Module 3 Mesh Generation

Transcription

1 Module 3 Mesh Generation 1

2 Lecture 3.1 Introduction 2

3 Mesh Generation Strategy Mesh generation is an important pre-processing step in CFD of turbomachinery, quite analogous to the development of solid modeling that has been discussed in the earlier module for building the physical model of the computational domain. Two contrasting methodologies are developed for mesh generation: one, the multi-block structured mesh and the other, fully unstructured mesh using tetrahedra, hexahedra, prisms and pyramids. The former method of structured mesh generation produces the highest quality meshes from the point of view of solver accuracy but does not scale well on PC clusters. By contrast, fully unstructured meshes are fast to generate and automate the scale well on clusters, but do not allow solvers to 3 deliver their highest quality solutions.

4 Further, numerical tolerancing issues arise within the CAD system and are often exacerbated while imported from the modeler to the mesh generating tool. In the process, due to greatly differing scales within the geometry and lack of numerical compatibility between various geometrical representations, the model looses water-tightness and necessitates substantial cleaning. The CSG and BREP paradigms discussed in the previous module are also applicable while developing mesh generation algorithms and provide the required water-tightness to the geometry. Most CFD analysis codes, whether commercially available or developed in-house, follow the same (BREP) paradigm. 4

5 In order to solve the differential equations numerically, the continuous physical domain needs to be identified with a large set of discrete locations called nodes. The number of these discrete data points should be so large that the characteristic variations in the flow properties, determined after solving the differential equations by the numerical method, should be as close to the exact solution or bench-mark solution as possible. A method should be developed to mark the nodes in a fashion that is demanded by the numerical method that is to be used for solving the differential equation. The popular mesh generation methods are: structured, unstructured and hybrid. 5

6 Structured Mesh Generation For the implementation of numerical methods such as the finite difference, each node in the computational domain must have easily identifiable neighboring nodes. A grid or mesh that satisfies this demand is the structured mesh. Implementation of numerical methods on structured meshes using Cartesian or cylindrical polar grid system is possible only for simple rectangular or axi-symmetric geometries. In general, the generation of structured mesh for a complex flow domain involves automatic discretization methodology with boundary fitting coordinates and with coordinate transformations as discussed. 6

7 The basic steps in the methods of generating structured meshes for complex geometries are: mapping of the complex physical domain on to a simple computational domain; usage of body fitting coordinates transformation of lengths, areas, volumes and all vector quantities(e.g. velocity). 7

8 The mapping transformations should preferably be smooth conformal and controlled for grid spacing. Iso-parametric mapping of sub-domains enables creation of multi-block structured grids. The sequence of mapping determines whether the final mesh is a pseudo rectangular, O- type, C-type or H-type. 8

9 Figure demonstrates the method of generating the pseudo rectangular mesh for a physical region ABCD, bounded by lines x = 0.5 and y = (1-x 2 ). The co-ordinates ξ and η are body conforming. FIG Pseudo rectangular mesh 9

10 Using the transformation, y and x (1 x ) the domain ABCD in x-y plane (3.1.1 (a)) is mapped on to ξ-η plane as a unit square. Note that this transformation is not unique and we may have used suitable alternative transformations as well. The grid formed by the intersection of ξ = constant and η=constant lines in the physical domain shows the body conforming nature of these coordinates (Fig (b)). As we noticed in Lecture 1.2 (refer Fig ), the turbomachinery flow geometries are multiply connected domains, for which three basic grid configurations: O-type, C- type and H-type are widely used. For a given geometry, any one of these configurations can be obtained by suitable mapping. 10

11 Consider the multiply connected domain shown in Fig For the same geometry, different grid configurations (O, C or H) are generated by adopting slightly different methodologies. This is described in the following. O-type Meshing Introduce a branch cut and identify points (A,B,C,D) on either side of the branch cut as shown in Fig (a). Then, by mapping AB on A B, BC onto B C, CD on to C D and DA onto D A, O-type grid is obtained. The object boundary (AB ) and the external boundary (CD) become opposite sides of the transformed domain. The two sides of the branch-cut (BC and AD) are also mapped onto two opposite sides of the rectangular domain. Now, a grid constructed by ξ = constant and η=constant lines in the physical domain is O-type, as shown in Fig (c). The O-type meshes generated by this method for NACA airfoil and a turbomachinery blade are given in Figs and respectively.

12 (a) (b) Fig O-type Grid Generation, (a) basic branch-cut scheme, (b) Cartesian 12 grid in ξ-η plane (c) O-grid in the physical plane

13 An O-type mesh for symmetric NACA aerofoil is shown in Fig Fig O-type mesh for symmetric NACA aerofoil 13

14 Fig : O type mesh for a turbomachinery blade 14

15 C-type Meshing For C-type meshing of the same multiply connected domain, a branch-cut, as shown in Fig (a) is introduced and two points A and B are identified where the branch-cut meets the outer boundary. Points C and D are suitably selected on the external boundary and mapping is carried out with AB onto A B, BC onto B C, CD onto C D and DA onto D A. Note that the forward sweep of the branch-cut (AP), the object surface (PQ) and the reverse sweep of the branch-cut (QB) comprise one side A B of the transformed region. The object surface is mapped onto the patch P Q on this side. It can be seen that in this transformation, the η-constant lines in the interior envelop the object and the branch-cut, thus forming a C-type configuration. Figure shows a C-type mesh for a turbomachinery blade. 15

16 Fig C-type Mesh Generation, (a) basic branch-cut scheme, (b) Cartesian 16 grid in ξ-η plane (c) C-grid in the physical plane

17 Fig : C-type mesh for turbomachinery blade 17

18 H-type Meshing Figure shows a H-type mesh for a turbomachinery 18 blade. For H-type configuration, two branch-cuts are introduced on either side of the object and the upper and lower portions (ABCD and EFGH) are separately mapped onto A B C D and E F G H in the proper sequence (Fig ). Here, the object reduces to a line P Q in the middle of the transformed domain. It is evident form the examples that by choosing the mapping configuration, different types of grids can be generated for the same geometry. The appropriate choice depends on the nature of the problem to be solved. For complex domains with many objects, it may be necessary to map different regions separately, using local transformations. A variety of grid layouts such as the overlaid grids and embedded grids can be achieved through such procedures (Fig ).

19 Fig H-type Mesh Generation, (a) basic branch-cut scheme, (b) Cartesian 19 grid in ξ-η plane (c) H-grid in the physical plane

20 Fig : H type mesh for turbomachinery blade 20

21 Summary of Lecture 3.1 Mesh generation strategies for structured mesh are discussed. The methods for different types of meshes such as O, C and H type grids are presented. END OF LECTURE