Question

What is the relationship between \(\frac{\phi_c-\phi_w}{\phi_E-\phi_w}\) and the Peclet number (Pe) when the grid is uniform?

(a) \(\frac{\phi_c-\phi_w}{\phi_E-\phi_w} = \frac{1}{2}(1-\frac{Pe}{2}) \)

(b) \(\frac{\phi_c-\phi_w}{\phi_E-\phi_w} = \frac{1}{2}(1+\frac{Pe}{2}) \)

(c) \(\frac{\phi_c-\phi_w}{\phi_E-\phi_w} = \frac{1}{2}(\frac{Pe}{2}-1) \)

(d) \(\frac{\phi_c-\phi_w}{\phi_E-\phi_w} = (\frac{Pe}{4}) \)

This question was posed to me in an international level competition.

I need to ask this question from Convection-Diffusion Problems in division Convection-Diffusion Problems of Computational Fluid Dynamics

Answer 1

Right choice is (a) \(\frac{\phi_c-\phi_w}{\phi_E-\phi_w} = \frac{1}{2}(1-\frac{Pe}{2}) \)

The explanation is: For the central difference scheme applied to the convection-diffusion problems

\(\frac{\phi_c-\phi_w}{\phi_E-\phi_w}=\frac{a_E}{a_W+a_E}\)

Applying this to

\(\frac{\phi_c-\phi_w}{\phi_E-\phi_w} = \frac{1}{2}(1-\frac{Pe}{2}) \).