l14 - Lecture 14: Diffusion on Unstructured Meshes (Contd)...

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1 Lecture 14: Diffusion on Unstructured Meshes (Cont’d)
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2 Last Time… We looked at 2D steady diffusion on orthogonal and non-orthogonal unstructured meshes Saw that we needed a coordinate transformation to local coordinates to be able to discretize conveniently
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3 This Time We will Figure out how to use a local coordinate system to complete our discretization Figure out how to compute secondary gradients
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4 Non-Orthogonal Mesh Orthogonal Non-Orthogonal
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5 Cell Balance Governing equation: Manipulating as before: Here, face area vector Therefore:
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6 Area Vector is outward - pointing face normal is face area f f f f f A A An n
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7 Coordinate Transformation Writing the face flux as is not useful for discretization because the derivative cannot naturally be written in terms of cell-centroid values Transform to ( , ) coordinates Note: not orthogonal!
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8 Coordinate Transformation (cont’d) Using chain rule: Solve for x , y : Here, Jacobian J
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9 Face Flux Using coordinate transformation: How to find geometric factors?
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10 Geometric Factors
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11 Geometric Factors (cont’d) Furthermore:
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12 Face Flux Terms Recall face flux expression Let’s look at y
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13 Face Flux Terms (cont’d) Similarly
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14 Overall Face Flux Term Overall term: Secondary Gradient: Primary Gradient Secondary Gradient:
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This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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l14 - Lecture 14: Diffusion on Unstructured Meshes (Contd)...

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