KT MHENDSLK FAKLTES MAKNA MHENDSL BLM MM 3019

KTÜ MÜHENDİSLİK FAKÜLTESİ MAKİNA MÜHENDİSLİĞİ BÖLÜMÜ MM 3019 BİLGİSAYAR DESTEKLİ AKIŞ ANALİZİ Doç. Dr. Yücel ÖZMEN Makina Mühendisliği Bölümü Karadeniz Teknik Üniversitesi yozmen@ktu. edu. tr http: //yozmen. ktu. edu. tr http: //aves. ktu. edu. tr/yozmen

DERSİN İÇERİĞİ ü SAYISAL AKIŞKANLAR DİNAMİĞİNE GİRİŞ ü GENEL KORUNUM DENKLEMLERİ üSONLU FARKLAR YÖNTEMİ üSONLU HACİMLER YÖNTEMİ ü AYRIKLAŞTIRMA ü GAMBİT PROGRAMI İLE AĞ ÜRETİMİ ü SINIR VE BAŞLANGIÇ ŞARTLARI ü LAMİNER AKIŞ ÖZELLİKLERİ ü TÜRBÜLANSLI AKIŞ ÖZELLİKLERİ ü BASİT TÜRBÜLANS MODELLERİ ü FLUENT PAKET PROGRAMI İLE AKIŞ VE ISI TRANSFERİ PROBLEMLERİNİN ÇÖZÜMÜ

DERS KAPSAMINDA YAPILACAK UYGULAMALAR • LAMİNER BORU AKIŞI • TÜRBÜLANSLI BORU AKIŞI • SİLİNDİR ETRAFINDAN AKIŞ • BASAMAK AKIŞI • UÇAK KANADI ETRAFINDAN AKIŞ • KAVİTE ÜZERİNDEN AKIŞ • DÜZ BİR LEVHA ÜZERİNDEN AKIŞ VE ISI TRANSFERİ • ÇARPAN JET AKIŞI

Types of Errors and Problems Types of Errors: Ø Modeling Error. Ø Discretization Error. Ø Convergence Error. Reasons due to which Errors occur: Ø Stability. Ø Consistency. Ø Conservedness and Boundedness.

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

Examples Structured grid Unstructured grid

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.

C-O Mesh for wing

Block structured grid

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)

Unstructured grid

Mesh adaptation Original Mesh Adapted Mesh

Mesh generation software Gridgen, Gambit, ICEM CFD, Grid. Pro Generating really good grids is an art.

AĞ ÜRETİMİ Meshing Algorithms

The Simulation Process Geometry Basics The Mesh Generation Process Meshing Algorithms Tri/Tet Methods Quad/Hex Methods Hybrid Methods Surface Meshing Mesh Data Representation Mesh Post Processing Smoothing Topology Improvement Adaptive Mesh Size Control

2 2 k. N 3 1. Build CAD Model 2. Mesh 3. Apply Loads and Boundary Conditions 4. Computational Analysis 5. Visualization

Tri/Tet Methods Octree Advancing Front Delaunay http: //cubit. sandia. gov/ http: //www. simulog. fr/mesh/gener 2. htm

Quad/Hex Methods Structured • Requires geometry to conform to specific characteristics • Regular patterns of quads/hexes formed based on characteristics of geometry • Internal nodes always attached to same number of elements Unstructured • No specific requirements for geometry • quads/hexes placed to conform to geometry. • No connectivity requirement (although optimization of connectivity is beneficial)

Structured Geometry Requirements • 6 topological surfaces • opposite surfaces must have similar mapped meshes 3 D Mapped Meshing

Structured Block-Structured http: //www. gridpro. com/gridgallery/tmachinery. html Mapped Meshing http: //www. pointwise. com/case/747. htm

Structured Gambit, Fluent Inc. Sweeping

Structured Sweeping Geometry Requirements • source and target surfaces topologicaly similar • linking surfaces mapable or submapable

Structured linking surfaces target source Sweeping Geometry Requirements • source and target surfaces topologicaly similar • linking surfaces mapable or submapable

Structured Sweeping Geometry Requirements • source and target surfaces topologicaly similar • linking surfaces mapable or submapable

Unstructured-Hex Plastering • 3 D extension of “paving” • Row-by row or element-by-element

Unstructured-Hex • 3 D extension of “paving” • Row-by row or element-by-element

Unstructured-Hex Exterior Hex mesh Remaining Void Ford Crankshaft Plastering+Tet Meshing “Hex-Dominant Meshing”

Unstructured-Quad Q-Morph

Hybrid Methods CFD Meshing

Hybrid Methods

Mesh Representation mass quadrilateral beam triangle 0 D 1 D 2 D wedge hexahedra 3 D pyramid

Grid generation l Grids can either be structured (hexahedral) or unstructured (tetrahedral). Depends upon type of discretization scheme and application – Scheme l Finite differences: structured l Finite volume or finite element: structured or unstructured – Application l Thin boundary layers best resolved with highly-stretched structured grids l Unstructured grids useful for complex geometries l Unstructured grids permit automatic adaptive refinement based on the pressure gradient, or regions of interest (FLUENT)

Grid Resolution

Grid generation and transformation Grids designed to resolve important flow features which are dependent upon flow parameters (e. g. , Re) Commercial codes such as Gridgen, Gambit For research code, grid generated by one of several methods (algebraic vs. PDE based, conformal mapping) For complex geometries, body-fitted coordinate system will have to be applied (next slide). Grid transformation from the physical domain to the computational domain will be necessary Sample grid established by Gambit of FLUENT

Grid transformation y o x Physical domain l. Transformation between physical (x, y, z) and computational (x, h, z) domains, important for body-fitted grids. The partial derivatives at these two domains have the relationship (2 D as an example) o Computational domain