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Does solenoidal field orthogonal to dilatational field? 

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April 15, 2019, 19:20 

#41  
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Anyway, for w = 33, the recovered function yp is perfectly fine; the derivative has serious errors (can see in the figure below; the exact derivative is ye = w * cos(w*x) ); the results are off by a negative sign. However, I don't see significant imaginary parts in the derivative (dydx). What do you think? Screen Shot 20190415 at 3.18.57 PM.jpg Code:
clearvars; clc; close all N = 64; L = 2*pi; dx = L / N; x = 0 : dx : L; w = 2; y = sin(w * x); % FFT/iFFT yk = fft(y(1:end1)); yp = ifft(yk); % derivative k = [0:N/21, N/2:1]; dydxhat = 1i * k .* yk; dydx = ifft(dydxhat); dydxe = w * cos(w*x); % exact derivative plot(x(1:end1), dydxe(1:end1), 'b', x(1:end1), real(dydx), 'r:', 'linewidth', 2) l = legend('exact', 'numerical'); legend boxoff set(l, 'fontsize', 14); 

April 16, 2019, 04:15 

#42  
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Filippo Maria Denaro
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The effect of wavenumbers that are beyond the Nyquist frequency is clearly illustrated at page 10 of the note of Trefethen I poste above. The appearence of aliasing is worsened by the multiplication by k in the derivative 

April 16, 2019, 19:09 

#43 
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If i understand you correctly, there are two possibilities:
Is this correct? 

April 16, 2019, 22:46 

#44 
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As for the projection operator, I think I mentioned above, but I just write it again. I follow what Pope says in his Turbulence book where kappa is the wavenumber. Apply this operator on the velocity will give back which is the solenoidal part. Apparently it's a realnumber tensor, and I think it satisfies the Hermitian. 

April 17, 2019, 02:23 

#45 
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Yes i also think the second one is the correct one, if your operator is not Hermitian. The first one is wrong. You claim that your inverse FFT should be only real or that your operator should be Hermitian. I don't have such an overview of the paper inside right now.
Can you explain why you think this should hold? Last edited by Eifoehn4; April 17, 2019 at 03:25. 

April 17, 2019, 21:08 

#46  
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Since the main idea is to separate the velocity field, I think as long as everything is implemented correctly, the decomposition should return realvalue solenoidal/dilatational fields. Since they are physical variables, they should be real. Of course, even if the resulting imaginary part is not perfectly zero, as long as they are close to machine precision, I believe it's fine. For the projection tensor, as I mentioned above, it's a realsymmetric tensor, I believe it satisfies being Hermitian. 

April 18, 2019, 02:56 

#47  
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In my opinion it doesnt matter. Like FMDenaro already said, you can evaluate your polynomial basis directly in modal space with coefficients . It is possible to super sample your whole computational domain by evaluating the modal basis, even on non equidistant points. And you wont even loose information or even have to consider what my data looks like by transformating it in a Langrange (like) representation. Last edited by Eifoehn4; April 18, 2019 at 11:35. 

Tags 
compressible flow, dilatational, helmholtz decomposition, solenoidal 
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