多维SUPG格式不可压流动高精度算法研究

在传统流线迎风Petrov-Galerkin (SUPG)有限元法基础上,通过对稳定因子关键参数进行分析,提出了基于流向投影的最优特征高度参数确定方法,同时针对不可压流动引入变量分裂算法,发展了一种可用于高雷诺数不可压流动计算的高精度稳定化SUPG方法．详细地给出了三角形单元基于流向的最优特征高度确定方法的分析过程,并给出了基于分裂算法的有限元计算步骤和公式．采用该方法对典型的方腔拖曳、圆柱绕流流动问题进行了分析,在网格较稀疏,且雷诺数较大的情况下,依然可以得到稳定的计算结果,从而验证了该方法的稳定性、有效性,对实际工程应用具有积极的意义．     

Three-dimensional vortex wake structure of a flapping-wing micro aerial vehicle in forward flight configuration

The unsteady flow field past a backward-facing step in a rectangular duct is investigated by adopting time-resolved particle image velocimetry (PIV) in the Reynolds number range of 2,640–9,880 based on step height and the inlet average velocity. The PIV realizations are subjected to post-processing techniques, namely, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). At low Reynolds numbers, the second spatial POD modes indicate the presence of the shear layer mode, whereas this feature shifts to higher modes at higher Reynolds numbers. The corresponding temporal modes are Fourier-transformed to obtain the dominant frequency, whose Strouhal number corroborates the above observation. Short-time windows in the transverse velocity component along the shear layer are selected to investigate the temporal stability of the flow field by DMD to quantify the growth rate of the shear layer mode. The higher harmonics of this mode are also observed to grow, albeit at lesser rate. By relating to POD analysis, the most energetic structures were found to correspond to the unstable modes. The correlation between these unstable DMD modes and the Fourier-filtered flow fields for the same frequencies indicate better match for the lower operating Reynolds number case as compared to higher ones. The spatial stability analysis demonstrates the growth of the shear layer vortices, which is combined with the temporal stability analysis to evaluate the phase velocity of the identified shear layer structures. The calculated phase velocity magnitude of the shear layer is found to be reasonably below the local velocity as expected.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

ADAPTIVE LINEARIZATION AND GRID ITERATIONS WITH THE TRI-TREE MULTIGRID REFINEMENT–RECOARSEMENT ALGORITHM FOR THE NAVIER–STOKES EQUATIONS

The tri-tree algorithm for refinements and recoarsements of finite element grids is explored. The refinement–recoarsement algorithm not only provides an accurate solution in certain parts of the grid but also has a major influence on the finite element equation system itself. The refinements of the grid lead to a more symmetric and linear equation matrix. The recoarsements will ensure that the grid is not finer than is necessary for preventing divergence in an iterative solution procedure. The refinement–recoarsement algorithm is a dynamic procedure and the grid is adapted to the instant solution. In the tri-tree multigrid algorithm the solution from a coarser grid is scaled relatively to the increase in velocity boundary condition for the finer grid. In order to have a good start vector for the solution of the finer grid, the global Reynolds number or velocity boundary condition should not be subject to large changes. For each grid and velocity solution the element Reynolds number is computed and used as the grid adaption indicator during the refinement–recoarsement procedure. The iterative tri-tree multigrid method includes iterations with respect to the grid. At each Reynolds number the same boundary condition s are applied and the grid is adapted to the solution iteratively until the number of unknowns and elements in the grid becomes constant. In the present paper the following properties of the tri-tree algorithm are explored: the influence of the increase in boundary velocities and the size of the grid adaption indicator on the amount of work for solving the equations, the number of linear iterations and the solution error estimate between grid levels. The present work indicates that in addition to the linear and non-linear iterations, attention should also be given to grid adaption iterations. © 1997 by John Wiley & Sons, Ltd.     

Research on stability of moving jet containing dense suspended solid particles

Ｎｏｍｅｎｃｌａｔｕｒｅαｊ:ｐｈａｓｅｊｖｏｌｕｍｅｆｒａｃｔｉｏｎ;ｕｊ:ｐｈａｓｅｊｖｅｌｏｃｉｔｙｖｅｃｔｏｒ;Ｕ :ｍｅａｎｖｅｌｏｃｉｔｙｖｅｃｔｏｒ;Ｕ :ｍｅａｎａｘｉａｌｖｅｌｏｃｉｔｙ;Ｕｏ:ｍｅａｎｘ＿ａｘｉａｌｖｅｌｏｃｉｔｙｏｆｔｈｅｅｘｉｔ;Ｕ∞ :ｖｅｌｏｃｉｔｙｏｆｊｅｔｔｉｎｇｄｅｖｉｃｅ;ΔＵ :ｊｕｍｐｉｎｇｖｅｌｏｃｉｔｙｏｆｍｏｖｉｎｇｊｅｔ,ｃｈａｒａｃｔｅｒｉｓｔｉｃｖｅｌｏｃｉｔｙｏｆｆｌｏｗ＿ｆｉｅｌｄ ,ΔＵ =Ｕｏ-Ｕ∞ ;ｐ:ｐｒｅｓｓｕｒｅ;Ｐ :ｅｉｇｅｎｐｒｅｓ…     

A high‐order accurate discontinuous Galerkin finite element method for laminar low Mach number flows

In this paper we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of laminar flow simulations at low Mach numbers using an implicit scheme. The algorithm is based on the flux preconditioning approach, which modifies only the dissipative terms of the numerical flux. This formulation is quite simple to implement in existing implicit DG codes, it overcomes the time‐stepping restrictions of explicit multistage algorithms, is consistent in time and thus applicable to unsteady flows. The performance of the method is demonstrated by solving the flow around a NACA0012 airfoil and on a flat plate, at different low Mach numbers using various degrees of polynomial approximations. Computations with and without flux preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers. The time accurate solution of unsteady flow is also demonstrated by solving the vortex shedding behind a circular cylinder at the Reynolds number of 100. Copyright © 2012 John Wiley & Sons, Ltd.     

The AGE method for direct numerical simulation of turbulent shear flow

The advected grid explicit (AGE) method for direct numerical simulation of ‘incompressible’ turbulent shear flows is presented. The Navier–Stokes equations are used for momentum in a velocity–pressure formulation. Mass continuity and an equation of state link pressure with density (which is not assumed identically constant). Time advancement is entirely explicit, and spatial representation is localized (e.g. finite difference) and centred. Magnitudes of non-linear terms are reduced on advected grid(s), and numerical instabilities are efficiently reduced by ‘targeted diffusion’. Computation time scales directly on the number of grid points (virtual memory issues aside), and is very short for a DNS method. A spatially developing two-stream mixing layer was simulated as an example, reaching a vorticity thickness Reynolds number >20 000. Comparison with experimental results from self-similar mixing layers is satisfactory in terms of growth rate and Reynolds stress profiles. Turbulent vortical structures are visualized by means of pressure surfaces. © 1998 John Wiley & Sons, Ltd.     

Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes

New implicit finite difference schemes for solving the time-dependent incompressible Navier-Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimates for the discrete solution of the methods are obtained. Employing the operator approach, some requirements on the difference operators of the scheme are formulated in order to derive a scheme which is essentially consistent with the initial differential equations. The operators of the scheme inherit the fundamental properties of the corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To derive the consistent scheme, special approximations for convective terms and div and grad operators are employed. Two variants of time discretization by the operator-splitting technique are considered and compared. It is shown that the derived scheme has a very weak restriction on the time step size. A lid-driven cavity flow has been predicted to examine the stability and accuracy of the schemes for Reynolds number up to 3200 on the sequence of grids with 21 × 21, 41 × 41, 81 × 81 and 161 × 161 grid points.     

Prediction of asymmetric vortical flows around slender bodies using Navier-Stokes equations

Steady and unsteady asymmetric vortical flows around slender bodies at high angles of attack are solved using the unsteady, compressible, this-layer Navier-Stokes equations. An implicit, upwind-biased, flux-difference splitting, finite-volume scheme is used for the numerical computations. For supersonic flows past point cones, the locally conical flow assumption has been used for efficient computational studies of this phenomenon. Asymmetric flows past a 5° semiapex-angle circular cone at different angles of attack, free-stream Mach numbers, and Reynolds numbers has been studied in responses to different sources of disturbances. The effects of grid fineness and computational domain size have also been investigated. Next, the responses of three-dimensional supersonic asymmetric flow around a 5° circular cone at different angles of attack and Reynolds numbers to short-duration sideslip disturbances are presented. The results show that flow asymmetry becomes stronger as the Reynolds number and angles of attack are increased. The asymmetric solutions show spatial vortex shedding which is qualitatively similar to the temporal vortex shedding of the unsteady locally conical flow. A cylindrical afterbody is also added to the same cone to study the effect of a cylindrical part on the flow asymmetry. One of the cases of flow over a cone-cylinder configuration is validated fairly well by experimental data.     

A multigrid solver for the vorticity-velocity Navier-Stokes equations

This paper provides a multigrid incremental line-Gauss-Seidel method for solving the steady Navier-Stokes equations in two and three dimensions expressed in terms of the vorticity and velocity variables. The system of parabolic and Poisson equations governing the scalar components of the vector unknowns is solved using centred finite differences on a non-staggered grid. Numerical results for the two-dimensional driven cavity problem indicate that the spatial discretization of the equation defining the value of the vorticity on the boundary is extremely critical to obtaining accurate solutions. In fact, a standard one-sided three-point second-order-accurate approximation produces very inaccurate results for moderate-to-high values of the Reynolds number unless an exceedingly fine mesh is employed. On the other hand, a compact two-point second-order-accurate discretization is found to be always satisfactory and provides accurate solutions for Reynolds number up to 3200, a target impossible heretofore using this formulation and a non-staggered grid.     

The three-dimensional prolonged adaptive unstructured finite element multigrid method for the Navier–Stokes equations

This paper presents the development of the three- dimensional prolonged adaptive finite element equation solver for the Navier–Stokes equations. The finite element used is the tetrahedron with quadratic approximation of the velocities and linear approximation of the pressure. The equation system is formulated in the basic variables. The grid is adapted to the solution by the element Reynolds number. An element in the grid is refined when the Reynolds number of the element exceeds a preset limit. The global Reynolds number in the investigation is increased by scaling the solution for a lower Reynolds number. The grid is refined according to the scaled solution and the prolonged solution for the lower Reynolds number constitutes the start vector for the higher Reynolds number. Since the Reynolds number is the ratio of convection to diffusion, the grid refinements act as linearization and symmetrization of the equation system. The linear equation system of the Newton formulation is solved by CGSTAB with coupled node fill-in preconditioner. The test problem considered is the three-dimensional driven cavity flow. © 1997 John Wiley & Sons, Ltd.     

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

A high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22 : 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.     

Forced‐convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers

This paper presents the results of a numerical study on the flow characteristics and heat transfer over two equal square cylinders in a tandem arrangement. Spacing between the cylinders is five widths of the cylinder and the Reynolds number ranges from 1 to 200, Pr=0.71. Both steady and unsteady incompressible laminar flow in the 2D regime are performed with a finite volume code based on the SIMPLEC algorithm and non‐staggered grid. A study of the effects of spatial resolution and blockage on the results is provided. In this study, the instantaneous and mean streamlines, vorticity and isotherm patterns for different Reynolds numbers are presented and discussed. In addition, the global quantities such as pressure and viscous drag coefficients, RMS lift and drag coefficients, recirculation length, Strouhal number and Nusselt number are determined and discussed for various Reynolds numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Spatial Instabilities of the Incompressible Attachment-Line Flow Using Sparse Matrix Jacobi–Davidson Techniques

We consider the linear stability of incompressible attachment-line flow within the spatial framework. No similarity or symmetry assumptions for the instability modes are introduced and the full two-dimensional representation of the modes is used. The perturbation equations are discretized on a two-dimensional staggered grid. A high order finite difference scheme has been developed which gives rise to a large, sparse, quadratic, eigenvalue problem for the instability modes. The benefits of the Jacobi–Davidson method for the solution of this eigenvalue system are demonstrated and the approach is validated in some detail. Spatial stability results are presented subsequently. In particular, instability predictions at very high Reynolds numbers are obtained which show almost equally strong instabilities for symmetric and antisymmetric modes in this regime.     

Simulation of Richtmyer–Meshkov instability by sixth-order filter methods

Simulation of a 2-D Rightmyer–Meshkov instability (RMI), including inviscid, viscous and magnetic field effects was conducted comparing recently developed sixth-order filter schemes with various standard shock-capturing methods. The suppression of the inviscid gas dynamics RMI in the presence of a magnetic field was investigated by Samtaney and Wheatley et al. Numerical results illustrated here exhibit behavior similar to the work of Samtaney. Due to the different amounts and different types of numerical dissipation contained in each scheme, the structures and the growth of eddies for the chaotic-like inviscid gas dynamics RMI case are highly grid size and scheme dependent, even with many levels of refinement. The failure of grid refinement for all studied numerical methods extends to the viscous gas dynamics case for high Reynolds number. For lower Reynolds number, grid convergence has been achieved by all studied methods. To achieve similar resolution, standard shock-capturing methods require more grid points than filter schemes and yet the CPU times using the same grid for all studied methods are comparable. This paper is based on work that was presented at the 17th International Shock Interaction Symposium (ISIS17), Rome, Italy, 4–8 September 2006.     

The Absolute Instability of Joukowski-Type Airfoils

In this paper an investigation is undertaken to explore the nature of the flow in the vicinity of the trailing edge of Joukowski-type airfoil configurations. Making use of the asymptotic interactive boundary layer theory, the basic flow profiles in the attached and detached flow regions are computed numerically through integrating the interactive boundary layer equations governing the flow motion for sufficiently large Reynolds numbers. Employing a Spectral Chebyshev collocation numerical integration scheme, boundary layer features corresponding to a number of thickness-to-chord ratio parameter cases are produced. The analysis carried out over the interaction region of the trailing edge shows that flow separation always takes place beyond certain critical value of the thickness-to-chord ratio parameter under the action of a self-induced pressure gradient. In addition, reversed flow regions of a sufficiently large size are found to be absolutely unstable, within the framework of linear spatio-temporal stability analysis in combination with the Briggs--Bers branch point criterion. Received 10 January 2001 and accepted 15 August 2001     

Numerical simulation of laminar flow and heat transfer over banks of staggered cylinders

A study is conducted to investigate forced convective flow and heat transfer over a bank of staggered cylinders. Using a novel numerical formulation based on a non‐orthogonal collocated grid in a physical plane, the effects of Reynolds number and cylinder spacing on the flow and heat transfer behaviour are systematically studied. It is observed that both the Reynolds number and cylinder spacing influence the recirculatory vortex formation and growth in the region between the cylinders; in turn, the rates of heat transfer between the fluid and the staggered cylinders are affected. As the cylinder spacing decreases, the size and length of eddies reduce. For sufficiently small spacings, eddy formation is completely suppressed even at high Reynolds number. Pressure drop and Nusselt number predictions based on numerical study are in excellent agreement with available correlations. The study provides useful insight on the detailed flow and heat transfer phenomena for the case of a bank of staggered cylinders. Copyright © 2002 John Wiley & Sons, Ltd.     

Stability analysis in spatial mode for channel flow of fiber suspensions

Different from previous temporal evolution assumption, the spatially growing mode was employed to analyze the linear stability for the channel flow of fiber suspensions. The stability equation applicable to fiber suspensions was established and solutions for a wide range of Reynolds number and angular frequency were given numerically . The results show that, the flow instability is governed by a parameter H which represents a ratio between the axial stretching resistance of fiber and the inertial force of the fluid. An increase of H leads to a raise of the critical Reynolds number, a decrease of corresponding wave number, a slowdown of the decreasing of phase velocity , a growth of the spatial attenuation rate and a diminishment of the peak value of disturbance velocity. Although the unstable region is reduced on the whole, long wave disturbances are susceptible to fibers.     

A technique of flux reconstruction at the interfaces of nonconforming grids for aeroacoustic simulations

A flux reconstruction technique is presented to perform aeroacoustic computations using implicit high-order spatial schemes on multiblock structured grids with nonconforming interfaces. The use of such grids, with mesh spacing discontinuities across the block interfaces, eases local mesh refinements, simplifies the mesh generation process, and thus facilitates the computation of turbulent flows. In this work, the spatial discretization consists of sixth-order finite-volume implicit schemes with low-dispersion and low-dissipation properties. The flux reconstruction is based on the combination of noncentered schemes with local interpolations to define ghost cells and compute flux values at the grid interfaces. The flow variables in the ghost cells are calculated from the flow field in the grid cells using a meshless interpolation with radial basis functions. In this study, the flux reconstruction is applied to both plane and curved nonconforming interfaces. The performance of the method is first evaluated by performing two-dimensional simulations of the propagation of an acoustic pulse and of the convection of a vortex on Cartesian and wavy grids. No significant spurious noise is produced at the grid interfaces. The applicability of the flux reconstruction to a three-dimensional computation is then demonstrated by simulating a jet at a Mach number of 0.9 and a diameter-based Reynolds number of 4×10⁵ on a Cartesian grid. The nonconforming grid interface located downstream of the jet potential core does not appreciably affect the flow development and the jet sound field, while reducing the number of mesh points by a factor of approximately two.