Engineering Questions with Answers - Multiple Choice Questions

# Computational Fluid Dynamics – Transient Flows – Second Order Finite Volume Schemes

1 - Question

What is the equivalent of (ρC ΦC)t+Δt/2 using the Crank-Nicolson scheme for finite volume approach?
a) 12(ρC ΦC)t+12(ρC ΦC)t+Δ t
b) (ρC ΦC)t+(ρC ΦC)t+Δt
c) (ρC ΦC)t-(ρC ΦC)t+Δt
d) 12(ρC ΦC)t–12(ρC ΦC)t+Δt
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=12(ρC ΦC)t+12(ρC ΦC)t+Δt.

2 - Question

Which of these time-steps are used to approximate the value at time-step t-Δt2 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-Δt2
d) t and t+Δt2
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-Δt2 . It is shared by the faces t and t-Δ t. So, the scheme uses both of these values for the approximation.

3 - Question

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
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 - Question

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
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 - Question

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
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 - Question

What is the equivalent of (ρC ΦC)t+Δt/2 using the second-order upwind Euler scheme for finite volume approach?
a) 32 (ρC ΦC)t+(ρC ΦC)t-Δt
b) (ρC ΦC)t+12 (ρC ΦC)t-Δt
c) 32 (ρC ΦC)t+12 (ρC ΦC)t-Δ t
d) 12(ρC ΦC)t+12 (ρC ΦC)t-Δ t
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=32(ρC ΦC)t+12(ρC ΦC)t-Δt.

7 - Question

Which of these time-steps are needed to approximate the value at time-step Δt2 using the second-order upwind Euler scheme for finite volume approach?
a) t-Δt2 and t-2Δ t
b) t and t-Δ t
c) t-Δ t and t-2Δ t
d) t and t-2Δ t
Explanation: The second-order upwind Euler scheme uses two upwind nodes to approximate the values. The immediate upwind and the far upwind of t-Δt2 are t-Δ t and t-2Δ t. The vales at these two nodes are used for the approximation.

8 - Question

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
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 - Question

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
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 - Question

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) 3ρoCVC2Δt
b) −3ρoCVC2Δt
c) 2ρoCVCΔt
d) −2ρoCVCΔt