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Correct ANSWER is (d) less than 10^-5

For explanation I would say: 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.

Right OPTION is (a) Different variables result in different level of RESIDUALS

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.

Right choice is (b) calculated from the PREVIOUS iteration’s values

The explanation is: As the source TERM is a FUNCTION of the flow VARIABLE, it can be calculated using the available Φ values. In an ITERATIVE process, the available values are the previous iteration’s flow variable values.

The correct answer is (c) zero

The best I can explain: 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.

Correct answer is (B) FLOW VARIABLES, corrections

The best 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.

The correct ANSWER is (d) the cell volume

For explanation I would SAY: 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.

The correct option is (a) convergence

For EXPLANATION I WOULD say: 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.

Right choice is (a) E=\(\FRAC{1}{1-\lambda}\)

To explain: 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.

Correct answer is (c) Explicit under-relaxation

The EXPLANATION is: 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.

Right answer is (a) STABILITY is DECREASED

The EXPLANATION: 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.

The correct option is (B) The diagonal coefficients are modified

To explain: Patankar’s under-relaxation modifies the VALUE from the previous iteration, the diagonal coefficients and the NEIGHBOURING coefficients. The WHOLE equation REMAINS unmodified.

Right choice is (b) COMPARE with the vanishing value

The 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.

The correct choice is (d) Numerical errors are less

For 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.

Correct choice is (b) SLOW down the speed of CONVERGENCE

Explanation: 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.

Right CHOICE is (b) Treat sinks implicitly and SOURCES explicitly

Explanation: If the source TERM is properly TREATED, the robustness of the problem will increase. In general, it is better to treat the sinks (negative values) implicitly and the sources (positive values) explicitly.

The correct option is (c) effect of the neighbouring elements and the SOURCE terms

The explanation: 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.

Correct choice is (c) STABILITY

The explanation is: The physics of the problem is greatly affected by the SOURCE TERM. Moreover, talking from the analysis SIDE, the stability condition of the problem differs for the problems with sources and the problems WITHOUT sources.

The CORRECT answer is (c) a function of the flow variable

The EXPLANATION is: The SOURCE term QC is a function of the DEPENDENT variable which is the flow variable. GIVING this mathematically,

QC=Q(ΦC).

Correct ANSWER is (a) The discretized equations

Easy 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.

Correct answer is (B) Series EXPANSION

The best I can explain: To reduce the effects of the highly VARYING SOURCE term on the process, the term should be linearized. For this linearization, a series expansion like the Taylor series expansion is USED.

The correct option is (d) RATE of convergence

For EXPLANATION I would say: When the source term is a constant or small, the analysis is not AFFECTED by it. But, when the variation of the source term is large, the rate of convergence of the numerical PROCESS is GREATLY affected.

The correct option is (d) zero

The best explanation: 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.