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

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Computational Fluid Dynamics Lecture 13 Solving for Pressure () 2 ,, p Rxyz ∇= 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. 222 Real Half complex ˆ First FFT in , , , ˆˆˆ ˆ ˆ ˆˆ then FFT in direction, , , ˆ ˆ xp pkyz ppp Rkyz xyz yp p k Rk z ∂∂∂ ++= A A z This second transform is a complex-to-complex transform. Finally, FFT in z -direction (complex-to- complex). ˆ p pk m A 1
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The PDE has been transformed into nx ny nz × × uncoupled algebraic equations. () 222 2 2 2 22 2 ˆˆˆ ˆ ˆ ˆ ,, then using in wave number space. ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ which is solved for ˆ ˆ ˆ ˆ ˆ ˆ for each , , and ; and then inverse transformi ppp Rk m xyz y kp p mp R p R pk m km ∂∂∂ ++= =− −−− = = ++ A A A A A ng in the reverse order ˆ ˆˆ pppp x y z →→→ The pressure field is spectral accurate. If only one -or two- flow directions are periodic, this is still
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This note was uploaded on 06/08/2011 for the course EOC 6850 taught by Professor Slinn during the Spring '07 term at University of Florida.

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

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