the method is used to simulate the sedimentation of two circular particles in a channel. The results agree very well with previous experimental and numerical data, and are more accurate than the conventional direct-forcing method, especially in the vicinity of a moving boundary.     

Convection regime flow in a vertical slot: continuum of solutions from capped to open ends

In a recent publication Bühler (Heat Mass Transfer 39:631–638, 2003) reported new results for conduction regime flow between vertical differentially-heated walls that provide a continuum of solutions between capped and open ends. In this paper we extend Bühler’s work to realize a continuum of solutions of convection regime flow using empirical results for the vertical temperature gradient that develops in tall aspect ratio geometries. The mass flux is determined analytically for this three-parameter family of solutions. Identical viscous and thermal boundary layers exist at the opposing walls when the cavity is capped. However, as the flow evolves to one with open ends, there is an intensification (attenuation) of the boundary layers near the hot (cold) walls. In the limit corresponding to an open-ended cavity, the boundary layer at the cold wall vanishes altogether.     

Vortex‐in‐cell method combined with a boundary element method for incompressible viscous flow analysis

In this study, an immersed boundary vortex‐in‐cell (VIC) method for simulating the incompressible flow external to two‐dimensional and three‐dimensional bodies is presented. The vorticity transport equation, which is the governing equation of the VIC method, is represented in a Lagrangian form and solved by the vortex blob representation of the flow field. In the present scheme, the treatment of convection and diffusion is based on the classical fractional step algorithm. The rotational component of the velocity is obtained by solving Poisson's equation using an FFT method on a regular Cartesian grid, and the solenoidal component is determined from solving an integral equation using the panel method for the convection term, and the diffusion term is implemented by a particle strength exchange scheme. Both the no‐slip and no‐through flow conditions associated with the surface boundary condition are satisfied by diffusing vortex sheet and distributing singularities on the body, respectively. The present method is distinguished from other methods by the use of the panel method for the enforcement of the no‐through flow condition. The panel method completes making use of the immersed boundary nature inherent in the VIC method and can be also adopted for the calculation of the pressure field. The overall process is parallelized using message passing interface to manage the extensive computational load in the three‐dimensional flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

A weakly compressible MPS method for modeling of open‐boundary free‐surface flow

A mesh‐free particle method, based on the moving particle semi‐implicit (MPS) interaction model, has been developed for the simulation of two‐dimensional open‐boundary free‐surface flows. The incompressibility model in the original MPS has been replaced with a weakly incompressible model. The effect of this replacement on the efficiency and accuracy of the model has been investigated. The new inflow–outflow boundary conditions along with the particle recycling strategy proposed in this study extend the application of the model to open‐boundary problems. The final model is able to simulate open‐boundary free surface flow in cases of large deformation and fragmentation of free surface. The models and proposed algorithms have been validated and applied to sample problems. The results confirm the model's efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient edge‐based level set finite element method for free surface flow problems

We present an efficient technique for the solution of free surface flow problems using level set and a parallel edge‐based finite element method. An unstructured semi‐explicit solution scheme is proposed. A custom data structure, obtained by blending node‐based and edge‐based approaches is presented so to allow a good parallel performance. In addition to standard velocity extrapolation (for the convection of the level set function), an explicit extrapolation of the pressure field is performed in order to impose both the pressure boundary condition and the volume conservation. The latter is also improved with a modification of the divergence free constrain. The method is shown to allow an efficient solution of both simple benchmark cases and complex industrial examples. Copyright © 2012 John Wiley & Sons, Ltd.     

幂律流体圆柱绕流的格子波尔兹曼模拟

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明:插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.      

Symmetry techniques for the numerical solution of the 2D Euler equations at impermeable boundaries

The implementation of boundary conditions at rigid, fixed wall boundaries in inviscid Euler solutions by upwind, finite volume methods is considered. Some current methods are reviewed. Two new boundary condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique are then presented. Their behaviour in relation to the problem of the subsonic flow about blunt and slender elliptic bodies is analysed. The subsonic flow inside the Stanitz elbow is then computed. The symmetry technique is proven to be as accurate as one of the current methods, second-order pressure extrapolation technique. Finally, for arbitrary curved geometries, dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown. © 1998 John Wiley & Sons, Ltd.     

Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem

Viscous flow in elastic (collapsible) tubes is a large-displacement fluid-structure interaction problem frequently encountered in biomechanics. This paper presents a robust and rapidly converging procedure for the solution of the steady three-dimensional Stokes equations, coupled to the geometrically non-linear shell equations which describe the large deformations of the tube wall. The fluid and solid equations are coupled in a segregated method whose slow convergence is accelerated by an extrapolation procedure based on the scheme's asymptotic convergence behaviour. A displacement control technique is developed to handle the system's snap-through behaviour. Finally, results for the tube's post-buckling deformation and for the flow in the strongly collapsed tube are shown. © 1998 John Wiley & Sons, Ltd.     

Computational analysis and flow structure of a transitional separated-reattached flow over a surface mounted obstacle and a forward-facing step

Large-eddy simulation (LES) of transitional separating-reattaching flow on a two-dimensional square surface mounted obstacle and a forward facing step has been performed using a dynamic sub-grid scale model. The Reynolds number based on the uniform inlet velocity and the obstacle/step height is 4.5 × 10³. The mean LES results for both the obstacle and step flow compare reasonably well with the available experimental and DNS data.

The flow structures upstream of the surface-mounted obstacle (referred to hereafter as obstacle) and the forward-facing step (referred to hereafter as FFS) consist of unstable two-dimensional structures and coherent rib-shaped structures. These structures with the aid of 3D streamline visualisation strongly indicate that the upstream separation bubble is a closed one rather than an open one in the sense that there is little evidence to suggest that there is fluid injection from the upstream separation region into the downstream separated region for the two geometries. The spectra and time history for the velocities and pressure fields at locations immediately upstream of the obstacle and FFS (including the recirculation region) were analysed using both the Fourier and wavelet transforms and revealed the unsteady nature of the recirculation region upstream of the obstacle and FFS.

The transition process has been elucidated using both 2D and 3D flow visualisation of the flow. In both geometries (obstacle and FFS), the separated boundary layer downstream of the leading edge shows 2D nature and roll-up shortly downstream of the separation line leading to 2D K-H rolls to be shed from the leading edge. Coherent structures such as the λ-shaped and rib-like vortices commonly associated with a flat plate boundary layer and also found in the separated-reattached flow of a blunt leading edge plate aligned horizontally to a flow are not common in the separated-reattached flow over the obstacle and FFS.     

Numerical analysis of turbulent structure in compound meandering open channel by algebraic Reynolds stress model

The turbulent flow in a compound meandering channel with a rectangular cross section is one of the most complicated turbulent flows, because the flow behaviour is influenced by several kinds of forces, including centrifugal forces, pressure‐driven forces and shear stresses generated by momentum transfer between the main channel and the flood plain. Numerical analysis has been performed for the fully developed turbulent flow in a compound meandering open‐channel flow using an algebraic Reynolds stress model. The boundary‐fitted coordinate system is introduced as a method for coordinate transformation in order to set the boundary conditions along the complicated shape of the meandering open channel. The turbulence model consists of transport equations for turbulent energy and dissipation, in conjunction with an algebraic stress model based on the Reynolds stress transport equations. With reference to the pressure–strain term, we have made use of a modified pressure–strain term. The boundary condition of the fluctuating vertical velocity is set to zero not only for the free surface, but also for computational grid points next to the free surface, because experimental results have shown that the fluctuating vertical velocity approaches zero near the free surface. In order to examine the validity of the present numerical method and the turbulent model, the calculated results are compared with experimental data measured by laser Doppler anemometer. In addition, the compound meandering open channel is clarified somewhat based on the calculated results. As a result of the analysis, the present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent flow in a compound meandering open channel. Copyright © 2005 John Wiley & Sons, Ltd.     

Large-eddy simulation study of upstream boundary conditions influence upon a backward-facing step flowÉtude par simulation des grandes échelles de l'influence des conditions aux limites amont sur l'écoulement derrière une marche descendante

We use Large Eddy Simulation to investigate the influence of upstream boundary conditions on the development of a backward facing step flow. The first inlet condition consists of a mean turbulent boundary layer velocity profile perturbed by a white noise. The second relies upon a precursor calculation where the development of a quasi-temporal turbulent boundary layer is simulated. In this case, the quasi-longitudinal vortices in the upstream turbulent boundary-layer trigger the destabilization of the shear layer just behind the step, resulting in a shortening of the recirculation length and an increase of the characteristic frequency associated to the Kelvin–Helmholtz vortices. The mean flow and the characteristic frequencies of pressure fluctuations are strongly dependent of the upstream flow. It demonstrates the importance of realistic boundary conditions for the simulation of complex 3D flows or for flow control simulations. To cite this article: J.-L. Aider, A. Danet, C. R. Mecanique 334 (2006).