This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Relaxation – Residuals”.

1. Which of these equations can be written in the residual form?

a) The discretized equations

b) The governing equations

c) The PDEs

d) The ODEs

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Explanation: The residual form can be taken by the discretized algebraic equation. While the other form of the equation is called the standard or direct form, the residual form can also be called correction form.

2. The standard form of the equations is solved for __________ and the residual form is solved for _________

a) flow variables, residuals

b) flow variables, corrections

c) corrections, flow variables

d) residuals, flow variables

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Explanation: The standard form of the equations is solved for the flow variables so that the algebraic equations are satisfied. The residual form of the discretized equations is solved for the corrections needed to satisfy those equations.

3. When the field of the flow variable is exact, the residual will be __________

a) negative

b) positive

c) zero

d) one

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Explanation: When the solution field is exact, it means that the solutions converged. Therefore, the residual will be zero. The correction will also be zero in this case. But, in real, the solution will not reach the exact solution. Therefore, the residual will never be zero.

4. Why is the residual form of the discretized equations used?

a) It increases the rate of convergence

b) It increases the stability

c) It increases the accuracy

d) Numerical errors are less

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Explanation: The residual form of the discretized equations are mathematically equivalent to the standard form. In this form, the numerical errors are less when the variations are small for large flow variable values.

5. The absolute residual is calculated __________

a) to overcome high values of residuals

b) to decelerate the convergence process

c) to overcome the sign problems in the actual residual value

d) to increase the rate of convergence

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Explanation: The residual may be a positive or a negative quantity. As the sign is not needed, the absolute value of the residual is found. If the absolute residual decreases with iterations, the solution converges.

6. The maximum value of residual is used to __________

a) decrease the importance of sign

b) compare with the vanishing value

c) include the sign of the residual

d) increase the importance of sign

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Explanation: The maximum value of residual is found by comparing the values of the absolute residuals. If this maximum value becomes less than the vanishing quantity (provided by the user), then the result is converged.

7. Which of these statements states the need for the normalization of the residual?

a) Different variables result in different level of residuals

b) Different variables result in the same level of residuals

c) Different variables result in different residuals

d) Different variables result in same residuals

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Explanation: When the influence of the sign of the residual can be ruled out by the absolute residual, the level of the absolute residual varies with the value of the flow variables. To overcome this problem, the residuals are normalized.

8. Which of these values is used to find the normalized residual?

a) The minimum of all the residuals

b) The maximum of all the residuals

c) The sum of all the residuals

d) The product of all the residuals

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Explanation: The normalized residual is obtained by using the absolute residual divided by the maximum of all the residuals. When the maximum scaled value is less than the vanishing value, the solution is said to be converged.

9. Which of these is the denominator while finding the root mean square residual?

a) Sum of the absolute residuals

b) Sum of the actual residuals

c) Number of elements

d) Sum of the scaled residuals

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Explanation: The root mean square is the ratio of the square root of the sum of squares of the absolute residuals to the number of residuals. The number of residuals is the same as the number of elements.

10. When the residuals are scaled, the vanishing value is ___________

a) less than 10

b) less than 10

c) more than 10^{-5}

d) less than 10^{-5}

View Answer

Explanation: The common range of the vanishing value is from 10

^{-3}to 10

^{-5}for the scaled residuals. I can also range below this level. The integrated quantities are also used to ensure the convergence of the solution.

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