25 Hybrid Grids and Their Applications 25.1 25.2

Introduction Underlying Principles The Structured Marching Method for Prisms • The Octree-Advancing Front Methods for Tetrahedra

25.3

Best Practices High Speed Civil Transport (HSCT) Aircraft • Adapted Hybrid Mesh • Resolution of Multiple Wakes • Deforming Hybrid Mesh in 2D • Turbomachinery Blade with Tip Clearance • ABB Burner Case

Yannis Kallinderis

25.4

Research Issues and Summary

25.1 Introduction There is an ever-increasing demand to perform flow simulations that incorporate the complete details of geometry as well as sophisticated field physics. The success of numerical flow simulators depends to a great extent on the computational grid that is employed. As a consequence, grid generation has become a task of primary importance. Books and surveys on grid generation include [1–6]. Structured meshes consisting of blocks of hexahedra and unstructured grids consisting of tetrahedra have been the traditional means of discretizing 3D flow domains [2, 3]. Hybrid grids usually consist of prisms and tetrahedra in 3D, and correspondingly quadrilaterals and triangles in 2D. Layers of prisms are employed to resolve boundary layers and wakes, while tetrahedra cover the rest of the domain. Hybrid meshes are intended to provide flexibility by combining essential features of the two broad types of meshes, namely the structured and the unstructured grids [7–15]. Hybrid meshes consisting of triangles and quadrilaterals have been employed in two dimensions in [16–26]. Other hybrid mesh techniques involve generating a mesh made up of tetrahedral and prismatic elements and then destructuring the prisms to form tetrahedra [27, 28]. Adaptation and load balancing for parallel computation of hybrid grids have been presented in [29, 30]. There are a number of issues to be addressed when dealing with turbulent flow simulations involving complex geometries. These considerations include: (1) the different orientation of the viscous flow features, (2) the disparate length scales that need to be resolved within the same domain, (3) the requirements of the Navier–Stokes solvers, (4) the grid generation time, (5) the required user expertise, as well as (6) the university of application of the grid generator. The main features that are encountered in flow fields include boundary layers, wakes, shock waves, and vortices. These features have different orientations that make generation of a single grid that conforms to them very difficult. In addition, the mesh has to follow the boundaries of the computational domain. A hybrid grid that combines elements of different orientation appears to be much more flexible in

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conforming to the flow features. The prisms are assigned the task of capturing the features that are following the body surface, while the tetrahedra are used for the features that are away (e.g., shocks and vortices). The different spatial scales encountered in viscous flows vary by orders of magnitude from each other. These scales are imposed by the flow features and the geometry. The laminar sublayer requires placement of grid points at distances away from the wall of the order of 10–6 times the scale of the geometry, while the points at the farfield may be at a distance of order 1 from one another. Shock waves and vortices have very different scales as well. Furthermore, the details of the geometry frequently impose scales on the grid generator. The gaps between the main wing and the flap and the tip clearances in turbomachinery geometries are typical examples of small scales. The issue becomes even more complex when taking into account the directionality of the different scales. The small scale required in the boundary layers is in the direction normal to the surface, while much larger sizes of the mesh are sufficient in the lateral directions. Similar directionality also exists in wakes and shock waves. This directionality leads to the issue of generating high aspect ratio grid cells. Generation of thin prismatic grids for the boundary layers and wakes has the advantages of being feasible and fast, and also results in a smaller number of elements compared to tetrahedra. On the other hand, the isotropic nature of tetrahedra appears to be appropriate for the vortices and other regions of the domain where the flow is changing equally in all directions. Navier–Stokes solvers place strict requirements on the mesh. Accuracy and stability of the numerical methods depend crucially on the local resolution and the uniformity of the grid. Smooth transition of element sizes at the prism/tetrahedra interface is important for accuracy and robustness of Navier–Stokes numerical methods [8, 9]. Furthermore, computing resources in terms of CPU time and memory storage are dictated by the number of grid elements. These facts place several requirements on mesh generation. Employment of the thin semistructured prismatic elements in the regions of shear layers results in sufficient accuracy with significantly reduced computing resources compared to all-tetrahedral meshes. The flow field on the body surface usually contains regions of strong flow directionality such as the leading and trailing edges of a wing. Generation of anisotropic surface grid elements results in significant savings in the number of elements without sacrificing accuracy. Minimum user expertise and universal application are also primary considerations placed on grid generators. A generation method must use a relatively small number of control parameters whose effects are obvious even to an inexperienced user. It is highly desired that a grid generation method be applicable to a great variety of geometries without modification. Furthermore, the setup time to apply the generator should be kept to a minimum.

25.2 Underlying Principles The hybrid grid generator consists of two major parts: (1) the prisms generator, which is an algebraic, marching-type technique, and (2) the tetrahedra generator which is an advancing front type of method (see Chapter 19). Details of the two techniques can be found in [6, 8, 9].

25.2.1 The Structured Marching Method for Prisms An unstructured triangular grid is employed as the starting surface to generate a prismatic mesh. This grid, covering the body surface, is marched away from the body in distinct steps, resulting in generation of semistructured prismatic layers in the marching direction (Figure 25.1). The process can be visualized as a gradual inflation of the body’s volume. A major issue with marching methods is the avoidance of crossing of the grid lines. There are three main aspects of the algebraic grid generation process: (1) determination of the directions along which the nodes will march (marching vectors), (2) determination of the distance by which the nodes will march along the marching vectors, and (3) smoothing operations on positioning of the nodes on the new layer.

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FIGURE 25.1

FIGURE 25.2

Generation of a prismatic grid from a triangular boundary surface.

Example of the manifold of point Pi and its corresponding visibility region.

25.2.1.1 Determination of the Marching Vectors Each node on the marching surface is advanced along a marching vector. The marching direction is based on the node-manifold, which consists of the group of faces sharing the node to be marched. The primary criterion to be satisfied when marching is that the new node should be visible from all the faces on the manifold (visibility condition) [7]. An example of a manifold and its corresponding visibility region is shown in Figure 25.2. The dark-shaded region is the manifold of node Pi, and the polyhedral cone above the node is the visibility region. The vector, Vi, is one possible node-normal satisfying the visibility constraint. The node-normal vector lies on the bisection plane of the two faces on the manifold that form the wedge with the smallest angle. This process has yielded consistently valid normal vectors at the nodes by constructing the vector most normal to the most acute face planes. Essentially, it does this by

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maximizing the minimum angle between the node-normal and all the surrounding face normals. This vector is then used as the marching direction for the nodes on the surface to form the new layer. A more detailed description of the marching procedure can be found in [7]. 25.2.1.2 Marching Step Size Determination of marching distances is based on the characteristic angle βave of the manifold of each node to be marched. This angle is computed using the average dot product between the pairs of faces forming the manifold. The marching distance is a linear function of βave. It yields relatively large marching steps in the concave regions, and small steps in the convex areas of the marching surface. Specifically, the distance ∆n is:

∆n = (1 + α )∆nave ,

(25.1)

where ∆nave is the averaging marching step for the layer, and α is a linear function of the manifold angle βave. The sign of α is positive for concave regions and negative for convex regions. The average marching step for each layer (j), ∆nave is computed based on a user-specified initial marching step ∆no on the body surface and a stretching factor st, as follows: j ∆nave = ∆no × st ( j −1) .

(25.2)

25.2.1.3 Smoothing Steps The initial marching vectors are the normal vectors. However, this may not provide a valid grid, since overlapping may occur, especially in concave regions of the grid surface with closely spaced nodes. To prevent overlapping, the directions of the marching vectors must be altered. Altering of the directions should not end abruptly in the local neighborhood of the nodes involved, since this may cause overlapping in nearby regions. A gradual reduction of the magnitude of the change in the vector direction is accomplished via a number of weighted Laplacian type smoothing operations over the marching vectors of all nodes. Typically, ten smoothing passes are performed. These smoothing steps rotate each original marching vector based on the normal vectors of its surrounding manifold nodes as follows:

r r Vi = ωVi ′+ (1 − ω )

1 ∑ j 1 dij

r

∑ (1 d )V , ij

j

(25.3)

j

r r r where Vi ′ and Vi are the initial and final marching vectors of node i, while Vj are the marching vectors of the surrounding nodes j that belong to the manifold of node i. The weighting factor ω is a function of the manifold characteristic angle βave. It has small values in concave regions, and relatively large ones in convex areas. The averaging of the marching vectors of the neighboring nodes is distance-weighted with dij denoting the distance between nodes i and j. A similar procedure is employed for the smoothing of the marching steps ∆n to eliminate abrupt changes in cell sizes.

25.2.1.4 Constraints Imposed to Enhance Quality Typical Navier–Stokes integration methods impose restrictions on the spacing of the points along the marching lines and on the smoothness of these lines. In other words, the prismatic grid should not be excessively stretched or skewed. Constraints are imposed on the lateral and normal distribution of marching step sizes and the deviation of the direction of the marching vectors from one layer to the next. The lateral distribution of cell sizes are constrained so that any node on the current marching surface cannot have a step size (∆ni) that is very different from the size (∆nj) corresponding to any of its surrounding nodes. Specifically,

0.5 × ∆n j < ∆ni < 2.0 × ∆n j , ©1999 CRC Press LLC

(25.4)

The constraints on the step size variation along each marching line are applied in a similar manner. A node on the prismatic layer j cannot have a step size that is smaller than that on the previous layer (j – 1). Also, it cannot exceed the size of the previous step by more than a factor of stmax (usually set to 1.3). Specifically,

∆n j −1 < ∆n j < stmax × ∆n j −1 .

(25.5)

r r Another constraint limits the deviation between two consecutive marching vectors Vj−1 and Vj to be less than a specified angle (typically 30°). The above-mentioned constraints reduce “kinks” in the marching vector directions as well as abrupt changes in step sizes, thus providing a smooth mesh suitable for viscous flow computations. Since the visibility criterion is the ultimate test for the validity of the mesh, this criterion is the final constraint imposed on the grid.

25.2.1.5 Automatic Adjustment of the Prism Layer Thickness Treatment of narrow gaps and cavities in regions such as wing–engine configurations and in between different bodies in multiply connected domains has been a major concern for structured and semistructured mesh generators. The structured nature of prisms prohibits filling such complex geometries without overlapping layers if special measures are not taken. A method has been developed that adjusts the marching step of the prism layers for the treatment of such gaps [8]. The technique allows entirely automatic generation of single-block, nonoverlapping prismatic meshes. Two key features of the method are no user interaction and universality of its application to different geometries. The nodes in the vicinity of a cavity are detected by a special algorithm. The marching distances of these flagged nodes are reduced so that the mesh does not overlap. This may result in prismatic meshes of significantly varying local thickness. Smooth variation of the thickness is attained via lateral smoothing of the size of the marching steps. The local thickness of the prism layer in the cavity or gap region is reduced to avoid overlapping prism layers. This is done by recomputing the initial marching distance ∆no for all the flagged nodes according to the following equation:

∆no =

C1 × dG , ∑ jst ( j −1)

(25.6)

where dG denotes the gap distance computed by a special gap-detection algorithm, C1 is a user- specified constant controlling the extent of reduction (usually chosen to be 0.25), st is the stretching factor, and j is the prism layer index. Thus, the total thickness of the prism layers in the vicinity of the gap is approximately C1 × dG, with slight variations depending on the local curvature of the marching surface. The exact step size for every node on each layer is then determined by Eqs. 25.1 and 25.2. In order to avoid abrupt changes in the thickness of the prism layers due to the local receding, the unflagged nodes in the neighborhood of the cavity are also receded to a certain extent. This extent gradually reduces to zero as the nodes get farther away from the cavity or gap.

25.2.2 The Octree-Advancing Front Methods for Tetrahedra A combined octree-advancing front method is used to generate the unstructured grid [9]. Advancing front type methods require specification by the user of the distribution of three parameters over the entire domain to be gridded. These field functions are (1) are node spacing, (2) the grid stretching, and (3) the direction of the stretching. Using the octree-advancing front method, these parameters do not need to be specified. Instead, they are determined via an automatically generated octree. The octree is constructed via a divide-and-conquer process, which starts with a master hexahedron that contains the body. This hexahedron is recursively subdivided into eight smaller hexahedra called octants. ©1999 CRC Press LLC

Any octant that intersects the body is a boundary octant and is subdivided further (inward refinement). The subdivision of a boundary octant ceases when its size matches the local length scale of the geometry. The choice of the local length scale depends on the particular application of the octree. The length scale can be chosen to be local prism thickness, edge length, or curvature. This flexibility allows the same octree creation technique to be used for many different unstructured applications. Then, the hexahedral grid is further refined in a balancing process (outward refinement) to prevent neighboring octants whose depth differs by more than one. Outward refinement is performed to ensure that the final octree varies smoothly in size away from the original surface. The sole criterion for outward refinement is a depth difference greater than one between the octant itself and any of its neighbors. The outward refinement continues until no octants meet the refinement criterion. Typically, five sweeps are performed to produce a balanced octree. The octree data structure is similar to earlier data structures used for search operations during the grid generation process [31] (see Section 14.4.2.1 of Chapter 14). Two important features of the octree-advancing front method are its capability to match disparate length scales and its geometry independence. The octree is able to insure a smooth size transition over the large range of length scales which are present in a “viscous” mesh. The octree is also able to be used for many different types of geometries with minimal user interaction. 25.2.2.1 Length Scales Octree refinement is terminated when the size of a boundary octant is the same size as the local length scale of the geometry. This local length scale depends on the application. Three different applications are considered, namely, surface mesh generation, tetrahedral mesh generation for hybrid grids, and all tetrahedral mesh generation. For surface mesh generation, the local length scale is determined by the local curvature of the geometry. This length scale is small in areas where the curvature is large, i.e., the trailing edge of a wing, and large where the geometry is flat. The distance between surfaces is another length scale used for surface mesh generation. The local length scale is proportional to this distance. This allows for automatic clustering in regions where surfaces are in close proximity. For hybrid prismatic/tetrahedral mesh generation, the local length scale is simply the local thickness of the last prismatic layer. This will ensure that the size of the tetrahedra in the direction normal to the outer prismatic surface is the same as the height of the neighboring prisms. This smooth transition in size from the prisms to the tetrahedra is important for accuracy of the numerical method. Finally, for an all tetrahedral mesh, the local length scale is the local edge length of the original triangulated surface. The octree-advancing front method can also be used to create meshes for inviscid simulations. Given an initial surface triangulation, the octree is refined until the boundary octants match the size of the local surface triangulation. Figure 25.3 shows plane cuts of the octree for two different geometries. The first case corresponds to the High Speed Civil Transport (HSCT) aircraft, while the second to a two-element wing. A plane cut of the prismatic part of the hybrid mesh is also shown. The size of the octants intersecting the outermost prismatic surface matches the thickness of the last prismatic layer, even in the region of the engine where the thickness of the prisms is several orders of magnitude smaller than their thickness away from the engine. The same observations apply to the second case of the two-element wing. 25.2.2.2 Octree Guides Advancing Front Mesh Generation The advancing front volume grid generation starts from the surface of the body or the outermost prismatic surface for the case of a hybrid grid. The triangular faces of this surface form the initial front list. A face from this list is chosen to start the tetrahedra generation. Then, a list of points is created that consists of a new node, as well as of “nearby” existing points of the front. One of these points is chosen to connect to the vertices of the face. Following the choice of the point, a new tetrahedron is formed. The list of the faces, edges, and points of the front is updated by adding and/or removing elements [32]. The method requires a data structure that allows for efficient addition/removal of faces, edges, and points, as well as for fast identification of faces and edges that intersect a certain region. The alternating digital tree (ADT) algorithm is employed for these tasks [33] (see Section 14.25.4.3 of Chapter 14).

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FIGURE 25.3 Plane cuts of octree meshes. The top figure shows a plane cut of an octree mesh for the HSCT aircraft. The bottom figure shows a plane cut of an octree mesh for the partial-flap high-lift wing at mid span of the flapped region. Both figures show how the octant sizes match the local thickness of the final prismatic layer. Every third layer of the prismatic mesh is shown for clarity of the figure.

The tetrahedra that are generated using this octree method grow in size as the front advances away from the original surface. Their size, the rate of increase of their size, as well as the direction of the increase, are all given from the octree. The octants are progressively larger with distance away from the body. Their sizes determine the characteristic size of the tetrahedra that are generated in their vicinity. This method is flexible and can be used to generate tetrahedra around different types of geometry. The surface mesh generation proceeds in the same manner as the tetrahedral mesh generation, except that surface triangles are generated from an initial front made up of edges [32]. The surface geometry is treated as a patchwork of CAD panels (see Chapter 19 and Part III). An interface is required between the CAD representation and the surface grid generator. The interior of each panel is filled with triangles using the same octree for each panel to insure smooth size transitions across panel boundaries. New triangles are generated using either already existing points, or new points generated on the surface using information from the octree. The octree allows for a smooth transition in size on the surface from areas where the triangles are small (i.e., trailing edge) to areas where the triangles are larger. The advancing front method creates a new element by connecting each face or edge of the current front to either a new or an existing node. This new point is found by using a characteristic distance δ calculated from the size of the local octant to which the face of the front belongs. Specifically,

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δ = αst ( t ) , l −l

(25.7)

where α is a scaling factor, st is the stretching parameter, lt is the total number of octant levels, and l is the level of the local octant. The value of st controls the rate of growth of the mesh. The lower the value of st, the less the mesh increases in size away from the body. A typical value of the stretching parameter st is 1.8. The level l of the local octant is the number of subdivisions of the master octant required to get to the size of the local octant. For hybrid mesh generation, smooth transition in size from the prisms to the tetrahedra is important for accuracy of the numerical methods. The value of the scaling factor α is calculated so that the initial marching size (δ ) of the tetrahedra equals the local thickness of the outermost prismatic layer. For surface mesh generation, α can be varied to generate different meshes using the same octree. Higher values of α result in coarser meshes, while lower values of α yield finer meshes. Both the coarse and fine meshes will have similar local variation of the sizes of the surface triangles. 25.2.2.3 Anisotropic Surface Meshes The octree-advancing front method can also create anisotropic surface meshes. Anisotropic meshes are useful in reducing the number of triangular faces needed to capture all the flow features in a simulation. Allowing high aspect ratio triangles aligned with geometry and flow features in regions that exhibit strong directionality enables a substantial savings in number of both surface and volume grid elements. A user only needs to specify the following: (1) a line segment that defines the direction of the stretching of the mesh, (2) the aspect ratio (AR) of the triangles desired along that line segment, and (3) the area of influence (dmax) of the line segment. Examples of such line segments include the leading edges, trailing edges and engine inlets. The method for generating anisotropic meshes starts with the size, δoct , given by the octree and augments it with the perpendicular distance, d from the user-specified line segment. The local mesh size is now characterized by three sizes, δ 1, δ 2 and δ 3 given by

δ1 = c × δ oct δ 2 = δ oct δ 3 = δ oct ,

(25.8)

with

c=

AR − 1 d + AR, dmax

(25.9)

and δ 1 is the size of the mesh in the direction of the line segment, while δ 2 and δ 3 are the sizes of the mesh in directions perpendicular to the line segment and perpendicular to each other. The method is flexible and robust using multiple line segments at different locations and directions to define directionality on different parts of the surface. Furthermore, it provides a smooth transition between regions of different directionality. Figure 25.4 shows both an isotropic surface mesh for the M6 wing and an anisotropic mesh created with line segments extending over the entire leading and trailing edges with the same aspect ratios as the previous mesh. The isotropic mesh has 39,290 faces while the anisotropic mesh has 6333 faces while maintaining the same chord-wise point density obtained from the same octree. These meshes show the 6.2:1 reduction in the number of generated faces when an anisotropic method is used. This reduction in faces leads to a substantial reduction in the number of elements of the corresponding volume mesh. 25.2.2.4 Automatic Partial Remeshing Grids generated using an advancing front type scheme can contain regions of low quality within the mesh domain. These low-quality regions must be altered before the mesh can be used with a flow solver. A method for improving low quality regions has been developed [9]. This method removes low quality regions from the mesh and fills the resulting cavities using the same advancing front generator on the new front defined by the surface of these holes. ©1999 CRC Press LLC

FIGURE 25.4 Significant savings in number of triangles are realized due to the use of leading- and trailing-edge line segments for the ONERA M6 wing. The top mesh is an isotropic mesh with 39,290 faces. The bottom mesh is an anisotropic mesh with only 6,333 faces. Note that even though the isotropic mesh has six times the number of faces, the anisotropic mesh has the same chordwise point distribution.

In order to properly define the low quality regions of the mesh, the quality of a given region must be quantified. There are several measures of mesh quality. One such indicator that has been used is the volume ratio of the two tetrahedra sharing each face, R = Volmax/Volmin. Large values of R indicate a very stretched mesh. If R = 1, the mesh is locally uniform. Once the low quality regions of the mesh have been located using the quality measure R, these regions must be removed from the mesh. For each face with a value of R greater than a user-specified value, Rsp, a cavity is opened around the low quality region by removing tetrahedra. The radius of the opened cavity is dependent on the local length scale of the mesh. After cavities have been formed around each of the low quality regions of the mesh, the exposed triangular faces inside the cavities are put together to form a new initial front. Then, the advancing front generator refills the cavities with better quality tetrahedra. This process of cavity definition and cavity remeshing is repeated until a specific level of quality is reached. The entire process of cavity definition and remeshing is performed automatically with no user intervention. The remeshing process is efficient and typically takes a quarter of the time that the initial tetrahedral generation requires.

25.3 Best Practices This section presents applications of hybrid grids that include both external and internal geometries. The cases are chosen in order to demonstrates the suitability of the hybrid grids for complex geometries, as

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FIGURE 25.5 Anisotropic surface mesh for the HSCT with engines. The figure shows the anisotropic regions near the leading and trailing edge of the wing. The mesh has 30,189 faces, while a similarly spaced isotropic mesh would have 60,583 faces.

well as the robustness and generality of the generator to yield meshes for very different topologies. The specific cases are: (1) an aircraft configuration, (2) an adapted hybrid mesh, (3) resolution of multiple wakes past a wing, (4) a deformable hybrid grid in two dimensions, (5) a turbomachinery blade with tip clearance, and (6) a burner.

25.3.1 High Speed Civil Transport (HSCT) Aircraft The High Speed Civil Transport (HSCT) aircraft is a next-generation aircraft being designed to travel at supersonic speeds. It has a double-delta wing configuration emerging from the nose. The cavity between the engine and the wing presents a challenge in grid generation. An example of a locally directional surface mesh for the aircraft is illustrated in Figure 25.5. It is observed that the method generates a reduced number of points on the wing in the spanwise direction while maintaining a large number of nodes in the chordwise direction. A strongly directional mesh has been generated primarily in the leading and trailing edge regions of the wing. A view of the hybrid mesh is shown in Figure 25.6. The third is shown on two surfaces that are perpendicular to each other. The first is the symmetry plane with the quadrilateral faces corresponding to the prisms and the triangular faces corresponding to the tetrahedra. The second surface is a field cut intersecting the fuselage and engine. A field cut is a cut through the discretized grid showing all the cells that intersect the plane cut, thus emphasizing the 3D nature of the grid. The prisms are assigned the task of capturing the features that are following the aircraft surface, while the tetrahedra are used for the features that are away (e.g., shocks and vortices). Figure 25.7 illustrates the widely varying length scales of the hybrid grid. The field cut shows portion of the fuselage, the wing, as well as the engine. Note the varying thickness of the prismatic layer which is dictated not only by the thickness of the boundary layer, but also by the size of the cavity between the engine and the wing. The tetrahedral part of the mesh is very dense in the cavity area in order to match the sizes of the local prisms and becomes isotropic away from the cavity.

25.3.2 Adapted Hybrid Mesh The case of an adaptively embedded hybrid mesh is presented next for the same HSCT aircraft geometry. Turbulent flow is simulated with Mach number (M∞) equal to 3, angle of attack (α ) equal to 5° and Reynolds number (Re) equal to 6.3 × 106. The grid is locally embedded according to the magnitude of flow gradients [29]. Figure 25.8 shows a plane cut of the adapted hybrid grid employed for simulation of turbulent supersonic flow around the aircraft. A view of the solution via entropy contours on the initial coarse grid and the corresponding locally refined hybrid mesh is shown. The right hand side of

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FIGURE 25.6 View of the hybrid mesh around the HSCT aircraft with engines on two different planes that are perpendicular to each other. The first plane is that of the symmetry while the second is a field cut intersecting the fuselage and engine.

FIGURE 25.7 Close up of the hybrid grid for the HSCT aircraft around the engine cavity. The tetrahedral mesh is very dense here compared with other regions so as to match the thin local prism cell sizes.

©1999 CRC Press LLC

FIGURE 25.8 View of the solution (entropy contours) on the coarse grid and the corresponding adapted grid for the HSCT configuration. The right-hand side of the figure shows the initial mesh superimposed with entropy contours. The adapted hybrid grid (left side) has been refined in the vicinity of the vortex, and near the wing/fuselage junction. Case of turbulent flow with M∞ = 3, α = 5° and Re = 6.3 × 106.

the figure illustrates the initial grid superimposed with entropy contours of the solution. Two are the main flow features here. The boundary layer conforms to the surfaces of the fuselage and wing, while the vortex has a totally independent orientation. The prismatic mesh used follows the shape of the boundary layer, while the tetrahedral grid appears to be more appropriate for the vortex. Furthermore, local refinement has been applied by the adaptation algorithm in the region of the vortex.

25.3.3 Resolution of Multiple Wakes The ability of the prismatic elements to capture multiple wakes is illustrated by generating a hybrid grid about a generic two-element wing. The approach used here extends fictitious surfaces past the trailing edges of both the blades in the direction of the wakes. Then, prisms are generated marching away from both the wing-surface as well as the fictitious surfaces to capture the viscous effects at the wall and the wake region. The grid consists of 9000 boundary nodes of which 5300 are on the surface of the main wing and flap (the rest are on the fictitious surfaces extended into the wakes). A view of the hybrid mesh is shown in the field cut in Figure 25.9. The grid consists of seven prism layers and 53,000 tetrahedra. A completely unstructured mesh in the wake region would require a very large number of tetrahedra. The prisms in between the main wing and the flap have been receded by the procedure described in Section 25.2.1.5 to prevent grid overlapping. Note the grid clustering in the wake and the smooth transition in cell sizes across the domain.

25.3.4 Deforming Hybrid Mesh in 2D Deformation of a hybrid mesh is now demonstrated via an example of a two-dimensional grid about two circular discs aligned in the tandem direction. This grid has been employed for simulation of vortexinduced vibrations to the two bodies. Figure 25.10a shows the mesh when both cylinders are in their initial position. The thick horizontal and vertical lines are included as a point of reference indicating the

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FIGURE 25.9 Field cut of the hybrid grid for the two-element wing. The grid comprises 9K boundary nodes (including those on the fictitious surface), seven prism layers and 53K tetrahedra. The prisms provide adequate grid clustering in the wake region with fewer cells compared with an all-tetrahedral mesh.

equilibrium position of each cylinder. Figure 25.10b shows the resulting deformed mesh when the two cylinders move away from each other in the transverse direction. Note that the significant displacement of the two cylinders is nicely accommodated by the triangular elements, and connectivity of the mesh is preserved.

25.3.5 Turbomachinery Blade with Tip Clearance The next case considered is an internal geometry. It is a turbine blade with narrow tip clearance. Figure 25.11 shows two perpendicular field cuts of the hybrid mesh around the blade. The surface was composed of 13,663 triangular faces. The hybrid grid consists of 14 layers of prisms (191,282 prismatic cells), and 415,086 tetrahedral cells. The tetrahedra were able to easily match the prismatic thickness everywhere, including the small gap between the tip of the blade and the shroud. Also, the surface mesh is much finer in the tip region adapting to the features of the geometry. It is important to note that the grid generation scheme was able to mesh an internal geometry as easily as the external geometries presented in the previous sections.

25.3.6 ABB Burner Case The final case corresponds to flow through a burner, which consists of an annulus diffuser, a swirl producer, and a combustion chamber. This case has been provided by ABB. The geometry has various complexities such as the fuel injection holes, severe cavities, twisted blades that produce the swirl, and vastly different length scales. The geometry has periodic boundaries, and only one burner is being modeled. Figure 25.12 shows a close-up of the surface triangulation for the swirl producing section. The surface consists of approximately 75,000 triangles. A hybrid mesh of the burner is seen in Figure 25.13, which is a two-dimensional cut along the axis. The view shows that the hybrid grid generator was capable of capturing all the fine features of the geometry, and also clustered points downstream of the swirl producing section. The mesh consists of 415,000 nodes, 521,000 prisms, and 748,000 tetrahedra. A cut

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(a)

(b) FIGURE 25.10 Deforming hybrid grids about a tandem cylinder geometry for (a) initial cylinder configuration, (b) cylinders displaced in the transverse direction.

across the swirl producing blades is shown in Figure 25.14. The view shows the hybrid nature of the mesh, and demonstrates the smooth transition in cell sizes even across different element types.

25.4 Research Issues and Summary Employment of hybrid grids for complex geometries was demonstrated. The prism covered regions of strong flow directionality, such as boundary layers and wakes, while tetrahedra were created elsewhere. The hybrid grid generator consists of two major parts: (1) a special marching method for generation of the prismatic elements, and (2) a combined octree-advancing front technique for generation of the tetrahedra. Narrow gaps and cavities, very disparate length scales, body and flow-field conformity of the mesh, as well as automation were the primary issues that guided the development of the generator. The use of hybrid grids was demonstrated through complex geometries. The hybrid mesh generator was successful in handling severe cavities and capturing widely varying length scales. Applications included the two main categories of topologies: (1) external and (2) internal. The marching-vectors procedure to generate the prisms proved to be robust (in avoiding overlapping of prism layers) and efficient. The smoothing operations and the imposition of constraints eliminated

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FIGURE 25.11 Surface and volume hybrid mesh for a turbine blade with narrow tip clearance. The volume grid is shown via two field cuts that are on surfaces perpendicular to each other intersecting the blade.

FIGURE 25.12 A close-up view of the swirl-producing section of the ABB Burner geometry. The surface is made up of 75,000 triangles.

surface “ripples” and avoided excessively stretched and skewed meshes. The automatic adjustment of the thickness of the prismatic layer allowed the generation of a single-block, nonoverlapping prismatic mesh even when the surface geometry contained narrow gaps and cavities. The mesh generator allowed for marching along arbitrary parametric surfaces, and was also capable of generating periodic meshes. ©1999 CRC Press LLC

FIGURE 25.13 A close-up view of the hybrid cut along the axis of the burner geometry. The grid generator was capable of automatically handling the small length scales and the severe cavities.

FIGURE 25.14 A cut across the swirl-producing blades of the burner geometry. The hybrid nature of the mesh and the smooth transition in cell sizes are visible.

The octree-advancing front approach provided an automatic method for generating unstructured meshes. The method was effective in generating surface triangulations for different complex geometries including a burner surface. The octree allowed surface triangulations to be generated that captured all of the geometry features. The octree also provided for a smooth variation of grid size over the entire surface mesh. ©1999 CRC Press LLC

Anisotropic surface meshes were generated using the octree and minimal user input. These anisotropic meshes resulted in a significant reduction in the number of faces generated. Smooth transition between the different regions of directionality was also accomplished. Generation of tetrahedra via the advancing front method was also made simpler and more automatic by eliminating the traditional user-defined background mesh for determination of mesh spacing. An automatically generated octree guided the growth of the tetrahedra and enabled a smooth transition of the mesh from the prisms to the tetrahedra in a hybrid mesh. The universality of the octree-advancing front method was demonstrated through its application to different complex geometries. The HSCT aircraft configuration demonstrated that the method is flexible enough to adapt to 200:1 size variations in the local length scale. Local remeshing of the tetrahedral mesh proved very effective in removing areas of abrupt changes in size of the tetrahedra.

Further Information Additional sources of information on hybrid grids and grid generation, in general, can be found in the proceedings and papers of the following conferences: • International Conference on Numerical Grid Generation, held every two years • AIAA Computational Fluid Dynamics Conference, held every two years • AIAA Aerospace Sciences Meeting, held in Reno, NV every year • AIAA Applied Aerodynamics Conference, held every year • International Conference on Finite Elements in Fluids, held every two years • International Meshing Roundtable, sponsored by Sandia National Labs • NASA Conference Proceedings on “Unstructured Grid Generation Techniques” (NASA LaRC, CP

10119, Sept. 1993) and “Surface Modeling, Grid Generation, and Related Issues in Computational Fluid Dynamics Solutions” (NASA LeRC, CP 3291, May 1995)

References 1. Thompson, J.F., Warsi, Z.U.A., Mastin, C.W., Numerical Grid Generation, North-Holland, New York, 1985. 2. Thompson, J.F. and Weatherill, N.P., Aspects of numerical grid generation: Current Science and Art, AIAA Paper 93-3539, 1993. 3. Baker, T.J., Developments and trends in three dimensional mesh generation, Applied Numerical Mathematics. 1989, Vol. 5, pp 275–304. 4. Baker, T.J., Mesh generation for the computation of flowfields over complex aerodynamic shapes, Computers Math. Applic. 1992, Vol. 24, No. 5/6, pp 103–127. 5. Eiseman, P. R. and Erlebacher, G., Grid Generation for the solution of partial differential equations, NASA CR 178365 and ICASE Report No. 87–57, August 1987. 6. Von Karman Institute (VKI) Lecture series in computational fluid dynamics, LS 1996-06, March 25–29, 1996. 7. Kallinderis, Y. and Ward, S., Prismatic grid generation for 3D complex geometries, Journal of the American Institute of Aeronautics and Astronautics. October 1993, Vol. 31, No. 10, pp. 1850–1856. 8. Kallinderis, Y., Khawaja, A., McMorris, H., Hybrid prismatic/tetrahedral grid generation for viscous flows around complex geometries, AIAA Journal. February 1996, Vol. 34, No. 2, pp 291–298. 9. McMorris, H. and Kallinderis, Y., Octree-advancing front method for generation of unstructured surface and volume meshes, AIAA Journal. June 1997, Vol. 35, No. 6, pp 976–984. 10. Nakahashi, K. and Obayashi, S., FDM-FEM approach for viscous flow computations over multiple bodies, AIAA-87-0605, 1987. 11. Karman, S. L., SPLITFLOW: A 3D unstructured Cartesian/prismatic grid CFD code for complex geometries, AIAA-95-0343. Reno, NV, January 1995.

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12. Sharov, D. and Nakahashi, K., Hybrid prismatic/tetrahedral grid generation for viscous flow applications, AIAA-96-2000, Proc. of the 27th AIAA Fluid Dynamics Conf. New Orleans, LA, June 1996. 13. Van der Burg, J., Maseland, J., Oskam, B., Development of a fully automated CFD system for threedimensional flow simulations based on hybrid prismatic-tetrahedral grids, Proc. of the 5th Int. Conf. on Numerical Grid Generation in Computational Field Simulations. Mississippi State University, April 1–5, 1996, pp 557–566. 14. Chappell, J., Shaw, J., Leatham, M., The generation of hybrid grids incorporating prismatic regions for viscous flow calculations, Proc. of the 5th Int. Conf. on Numerical Grid Generation in Computational Field Simulations, pp 537–546, Mississippi State University, April 1–5, 1996. 15. Noack, R., Steinbrenner, J., Bishop, D., A three-dimensional hybrid grid generation technique with application to bodies in relative motion, Proc. of the 5th Int. Conf. on Numerical Grid Generation in Computational Field Simulations. Mississippi State University, April 1–5, 1996, pp 547–556. 16. Kallinderis, Y. and Nakajima, K., Finite element method for incompressible viscous flows with adaptive hybrid grids, AIAA Journal. August 1994, Vol. 32, No. 8, pp 1617–1625. 17. Hufford, G.S. and Mitchell, C.R., The generation of hybrid and unstructured grids using curve and area sources, AIAA-95-0215. Reno, NV, January 1995. 18. Spragle, G.S., Smith, W.A., Weiss, J. M., Hanging node solution adaption on unstructured grids, AIAA-95-0216, Reno, NV, January 1995. 19. Kao, K.H. and Liou, M.S., Direct replacement of arbitrary grid-overlapping by nonstructured grid, AIAA-95-0346. Reno, NV, January 1995. 20. Nakahashi, K., FDM-FEM Zonal approach for computations of compressible viscous flows, Lecture Notes in Physics. 1986, Springer, Vol. 264, pp 494–498. 21. Weatherill, N.P., Mixed structured–unstructured meshes for aerodynamics flow simulation, The Aeronautical Journal. Vol. 94, 134, pp 111–123. 22. Soetrisno, M., Imlay, S.T., Roberts, D.W., A zonal implicit procedure for hybrid structured-unstructured grids, AIAA-94-0645, Reno, NV, January 1994. 23. Koomullil, R.P., Soni, B.K., Huang, C.-T., Navier–Stokes Simulation on hybrid grids, AIAA Paper 96-0768, Reno, NV, January 1996. 24. Hwang, C.J. and Wu, S.J., Adaptive finite volume approach on mixed quadrilateral-triangular meshes, AIAA Journal. January 1993, Vol. 31, No. 1, pp 61–67. 25. Banks, D., Mueller, J.-D., VankeirsBilck, P., An Object oriented approach to hybrid structured/unstructured grid generation, AIAA Paper 96-0032. Reno, NV, January 1996. 26. Coirier, W. and Jorgenson, P., A Mixed volume grid approach for the Euler and Navier–Stokes equations, AIAA Paper 96-0762. Reno, NV, January 1996. 27. Connell, S.D. and Braaten, M.E., Semistructured mesh generation for 3D Navier–Stokes calculations, AIAA-95-1679-CP. San Diego, CA, June 1995. 28. Pirzadeh, S., Viscous unstructured three-dimensional grids by the advancing-layers method, AIAA Paper 94-0417. Reno, NV, January 1994. 29. Parthasarathy, V. and Kallinderis, Y., Adaptive prismatic-tetrahedral grid refinement and redistribution for viscous flows, AIAA Journal. April 1996, Vol. 34, No. 4, pp 707–716. 30. Minyard, T. and Kallinderis, Y., Octree partitioning of hybrid grids for parallel adaptive viscous flow simulations, Int. J. for Num. Meth. in Fluids. January, 1998, Vol. 26, pp 1–22. 31. Lohner, R., Some useful data structures for the generation of unstructured grids, Communications in Applied Numerical Methods. 1988, Vol. 4, pp 123–135. 32. Peraire, J., Morgan, K., Peiro, J., Unstructured Finite element mesh generation and adaptive procedures for CFD, in Application of Mesh Generation to Complex 3D Configurations, AGARD Conference Proceedings No. 464, 1990, pp 18.1–18.12. 33. Bonet, J. and Peraire, J., An Alternating Digital Tree (ADT) algorithm for 3D geometric searching and intersection problems, Int. J. for Numerical Methods in Engineering. 1991, Vol. 31, pp 1–17.

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