Adaptive Euler simulations of airfoil–vortex interaction

A grid redistribution method is used together with an improved spatially third‐order accurate Euler solver to improve the accuracy of direct Euler simulations of airfoil–vortex interaction. The presented numerical results of two airfoil–vortex interaction cases indicate that with combination of the two methods, the numerical diffusion of vorticity inherent in the direct Euler simulations is drastically reduced without increasing the number of grid points. With some extra works due to grid redistribution, the predicted vortex structure is well preserved after a long convection and much sharper acoustic wave front resulting from airfoil–vortex interaction is captured. Copyright © 2006 John Wiley & Sons, Ltd.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Adaptive strategy of transonic flows over vibrating blades with interblade phase angles

An error indicator and a locally implicit scheme with anisotropic dissipation model on dynamic quadri‐ lateral–triangular mesh are developed to study transonic flows over vibrating blades with interblade phase angles. In the Cartesian co‐ordinate system, the unsteady Euler equations with moving domain effects are solved. The error indicator, in which unified magnitudes of dynamic grid speed, substantial derivative of pressure, and substantial derivative of vorticity magnitude are incorporated to capture the unsteady wave behaviours and vortex‐shedding phenomena due to unsteadiness. To assess the accuracy of the locally implicit scheme with anisotropic dissipation model on quadrilateral–triangular mesh, two flow calculations are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high‐resolutional result on the adaptive mesh, the unsteady pressure wave, shock and vortex‐shedding behaviours are clearly demonstrated. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical diffusion for flow‐aligned unstructured grids with application to estuarine modeling

The benefits of unstructured grids in hydrodynamic models are well understood but in many cases lead to greater numerical diffusion compared with methods available on structured grids. The flexible nature of unstructured grids, however, allows for the orientation of the grid to align locally with the dominant flow direction and thus decrease numerical diffusion. We investigate the relationship between grid alignment and diffusive errors in the context of scalar transport in a triangular, unstructured, 3‐D hydrodynamic code. Analytical results are presented for the 2‐D anisotropic numerical diffusion tensor and verified against idealized simulations. Results from two physically realistic estuarine simulations, differing only in grid alignment, show significant changes in gradients of salinity. Changes in scalar gradients are reflective of reduced numerical diffusion interacting with the complex 3‐D structure of the transporting flow. We also describe a method for utilizing flow fields from an unaligned grid to generate a flow‐aligned grid with minimal supervision. Copyright © 2013 John Wiley & Sons, Ltd.     

A vortex axis and vortex core border grid adaptation algorithm

In the design process of hydrodynamical and aerodynamical technical applications, the numerical simulation of massively separated vortical flow is crucial for predicting, for example, lift or drag. To obtain reliable numerical results, it is mandatory to accurately predict the physical behavior of vortices. Thus, the dominant vortical flow structures have to be resolved in detail, which requires a local grid refinement and certain adaptation techniques. In this paper, a vortex flow structure adaptation algorithm is presented, which is particularly designed for local grid refinement at vortex axes positions and associated vortex core border locations. To this end, a fast and efficient vortex axis detection scheme is introduced and the algorithm for the vortex core border determination is explained. As the interaction between vortices makes the assignment of grid points to a certain vortex axis difficult, a helicity‐based vortex distinction approach in combination with a geometrical rotational sensor is developed. After describing the combined different techniques in detail, the vortex feature adaptation algorithm is applied to analytical and more realistic examples, which show that the described grid adaptation algorithm is able to enhance the grid cell resolution locally such that all significant vortical flow phenomena are resolved. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

Thin‐tube vortex simulations for sinusoidal instability in a counter‐rotating vortex pair

A thin‐tube vortex method is developed to investigate the intrinsic instability within a counter‐rotating vortex pair system and the effects from the core size and the wavenumbers (or wavelengths). The numerical accuracy and the advantages of the scheme are theoretically estimated. A nearest‐neighbour‐image method is employed in this three‐dimensional vortex simulation. Agreement with Crow's instability analysis has been achieved numerically for the long‐wave cases. A short‐wave instability for the zeroth radial mode of bending instability has also been found using the thin‐tube vortex simulations. Then, the combinations of long‐ and short‐wave instability are investigated to elucidate the non‐linear effects due to the interactions of two different modes. It is shown that instability is enhanced if both long‐ and short‐wave instabilities occur simultaneously. Although the method used in the paper is not capable of including effects such as axial flow, vortex core deformation and other complicated viscous effects, it effectively predicts and clarifies the first‐order factor that dominates the sinusoidal instability behaviour in a vortex pair. Copyright © 2002 John Wiley & Sons, Ltd.     

基于反馈力浸入边界法模拟复杂动边界流动

浸入边界法是模拟流固耦合的重要数值方法之一。本文采用反馈力浸入边界方法,对旋转圆柱和水轮机活动导叶旋转摆动绕流后的动边界流场进行数值模拟。其中,固体边界采用一系列离散的点近似代替,流体为不可压缩牛顿流体,使用笛卡尔自适应加密网格,利用有限差分法进行求解。固体对流场的作用通过构造适宜的反馈力函数实现。本文首先通过旋转圆柱绕流的计算结果同实验结果进行对比,吻合较好,验证了该计算方法的可靠性。然后针对水电站水力过渡过程中水轮机活动导叶旋转摆动绕流后的动边界流场进行数值模拟,得到导叶动态绕流后的流场分布特性和涡结构的演化特性。     

Numerical simulation on an interaction of a vortex street with an elliptical leading edge using an unstructured grid

An algorithm for a time accurate incompressible Navier–Stokes solver on an unstructured grid is presented. The algorithm uses a second order, three‐point, backward difference formula for the physical time marching. For each time step, a divergence free flow field is obtained based on an artificial compressibility method. An implicit method with a local time step is used to accelerate the convergence for the pseudotime iteration. To validate the code, an unsteady laminar flow over a circular cylinder at a Reynolds number of 200 is calculated. The results are compared with available experimental and numerical data and good agreements are achieved. Using the developed unsteady code, an interaction of a Karman vortex street with an elliptical leading edge is simulated. The incident Karman vortex street is generated by a circular cylinder located upstream. A clustering to the path of the vortices is achieved easily due to flexibility of an unstructured grid. Details of the interaction mechanism are analysed by investigating evolutions of vortices. Characteristics of the interactions are compared for large‐ and small‐scale vortex streets. Different patterns of the interaction are observed for those two vortex streets and the observation is in agreement with experiment. Copyright © 2004 John Wiley & Sons, Ltd.     

Study of water waves with submerged obstacles using a vortex method with Helmholtz decomposition

The present study develops a 2‐D numerical scheme that combines the vortex method and the boundary integral method by a Helmholtz decomposition to investigate the interaction of water waves with submerged obstacles. Viscous effects and generation of vorticity on the free surface are neglected. The second kind of Fredholm integral equations that govern the strengths of vortex sheets along boundaries are solved iteratively. Vorticity is convected and diffused in the fluid via a Lagrangian vortex (blob) method with varying cores, using the particle strength exchange method for diffusion, with particle redistribution. A grid‐convergence study of the numerical method is reported. The inviscid part of the method and the simulation of the free‐surface motion are tested using two calculations: solitary wave propagation in a uniform channel and a moving line vortex in the fluid. Finally, the full model is verified by simulating periodic waves travelling over a submerged rectangular obstacle using nonuniform vortex blobs with a mapping of the redistribution lattice. Overall, the numerical model predicts the vortices' evolution and the free‐surface motion reasonably well. Copyright © 2008 John Wiley & Sons, Ltd.     

网格自适应技术在复杂外形流场模拟中的应用

建立了一套适用于非结构混合网格自适应方法,针对激波和涡的不同特征采用不同加密探测器,各向异性加密棱柱单元并沿物面法向方向剖分所有棱柱层,各向异性剖分四面体单元,并保证四面体与棱柱交界面上网格协调。构造Hermit插值近似投影物面新加网格点和基于Laplacian光滑方法对空间网格进行优化。通过网格自适应加密,使用Roe格式计算高超声速球头绕流的红玉现象得到明显减轻。F16飞机含激波和脱体涡的流场自适应计算表明,网格加密集中在激波面和涡核附近区域,激波和涡计算更准确。     

Modified Lagrangian vortex method with improved boundary conditions for water waves past a thin bottom‐standing barrier

Herein, the modified Lagrangian vortex method (LVM), a hybrid analytical‐numerical algorithm per se, is devised to simulate the process of vortex formation and shedding from the sharp edge of a zero‐thickness vertical plate under linear water‐wave attack. Application of the Helmholtz decomposition facilitates a convenient switch between the inviscid‐ and viscous‐flow models, thereby enabling easy incorporation of vorticity effects into the potential‐flow calculations for the viscous‐dominated region. In evaluating the potential‐flow component, making good use of the quickly convergent technique with singular basis functions, correctly capturing the singular behavior in velocity fields near the tip of the plate, leads to a considerable reduction of computational burdens and to 12‐decimal‐place accuracy. The viscous correction is carried out via the meshless LVM with improved boundary conditions. Comparisons with previously published results show good agreement. Simulations of vortex generation and evolution illuminate the ability of the present method, and provide a supplement to pertinent experimental works. The hybrid scheme proposed herein allows flexibility for the former LVM and convenience in the code development. Such a compromise fits particularly well for the high‐resolution modeling of sharp‐edged vortex shedding without heavy numerical developments. Copyright © 2014 John Wiley & Sons, Ltd.     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

A model study for the steady state error of numerical approximations of inviscid flow equations on Cartesian grid

The widely used locally adaptive Cartesian grid methods involve a series of abruptly refined interfaces. In this paper we consider the influence of the refined interfaces on the steady state errors for second‐order three‐point difference approximations of flow equations. Since the various characteristic components of the Euler equations should behave similarly on such grids with regard to refinement‐induced errors, it is sufficient enough to conduct the analysis on a scalar model problem. The error we consider is a global error, different to local truncation error, and reflects the interaction between multiple interfaces. The steady state error will be compared to the errors on smooth refinement grids and on uniform grids. The conclusion seems to support the numerical findings of Yamaleev and Carpenter (J. Comput. Phys. 2002; 181: 280–316) that refinement does not necessarily reduce the numerical error. Copyright © 2005 John Wiley & Sons, Ltd.     

The two‐grid stabilization of equal‐order finite elements for the stokes equations

This work presents a two‐grid stabilized method of equal‐order finite elements for the Stokes problems. This method only offsets the discrete pressure space by the residual of pressure on two grids to circumvent the discrete Babu?ka–Brezzi condition. The method can be done locally in a two‐grid approach without stabilization parameter by projecting the pressure onto a finite element space based on coarse mesh. Also, it leads to a linear system with minimal additional cost in implement. Optimal error estimates are obtained. Finally, some numerical simulations are presented to show stability and accuracy properties of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

Higher‐order and adaptive discontinuous Galerkin methods with shock‐capturing applied to transonic turbulent delta wing flow

Discontinuous Galerkin (DG) methods allow high‐order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight‐sided elements. DG discretizations with higher polynomial degrees must, however, be stabilized in the vicinity of discontinuities of flow solutions such as shocks. In this article, we device a consistent shock‐capturing method for the Reynolds‐averaged Navier–Stokes and k‐ ω turbulence model equations based on an artificial viscosity term that depends on element residual terms. Furthermore, the DG method is combined with a residual‐based adaptation algorithm that targets at resolving all flow features. The higher‐order and adaptive DG method is applied to a fully turbulent transonic flow around the second Vortex Flow Experiment (VFE‐2) configuration with a good resolution of the vortex system.Copyright © 2013 John Wiley & Sons, Ltd.     

HLLE++格式在高马赫数空腔流动模拟中的应用

空腔流动存在剪切层运动、涡脱落与破裂,以及激波与激波、激波与剪切层、激波与膨胀波和激波/涡/剪切层相互干扰等现象,流动非常复杂,特别是高马赫数(M>2)时,剪切层和激波更强,激波与激波干扰更严重,对数值格式的要求更高,既需要格式耗散小,对分离涡等有很高的模拟精度,又需要格式在激波附近具有较大的耗散,可以很好地捕捉激波,防止非物理解的出现。Roe和HLLC等近似Riemann解格式在高马赫数强激波处可能会出现红玉现象,而HLLE++格式大大改善了这种缺陷,在捕捉高超声速激波时避免了红玉现象的发生,同时还保持在光滑区域的低数值耗散特性。本文在结构网格下HLLE++格式的基础上,通过改进激波探测的求解,建立了基于非结构混合网格的HLLE++计算方法,通过无粘斜坡算例,验证了HLLE++格式模拟高马赫数流动的能力,并应用于高马赫数空腔流动的数值模拟,开展了网格和湍流模型影响研究,验证了方法模拟高马赫数空腔流动的可靠性和有效性。