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# Computational Fluid Dynamics – Discretization Aspects – Boundedness

1 - Question

A flow property Φ is bounded when ___________

a) the value of flow properties at a node is bounded by its boundary values

b) the error is bounded

c) the numerical solution approaches the exact solution

d) the solution does not change with further iterations

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Answer: aExplanation: The flow property is said to be bounded if the internal nodal values of the flow property do not cross the minimum and maximum values of the flow properties in the boundaries. Physically the flow properties will not go beyond the boundary values. This should be guaranteed by the numerical approximations also.

2 - Question

For a system to be bounded, the coefficients of the nodes should ___________

a) have the same sign

b) be the same

c) be different everywhere

d) have different signs everywhere

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Answer: aExplanation: All the coefficients of the discretized form of the governing equations should have the same sign for the system to be bounded. In most of the cases, the signs should be all positive. This means that an increase in the flow variable at one node increases the variables at the neighbouring nodes too.

3 - Question

___________ of the coefficient matrix is a desirable feature for boundedness.

a) Non-diagonal dominance

b) Singularity

c) Sparsity

d) Diagonal dominance

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Answer: dExplanation: A system is bounded if the Scarborough criterion is satisfied. This needs the coefficient matrix to be diagonally dominant. This is a desirable feature for boundedness of solutions. This depends upon the coefficients of the neighbouring nodes and the central nodes.

4 - Question

Boundedness of a system has a direct impact on ______________

a) stability

b) convergence

c) conservativeness

d) transportiveness

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Answer: bExplanation: Convergence of an iterative solution is when two consecutive iterations result in the same solution. If the discretized system does not satisfy boundedness, it is possible that the solution wiggles continuously without converging.

5 - Question

A coefficient matrix is diagonally dominant if ________

Note: anb → coefficients of the neighbouring nodes

ap → coefficient of the central node

a) ∣∣anb∣∣∣∣ap∣∣< 1

b) ∣∣ap∣∣∑∣∣anb∣∣ < 1

c) ∑∣∣anb∣∣∣∣ap∣∣ < 1

d) ∣∣ap∣∣∣∣anb∣∣ < 1

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Answer: cExplanation: For a coefficient matrix to be diagonally dominant, the diagonal elements should be larger than the non-diagonal elements of the row. So,∑∣∣anb∣∣∣∣ap∣∣ should be less than 1. This is the condition given by Scarborough criterion.

6 - Question

Which of these schemes often result in unbounded solutions?

a) Central difference schemes.

b) First order schemes

c) QUICK scheme

d) High-resolution schemes

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Answer: cExplanation: Higher order schemes result in unbounded solutions. This means that the solution is erroneous. This happens only when the grid is so coarse. The problem of boundedness can be overcome by refining the grids. The QUICK scheme is a higher order scheme.

7 - Question

Which of these schemes guarantee boundedness?

a) Central difference schemes.

b) Forward difference schemes

c) High-resolution schemes

d) First order schemes

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Answer: dExplanation: Irrespective of coarse or fine grids first order schemes always guarantee boundedness without any overshoots or undershoots. So, while choosing a coarse grid, it is better to use the first order schemes to get rid of errors.

8 - Question

Which of these conditions define boundedness?

ΦF → Flow property at boundary

Φi → Flow property at nodes

a) Φi≤max(ΦF)

b) min(ΦF )≤Φi≤max(ΦF)

c) Φi≥min(ΦF)

d) Φi≥max(ΦF)

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Answer: bExplanation: The value of flow property Φ at any node inside the domain must be smaller than the minimum value of ϕ at the boundaries. Similarly, Φ at any node must be greater than the maximum value of Φ at the boundaries. Representing this mathematically, min(ΦF)≤ Φi≤max(ΦF) holds true.

9 - Question

Applying the boundedness condition, which is correct?

ρi → Density at nodes

a) ρi < 1

b) ρi < 0

c) ρi > 0

d) ρi > 1

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Answer: cExplanation: According to boundedness condition, flow properties should lie in the range. So, the non-negative properties like density, temperature and kinetic energy should always be positive. Therefore, ρi > 0.

10 - Question

Flow properties are not bounded by the boundary values __________

a) in the absence of convection term

b) in the absence of source term

c) in the presence of source term

d) in the absence of convection term

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Answer: cExplanation: When there is a source or sink inside the domain of interest, the flow properties may increase or decrease drastically. This may lead to a value which does not lie in the range of the boundary values. So, a system with a source or sink will not be bounded.