Navier-Stokes computations of HSCT off-design aerodynamics using unstructured grids

This paper discusses the use of the vorticity confinement method coupled with the unstructured grid approach to simulate vortical flows. The method is evaluated by several vortical flow computations of the leading-edge separation vortices on delta wings and the wing tip vortices of NACA0012 wing. It is shown that the vorticity confinement can keep the vorticity away from the numerical diffusion effectively. Although further study to reduce the dependency of the confinement coefficient on the grid density is required, the present results indicate the possibility of accurate vortical flow computations by the vorticity confinement method coupled with unstructured and adaptive refinement grids.

A new agglomeration algorithm to generate coarse grids for multigrid methods on unstructured and hybrid grids is proposed in this paper. The algorithm, which we call a global coarsening algorithm, is based on the edge coloring of the grids so that it can agglomerate any type of grids. The agglomeration procedure is automatic and generates a globally optimum shape in contrast to the conventional greedy-type frontal algorithm. The method is applied to compute the 2D/3D Euler and 2D Navier-Stokes equations on triangular, tetrahedral and hybrid meshes, respectively. The present agglomeration multigrid method shows better convergence than the existing method owing to the better quality of the coarse grids.

A new two-dimensional hybrid grid generation method based on improved hole cutting

An implicit point Gauss-Seidel relaxation scheme is described for solution of the unsteady Euler equations on unstructured hybrid grids in three dimensions. The use of generalized hybrid grids is motivated by the need for flexible discretization of complex configurations and subsequent solution of the unsteady Navier-Stokes equations. The solver uses a cell-centered finite-volume implementation with an unstructured face-based grid connectivity, which allows each cell to have an arbitrary number of bounding faces, and each face to have an arbitrary number of bounding nodes. The method to compute cell and face geometry metrics for arbitrary element types is described. The numerical flux is defined by a hybrid flux-vector splitting scheme, and the upwind states at each cell face are computed with a second-order accurate extrapolation of the flow solution. The cell-averaged flow gradient is computed with the Green-Gauss integral theorem, which requires the face-averaged solution that is constructed from the solution at the forming nodes. The solution at each node is reconstructed by a second-order accurate interpolation of cell-averaged data in all cells connected to the node. Flow solutions on generalized hybrid grids are presented for a NACA 0012 airfoil section in two dimensions, and a ONERA M6 wing configuration in$hree dimensions.

Nonlinear aerodynamics significantly influence the flight loads and aeroelastic stability of current and proposed transport aircraft. Direct simulation of the nonlinear aerodynamics with Computational Fluid Dynamics codes, while technically possible, is economically impractical due to the large number of conditions which must be analyzed. While techniques exist for addresssing these nonlinear effects in production analysis for conventional configurations, the available techniques are sometimes inaccurate, and are not directly applicable to advanced configurations such as High Speed Civil Transport or Blended Wing Body designs. A technique is presented that uses a small number of CFD solutions or wind tunnel test results along with linear aerodynamic theory to generate linearized aerodynamic models for use in production aeroelastic analysis and design. The technique is applied to the steady aeroelastic loads analysis of a High Speed Civil Transport configuration, and to the flutter analysis of an isolated wing.

The objective of the Cranked-Arrow Wing Aerodynamics Project International (CAWAPI) was to allow a comprehensive validation of Computational Fluid Dynamics methods against the CAWAP flight database. A major part of this work involved the generation of high-quality computational grids. Prior to the grid generation an IGES file containing the air-tight geometry of the F-16XL aircraft was generated by a cooperation of the CAWAPI partners. Based on this geometry description both structured and unstructured grids have been generated. The baseline structured (multi-block) grid (and a family of derived grids) has been generated by the National Aerospace Laboratory NLR. Although the algorithms used by NLR had become available just before CAWAPI and thus only a limited experience with their application to such a complex configuration had been gained, a grid of good quality was generated well within four weeks. This time compared favourably with that required to produce the unstructured grids in CAWAPI. The baseline all-tetrahedral and hybrid unstructured grids has been generated at NASA Langley Research Center and the USAFA, respectively. To provide more geometrical resolution, trimmed unstructured grids have been generated at EADS-MAS, the UTSimCenter, Boeing Phantom Works and KTH/FOI. All grids generated within the framework of CAWAPI will be discussed in the article. Both results obtained on the structured grids and the unstructured grids showed a significant improvement in agreement with flight test data in comparison with those obtained on the structured multi-block grid used during CAWAP.

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

The results of numerical simulation and comparison with experimental data of the aerodynamic characteristics of a blunt cone are presented. Two authors’ computer codes were used, intended for the numerical integration of the Navier-Stokes and Euler equations on tetrahedral unstructured and multi-block structured computational grids. Alternative numerical models on unstructured grids are analyzed in detail.

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values.

and back portion of the airfoil is shown in Figs. 2b and 2c, respectively. The smooth transition of the grid from one type to another and the decency of the grid at sharp corners and between close surfaces are clearly shown, which indicate excellent compatibility and communication between the two grid strategies as controlled by a common background grid. The grid was generated using a Silicon Graphics IRIS 4D/210 VOX workstation. The code's performance is about 360 triangles/s on that workstation and 2350 triangles/s on a Cray Y-MP. The total turnaround time required for generating a grid around a complex geometry, including the setup time, is about 1 h. To examine the fidelity of the grids generated with the present method, a turbulent viscous-flow computation was performed on the grid shown in Fig. 2. The flow solution was obtained with an available node-based, upwind flow solver8 using the BaldwinEarth turbulence model at a Mach number of 0.2, an angle of attack of 16.2 deg, and a Reynolds number of 9 X 106. Comparisons for surface pressure distributions and velocity profiles are shown in Figs. 3a and 3b, respectively. Excellent agreement with experimental data confirms the viability of the generated grid. Concluding Remarks A new method of unstructured viscous grid generation has been introduced. The approach is conceptually simple but powerful, and capable of producing high-quality viscous unstructured grids for complex configurations with ease. Being based on a totally unstructured grid strategy, the method is fully automatic and flexible and, thus, alleviates the difficulties stemming from the structural limitations of the existing structured or semi-unstruct ured techniques. Because of an efficient grid-marching strategy and a simple front-detection algorithm, the method is highly efficient and operational on small workstations. The method is also self-sufficient for insertion of grid points in the boundary layer and beyond. The method has been applied to two-dimensional problems with satisfactory results. The basic elements of the technique, however, have been primarily designed for its subsequent extension for generating three-dimensional highly stretched tetrahedral grids which is currently in progress. The full benefit of the method will be realized for generation of three-dimensi onal viscous grids where the complexity of the problem becomes excessive.

ABSTRACTThis paper presents a novel numerical model for incompressible flows on unstructured hybrid grids by combining the pressure-implicit with splitting of operator (PISO) algorithm and volume-integrated average and point value-based multimoment (VPM) method. Implementing the spatial discretization of VPM to the PISO solution procedure results in a novel formulation that is unconditionally stable and superior in numerical accuracy and robustness in comparison with the conventional finite volume method. The present VPM/PISO formulation provides a numerical framework of great practical significance that well balances the numerical accuracy and algorithmic complexity. Numerical verifications demonstrate that the present model can significantly improve numerical accuracy. Moreover, the numerical dissipation is effectively suppressed, which shows a great potential for simulations of high-Reynolds number flows.

An experimental investigation into the physical phenomena associated with limit cycle wing rock on slender delta wings has been conducted. The model used was a slender flat plate delta wing with 80-deg leading edge sweep. The investigation concentrated on three main areas: motion characteristics obtained from time history plots, static and dynamic flow visualization of vortex position, and static and dynamic flow visualization of vortex breakdown. The flow visualization studies are correlated with model motion to determine the relationship between vortex position and vortex breakdown with the dynamic rolling moments. Dynamic roll moment coefficient curves reveal rate-dependent hysteresis, which drives the motion. Vortex position correlated with time and model motion show a time lag in the normal position of the upward moving wing vortex. This time lag may be the mechanism responsible for the hysteresis. Vortex breakdown is shown to have a damping effect on the motion.

The objective of this research project was to further develop and integrate numerical methods for the fast and accurate simulation of wave propagation problems in the time domain. In support of the long-term goal of creating high-quality software for simulating waves, we seek methods which are not only efficient, but which are reliable in that both their stability and the accuracy of the results are essentially guaranteed. In support of this goal we have developed i. convenient implementations of optimal local radiation boundary sequences for isotropic waves, with implementations in a wide variety of popular discretization schemes for Maxwells equations ii. extensions of these sequences to more complex systems arising in linear elasticity iii. new highly efficient energy-stable discretization schemes on structured grids - these include methods based on Hermite interpolation and compact difference schemes constructed using Galerkin techniques iv.stable coupling of the efficient structured grid methods with upwind discontinuous Galerkin methods defined on unstructured grids - using hybrid grids allows us to treat very complex geometry with efficency comparable to simple domains v. natural upwind discontinuous Galerkin discretizations for wave equations in second order form - using the second order form for complex systems results in fewer dependent variables.

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.