This set of Computational Fluid Dynamics Questions & Answers for Exams focuses on “Transient Flows – Second Order Finite Volume Schemes”.
1. What is the equivalent of (ρC ΦC)t+Δt/2 using the Crank-Nicolson scheme for finite volume approach?
a) \(\frac{1}{2}\)(ρC ΦC)t+\(\frac{1}{2}\)(ρC ΦC)t+Δ t
b) (ρC ΦC)t+(ρC ΦC)t+Δt
c) (ρC ΦC)t-(ρC ΦC)t+Δt
d) \(\frac{1}{2}\)(ρC ΦC)t–\(\frac{1}{2}\)(ρC ΦC)t+Δt
View Answer
Explanation: The Crank-Nicolson scheme gives equal weight to both the cells which share the face. The formula is given by,
(ρC ΦC)t+Δt/2=\(\frac{1}{2}\)(ρC ΦC)t+\(\frac{1}{2}\)(ρC ΦC)t+Δt.
2. Which of these time-steps are used to approximate the value at time-step t-\(\frac{\Delta t}{2}\) using the Crank-Nicolson scheme for finite volume approach?
a) t and t+Δ t
b) t and t-Δ t
c) t and t-\(\frac{\Delta t}{2}\)
d) t and t+\(\frac{\Delta t}{2}\)
View Answer
Explanation: The Crank-Nicolson scheme uses the cell centres of both the cells which share the face which is considered. The face considered here is t-\(\frac{\Delta t}{2}\)
. It is shared by the faces t and t-Δ t. So, the scheme uses both of these values for the approximation.
3. The stability of the Crank-Nicolson scheme for finite volume approach is constrained by ________
a) CFL number
b) Peclet number
c) Time-step size
d) Spatial grid size
View Answer
Explanation: The Crank-Nicolson scheme is an explicit scheme which uses the values before the face and after the face to find the value at the face. Therefore, its stability is constrained by the CFL or Courant number.
4. The results using the Crank-Nicolson scheme for finite volume approach can be reformulated using the ________
a) implicit first-order Euler scheme
b) implicit and explicit first-order Euler schemes
c) explicit first-order Euler scheme
d) central difference scheme
View Answer
Explanation: The finite volume approach using the Crank-Nicolson scheme can be reformulated using the first-order implicit Euler scheme and then the explicit Euler scheme which is modified and used in the form of extrapolation.
5. Which of these terms cause instability in the Crank-Nicolson scheme when used for finite volume approach?
a) Anti-diffusion term
b) Anti-dispersive term
c) Diffusion term
d) Dispersive term
View Answer
Explanation: By expanding the results of the Crank-Nicolson scheme using the Taylor series expansion, the scheme is proved to be a second-ordered scheme. The third-ordered term omitted here is a dispersive term causing instabilities.
6. What is the equivalent of (ρC ΦC)t+Δt/2 using the second-order upwind Euler scheme for finite volume approach?
a) \(\frac{3}{2}\) (ρC ΦC)t+(ρC ΦC)t-Δt
b) (ρC ΦC)t+\(\frac{1}{2}\) (ρC ΦC)t-Δt
c) \(\frac{3}{2}\) (ρC ΦC)t+\(\frac{1}{2}\) (ρC ΦC)t-Δ t
d) \(\frac{1}{2}\)(ρC ΦC)t+\(\frac{1}{2}\) (ρC ΦC)t-Δ t
View Answer
Explanation: The second-order upwind Euler scheme gives more importance to the immediate upwind than the far upwind. The formula is
(ρC ΦC)t+Δ t/2=\(\frac{3}{2}\)(ρC ΦC)t+\(\frac{1}{2}\)(ρC ΦC)t-Δt.
7. Which of these time-steps are needed to approximate the value at time-step \(\frac{\Delta t}{2}\) using the second-order upwind Euler scheme for finite volume approach?
a) t-\(\frac{\Delta t}{2}\) and t-2Δ t
b) t and t-Δ t
c) t-Δ t and t-2Δ t
d) t and t-2Δ t
View Answer
Explanation: The second-order upwind Euler scheme uses two upwind nodes to approximate the values. The immediate upwind and the far upwind of t-\(\frac{\Delta t}{2}\) are t-Δ t and t-2Δ t. The vales at these two nodes are used for the approximation.
8. How many numerical diffusion terms does the second-order upwind Euler scheme have?
a) Infinity
b) No diffusion term
c) One term
d) Two terms
View Answer
Explanation: The transient term discretization also has similar properties like the convection term discretization. The second-order upwind Euler scheme does not have any numerical diffusion or anti-diffusion terms. These terms are present for the first-order schemes only.
9. The numerical dispersion term of the second-order upwind Euler scheme is of ____________
a) third-order
b) second-order
c) first-order
d) no dispersion
View Answer
Explanation: When the terms in the discretized form of the transient term using the second-order upwind Euler scheme is further expanded with the Taylor series expansion, the dispersion term of the third order arises.
10. When the finite volume approach is used, if the general form is given as
FluxT=FluxC ΦC+FluxC° ΦC°+FluxV
The superscript o indicates the older time step, the value of FluxC° while using the second-order upwind Euler scheme is ________
a) \(\frac{3\rho_C^o V_C}{2 \Delta t}\)
b) \(-\frac{3\rho_C^o V_C}{2 \Delta t}\)
c) \(\frac{2\rho_C^o V_C}{\Delta t}\)
d) \(-\frac{2\rho_C^o V_C}{\Delta t}\)
View Answer
Explanation: When the values for the (ρC ΦC)t+Δt/2 and the (ρC ΦC)t-Δt/2 are substituted in the semi-discretized equation, the final form obtained is
FluxT=FluxC ΦC+FluxC° ΦC°+FluxV
Where FluxC°=\(-\frac{2\rho_C^o V_C}{\Delta t}\).
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