Introduction to Computational Fluid Dynamics

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Introduction to Computational Fluid Dynamics

• Course Notes (CFD 4)

Karthik Duraisamy Department of Aerospace
Engineering University of Glasgow
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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

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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

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Examples
Structured grid
Unstructured grid
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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 (?,?,?) 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.

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C-O Mesh for wing
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Block structured grid
Courtesy, Pointwise Inc.
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Chimera Grid
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Chimera grids
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Chimera Grid
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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.

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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)

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Unstructured grid
Courtesy, MetaComp Tech
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Unstructured grid
Courtesy, MetaComp Tech
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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.

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Transonic flow over Onera M6 wing
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Mesh adaptation
Original Mesh
Adapted Mesh
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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 2Assume 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

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Mesh generation software

• Gridgen, Gambit, ICEM CFD, GridPro
• Generating really good grids is an art.