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CFD lecture 13

CFD lecture 13 - Computational Fluid Dynamics Lecture 13...

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Computational Fluid Dynamics Lecture 13 Solving for Pressure ( ) 2 , , p R x y z = Elliptic equations such as the Poisson equation are sensitively dependent on Boundary Conditions. Choices of B.C.’s are: 1. Periodic (if you’re lucky) 2. Dirichlet: 0 B p g = 3. Neumman: 0 B p f z = The easiest method (computationally fast) is to use Fast Fourier transforms in any periodic directions. For example in 3-D if all 3 directions are periodic. ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 Real Half complex ˆ First FFT in , , , ˆ ˆ ˆ ˆ , , ˆ ˆ ˆ then FFT in direction, , , ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ , , x p p k y z p p p R k y z x y z y p p k p p p R k z x y z + + = + + = A A z This second transform is a complex-to-complex transform. Finally, FFT in z -direction (complex-to- complex). ( ) ˆ ˆ ˆ ˆ ˆ , , p p k m A 1

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The PDE has been transformed into nx ny nz × × uncoupled algebraic equations. ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ , , then using in wave number space. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ which is solved for ˆ ˆ ˆ ˆ ˆ ˆ for each , , and ; and then inverse transformi p p p R k m x y z y k p p m p R p R p k m k m + + = = − = = + + A A A A A ( )
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CFD lecture 13 - Computational Fluid Dynamics Lecture 13...

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