This set of Computational Fluid Dynamics Problems focuses on “Approaches for Non-uniform Time Steps”.
1. Discretization of the transient term is not affected by uniform or non-uniform grids when _________
a) the scheme is downwind
b) the scheme is upwind
c) the scheme is first-order
d) the scheme is second-order
View Answer
Explanation: Since the second-order schemes use a stencil with two time-steps in the same direction, only the second-order schemes are affected by the non-uniformity of the grids. The first-order schemes are not affected.
2. For which of these schemes is the interpolation profile need not be modified for the non-uniform transient grid?
a) Downwind scheme
b) Upwind scheme
c) Crank-Nicolson scheme
d) second-order schemes
View Answer
Explanation: For all the second-order schemes, the interpolation profile has to be modified when the transient grid is non-uniform. But this is not the case when the Crank-Nicolson scheme is used. There is no change in the interpolation profile needed here.
3. Which of these characteristics of the Crank-Nicolson scheme is affected by the non-uniform transient grids?
a) Consistency
b) Convergence
c) Stability
d) Accuracy
View Answer
Explanation: When the Crank-Nicolson scheme is used on the non-uniform transient grids, for each of the two steps, a different time-step is used. So, the spatial derivative is not at the centre of the temporal elements. Therefore, the accuracy is affected.
4. Which of these statements is correct about the variable time-steps?
a) The finite volume and finite difference schemes do not yield equivalent algebraic equations
b) The finite volume scheme yields equivalent algebraic equations irrespective of the non-uniformity
c) The finite difference scheme yields equivalent algebraic equations irrespective of the non-uniformity
d) All the second-order schemes result in the same algebraic equation when the grid is non-uniform
View Answer
Explanation: The finite volume and finite difference schemes yield equivalent algebraic equations when the transient grid is uniform. If the grid is non-uniform, the algebraic equations are not the same.
5. If the Crank-Nicolson scheme is used with the finite difference approach to get the discretized equation of the transient term with the non-uniform grid, what is the central coefficient of the previous time-step?
(Note: ρ is the density and V the volume).
a) \(\frac{\Delta t-\Delta t^o}{\Delta t+\Delta t^o}\rho_C V_C\)
b) \(\frac{\Delta t-\Delta t^o}{\Delta t+\Delta t^o}\rho_C^o V_C\)
c) \(\frac{\Delta t-\Delta t^o}{\Delta t+\Delta t^o}\rho_C^o\)
d) \(\frac{\Delta t+\Delta t^o}{\Delta t-\Delta t^o}\rho_C^o V_C\)
View Answer
Explanation: When the transient grid system is not uniform,
Δt≠Δ t°
Therefore, when the Crank-Nicolson scheme is used, density varies but the volume does not vary.
\(a_C^o=\frac{\Delta t-\Delta t^o}{\Delta t+\Delta t^o}\rho_C^o V_C\) This becomes zero when a uniform transient grid is used.
6. If the Crank-Nicolson scheme is used with the finite difference approach to get the discretized equation of the transient term with the non-uniform grid, what is the central coefficient of the second previous time-step?
(Note: ρ is the density and V the volume).
a) \(\frac{\Delta t^o}{\Delta t(\Delta t+\Delta t^o)}\rho_C^{oo}V_C\)
b) \(\frac{\Delta t(\Delta t+\Delta t^o)}{\Delta t×\Delta t^o}\rho_C^{oo}V_C\)
c) \(\frac{\Delta t+2\Delta t^o}{(\Delta t+\Delta t^o)}\rho_C^{oo}V_C\)
d) \(\frac{\Delta t}{\Delta t^o(\Delta t+\Delta t^o)}\rho_C^{oo}V_C\)
View Answer
Explanation: While using the Crank-Nicolson scheme, the previous and the next time steps are t+Δt and t-Δ t°. Using these in the semi-discretized equation, the coefficient of the second previous time-step is
\(a_C^{oo}=\frac{\Delta t}{\Delta t^o(\Delta t+\Delta t^o)}\rho_C^{oo}V_C\).
7. If the Adams-Moulton scheme is used with the finite difference approach to get the discretized equation of the transient term with the non-uniform grid, what is the central coefficient of the current time-step?
(Note: ρ is the density and V the volume).
a) \((\frac{1}{\Delta t^o}+\frac{1}{\Delta t+\Delta t^o}) \rho_C V_C\)
b) \((\frac{1}{\Delta t}+\frac{1}{\Delta t^{oo}+\Delta t^o}) \rho_C V_C\)
c) \((\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^{oo}})\rho_C V_C\)
d) \((\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^o})\rho_C V_C\)
View Answer
Explanation: The Adams-Moulton scheme uses the time-steps Δt-Δt° and Δt-Δt°. Applying these time-steps, the central coefficient of the current time-step is given by
\((\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^o})\rho_C V_C\).
8. Which of these is correct for the non-uniform time-steps?
(Note: δt is the distance between the centroids of two temporal elements. Δt is the size of a temporal element. The superscript o indicates the older time step).
a) \(\delta t=\frac{(\Delta t-\Delta t^o)}{2}\)
b) \(\delta t=\frac{(\Delta t+\Delta t^o)}{2}\)
c) δt=Δt
d) δt=Δt°
View Answer
Explanation: For the variable time-steps, \(\delta t=\frac{(\Delta t+\Delta t^o)}{2}\) as the size of the temporal element is not the same. For the uniform transient grids, this reduces to δt=Δt as Δ t=Δt°.
9. The general discretised form of the transient term using the finite volume approach is
FluxT=FluxC ΦC+FluxC°ΦC°+FluxV.
The superscript o indicates the older time step. If the Crank-Nicolson method is used with non-uniform grids, what is FluxV?
a) \(-\frac{\Delta t^{oo}}{\Delta t^o}\frac{\rho_c^{oo} V_C\phi_c^{oo}}{\Delta t}\)
b) \(\frac{\Delta t^{oo}}{\Delta t^o+\Delta t^{oo}}\frac{\rho_c^{oo} V_C \phi_c^{oo}}{\Delta t}\)
c) \(-\frac{\Delta t^{oo}}{\Delta t^o+\Delta t^{oo}}\frac{\rho_c^{oo} V_C \phi_c^{oo}}{\Delta t}\)
d) \(\frac{\Delta t^{oo}}{\Delta t^o}\frac{\rho_c^{oo} V_C \phi_c^{oo}}{\Delta t}\)
View Answer
Explanation: The Crank-Nicolson scheme uses the average of the two central values to get the values at the interface. Using this with the non-uniform time-steps,
FluxV=-\(-\frac{\Delta t^{oo}}{\Delta t^o+\Delta t^{oo}}\frac{\rho_c^{oo} V_C \phi_c^{oo}}{\Delta t}\).
10. Which of these statements is correct regarding the Adams-Moulton scheme used on the non-uniform grids?
a) The current central coefficient for the finite difference and the finite volume schemes are the same
b) The current central coefficient for the uniform and the non-uniform grid is the same
c) The variation in time-steps does not result in any change
d) There is no variation in the values when the grid is uniform
View Answer
Explanation: For the finite difference approach,
\(a_c=(\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^o})\rho_C V_C\)
For the finite volume approach, the central coefficient is
FluxC=\((\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^o})\rho_C V_C\)
These two are the same irrespective of the approach.
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