Computational Fluid Dynamics Multiple Choice Questions on “Under-Relaxation”.
1. Relaxation techniques are used to ensure __________
a) convergence
b) stability
c) accuracy
d) boundedness
Answer: a
Clarification: Relaxation techniques are used to increase the convergence of the solution by changing the values of the variables during the iterative process. It actually relaxes the conditions which should be satisfied to get the solutions.
2. The relaxation techniques slow down the __________
a) whole iterative process
b) effect of the sources
c) effect of the neighbouring elements and the source terms
d) effect of the neighbouring elements
Answer: c
Clarification: It slows down the effects of the neighbouring elements and the source term on the central element. It is because these two impose the constraint on the value of the central element.
3. The explicit under-relaxation method means that __________
a) the changes are made before each iteration
b) the changes are made before the iteration process
c) the changes are made after the iteration process
d) the changes are made after each iteration
Answer: c
Clarification: Relaxation can be performed either after obtaining the solution of each iteration (explicitly) or before the process of iteration at each step (implicitly). Implicit methods are most commonly used.
4. When the relaxation factor is less than one, which of these will occur?
a) Increase the speed of convergence
b) Slow down the speed of convergence
c) Decrease the stability
d) Increase the divergence
Answer: b
Clarification: The process is called under-relaxation when the relaxation factor is less than one. This decreases the rate of convergence of the results. But, the advantage is that the stability of the solution is increased.
5. There is no relaxation when the relaxation factor is __________
a) positive infinity
b) negative infinity
c) one
d) zero
Answer: d
Clarification: If the relaxation factor is zero, it means that the effect of the external factors is completely removed. There is no relaxation if the relaxation factor is one. The same effect as the original one is taken for consideration.
6. What is the disadvantage of over-relaxation?
a) Stability is decreased
b) Divergence is increased
c) Oscillations are increased
d) Convergence is increased
Answer: a
Clarification: When the relaxation factor is more than one, the process is called over-relaxation. This will lead to a higher rate of convergence and leads to faster convergence. But, the disadvantage is that stability will be decreased.
7. Which of these relaxation techniques is used in the SIMPLE algorithm?
a) Explicit over-relaxation
b) Implicit over-relaxation
c) Explicit under-relaxation
d) Implicit under-relaxation
Answer: c
Clarification: The very common algorithm which uses the relaxation technique is the SIMPLE algorithm. It uses the explicit under-relaxation technique to decrease the effect of the correction factors in the pressure and velocity correction equations.
8. Which of these statements about the Patankar’s under-relaxation is true?
a) The value form the previous iteration is modified
b) The diagonal coefficients are modified
c) The whole equation is modified
d) The neighbouring coefficient is modified
Answer: b
Clarification: Patankar’s under-relaxation modifies the value from the previous iteration, the diagonal coefficients and the neighbouring coefficients. The whole equation remains unmodified.
9. The time-step advancement of the E-factor relaxation is dependent on ___________
a) the convergence
b) the stability
c) the accuracy
d) the cell volume
Answer: d
Clarification: The time-step advancement of the E-factor relaxation depends on the grid size. The solution in the smaller element advances more slowly than the solution in the larger element.
10. What is the relation between the E of the E-factor relaxation and the relaxation factor λ?
a) E=(frac{1}{1-lambda})
b) E=(frac{lambda}{1-lambda})
c) E=(frac{lambda}{1+lambda})
d) E=(frac{1}{1+lambda})
Answer: a
Clarification: The relaxation factor is expressed in terms of E factors in the E-factor relaxation. In general, the E factors range from 4 to 10. This range corresponds to the under-relaxation factors of range 0.75 and 0.9.