Lecture 3.4 Differential Equation Based Schemes

Transcription

1 Lecture 3.4 Differential Equation Based Schemes 1

2 Differential Equation Based Schemes As stated in the previous lecture, another important and most widel used method of structured mesh generation is based on solving partial differential equations. These techniques ma be based on an one of the following schemes depending on the characteristics of the grid generation equations. The are: Hperbolic PDE based schemes Elliptic PDE based schemes Parabolic PDE based schemes 2

3 Elliptic PDE Based Methods In this lecture, Elliptic PDE based grid generation schemes will onl be discussed. These methods are particularl useful for confined phsical domains of turbomachiner flows. The elliptic PDEs for grid generation describe the variation of the bod fitting coordinates (ξ, η, ζ ) in the interior of the phsical domain, with prescribed values or slopes at the boundar. In the computational domain, however, the phsical coordinates (,, z) are treated as the unknown variables on the grid formed b the ξ=constant, η= constant and ζ=constant lines. The are determined b numericall solving the transformed grid generation equations. 3

4 Elliptic Solvers Consider the transformation functions, which are solutions of an elliptic Dirichlet boundar value problem. The mathematical problem is given b z z z P(,, z) Q(,, z) R(,, z) (3.4.1) with fied,, on the boundaries. P, Q and R are source functions, which can be used to grid point controlling. 4

5 Laplace Solvers The condition: P=Q=R=0, results in to uniform distribution of the points. The sstem then becomes the Laplacian. The Laplacian grid generation sstem satisfies the maimum and the minimum principles, i.e. both maimum and minimum for,, occur onl at the boundar. The solutions of the Laplacian operator,,, are either harmonic, sub-harmonic or super-harmonic. Therefore the are ver smooth functions. Further, the,, have continuous derivatives of all orders. This makes the solution of the transformed governing equation accurate 5

6 Two dimensional Transformation Functions Let us consider the bod fitting coordinate transformation, of the form = (,) and = (,). Therefore, one can write, (, ) d d d d (, ) d d d d (3.4.2) Considering the reverse transformation =(, ) and =(, ), (, ) d d (, ) d d (3.4.3) Combining the above two, we have 1 J 1 (3.4.4) 6

7 Two dimensional Transformation Functions One can therefore write the two dimensional transformation functions as f f ( f, ) (, ) (, ) (, ) (, f) (, ) (, ) (, ) ( f f ) ( f f ) (3.4.5) Note that the Jacobian, J, represents local area scaling factor and should not become zero. J J 7

8 Etension to Three Dimensions Etending the arguments to three dimensions, one can derive the relationships between the derivatives of the Cartesian coordinates (,,z) and the curvilinear coordinates (,, ) in the form: z z z z z z 1 (3.4.6) 8

9 Elliptic Solvers in 2D P 2 Q P 2 Q P (3.4.7) Q , 9

10 Demonstration Consider a planar region, as shown in Fig , in which a structured grid has to be generated Fig Phsical domain 10

11 Generate a rectangular (ξ, η) = (0,1)(0,1) uniforml discretized, as shown in Fig , (ξ i = i*δξ, η j =Δη, Δξ, Δη are the uniform step lengths in ξ, η directions, respectivel ) plane given b Fig Uniform grid in computation domain 11

12 Algorithm Map the boundaries (from phsical to computational) as shown in Fig Fig Mapping of phsical to computational domain Due to the mapping of the phsical boundaries over the boundaries of the computation domain,, are known along the boundaries of the computational domain. 12

13 Therefore, once, are computed in the interior of the computational domain, the required grid in the phsical domain is established. To obtain, in the interior of the computational domain, solve (using the boundar values of and as boundar conditions) P 2 Q P 2 Q (3.4.8) for, over the uniforml discretized computational domain 13

14 Multipl Connected Domain The algorithm described in the earlier slides works for simpl connected domains. For multipl connected domains, for eample like annular regions shown in Fig , artificial boundaries can be introduced to convert them in to simpl connected regions. Fig Introduction of artificial boundar 14

15 Boundar Conditions on the Artificial Boundaries Dirichlet boundar conditions over the artificial boundaries in the multipl-connected regions ma lead to non-smooth grid lines as shown in Fig Fig Non smooth grid lines over artificial boundar 15

16 Periodic Boundar Conditions on Artificial Cuts However Periodic boundar conditions over artificial cuts generate smooth grid lines as shown in Fig Fig Smooth grid over artificial boundaries 16

17 Grid Lines Attraction and Repulsion In general, grid points are attracted in the conve regions and repulsive in the concave regions as shown in the Fig Fig Grid point attraction and repulsion 17

18 Eercise Problems Repeat the eercise problems of Lecture 3.2 using PDE method. Generate uniform grid in square and cubical, rectangular and cubical region. Generate uniform grid in clindrical and spherical regions. Two-dimensional region bounded b circle of radius r = 1. Three-dimensional region bounded b sphere of radius r = 1. Annular region in 2D bounded b r = a and r = b with a < b. Annular region in 3D bounded b r = a and r = b with a < b. 18

19 Summar of Lecture 3.4 Mesh generation schemes b solving hperbolic, elliptic and parabolic partial differential equation methods are presented. The methods are eplained through eamples. END OF LECTURE