CFD lecture 7 - Computational Fluid Dynamics Lecture 7...

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Computational Fluid Dynamics Lecture 7 Artificial Dissipation Lack of dissipation in centered spatial differencing can be a disadvantage. Dispersed, small scale waves propagate in arbitrary directions without loss of amplitude. Adding scale selective dissipation is often useful. Non-linear problems can cause a cascade of energy to small scales and lead to numerical instabilities. Diffusive/Dissapative effects realized by even order derivatives: () 21 1 e.g. 2 j jj j t φ γφ +− =− + Where 2 γ is a constant that sets the magnitude of the filter. ( ) 2 2 solution 2 1 m ikx m m m DA tx A DA t k t ψψ ψ ∂∂ == ±²³ ²´ e Damping is largest for largest k (smallest wavelength) for the 2 nd order diffusion operator above, this leads to. 2 c o s 2 so 2 wave is damped the strongest. 2 A kxA t x x π λ ∆= Assume, for example, second ordered central differences: N 23 24 11 1 2 truncation error 1 2 26 jj j j x x xx x φφ αφ α σ  ∂−  ≈+ + = + ∆ − + ∂∆ 
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CFD lecture 7 - Computational Fluid Dynamics Lecture 7...

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