Question

Which of these equations is the discretized form of the transient term using the first-order implicit Euler scheme?

(a) \(\frac{(\rho_C\phi_C)^t-(\rho_C\phi_C)^{t+\Delta t}}{\Delta t} V_C+L(\phi_C^t)\)

(b) \(\frac{(\rho_C\phi_C)^t-(\rho_C\phi_C)^{t-\Delta t}}{\Delta t} V_C+L(\phi_C^t)\)

(c) \(\frac{(\rho_C\phi_C)^t+(\rho_C\phi_C)^{t+\Delta t}}{\Delta t} V_C+L(\phi_C^t)\)

(d) \(\frac{(\rho_C\phi_C)^t+(\rho_C\phi_C)^{t-\Delta t}}{\Delta t} V_C+L(\phi_C^t)\)

I got this question in an internship interview.

This key question is from Transient Flows topic in chapter Transient Flows of Computational Fluid Dynamics

Answer 1

Right choice is (b) \(\frac{(\rho_C\phi_C)^t-(\rho_C\phi_C)^{t-\Delta t}}{\Delta t} V_C+L(\phi_C^t)\)

The explanation is: The first-order implicit Euler scheme gives its terms using the older terms. Before using this scheme, the terms are

\(\frac{V_C(\rho_C \phi_C )^{t+\frac{\Delta t}{2}}}{\Delta t}-\frac{V_C(\rho_C\phi_C)^{t-\frac{\Delta t}{2}}}{\Delta t}+L(\phi_C^t)\)

When the scheme is applied to these equations,

\(\frac{(\rho_C\phi_C)^t-(\rho_C\phi_C)^{t-\Delta t}}{\Delta t} V_C+L(\phi_C^t)\).