Velocity gradient from finite different scheme

Lattice Boltzmann Method LBM: Theory
#1

Hello,

I want to calculate the velocity gradient using a finite difference scheme. The geometry looks like this:


       1     2    3
(wall) o..|..o....o (fluid)

Now I am interested in the velocity gradient at the last lattice node (number 2) in front of the wall (fullway bounceback boundary condition). Is is too naive to approximate it by


du/dz = [u_2 / (h / 2) + (u_3 - u_2) / h] / 2

?
h is the lattice constant. So basically, I calculate the gradient between nodes 2 and 3 and the gradient between the wall and node 2 and then just use the mean value.
This approach is only used for post-processing, i. e. comparing with another data set. It is not used within the LB code.

Thank you,
Timm

#2

I could imagine that what you want is an estimate of du/dz(z=2) which is second-order accurate with respect to the grid spacing, and thus compatible with the accuracy of the LB scheme. You can obtain this by Taylor-expanding the velocity around the spot of interest (z=2), using

an expansion of u up to the point z=3:


(a) u(z+h) = u(z) + h u'(z) + 1/2 h^2 u''(z) + O(h^3)

and an expansion of u back to the no-slip wall:


(b) u(z-h/2) = u(z) - h/2 u'(z) + 1/8 h^2 u''(z) + O(h^3)

Now, eliminate the O(h^2) term by computing (a)-4(b):


u(z+h)-4 u(z-h/2) = -3 u(z) + 3 h u'(z) + O(h^3),

which yields the estimate for the velocity derivative:


u'(z) = 1/h[ u(z) + 1/3u(z+h) - 4/3u(z-h/2) ] + O(h^3)

In your case, the velocity on the wall u(z-h/2) is zero, and your formula for the velocity is therefore


du/dz(z=2) = 1/h( u_2 + 1/3 u_3).

I may have made a arithmetics mistake, though, so please verify the calculations.

1 Like
#3

Hey Jonas,

this is a very nice answer to my question. Thanks a lot! I will immediately implement this approach.
By the way, can you recommend a review of finite difference methods used in LBM?

Cheers,
Timm