Fast Multiscale Algorithms for Wave Propagation in Heterogeneous Environments

The capability of an unstructured hybrid grid method to compute the compressible Navier-Stokes equations is discussed for vortical flows over slender delta wings at low subsonic, high-alpha conditions as a basic study of offdesign aerodynamics of HSCT (High-Speed Civil Transport) configurations. Computations of these flows require high resolutions of the boundary layers on the wing and of the vortical flow region above the wing. The unstructured hybrid grid, which is composed of prismatic grid in boundary layers and tetrahedral and pyramidal grids for the remaining region, is used to treat such viscous flows. The compressible Navier-Stokes equations are solved on the hybrid grid by a cell-vertex, upwind finite volume method. The LU-SGS implicit time integration scheme is used for the convergence acceleration to the steady state. The numerical accuracy of the present method is discussed by comparing with the experimental data of the same configurations. It is demonstrated that the present hybrid grid method is efficient and accurate enough to predict the vortical flow fields over the delta wings. The capability of the method is also shown for a HSCT configuration at a low subsonic, high alpha condition.

The harmonic balance method has proven to be an efficient tool for the study of aeroelastic phenomena associated with blades in turbomachinery components. In the turbomachinery industry it is used, for example, to predict the flutter susceptibility of blades using structured meshes. For the analysis of more complex geometries, like cavities or labyrinth seals, unstructured grids are easier to create than structured ones to accurately model geometric details of these components. With this application in mind, specifically the flutter analysis of labyrinth seals, an existing harmonic balance solver for the unsteady Reynolds averaged Navier-Stokes equations on structured grids has been extended to work efficiently on unstructured grids. Although the harmonic balance algorithm itself is agnostic to the spatial discretization method, as it is only related to the discretization in time, some adaptations are necessary for an efficient implementation. To verify the extensions for unstructured grids, the harmonic balance solver is applied to the flutter analysis of a NACA airfoil exhibiting an oscillating shock-induced separation region. To demonstrate the ability to handle unstructured mixed-element grids, a flutter analysis of a generic labyrinth seal test case on an unstructured grid is performed.

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.

We present a comprehensive study of velocity interpolation methods in polygons. These methods are often used as postprocessing procedures for numerical schemes that do not directly calculate the velocity field but only provide cell boundary flux conditions, such as the finite volume schemes. These methods extend the widely used velocity interpolation algorithms, such as the Pollock's algorithm, to more complex geometries such as perpendicular bisection (PEBI) grids, unstructured triangular grids and grids with local refinement. Once the velocity field is interpolated, streamline trajectories and time of flight along the streamlines can be calculated for reservoir simulation, model calibration and waterflood management, for instance. These velocity interpolation methods assume known lower‐order or higher‐order cell boundary fluxes, which satisfy global mass conservation and normal flux continuity. However, they differ in the interpolation of velocities within the interior of the cells. The interpolating velocity may be locally conservative or nonconservative, continuous or discontinuous, lower order or higher order. Results show that the interpolated velocity field has to be locally conservative in order to guarantee the correct volumetric transformation for the calculated streamlines and the time of flight. Velocity continuity is not as important as local conservation for the purpose of streamline applications. Compared to higher‐order interpolation for the streamline trajectories, lower‐order interpolation has the advantage of an analytic solution and an efficient implementation. Based on our analysis, we recommend a lower‐order locally conservative method for the most robust and numerically efficient calculation of streamline trajectories on unstructured grids.

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

The discretisation scheme presented for an unstructured triangular grid, is able to reflect the physical properties of stationary, nonlinear heat conduction even in the numerical case and leads to a stabilisation of iterative solution techniques for nonlinearities.

Comparison of structured and unstructured grids in marine controlled source electromagnetic inversions for offshore hydrocarbon exploration

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.

Robust event-driven particle tracking in complex geometries

Towards the Parallelisation of Pressure Correction Method on Unstructured Grids

In this paper we focus on the application of a higher-order finite volume method for the resolution of Computational Aeroacoustics problems. In particular, we present the application of a finite volume method based in Moving Least Squares approximations in the context of a hybrid approach for low Mach number flows. In this case, the acoustic and aerodynamic fields can be computed separately. We focus on two kinds of computations: turbulent flow and aeroacoustics in complex geometries. Both fields require very accurate methods to capture the fine features of the flow, small scales in the case of turbulent flows and very low-amplitude acoustic waves in the case of aeroacoustics. On the other hand, the use of unstructured grids is interesting for real engineering applications, but unfortunately, the accuracy and efficiency of the numerical methods developed for unstructured grids is far to reach the performance of those methods developed for structured grids. In this context, we propose the FV-MLS method as a tool for accurate CAA computations on unstructured grids.

A multi-moment constrained finite volume method on arbitrary unstructured grids for incompressible flows

In reservoir simulation the pressure is the solution of an elliptic equation. It follows from the maximum principle that this equation satisfies a monotonicity property. This property should be preserved when discretising the pressure equation. If this is not the case, the pressure solution may have false internal oscillations and extrema on no-flow boundaries. Thus, there is a need for sufficient conditions for the discretization methods to be monotone. These conditions will depend on the permeability tensor and the grid. Previously monotonicity for control volume methods has been studied on quadrilateral grids and on hexagonal grids. The former was done for general methods which reproduces linear potential fields, while the latter was done in a Control Volume Finite Element setting. These analyses have given sharp sufficient conditions for the discretisation methods to be monotone. However, for methods whose cell stencils include cells which do not have any edges common with the central cell in the discretisation scheme, no work has been done on unstructured grids. In this work, we study monotonicity on triangular grids for control volume methods which are exact for linear potential fields. We derive sufficient conditions for monotonicity of the MPFA-O and -L methods. The found monotonicity regions for the MPFA methods are also tested numerically. The tests are done both on uniform grids in homogeneous media, and on perturbed grids, which corresponds to heterogeneous media. The investigations are done for single phase flow only. However, the results are relevant for multiphase simulations. The results obtained in this work may be utilised in grid generation. In this way we can construct grids where the discretisation of the pressure equation is guaranteed to be monotone.

This paper describes the hierarchical hybrid grid framework, which is a storage scheme for the efficient representation and application of discrete operators and variables on globally unstructured but patch-wise regular grids. By exploiting patch-wise regularity it overcomes the performance penalty commonly associated with unstructured grids. Furthermore, it is well suited for distributed memory parallel architectures.

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.