CFD lecture 6

CFD lecture 6 - Computational Fluid Dynamics Lecture 6...

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Computational Fluid Dynamics Lecture 6 Space differencing errors. 0 C tx ψ ∂∂ += Seek traveling wave solutions. ( ) ik x t e ω k is wave number and is frequency. =kC is dispersion relation. where C is phase speed. C k = , true solution is non dipersive for constant C . note that the group velocity, g C , the speed of energy propagation is defined as g C k = , in this case g CC = 2 nd order spatial differences. () 11 2 2 2 2 2 0 2 seek discrete solutions. 2 2 sin sin sin Note that this is the sin jj j j x t j ik x ik x t t ik x ik x C C C C C C et ee ie C e x Ce e xi C kx x C kx C kk x c ωω φ φφ +− ∆− ∂− ∂∆ =  +   = =∆ = = sin sin function, , which has the nice property 1 as 0. xx →→ 1

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Which is a function of k , dispensive unlike solutions to the true advection equation. () 3 2 2 For good resolution 1 and Taylor series sin 6 1 1 6 C x kx x x CC k x ∆≈  ≈−∆   ± Phase speed error is time lagging and second order. Phase error larger for larger , worst case is: 2 2 , 2 sin 0 C k x x x x x π λ =− = == The phase speed of the 2 x wave is zero. Noise doesn’t propagate at all. Since 2 C ω is real, there is no change in wave amplitude with time, and no amplitude error. The group velocity of waves 2 cos C Ck k x = is approximate correct for small . But for poorly resolved waves, larger 2 , 0 2 x x <∆< ∆= cos 1 and the energy propagation for poorly resolved waves is 2 C C k = − The energy propagates backwards!
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CFD lecture 6 - Computational Fluid Dynamics Lecture 6...

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