Chemical Diffusion Using Fast Fourier Transform
Explore the computational analysis of chemical diffusion through the application of the 2D FFT method in Computational Earth Science. Learn about variables, conservation of mass principles, and the computational procedures involved in studying chemical diffusion. Discover the correlation between low and high concentrations, diffusivity, and the mathematical transformations utilized in the analysis.
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2023 EESC W3400 Lec 17: Chemical diffusion computed with the 2D FFT Computational Earth Science Bill Menke, Instructor Emily Glazer, Teaching Assistant TR 2:40 3:55
Today Additional Uses of the Fast Fourier Transform Chemical diffusion (but same as heat diffusion)
Variables ? (volume fraction) Concentration of chemical Source of chemical ? (volume fraction per unit time) Flux of chemical ? (volume fraction crossing unit surface per unit time)
Conservation of mass ??? + ? ??? + ? ??? ??? ??+??? ? ??? = + ? ? and ? ?? ???
Conservation of mass ??? + ? ??? + ? ??? ??? ??+??? ? ??? = + ? ? and ? ?? ??? ??= ?? Chemical diffusion ?? low C ? + ? high C ? ??= ?? ?? diffusivity ???
Conservation of mass ??? + ? ??? + ? ??? ??? ??+??? ? ??? = ??= ?? + ? ? and ? ?? ?? ??? ??= ?? Chemical diffusion ?? low C ? + ? high C ? ?2 ??2+ ?2 ??2? + ? ? ??? = ???
when ? = 0 ?2 ??2+ ?2 ??2? ?,?,? ? ??? ?,?,? = ? ??? ??,??,? = ?? 2+ ?? 2? ??,??,? FT If ?? ??= ?? then ? ??,??,? = ?0 exp ??
when ? = 0 ?2 ??2+ ?2 ??2? ?,?,? ? ??? ?,?,? = ? ??? ??,??,? = ?? 2+ ?? 2? ??,??,? 2+ ?? 2? ? ??,??,? = ?0 exp ?? with ?0??,?? = ? ??,??,? = 0
procedure Start with ?0?,? FT to ?0??,?? for each ? 2+ ?? 2?to get ? ??,??,? Multiply by exp ?? IFT to ? ?,?,?
procedure Start with ?0?,? FT to ?0??,?? for each ? 2+ ?? 2?to get ? ??,??,? Multiply by exp ?? IFT to ? ?,?,? Very similar to how we computed gravity anomalies
when ? 0 ?2 ??2+ ?2 ??2? ?,?,? + ? ?,?,? ? ??? ?,?,? = ? ??? ??,??,? = ?? 2+ ?? 2? ??,??,? + ? ??,??,?
when ? 0 ?2 ??2+ ?2 ??2? ?,?,? + ? ?,?,? ? ??? ?,?,? = ? ??? ??,??,? = ?? 2+ ?? 2? ??,??,? + ? ??,??,? ? ??? = ? ?,? We know how to solve this form numerically
when ? 0 ?2 ??2+ ?2 ??2? ?,?,? + ? ?,?,? ? ??? ?,?,? = ? ??? ??,??,? = ?? 2+ ?? 2? ??,??,? + ? ??,??,? ? = ? ??,??,? ? ??? = ? ?,? 2+ ?? 2? ??,??,? + ? ??,??,? ? ?,? = ?? ?0= ? ??,??,? = 0
Simple example ? ?,?,? = ? ?,? (constant in time)
procedure start with ?0?,?and? ?,? FT to ?0??,??and? ??,?? for each ??,?? 2+ ?? 2? ??,??,? + ? ??,?? numerically solve ? ??,??,? = ?? with initial condition ?0??,?? to get ? ??,??,? for each ? IFT to ? ?,?,?
# Fourier Transforms C0t = np.fft.fft2(C0); St = np.fft.fft2(S); FT to ?0??,??and? ??,?? # function for differential equation du/dt = f(u,t) # here u=C(kx,ky,t) and f=-k(kx**2+ky**2)+S def myfun( t, u ): f = np.zeros((1,),dtype=complex); f[0] = -b * u[0] + St[ikx,iky]; return f; for each ??,?? for ikx in range(Nx): kx0 = kx[ikx,0]; for iky in range(Ny): ky0 = ky[iky,0]; b = kappa*(kx0**2 + ky0**2); u0 = np.zeros((1,),dtype=complex); u0[0] = C0t[ikx,iky]; sol=solve_ivp(myfun, [t0, tmax], u0, method = 'RK45', t_eval=t); Et = np.sum( np.power((t-sol.t)/tmax,2) ); Ct[ikx,iky,0:Nt]=sol.y[0]; for i in range(Nt): C[0:Nx,0:Ny,i] = np.real(np.fft.ifft2( Ct[0:Nx,0:Ny,i] )); numerically solve for each ? IFT to ? ?,?,?
PART 1 of code dfilename='d.txt'; Make file of random noise Purpose: allow random pattern to be reused
PART 2 of code Make file of ?? Based on correlated random noise (as we did last week) dfilename='d.txt'; C0filename= C0.txt'; A = 1.0; # 0 < C0 < A Plus all the parameters from last week Purpose: allow ??to be reused
PART 3 of code Make file of ? Based on correlated random noise (as we did last week) dfilename='d.txt'; Sfilename= S.txt'; A = 1.0; # 0 < S < A Plus all the parameters from last week Purpose: allow ?to be reused
PART 4 of code ?2 ??2+ ?2 ??2? ?,?,? + ? ?,?,? ? ??? ?,?,? = Solve numerically C0filename= C0.txt'; Sfilename= S.txt'; kappa = 1.0; # diffusivity PLOTSTEP = 2; # plot every PLOTSTEP time slice
Group 1 Case 1: Gaussian ?0 with ? = 0 and ? with ? = 0.05 Case 2: exponential and same d.txt Compare how the concentration evolves with time In particular, how does you ability to distinguish the two cases vary with time?
Group 2 Case 1: Gaussian ?0 with ? = 1.0 and ? with ? = 0.0 and same d.txt Case 2: exponential Compare how the concentration evolves with time In particular, how does you ability to distinguish the two cases vary with time?
Group 3 ?0 with ? = 1.0 and some d.txt exponential ? with ? = 0.05 and a different d.txt Compare how the concentration evolves with time as you vary In particular, examine the variability of the time at which the ?0 pattern is no longer discernable
