Pseudospectral method using FFT to solve diff equation

[John_Blacktower]

Hello everyone, I'm trying to solve by FFT a convection diffusion eqaution on a 3 D box with an slit condition on z-axix and periodic conditions on x and y axis.
∂C/∂t=D∇[2]C-v⋅∇C (1)
v=vx + vy + vz
i have solved the velocity of fluid, i mean a really know what is the velocity of flow field.
But when i try to solve the eq. (1) i don't see the effect of the convection therm on my solution, this mean in the solution i only get a result like a pure diffusion problem. probably i have a wrong discretization scheme for convective term. but i don't nkow.
Please help me¡¡

now i show you what its my procedure:

1.
i perform a cranck nicholson scheme for eq. (1):
C[k+1]-1/2*DΔt(∇[2]C)[k +1]=C[k] +(1/2*DΔt(∇[2]C)-v⋅∇C)[2] (2)

2. Then i transform the convective term (a nonlinear term)
v⋅∇C=∇⋅(Cv)-C∇⋅v; ∇⋅v=0
∇⋅(Cv)→∇.f⇒f=Cv

3. i apply FFT for the terms on x and y-axis on eq. (2)
C'[k+1]-1/2*DΔt(-k[2]C'+d[2]C'/dz[2])[k +1]=C'[k] +(1/2*DΔt(-k[2]+d[2]C'/dx[2])-(-ik*f' + df'z/dz))[2] (3)
where; k=norm(k); k=k1 +k2; f'= f'x + f'y; C'=∫C(x,y,z)exp(-i*(k)⋅(r))dxdy; r=x + y.
f'x=∫C(x,y,z)vx(x,y,z)exp(-i*(k)⋅(r))dxdy
f'y=∫C(x,y,z)vy(x,y,z)exp(-i*(k)⋅(r))dxdy
f'z=∫C(x,y,z)vz(x,y,z)exp(-i*(k)⋅(r))dxdy

4. then i apply a finite difference discretization on z axis
Ci'[k+1]-1/2*DΔt(-k[2]C'i+(C'i+1-2C'i+C'i-1)/(Δz*Δz))[k +1]=C'[k] +(1/2*DΔt((-k[2]C'i+(C'i+1-2C'i+C'i-1)/(Δz*Δz))-(-ik1*f'xi -ik2*f'yi + (f'zi-f'zi-1)/Δz))[2] (5)

5. then on matriz notation i have:
A[k+1]⋅b[k+1]=A[k]⋅b0[k] - bfx[k] - bfy[k]- C[k]⋅bfz[k] (6)

i have been solved but the solution is very similar to pure diffusion problem.

Thanks for your time and sorry for my bad english

 

[jvicens]

.Hello,

Thank you for sharing your problem with us. It seems like you have a good understanding of the equations and have taken a rigorous approach in your solution procedure. However, there are a few things that I would like to point out which may help you in solving this problem.

Firstly, when you say that you have solved the velocity of the fluid, do you mean that you have obtained a numerical solution for the velocity field? If yes, then have you validated your solution with any experimental or analytical data? It is important to make sure that your velocity field is accurate before proceeding with the solution of the convection-diffusion equation.

Secondly, in your equation (2), you have a term (1/2*DΔt(∇[2]C)-v⋅∇C)[2] which is dependent on the concentration field C. This means that you are solving for C at two different time levels, which could lead to numerical instability. It would be better to use the same time level for both terms in this equation.

Thirdly, in equation (3), you have a term (-ik*f' + df'z/dz)[2] which is a Fourier transform of the convective term. This term should be multiplied by the concentration field C' and not just the velocity field f'. This could be one reason why you are not seeing the effect of the convection term in your solution.

Lastly, in equation (5), you are using a finite difference discretization in the z-direction. Have you considered using a spectral method in the z-direction as well? This could improve the accuracy of your solution.

I hope these suggestions will help you in solving your problem. Good luck!