Overset composite grids for the simulation of complex moving geometries

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1 EUA4X#21 Poster Session MASCOT06, IAC-CNR, Roma, Italy Overset composite grids for the simulation of complex moving geometries Joel Guerrero DICAT, University of Genoa, Via Montallegro 1, Genoa, Italy Abstract The overset composite grid method, also known as the Chimera overset grid technique, is reviewed and discussed in the context of a method for the solution and analysis of flows around complex moving geometries. We consider cases when the boundaries move according to a prescribed motion law and when the boundaries remain fixed. An important feature of the approach presented here is the efficient treatment of moving domains. Keywords: Chimera overset grids, overset composite grids, moving bodies. 1. Introduction Overset composite grids have long been recognized as an attractive approach for treating problems with complex geometries. The solution process uses a grid system that discretizes the problem domain by using separately generated but overlapping structured grids that periodically update and exchange boundary information through proper interpolation. This method has been used successfully over the last two decades, primarily to solve problems involving fluid flow in complex, often dynamically moving, geometries [1]. The first use of the overset composite grids method, also known as the Chimera overset grids technique (named like this after the composite monster of Greek mythology), was described by Volkov [2]. The method was developed further by Starius [3], and Chesshire & Henshaw [4]. Steger [5] and Benek [6], were among the first to introduce the Chimera overset grid technique into the CFD community. During this development, it became apparent that overlapping grids were also effective at handling moving grids problems [7]. The overlapping grids technique 1

2 has been used successfully to solve a wide variety of problems in aerodynamics, combustion, reactive flow with detonations, blood flow, visco-elastic flows and flows with deforming boundaries, to name a few. For example, transonic flow around the complete space shuttle vehicle with over 20 million grid points was simulated successfully using the overset composite grids methodology [8], and an unsteady flow around a tiltrotor helicopter under forward flight was solved with moving overlapped grids [9]. The use of adaptive mesh refinement in combination with overset composite grids has been considered by Brislawn, Chesshire et al. [10], Meakin [11], and Henshaw & Schwendeman [12]. Several numerical techniques have been developed to treat problems with moving boundaries. We mention some of these approaches and provide a nonexhaustive list of references. Of the techniques available, several may be grouped within a class of methods based on the use of a fixed underlying grid (usually a Cartesian grid). One such approach is the Cartesian-grid method. In this method, the boundary cuts the cells of the fixed grid creating irregularly shaped cells upon which the equations are discretized [13]. The immersedboundary method is another method which may be considered within this class. The technique imposes boundary conditions by introducing a body force into the equations and thus effectively smearing the interface [14]. In the immersed interface method and related approaches [15], the interface is kept sharp with special discretization stencils applied where the grid meets the boundary. Fictitious boundary methods [16] impose boundary conditions through the use of constraints and Lagrange multipliers, and may also be considered within this group. A second major class of methods is based upon the use of boundary-fitted conforming grids to represent moving boundaries. Approaches within this class include Arbitrary Lagrangian Eulerian (ALE) methods [17], moving mesh methods for unstructured meshes (r-refinement method) [18, 19] and global/local refinement/coarsening for unstructured meshes (h-refinement method) [18, 19]. Boundary-fitted conforming approaches are generally better suited for problems involving complex geometries and where boundary-layer phenomena are present. These methods also tend to be more expensive due to the cost of grid generation, and may be limited to moderate deformations of the boundaries. The approach discussed here, using overset composite grids, may be viewed as a synthesis of the two classes of methods. We use boundary-fitted conforming structured grids to achieve high-quality representations of boundaries. At the same time, the majority of grid points in a typical overset composite grid tends to belong to Cartesian grids so that the numerical efficiencies inherent with such grids can be exploited. The irregular boundary associated with standard Cartesian grid methods takes the form of the interpolation boundary between overlapping grids. The computational cost of grid generation for moving grids is small for the overset composite grids approach [1], and thus the total cost is similar to that required for a Cartesian-grid embedded-boundary method [12]. The remainder of this paper is organized as follows. Section 2 presents the overset composite grids methodology. Section 3 describes the extension of the methodology to moving boundary problems. Numerical results for a 2

3 fixed boundary problem are presented in Section 4. Here, we consider the flow around a multi element airfoil. In section 5, numerical results for a moving boundary problem are illustrated. Here, a heaving airfoil following a prescribed sinusoidal motion is simulated. The main purpose of these numerical results is to illustrate the use and accuracy of the methodology for a variety of geometries and prescribed motions. Finally, concluding remarks are given in Section Overview of the overset composite grids method The overset composite grids method is a way of assembling multiple grids and treating them as a single grid. Basically, this method consists in generating a set of structured component grids that cover the computational domain and overlap where they meet [20]. The geometry of the components can be defined individually and hence the grid around them can be generated separately. Boundary-fitted conforming grids are used near the components boundaries while one or more background Cartesian, O-type or C-type grids are used to handle the bulk of the domain. In this way, the method offers an effective way to reduce a geometrically complex problem into a set of simple component grids, allowing arbitrary movement of the component grids and scalability on parallel computer platforms [21]. Each component grid is structured and can have three types of grid points: discretization, interpolation, and hole points (fig. 1). The discretization points are used to discretize the PDE or the boundary conditions; the interpolation points interpolate their solution value from the overlapping component grid; and the hole points are disregarded during the discretization of the PDE. The hole points are either outside of the computational domain or are eliminated to reduce the total number of grid points in the overlapping grid. There are three major steps involved in the traditional overset composite grids scheme: (i) geometry and component grids generation, (ii) an algorithm for determining how to cut holes in the component grids which are overlapped, and (iii) an algorithm for interpolating field data between the various grids. Figure 3 shows various stages in the overset composite grids algorithm Geometry and component grids generation As a first step in the grid generation process, the geometry of the region to be discretized must be defined, that is, the surfaces that make up the boundary of the region must be described. A clean geometry description is created within the grid generation system itself or is exported to the grid generation system from a computer-aided-design (CAD) system. As mentioned before, the overset composite grid method makes use of boundary-fitted conforming grids (which are an essential asset to obtain viscous near-wall solutions accurately and economically), hence a set of structured grids must be created after the geometry definition, each grid defining a particular component of the overall geometry. One of these grids must cover the entire computational domain, this is the main grid. All other grids will be component grids (fig. 2, step 2). 3

4 Figure 1: A simple overset composite grid where the hole points are blanked. The solid squares symbols indicate interpolation points where the solution value is interpolated from the overlapping component grid. The remainder of the points are used to discretize the partial differential equation or the boundary conditions. 4

5 2.2. Hole cutting algorithm In this step, grid points are eliminated in both the main grid and the component grids. First, points in the component grids are eliminated, namely all points which are outside the computational domain and all points which overlap with other component grids and which are not needed for the solution interpolation. Next, all points in the main grid that are overlaid by component grids and which are not needed for the interpolation are eliminated. Doing so, a Chimera hole is created in the main grid (fig. 2, steps 3 and 4). For a detailed discussion of the different techniques used to effectively detect the extent of overlap for hole-cutting purposes and how to remove excess of interpolation points one can refer to [20, 22, 23, 24, 25] Interpolation algorithm The main difficulty in the use of overset composite grids methods is in the data transfer between overlapping grids and on how fluxes are treated at overlapping boundaries. Fig. 3 shows the inter-grid communication scheme for overlapping grids. Grid cell-centers are marked since we assume that the current scheme is a cell-centered finite volume scheme. The concepts of donor and receptor cells are used [26]. Consider two grids, Grid A and Grid B, that overlap with each other. A receptor point, say on Grid B, is a cell-center, i.e., a fringe point, which needs to receive flow information from Grid A to provide the boundary condition for Grid B. The donor cell for this receptor cell is identified as the cell on Grid A that contains the receptor point. A simple interpolation method consists in directly transfering the flow variables from the donor cell to the receptor cell. With little additional work, however, a more accurate trilinear interpolation scheme can be used, in which the eight or four (in two-dimensions) vertex points of the donor cell form the interpolation stencil points for the receptor point. The flow variables at the receptor point can then be easily obtained by using a trilinear interpolation over this set of stencil points. A trilinear interpolating scheme is fully compatible with second order finite volume solvers [21]. More sophisticated interpolation algorithms that maintain conservation are discussed in [27, 28] Data structures The hole cutting algorithm and interpolation algorithm explained before, involve frequent searches over the cells in different zones or component grids. These searches could be very expensive for large grids. In a steady state simulation, wherein the grid geometries remain invariant, hole cutting and interpolation stencil identification both need to be done only once. However, for a simulation that involves moving bodies, these procedures have to be done repetitively as the position of the overlapping grids relative to each other changes with time. Hence, it is imperative to expedite the search processes. These searches are highly optimized using an alternating digital tree (ADT) [23]. In ADT s, node points are organized in a hierarchical tree structure such that contiguous points 5

6 Figure 2: Overset composite grids method sequence; left to right, top to bottom. 1) Initial geometry. 2) Overlapping component grids. 3) Close up of the grid after cutting holes. Physical boundaries are used to cut holes in nearby grids. 4) Grid after removing all exterior points. The exterior points are easily swept out after the hole boundary has been marked, in this way a Chimera hole is created. 5) Grid after marking all (proper) interpolation points, indicated by solid squares symbols. 6) Final grid after removing excess interpolation points. 6

7 in the structure are actually neighboring points in the physical space. During an ADT search process, when an algorithm has to scan the grid for cells that match a given data, traversing down the tree enables the algorithm to get to the destination cells very efficiently. The ADT optimizes the search operations to O(log 2 N). Figure 3: Interpolation scheme for overlapping grids. 3. Extension of the methodology to moving boundary problems The presence of moving bodies changes the relative position of the overlapping grids continuously during the flow simulation. As the component grid (around a moving body) traverses through the computational domain, Chimera holes and interpolation stencils must be recomputed. The automation of hole cutting and interpolation procedures makes the present methodology a powerful tool for the simulation of flows with multiple moving bodies, since the grids do not have to be regenerated as the solution evolves. Only the Chimera holes and the locations of the interpolation points used to communicate information between the grids is recomputed at each time step, an operation which can be performed very efficiently. Without automation, such problems require the user to provide a priori all the input data that would enable hole cutting and interpolation stencil identification for any given configuration of the overset composite grid, a very time consuming operation. 7

8 4. Numerical results for a NHLP-2D three element airfoil Figure 4: Top, computational domain. Bottom, Mach number contours. A slotted NHLP-2D airfoil is analyzed and the results are compared with the results obtained using an unstructured mesh approach and results from the literature [29]. The method presented here is demonstrated to handle efficiently complex multi body geometries. The flow conditions for this case are: M = 0.198, α = 4.01 (air incidence angle). These results were obtained using a second order finite volume Euler solver with overlapping grids capability. In fig. 4 the computational domain is shown, which is made up of one main or background grid (BG) and three component grids (CGs). The BG (in black) is 8

9 made of elements. The airfoil CG (in red) consists of elements. The slat CG (frontward in blue) is made of 6879 elements and the flap CG (rearward in blue) has 9702 elements. The overall grid has elements. In figure 5, a comparison is displayed between the pressure coefficient obtained using the present approach, the results obtained with an unstructured flow solver and experimental results. We can notice that the results obtained with the current method show a similar trend to the experimental results [29], despite the fact that they are under/over predicted in some zones (mainly near the trailing and leading edge due to poor interpolation). Figure 5: Pressure coefficient comparison. Top figure correspond to the slat body, bottom left figure to the airfoil main body and bottom right figure to the flap body. The open square symbols correspond to experimental results, the open circle symbols correspond to a solution obtained on a unstructured mesh and the solid triangle symbols correspond to the solution obtained with the present approach. 9

10 5. Numerical results for a NACA 0012 heaving airfoil In this case, an unsteady flow computation is done as a NACA 0012 airfoil undergoes a periodic heaving motion. In fig. 6, the computational domain is shown, which is made up of one BG and one CG. The BG (in blue) is made up of elements; the CG (in green) consists of elements, the overall grid has elements. The flapping motion is defined by the following relationship: h = h a c cos(ω t), where c is the airfoil chord and h a the heaving amplitude (h a = 0.5); with a reduced frequency k = (ω c)/(2 U) = 0.5. The flow conditions for this case are: M = 0.3, Re = , α = 0. Time history of lift and drag coefficients is given on figure 7. It should be noticed that the flapping motion produces thrust (negative drag). The unsteady computation was carried out for more than 10 cycles of heaving motion. The results obtained agree remarkably well with those obtained in [30]. Figure 6: Computational domain. 10

11 Figure 7: Time history of lift and drag coefficients during flapping motion. Figure 8: Vorticity magnitude contours for non dimensional time t = 21.74, which correspond to the upward stroke. 11

12 6. Concluding remarks An overset composite grid methodology has been used in the present investigation. The current work demonstrates that the Chimera overset grids scheme is a promising and powerful tool to tackle complex flow problems involving multiple and moving bodies (for multiple bodies the grid generation process is greatly simplified by using the Chimera approach). Two test cases are presented to assess the accuracy of the present method; the results obtained agree very well with those from the literature in both test cases. 7. Acknowledgement The author would like to thank the European Atelier for Enginering and Computational Science (EUX4X) for the support provided. The research work of the author is supported by the Marie Curie FST project FLUBIO, through grant MEST-CT References [1] W. Henshaw, Overture: An Object-Oriented Framework for Overlapping Grid Applications, UCRL-JC , 2002 AIAA conference on Applied Aerodynamics, St Louis. [2] E. A. Volkov, The Method of Composite Meshes for Finite and Infinite Regions with Piecewise Smooth Boundaries, Proc. Steklov Inst. Math. 96 (1968) [3] G. Starius, Composite Mesh Difference Methods for Elliptic and Boundary Value Problems, Numer. Math. 28 (1977) [4] G. Chesshire, W. Henshaw, Composite Overlapping Meshes for the Solution of Partial Differential Equations, J. Comput. Phys. 90 (1990) [5] J. Steger, F. Dougherty, J. Benek, A Chimera Grid Scheme, ASME Mini- Symposium on Advances in Grid Generation, Houston, June [6] J. Benek, P. Buning, J. Steger, A 3D Chimera Grid Embedding Technique, AIAA Paper No , [7] F. C. Dougherty, J. Kuan, Transonic Store Separation Using a Three- Dimensional Chimera Grid Scheme, AIAA paper , [8] D. Pearce, S. Atanley, F. Martin, R. Gomex, G. Beau, P. Buning, Development of a Large Scale Chimera Grid System for the Space Shuttle Launch Vehicle, AIAA Paper No , [9] R. Meakin, Moving Body Overset Grid Methods for Complete Aircraft Tilt Rotor Simulations, AIAA Paper No CP,

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