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Introduction to Computational Fluid Dynamics

Introduction to Computational Fluid Dynamics. Course Notes (CFD 4). Karthik Duraisamy Department of Aerospace Engineering University of Glasgow. Contents. Introduction (1.5) Classification of PDE, Model equations (1.5) Finite difference methods:  Spatial discretization (3)

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Introduction to Computational Fluid Dynamics

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  1. Introduction to Computational Fluid Dynamics Course Notes (CFD 4) Karthik Duraisamy Department of Aerospace Engineering University of Glasgow

  2. Contents • Introduction (1.5) • Classification of PDE, Model equations (1.5) • Finite difference methods:  Spatial discretization (3)  Temporal discretization (2)  Convergence, Consistency, Stability (2) • Grids/Boundary conditions (1) • Euler equations (1) • RANS Equations and Turbulence modeling (2) • DNS/LES (1) • Best practices in CFD (1) • Case studies/Demonstrations (3) (.) – Approximate number of lectures

  3. Grids

  4. Introduction • Solution of PDEs require spatial and temporal discretization • Typically, the spatially-discretized domain is called the grid • The PDE (originally on a continuous domain) is solved on a discrete set of grid points • Typically, the PDE is reduced to a set of algebraic equations (that might or might not involve matrix inversion) • Types of grids:  Structured grid  Unstructured grid  Hybrid grids

  5. Examples Structured grid Unstructured grid

  6. Structured grids • Can be Cartesian or curvilinear • Cartesian grids cannot be used for complex geometries and hence, body-fitted curvilinear grids are used • Curvilinear grids: Equations need to be transformed from physical (x,y,z) space to computational (ξ,η,ζ) space • Need boundary conditions on each of the “boundaries” • Elements are usually quadrilaterals / hexahedra • Example of airfoil grid / boundary conditions • Since we are Aerospace Engineers, we might be interested in airfoil grids – C grid, O grid, • Block structured grid • Chimera grid [ease of mesh generation, relative motion] • Good structured grids are orthogonal and smooth.

  7. C-O Mesh for wing

  8. Block structured grid Courtesy, Pointwise Inc.

  9. Chimera Grid

  10. Chimera grids

  11. Chimera Grid

  12. Overset Grid Solution Compute solution in interior points and apply BCs 1. Determine chimera points (C1) of inner grid 2. Find donor cells (D2) in outer grid that contain C1 3. Interpolate from D2 to C1 4. Find a water tight chain of points (C2) in the outer grid enclosing the solid surface in the inner grid. • ‘Blank out’ points in outer grid that lie inside the boundary formed by connecting C2. These are hole point (H2) and wont be solved for • Just as in step 2, find donor cells (D1) in inner grid that contain C2 • Interpolate from D1 to C2.

  13. Unstructured grids • Unlike structured grid, do not have definite data structure • Elements are usually triangular/tetrahedral • Gives a lot of flexibility in mesh generation (multi-blocking not required, can tessellate CAD geometry). • Unstructured solvers are more expensive than structured solvers and getting high order of accuracy is more difficult. • Equations are solved in integral form (so no derivatives are required)

  14. Unstructured grid Courtesy, MetaComp Tech

  15. Unstructured grid Courtesy, MetaComp Tech

  16. Mesh adaptation • Accuracy goes as (dx)^n. Therefore, to get higher accuracy, either reduce dx or increase n. usually, it is hard to get very high order accurate methods to work properly (convergence, stability, software issues). Therefore, dx is reduced. This is called mesh adaptation. • Mesh adaptation is either done visually, or by identifying a variable in the solution. Example, shear layer  good idea might be to track vorticity.

  17. Transonic flow over Onera M6 wing

  18. Mesh adaptation Original Mesh Adapted Mesh

  19. Grid generation methods • Structured Grids:  Conformal mapping [complex variable transformations, restricted to 2D]  Algebraic [polynomials, trigonometric functions]  PDE Methods:  Step 1: Determine the grid point distribution on the boundaries of the physical space.  Step 2:Assume the interior grid point is specified by a differential equation that satisfies the grid point distributions specified on the boundaries and yields an acceptable interior grid point distribution. • Unstructured grids  Octree  Delaunay  Advancing front

  20. Mesh generation software • Gridgen, Gambit, ICEM CFD, GridPro • Generating really good grids is an art.

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