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

This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Transient Flows – First Order Finite Volume Schemes”.

1. 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

Explanation: 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. Consider the following equation representing the temporal integration over the time interval t-$$\frac{\Delta t}{2}$$ and t+$$\frac{\Delta t}{2}$$ at the spatial point C.
$$\int_{t-\Delta t/2}^{t+\Delta t/2}\frac{\partial(\rho_C\phi_C)}{\partial t}V_Cdt+\int_{t-\Delta t/2}^{t+\Delta t/2}L(\phi_C)dt=0$$
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) $$V_C (\rho_C\phi_C)^{t-\frac{\Delta t}{2}}+L(\phi_C^t )\Delta t$$
b) $$V_C (\rho_C\phi_C)^{t+\frac{\Delta t}{2}}-L(\phi_C^t )\Delta t$$
c) $$V_C (\rho_C\phi_C)^t+L(\phi_C^t )\Delta t$$
d) $$V_C (\rho_C\phi_C)^{t+\frac{\Delta t}{2}}-V_C(\rho_C \phi_C)^{t-\frac{\Delta t}{2}}+L(\phi_C^t)\Delta t$$

Explanation: The given equation is
$$\int_{t-\Delta t/2}^{t+\Delta t/2}\frac{\partial(\rho_C\phi_C)}{\partial t}V_Cdt+\int_{t-\Delta t/2}^{t+\Delta t/2}L(\phi_C)dt=0$$
Discretizing the first term using the difference of fluxes,
$$\int_{t-\Delta t/2}^{t+\Delta t/2}\frac{\partial(\rho_C\phi_C)}{\partial t}V_Cdt=V_C(\rho_C\phi_C)^{t+ \frac{\Delta t}{2}}-V_C(\rho_C\phi_C)^{t-\Delta\frac{\Delta t}{2}}$$
Discretizing the second term using the midpoint rule,
$$\int_{t-\Delta t/2}^{t+\Delta t/2}L(\phi_C)dt=L(\phi_C^t)\Delta t$$
Therefore, the final term is
$$V_C(\rho_C\phi_C)^{t+\frac{\Delta t}{2}}-V_C(\rho_C\phi_C)^{t-\frac{\Delta t}{2}}+L(\phi_C^t)\Delta t$$.

3. 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

Explanation: 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. 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-$$\frac{\Delta t}{2}$$
c) t+Δt
d) t-Δt

Explanation: 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+$$\frac{\Delta t}{2}$$ is replaced by the value at t.

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

Explanation: 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,
$$\frac{(\rho_C\phi_C)^t-(\rho_C\phi_C)^{t-\Delta t}}{\Delta t} V_C+L(\phi_C^t)$$.
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6. 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

Explanation: 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. 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

Explanation: 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. 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) $$\Delta t\frac{\partial^2(\rho\phi)}{\partial t^2}$$
b) $$-\Delta t\frac{\partial^2(\rho\phi)}{\partial t^2}$$
c) $$\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}$$
d) $$-\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}$$

Explanation: 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
$$-\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}$$.

9. According to the first-order explicit Euler scheme, the value at time-step t-$$\frac{\Delta t}{2}$$ is approximated to be equal to the value at __________
a) t+$$\frac{\Delta t}{2}$$
b) t
c) t-Δt
d) t+Δt

Explanation: The value at t-$$\frac{\Delta t}{2}$$ 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-$$\frac{\Delta t}{2}$$.

10. 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

Explanation: 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.

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