Question

Find the normalized functional relationship between φf and φC for a uniform grid while using the second-order upwind scheme?

(a) \(\tilde{\phi_f}=\frac{1}{2}\tilde{\phi_C}\)

(b) \(\tilde{\phi_f}=-\frac{1}{2}\tilde{\phi_C}\)

(c) \(\tilde{\phi_f}=\frac{3}{2}\tilde{\phi_C}\)

(d) \(\tilde{\phi_f}=-\frac{3}{2}\tilde{\phi_C}\)

I had been asked this question in final exam.

Asked question is from Convection-Diffusion Problems topic in section Convection-Diffusion Problems of Computational Fluid Dynamics

Answer 1

The correct answer is:

(a) ϕf ~= 1/2 ϕC

Explanation:

The question is related to the second-order upwind scheme for convection-diffusion problems. In this context, the goal is to find the normalized functional relationship between the face value (ϕf) and the cell-center value (ϕC) for a uniform grid.

In the second-order upwind scheme, the relationship between the face value and the cell-center value is derived based on the discretization of the convection term using an upwind scheme with higher-order accuracy. The scheme improves on the first-order upwind by considering information from two neighboring grid points, which leads to a more accurate approximation of the flux at the faces.

For a uniform grid in one-dimensional space, the second-order upwind scheme gives the following relationship:

ϕf≈12(ϕCprev+ϕCnext)\phi_f \approx \frac{1}{2} \left( \phi_C^{\text{prev}} + \phi_C^{\text{next}} \right)ϕf​≈21​(ϕCprev​+ϕCnext​)

Where ϕCprev\phi_C^{\text{prev}}ϕCprev​ and ϕCnext\phi_C^{\text{next}}ϕCnext​ are the values at the adjacent cell centers, and the relationship between ϕf\phi_fϕf​ (the value at the face) and ϕC\phi_CϕC​ (the value at the cell center) is approximately linear, with a factor of 1/2 for a uniform grid.