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# Computational Fluid Dynamics – Convection-Diffusion Problems – Error Sources

1 - Question

Numerical diffusion causes __________

a) smearing of sharp gradients

b) oscillations

c) undershoots and overshoots

d) inaccuracy

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Answer: aExplanation: The sources of numerical errors caused by the convection flux can be divided into two – numerical diffusion and numerical dispersion. The numerical diffusion leads to smearing of sharp gradients.

2 - Question

Which of these ways can be used to overcome stream-wise numerical diffusion?

a) Decreasing the order of interpolation

b) Increasing the order of interpolation

c) Increasing the number of neighbours considered

d) Decreasing the number of neighbours considered

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Answer: bExplanation: The numerical diffusion error can be further divided into two – stream-wise and cross-stream numerical diffusions. The stream-wise numerical diffusion can be reduced by using a higher-order interpolation profile.

3 - Question

What is the cause of cross-stream numerical diffusion?

a) One-dimensional nature of the assumed profiles

b) Multi-dimensional nature of the assumed profiles

c) Higher order of accuracy

d) Higher order of interpolation

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Answer: aExplanation: One of the types of numerical diffusion errors is the cross-stream numerical diffusion. Cross-stream numerical diffusion is caused by the one-dimensional nature of interpolation when the grid is multi-dimensional. This is caused by cross-flow diffusion or false diffusion.

4 - Question

Which of these methods cannot be used to reduce the errors due to cross-stream numerical diffusion?

a) Interpolation in the direction of flow

b) Higher order interpolation profile

c) Multi-dimensional interpolation profiles

d) Changing the direction of interpolation

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Answer: dExplanation: The cross-stream numerical diffusion can be decreased by either interpolating in the direction of the flow which means multi-dimensional interpolation profiles or using a one-dimensional higher-order interpolation profile.

5 - Question

Numerical dispersion error causes __________

a) convergence problems

b) accuracy problems

c) boundedness problems

d) stability problems

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Answer: cExplanation: One of the types of sources of the errors is numerical dispersion. This is shown out through oscillations in the resulting profiles in the presence of large gradients in the profile resulting in an unbounded solution.

6 - Question

Which of these methods can be used to evaluate the errors in convection-diffusion schemes?

a) Using the exact solution of the source-free problem

b) Using the exact solution of the source and diffusion-free problem

c) Using the exact solution of the diffusion-free problem

d) Using the first-order schemes

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Answer: bExplanation: To evaluate the errors, a simplified version of the problem where there is no source and diffusion is taken and further velocity and density fields are assumed to be constants. The exact solution of this problem is used for the analysis.

7 - Question

Which is correct regarding the upwind scheme?

a) Neither numerical dispersion nor numerical diffusion error arises

b) Only numerical diffusion error arises

c) Both numerical dispersion and numerical diffusion errors arise

d) Only numerical dispersion error arises

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Answer: cExplanation: In the analysis for errors, while comparing the exact and numerical solutions, the first-order upwind scheme gives a complex k-value. Therefore, it will have both numerical dispersion and numerical diffusion problems.

8 - Question

Which of these is correct for the central difference scheme?

a) Neither numerical dispersion nor numerical diffusion error arises

b) Only numerical diffusion error arises

c) Both numerical dispersion and numerical diffusion errors arise

d) Only numerical dispersion error arises

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Answer: dExplanation: For the central differencing scheme, there is no problem of numerical diffusion. Only numerical dispersion error arises here. This is because, the k-value is completely imaginary and free from a real part.

9 - Question

What is the problem of numerical diffusivity?

a) The simulated model has a higher diffusivity than the actual flow

b) The simulated model has a lower diffusivity than the actual flow

c) The simulated model has a different diffusivity than the actual flow

d) The simulated model has a zero diffusivity

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Answer: aExplanation: Numerical diffusivity is a problem which results in a simulated solution with a higher diffusivity than the actual flow problem. This becomes an important criteria to be considered when the flow is free from diffusion.

10 - Question

Numerical dispersion is a result of __________

a) higher-order interpolation profile

b) unphysical behaviour of assumed interpolation profile

c) first-order interpolation profile

d) quadratic interpolation profile

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Answer: bExplanation: Numerical dispersion error arises with all interpolation profiles except the upwind scheme. It is the result of the unphysical behaviour of the assumed interpolation profile. It leads to unphysical dispersion.