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

1 - Question

The discretization of the transient term using the finite volume approach is more like the spatial discretization of __________

a) the convection term

b) the diffusion term

c) the source term

d) the anti-diffusion term

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Answer: aExplanation: The finite volume approach for the discretization of the rate of change per unit time is more similar to the discretization of convection term while doing a spatial discretization. The only difference is that the first case is based on time and the second is based on spatial coordinates.

2 - Question

Consider the following equation representing the temporal integration over the time interval t-Δt2 and t+Δt2 at the spatial point C.

∫t+Δt/2t−Δt/2∂(ρCϕC)∂tVCdt+∫t+Δt/2t−Δt/2L(ϕC)dt=0

If the first term is discretized using the difference of fluxes and the second term is evaluated using the midpoint rule, what is the discretized form?

a) VC(ρCϕC)t−Δt2+L(ϕtC)Δt

b) VC(ρCϕC)t+Δt2−L(ϕtC)Δt

c) VC(ρCϕC)t+L(ϕtC)Δt

d) VC(ρCϕC)t+Δt2−VC(ρCϕC)t−Δt2+L(ϕtC)Δt

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Answer: dExplanation: The given equation is ∫t+Δt/2t−Δt/2∂(ρCϕC)∂tVCdt+∫t+Δt/2t−Δt/2L(ϕC)dt=0 Discretizing the first term using the difference of fluxes, ∫t+Δt/2t−Δt/2∂(ρCϕC)∂tVCdt=VC(ρCϕC)t+Δt2−VC(ρCϕC)t−ΔΔt2 Discretizing the second term using the midpoint rule, ∫t+Δt/2t−Δt/2L(ϕC)dt=L(ϕtC)Δt Therefore, the final term is VC(ρCϕC)t+Δt2−VC(ρCϕC)t−Δt2+L(ϕtC)Δt.

3 - Question

Which of these changes should be made in the semi-discretized equation to get the fully discretized equation?

a) Express the face values in terms of the neighbouring face values

b) Express the face values in terms of the cell values

c) Express the cell values in terms of the face values

d) Express the cell values in terms of the neighbouring cell values

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Answer: bExplanation: While discretizing the transient term, the semi-discretized equation contains the values at the cell faces. If these face values are expressed in terms of the cell values, the complete discretized form of the equation can be obtained.

4 - Question

If the first-order implicit Euler scheme is used, the value at t+Δt/2 is replaced by the value at _________

a) t

b) t-Δt2

c) t+Δt

d) t-Δt

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Answer: cExplanation: In the first-order implicit Euler scheme, the values at the cell faces are approximated by the values at cell centres of the backward direction. Therefore, the value at t+Δt2 is replaced by the value at t.

5 - Question

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

a) (ρCϕC)t−(ρCϕC)t+ΔtΔtVC+L(ϕtC)

b) (ρCϕC)t−(ρCϕC)t−ΔtΔtVC+L(ϕtC)

c) (ρCϕC)t+(ρCϕC)t+ΔtΔtVC+L(ϕtC)

d) (ρCϕC)t+(ρCϕC)t−ΔtΔtVC+L(ϕtC)

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Answer: bExplanation: The first-order implicit Euler scheme gives its terms using the older terms. Before using this scheme, the terms are VC(ρCϕC)t+Δt2Δt−VC(ρCϕC)t−Δt2Δt+L(ϕtC) When the scheme is applied to these equations, (ρCϕC)t−(ρCϕC)t−ΔtΔtVC+L(ϕtC).

6 - Question

The first-order implicit Euler schemes to discretize the transient term creates ________

a) cross-flow diffusion

b) cross-diffusion

c) numerical anti-diffusion

d) numerical diffusion

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Answer: dExplanation: As the transient term behaves like the convection term while discretizing, numerical diffusion is produced by the first-order implicit Euler schemes. The value of the numerical diffusion can be obtained using the Taylor series expansion.

7 - Question

When the first-order implicit Euler scheme is unconditionally stable, the solution is ________

a) stationary for large time-steps

b) oscillatory for large time-steps

c) stationary for small time-steps

d) oscillatory for small time-steps

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Answer: aExplanation: A numerical diffusion term scales with the time-step in a similar fashion to the upwind scheme for the advection term. Therefore, when this scheme is unconditionally stable, the solution using this scheme is stationary for large steps.

8 - Question

The extra term added while discretizing the transient term of a flow with density ρ and flow variable φ using the first-order explicit Euler scheme is _________

a) Δt∂2(ρϕ)∂t2

b) −Δt∂2(ρϕ)∂t2

c) Δt2∂2(ρϕ)∂t2

d) −Δt2∂2(ρϕ)∂t2

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Answer: dExplanation: While using the first-order explicit Euler scheme, an extra term called the numerical anti-diffusion occurs. This term can be obtained by using the Taylor series expansion. The term is −Δt2∂2(ρϕ)∂t2.

9 - Question

According to the first-order explicit Euler scheme, the value at time-step t-Δt2 is approximated to be equal to the value at __________

a) t+Δt2

b) t

c) t-Δt

d) t+Δt

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Answer: bExplanation: The value at t-Δt2 is at the interface of two cells. One has the cell centre t and the other has the cell centre t-Δ t. The first-order explicit Euler scheme is downstream biased. Therefore, the value at t is taken to approximate the value at t-Δt2.

10 - Question

The numerical diffusion and numerical anti-diffusion terms are equal for the first-order Euler scheme are equal in magnitude when __________

a) the courant number of diffusion is equal to one

b) the courant number of diffusion is equal to two

c) the courant number of convection is equal to one

d) the courant number of convection is equal to two

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Answer: cExplanation: The numerical diffusion term of the first-order implicit Euler scheme and numerical anti-diffusion term of the first-order explicit Euler scheme are equal in magnitude and opposite in sign when the courant number of convection is equal to one.