Grid and Mesh Generation. Introduction to its Concepts and Methods

Transcription

1 Grid and Mesh Generation Introduction to its Concepts and Methods

2 Elements in a CFD software sstem

3 Introduction What is a grid? The arrangement of the discrete points throughout the flow field is simpl called a grid. What is grid generation? The wa that a grid is determined is called grid generation.

4 Introduction Wh is grid generation necessar? The standard finite difference methods require a uniforml spaced rectangular grid. If a rectangular grid is used, few grid points fall on the surface. Flow close to the surface being ver important in terms of forces, a rectangular grid will give poor results in such regions. The rectangular grid is not appropriate for solution of the flow field.

5 Introduction How to overcame those problems? Use a nonuniform, curvilinear grid to mae the grid points naturall fall on the airfoil surface. The grid is not rectangular and is not uniforml spaced. The conventional difference equations are difficult to use. Need to transform the curvilinear grid in phsical space (,) to a rectangular grid in computational space (, ).

6 Introduction The procedure is as follows: Establish the transformation relations between the phsical space and the computational space Transform the governing equations and the boundar conditions into the computational space. Solve the equations in the computational space using the uniforml spaced rectangular grid. Perform a reverse transformation to represent the flow properties in the phsical space.

7 Outline General transformation Grid Generation Stretched (compressed ) grids Boundar-fitted coordinate sstem Adaptive grids Some modern development in grid generation Mesh Generation for Finite-Volume Method Unstructured Meshes Cartesian Meshes

8 Phsical Space & Computational Space If the airfoil is cut and the surface straightened out, it would form the -ais. Similarl, the outer boundar would become the top boundar of the computational domain. The left and right boundaries of the computational domain would represent the cut surface. Note the locations of points a, b, and c in the two figures.

9 General transformation The basic idea behind grid generation is the creation of the transformation laws between the phsical space and the computational space. There is a one-to-one correspondence between the phsical space and the computational space. Each point in the computational space represents a point in the phsical space. These laws are nown as the metrics of the transformation.

10 Transformation Relationship ),, ( ),, ( ),,, ( z and z z ζ ζ = = = = z z z w w w v v v u u u z w w w z v v v z u u u ζ ζ ζ ζ ζ ζ Jacobin matri = ζ ζ ζ z z z J 1 = z z z J ζ ζ ζ

11 Evaluation of the Transformation Parameters Tpicall the mapping is onl defined at grid and the transformation parameters must be evaluated numericall. The same means of discretization should be used for the evaluation of the transformation parameters and the derivative in the governing equations j j j j 1, 1, 1, 1, + + 1, 1, 1, 1, + + j j j j j j j j 1, 1, 1, 1, + + 1, 1, 1, 1, + + j j j j 2 1,,, = + j j j 2 1,, 1, 2 + = + j j j

12 The governing equations in generalized coordinates The continuit equation in Cartesian coordinates The continuit equation in generalized coordinates 0 ) ( ) ( ) ( = + + ρ ρ ρ J V J U J c c t v u U c = + where v u V c = + = v u t ρ ρ ρ

13 Formulation of grid generation problems In the view of phsics Find out topological correspondence between the phsical and computational domains In the view of mathematics Given = b (, ) and = b (, ) on the boundar R. Generate = (, ) and = (, ) in the region R bounded b R

14 Grid Generation Methods Algebraic methods Elliptic Grid Generation Adaptive grids Some eamples of modern development in grid generation 3-D dimensional boundar-fitted coordinate + adaptive grid Zonal grids Hbrid scheme Overlapping bloc

15 Algebraic Methods Concepts Known functions are used to map irregular phsical domain into rectangular computational domains. Eamples Stretched (compressed ) grids Boundar Fitted Coordinate Sstem

16 Eample: Stretched (compressed ) grids Grid stretching ma be necessar for some problems such as flow with boundar laers. Consider the transformation: = = ln( +1) Inverse transformation = = e 1

17 Eample: Stretched (compressed ) grids The following derivatives are used in the transformation = = ln( +1) = 1, = 0 1 = 0, = 1+ = = e 1 = 1, = 0 = 0, = e

18 Eample: Stretched (compressed ) grids The relation between increments and d = e d = e d = e d Therefore as increases, increases eponentiall. Thus we can choose constant and still have an eponential stretching of the grid in the -direction.

19 Eample: Stretched (compressed ) grids Transform the continuit equation Note there is relation between (,) plane and (,) plane

20 Eample: Stretched (compressed ) grids Substitute for the derivatives in Eq. (5.54) to get Eq. (5.57) is the continuit equation in the computational domain. Thus we have transformed the continuit equation from the phsical space to the computational space.

21 Boundar Fitted Coordinate Sstem Here we consider the flow through a divergent duct de is the curved upper wall fg is the centerline. Let s = f() be the function that represents the upper wall. The following transformation will give rise to a rectangular grid. = (5.65) = / (5.66) ma

22 Boundar Fitted Coordinate Sstem Eample: = ma =, = = ma 2 = 1, = 0 = = = 1 1 = 2 = 2

23 Elliptic Grid Generation The grid generation problem can be considered as a boundar-value problem, where the boundar conditions (namel, values of and ) are nown everwhere along the boundar. Given = b (, ) and = b (, ) on the boundar R. Generate = (, ) and = (, ) in the region R bounded b R. The transformation can be defined b an elliptic partial differential equation.

24 Elliptic Grid Generation Simplest elliptic equations Comments The elliptic equations are chosen to relate ζ and to and and hence constitute a transformation ( one-to-one correspondence of grid points) from the phsical plane to the computational plane = + = +

25 Elliptic Grid Generation An Eample:

26 Adaptive Grids Motivation Grid generation is important for the solution of CFD

27 Adaptive Grids Motivation - Wh need adaptive grids? We should put more grid points in the flow field with large gradients, and put less grid points in the flow field with small gradients. How do we now in advance where the major action is going to occur in the flow without actuall solving the problem? An adaptive grid is a solution!

28 Adaptive Grids What is an adaptive grid? A grid networ that automaticall clusters grid points in the regions of high flow-field gradient; It uses the solution of the flow-field properties to locate the grid points in phsical plane. It is intimatel lined to the flow-field solution and changes as the flow field changes. During the course of the solution, the grid points in the phsical plane move in such a fashion to adapt to regions of large flow-field gradients as these gradients evolve with time. It became stationar onl when the flow solution approach a stead state.

29 Eample Adaptive Grids

30 Some Modern Developments 3-D dimensional boundar-fitted coordinate + adaptive grid F-20

31 Some Modern Developments Zonal grids (Multi-bloc methods) The grid consists of two or more blocs Each bloc is a separate grid different from the others.

32 Some Modern Developments A zonal grid wrapped an F-16 airplane. Surface grid is shown as part of a 20-bloc grid.

33 Some Modern Developments in Finite- Volume Mesh Generation What is a structured mesh? The grid lines in phsical space pertain to constant coordinate values ζ, and ζ in the transformed space. A given famil of coordinate lines do not intersect. What is an unstructured mesh? Nonuniform grids in phsical space, not necessaril having the features of a structured mesh. The finite-volume calculations can be made directl in the phsical plane on a nonuniform mesh.

34 Some Modern Developments in Finite- Volume Mesh Generation Eamples for structured meshes and unstructured meshes structured meshes unstructured meshes

35 Return to Cartesian Meshes Ordinar Cartesian Meshes Eas to generate rectangular cells Not accurate in the region adjacent to the bod surface Modified Cartesian Meshes The mesh cells awa from the bod can be rectangular Those cells adjacent to the bod can be modified in the shape such that one side of each cell is along the bod surface.

36 Return to Cartesian Meshes Schematic diagram of modified Cartesian meshes

37 Return to Cartesian Meshes Modified Cartesian meshes + Adaptive concept A Cartesian mesh for the calculation of the subsonic flow over multielement airfoil

38 Return to Cartesian Meshes Modified Cartesian meshes + Adaptive concept A Cartesian mesh for the calculation of the hpersonic flow over a double ellipsoid, a configuration somewhat lie the space shuttle

39 Overlapping bloc This approach consists of building partiall overlapping blocs. Boundar conditions need to be echanged at the interface between domains and this is usuall done through some form of interpolation.

40 Hbrid scheme The hbrid scheme taes advantage of both unstructured and structured methods b appling structured bod fitted coordinates to the bod and unstructured networs in the outer boundaries.

41 Final Remars A transformation is required for finite-difference methods, because the finite-difference epressions are evaluated on the uniform grid. A transformation is inherentl not required for finitevolume methods, because it can deal directl with a nonuniform mesh in the phsical plane.