This set of Computational Fluid Dynamics Question Bank focuses on “Diffusion Problem – Discretization Equation Rules”.

1. The zero sum rule and the opposite signs rule are applicable to _______________

a) each discretized equation

b) the global matrix

c) the coefficients of each discretized equation

d) the coefficient matrix

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Explanation: A proper discretization should result in a discretized algebraic equation that reflects the characteristics of the original conservation equation. The coefficients of each discretized equation should satisfy the zero sum and the opposite signs rules.

2. Which of these is a sufficient condition for a discretized equation?

a) Neither the opposite sign rule nor the zero sum rule

b) Both the opposite sign and the zero sum rules

c) The opposite signs rule

d) The zero sum rule

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Explanation: For of the discretized equations to be bounded, the sufficient condition is the opposite signs rule. Sufficient condition means that the rule will be enough to make sure that the equation is bounded.

3. Which of these assumptions is made regarding the variation of Φ over a domain?

a) Linear profile

b) Central differencing

c) Quadratic profile

d) Downwind differencing

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Explanation: The major approximation made while discretizing a governing equation is “variation of the flow variable is considered to be linear”. The linear interpolation method is used to get the unknown values near the known values.

4. Why a parabolic profile is not used to model the variation of Φ?

a) Accuracy comes at the cost of computation

b) Exact results

c) Unphysical results

d) Difficult to use

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Explanation: Consider a parabolic (second-order) profile is used to find the value of a flow variable at the face in between two centroids. It will lead to a value higher or lower than the values at the centroids which will be unphysical.

5. In the absence of any source or sink, the steady-state diffusion problem is governed by _______________

a) Fourier series

b) Linear interpolation

c) Taylor series

d) Second order interpolation

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Explanation: For a source-less steady state diffusion problem, the transfer of the flow variable (Φ) occurs only by diffusion. So, the transfer of Φ will be in the direction opposite to the increasing Φ. This is governed by Fourier series.

6. Consider the general discretized equation given by a_{P} Φ_{P}+∑_{F~NB(P)}a_{F} Φ_{F} =0. Which of these statements is correct?

a) When the value of Φ_{F} is increased, the value of Φ_{P} remains the same

b) When the value of Φ_{F} is increased, the value of Φ_{P} increased

c) When the value of Φ_{F} is increased, the value of Φ_{P} decreases

d) When the value of Φ_{F} is decreased, the value of Φ_{P} decreases

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Explanation: The value of Φ

_{P}varies with the variation of the values of Φ

_{F}. So, it will not remain the same. When the value of Φ

_{F}is increased, the value of Φ

_{P}decreases. This is the statement of the opposite signs rule taken physically.

7. Boundedness is ensured in the steady-state diffusion problem _______________

a) only when the source term is non-negative

b) only when the source term is negative

c) only when the source term is non-zero

d) only when the source term is zero

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Explanation: When there is a source term, the linear profile of the flow variable may fail. The source or sink term may lead to increased or decreased values than that guesses by the linear interpolation. So, in this case, boundedness is not possible.

8. Consider the general discretized equation given by a_{P} Φ_{P}+∑_{F~NB(P)}a_{F} Φ_{F} =0. Which of these statements is correct according to the opposite signs rule?

a) a_{P} and a_{F} are of opposite signs

b) a_{P} and Φ_{P} are of opposite signs

c) a_{F} and Φ_{F} are of opposite signs

d) Φ_{P} and Φ_{F} are of opposite signs

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Explanation: The boundedness property is enforced only when the coefficients a

_{P}and a

_{F}are of opposite signs. So, the opposite sign rule says that the coefficients of the flow variables Φ

_{P}and Φ

_{F}are of opposite signs.

9. Consider the general discretized equation given by a_{P} Φ_{P}+∑_{F~NB(P)}a_{F} Φ_{F} = 0. According to the zero sum rule, which of these is correct?

a) a_{P}+∑_{F~NB(P)}a_{F} = ∞

b) a_{P}+∑_{F~NB(P)}a_{F} = 0

c) a_{P}+∑_{F~NB(P)}a_{F} = 1

d) a_{P}+∑_{F~NB(P)}a_{F} = -1

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Explanation: A consistent discretization method should yield an equation which incorporates the property of the overall domain – conservation. To ensure this, the equation should satisfy the condition a

_{P}+∑

_{F~NB(P)}a

_{F}= 0.

10. Consider the general discretized equation given by a_{P} Φ_{P}+∑_{F~NB(P)}a_{F} Φ_{F} =0. According to the zero sum rule, which of these is correct?

a) ∑_{F~NB(P)}\(\frac{a_F}{a_P}\) = ∞

b) ∑_{F~NB(P)}\(\frac{a_F}{a_P}\) = 1

c) ∑_{F~NB(P)}\(\frac{a_F}{a_P}\) = -1

d) ∑_{F~NB(P)}\(\frac{a_F}{a_P}\) = -∞

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Explanation: From the zero sum rule,

\(a_P+\sum_{(F \sim NB(P))}a_F = 0\)

Divided by a

_{P}, the equation becomes

\(1+\frac{\sum_{F \sim NB(P)}a_F}{a_P} = 0 \)

\(\frac{\sum_{F \sim NB(P)}a_F}{a_P} =-1\)

This can be written as

\(\sum_{F \sim NB(P)}\frac{a_F}{a_P} =-1\).

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