可压流数值模拟中当地DFD方法的改进和应用

阐述了求解守恒型Euler 方程的当地DFD (Domain-Free Discretization) 方法的改进和应用。DFD 离散策略的核心,是解域内点上控制方程的离散形式可与解域外的一些点相关。通过边界附近解的近似形式,外部相关点上的流动变量值得到确定并强加相应的边界条件。与最初的当地DFD方法不同,在解的近似形式构建中,采用了CCST技术 (Curvature-Corrected Symmetry Technique),因此外部相关点上的密度和切向速度分别由等熵和等总焓关系确定。空间离散采用Galerkin 有限体积格式。最后,给出了固定和运动物体可压缩绕流的数值模拟结果,以验证改进的当地DFD方法的可靠性和数值解精度的提高。     

A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

In this paper, the domain‐free discretization method (DFD) is extended to simulate the three‐dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior‐dependent points are updated at each time step by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of ‘osculating plane’ is adopted, with which the local DFD can be easily implemented for the three‐dimensional case. Geometry‐adaptive tetrahedral mesh is employed for three‐dimensional calculations. Finally, we validate the DFD method for three‐dimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary‐conforming grid was displayed and the results show that the present DFD results compare very well with the reference data. Copyright © 2009 John Wiley & Sons, Ltd.     

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

This paper presents a local domain‐free discretization (DFD) method for the simulation of unsteady flows over moving bodies governed by the incompressible Navier–Stokes equations. The discretization strategy of DFD is that the discrete form of partial differential equations at an interior point may involve some points outside the solution domain. All the mesh points are classified as interior points, exterior dependent points and exterior independent points. The functional values at the exterior dependent points are updated at each time step by the approximate form of solution near the boundary. When the body is moving, only the status of points is changed and the mesh can stay fixed. The issue of ‘freshly cleared nodes/cells’ encountered in usual sharp interface methods does not pose any particular difficulty in the presented method. The Galerkin finite‐element approximation is used for spatial discretization, and the discrete equations are integrated in time via a dual‐time‐stepping scheme based on artificial compressibility. In order to validate the present method for moving‐boundary flow problems, two groups of flow phenomena have been simulated: (1) flows over a fixed circular cylinder, a harmonic in‐line oscillating cylinder in fluid at rest and a transversely oscillating cylinder in uniform flow; (2) flows over a pure pitching airfoil, a heaving–pitching airfoil and a deforming airfoil. The predictions show good agreement with the published numerical results or experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive mesh refinement‐enhanced local DFD method and its application to solve Navier–Stokes equations

Recently, the domain‐free discretization (DFD) method was presented to efficiently solve problems with complex geometries without introducing the coordinate transformation. In order to exploit the high performance of the DFD method, in this paper, the local DFD method with the use of Cartesian mesh is presented, where the physical domain is covered by a Cartesian mesh and the local DFD method is applied for numerical discretization. In order to further improve the efficiency of the solver, the newly developed solution‐based adaptive mesh refinement (AMR) technique is also introduced. The proposed methods are then applied to the simulation of natural convection in concentric annuli between a square outer cylinder and a circular inner cylinder. Numerical experiments show that the present numerical results agree very well with available data in the literature, and AMR‐enhanced local DFD method is an effective tool for the computation of flow problems. Copyright © 2005 John Wiley & Sons, Ltd.     

An immersed boundary method for simulation of inviscid compressible flows

In this paper, an immersed boundary method for simulating inviscid compressible flows governed by Euler equations is presented. All the mesh points are classified as interior computed points, immersed boundary points (interior points closest to the solid boundary), and exterior points that are blanked out of computation. The flow variables at an immersed boundary point are determined via the approximate form of solution in the direction normal to the wall boundary. The normal velocity is evaluated by applying the no‐penetration boundary condition, and therefore, the influence of solid wall in the inviscid flow is taken into account. The pressure is computed with the local simplified momentum equation, and the density and the tangential velocity are evaluated by using the constant‐entropy relation and the constant‐total‐enthalpy relation, respectively. With a local coordinate system, the present method has been extended easily to the three‐dimensional case. The present work is the first endeavor to extend the idea of hybrid Cartesian/immersed boundary approach to compressible inviscid flows. The tedious task of handling multi‐valued points can be eliminated, and the overshoot resulting from the extrapolation for the evaluation of flow variables at exterior points can also be avoided. In order to validate the present method, inviscid compressible flows over fixed and moving bodies have been simulated. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of local DFD method to simulate unsteady flows around an oscillating circular cylinder

In this paper, the recently proposed local domain‐free discretization (DFD) method is applied to simulate incompressible flows around an oscillating circular cylinder. It is found that it is very easy for the local DFD method to handle such moving boundary flow problems. This is because it does not need to move the mesh, which is indeed needed in traditional methods. Numerical experiments show that the present numerical results agree very well with the available data in the literature, and that the local DFD method is an effective tool for the computation of moving boundary flow problems. Copyright © 2008 John Wiley & Sons, Ltd.     

An upwind scheme for the three-dimensional boundary layer equations

The development of a calculation method to solve the compressible, three-dimensional, turbulent boundary layer equations is described. An implicit finite difference solution procedure is adopted involving local upwinding of convective transport terms. A consistent approach to discretization and linearization is taken by casting all equations in a similar form. The implementation of algebraic, one-equation and two-equation turbulence models is described. An initial validation of the method is made by comparing prediction with measurements in two quasi-three-dimensional boundary layer flows. Some of the more obvious deficiencies in current turbulence-modelling standards for three-dimensional flows are discussed.     

Application of the finite point method to high‐Reynolds number compressible flow problems

In this work, the finite point method is applied to the solution of high‐Reynolds compressible viscous flows. The aim is to explore this important field of applications focusing on two main aspects: the easiness and automation of the meshless discretization of viscous layers and the construction of a robust numerical approximation in the highly stretched clouds of points resulting in such domain areas. The flow solution scheme adopts an upwind‐biased scheme to solve the averaged Navier–Stokes equations in conjunction with an algebraic turbulence model. The numerical applications presented involve different attached boundary layer flows and are intended to show the performance of the numerical technique. The results obtained are satisfactory and indicative of the possibilities to extend the present meshless technique to more complex flow problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Extension of the local domain-free discretization method to large eddy simulation of turbulent flows

In this work, an immersed boundary method, called the local domain-free discretization (DFD) method, is extended to large eddy simulation (LES) of turbulent flows. The discrete form of partial differential equations at an interior node may involve some nodes outside the solution domain. The flow variables at these exterior dependent nodes are evaluated via linear extrapolation along the direction normal to the wall. To alleviate the requirement of mesh resolution in the near-wall region, a wall model based on the turbulence boundary layer equations is introduced. The wall shear stress yielded by the wall model and the no-penetration condition are enforced at the immersed boundary to evaluate the velocity components at an exterior dependent node. For turbulence closure, a dynamic subgrid scale (SGS) model is adopted and the Lagrangian averaging procedure is used to compute the model coefficient. The SGS eddy viscosity at an exterior dependent node is set to be equal to that at the outer layer. To maintain the mass conservation near the immersed boundary, a mass source/sink term is added into the continuity equation. Numerical experiments on relatively coarse meshes with stationary or moving solid boundaries have been conducted to verify the ability of the present LES-DFD method. The predicted results agree well with the published experimental or numerical data.     

Application of the method of lines to unsteady compressible Euler equations

In the sense of method of lines, numerical solution of the unsteady compressible Euler equations in 1D, 2D and 3D is split into three steps: First, space discretization is performed by the first‐order finite volume method using several approximate Riemann solvers. Second, smoothness and Lipschitz continuity of RHS of the arising system of ordinary dimensional equations (ODEs) is analysed and its solvability is discussed. Finally, the system of ODEs is integrated in time by means of implicit and explicit higher‐order adaptive schemes offered by ODE packages ODEPACK and DDASPK, by a backward Euler scheme based on the linearization of the RHS and by higher‐order explicit Runge–Kutta methods. Time integrators are compared from several points of view, their applicability to various types of problems is discussed, and 1D, 2D and 3D numerical examples are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

亚、跨、超音速及不可压流动的数值分析方法的研究

为了对亚、跨、超音速及不可压无粘流动进行数值模拟,将LU－SGS方法与预处理方法结合,给出了PLU－SGS方法。方程离散基于有限体积法,采用高阶精度AUSMPW格式。方程求解采用了特征边界条件。通过典型算例的数值试验对比分析,表明PLU－SGS方法可以有效地对亚、跨、超音速及不可压流动进行数值模拟,并具有较高的计算精度和收敛速度。     

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided. Copyright © 2013 John Wiley & Sons, Ltd.     

A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A complete boundary integral formulation for compressible Navier–Stokes equations

A complete boundary integral formulation for compressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for wall pressure and wall skin friction of two‐dimensional compressible laminar viscous flow around airfoils are in good agreement with field numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

A promising boundary element formulation for three‐dimensional viscous flow

In this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.     

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

In this paper, we present an approach of dynamic mesh adaptation for simulating complex 3‐dimensional incompressible moving‐boundary flows by immersed boundary methods. Tetrahedral meshes are adapted by a hierarchical refining/coarsening algorithm. Regular refinement is accomplished by dividing 1 tetrahedron into 8 subcells, and irregular refinement is only for eliminating the hanging points. Merging the 8 subcells obtained by regular refinement, the mesh is coarsened. With hierarchical refining/coarsening, mesh adaptivity can be achieved by adjusting the mesh only 1 time for each adaptation period. The level difference between 2 neighboring cells never exceeds 1, and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases. A predictor‐corrector scheme is introduced to eliminate the phase lag between adapted mesh and unsteady solution. The error caused by each solution transferring from the old mesh to the new adapted one is small because most of the nodes on the 2 meshes are coincident. An immersed boundary method named local domain‐free discretization is employed to solve the flow equations. Several numerical experiments have been conducted for 3‐dimensional incompressible moving‐boundary flows. By using the present approach, the number of mesh nodes is reduced greatly while the accuracy of solution can be preserved.     

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations

The paper deals with the use of the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of viscous compressible flows. We start with a scalar convection–diffusion equation and present a discretization with the aid of the non‐symmetric variant of DGFEM with interior and boundary penalty terms. We also mention some theoretical results. Then we extend the scheme to the system of the Navier–Stokes equations and discuss the treatment of stabilization terms. Several numerical examples are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 John Wiley & Sons, Ltd.