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# Computational Fluid Dynamics – High Resolution Schemes – Downwind and Normalized Weighing Factor

. The Downwind Weighing Factor in the normalized form is given by __________

a) ϕf~−ϕc~1−ϕc~

b) ϕc~−ϕf~1−ϕc~

c) ϕf~−ϕc~1−ϕf~

d) ϕc~−ϕf~1−ϕf~

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Answer: a

Explanation: The Downwind Weighing Factor is given by

DWFf=ϕf−ϕcϕd−ϕc

Normalizing this, we get

DWFf=ϕf~−ϕc~ϕd~−ϕc~

But, the value of ϕd~ is 1. So,

DWFf=ϕf~−ϕc~1−ϕc~

The value of the Downwind Weighing Factor (DWF) lies between ___________

a) 0≤DWF≤∞

b) DWF≥0

c) 0≤DWF≤1

d) DWF≤1

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Answer: c

Explanation: By using DWF, the high-resolution estimate of ϕf~orϕf is redistributed between the upwind and the downwind nodes. As the value of Φ_{f} computed using Φ_{c} and Φ_{c}. The value of DWF always lies between 0 and 1.

The value of DWF for the downwind scheme is __________

a) 0

b) 1

c) 2

d) 3

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Answer: b

Explanation: The relation between the DWF formulation and the TVD formulation is given by

DWF_{f}=12 ψ(r_{f})

The ψ(r_{f}) value for downwind scheme is 2. Therefore, the DWF_{f} value is 1.

DWF_{f} for the FROMM scheme is ___________

a) 12(1−ϕc~)

b) 14(1−ϕc~)

c) 12

d) 14

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Answer: b

Explanation: For FROMM scheme,

ϕf~=ϕc~+14

Therefore,

DWF_{f}=(ϕc~+1/4)−ϕc~1−ϕc~

DWF_{f}=14(1−ϕc~).

For a scheme modelled using the DWF method, the diagonal coefficient becomes zero when ___________

a) DWF_{f} > 0

b) DWF_{f} > 1

c) DWF_{f} > 0.5

d) DWF_{f} > 2

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Answer: c

Explanation: For values of DWF_{f} larger than 0.5, results in a system with negative diagonal coefficients. So, the system becomes unsolvable by iterative methods. This happens whenever Φ_{f} > 0.5(Φ_{c}+Φ_{d}).

The value of DWFf for the central difference scheme is __________

a) 1

b) 13

c) 14

d) 12

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Answer: dExplanation: For the central difference scheme,

ψ(r

_{f})=1

So, the value of DWF

_{f}for this scheme is ½.

The deferred correction source term of the NWF method using he normalized interpolation profile ϕf~=lϕc~+k is _________

a) (1-l-k)Φ_{u}

b) (k)Φ_{u}

c) (-l)Φ_{u}

d) (l-k)Φ_{u}

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Answer: a

Explanation: We have the equation

ϕf~=lϕc~+k

This can be expanded as

ϕf−ϕuϕd−ϕu=lϕc−ϕuϕd−ϕu+k

ϕf−ϕuϕd−ϕu=lϕc−ϕuϕd−ϕu+kϕd−ϕuϕd−ϕu

Φ_{f}=l(Φ_{c}-Φ_{u})+k(Φ_{d}-Φ_{u})+Φ_{u}

Φ_{f}=l(Φ_{c}))+k(Φ_{d}))+(1-l-k)Φ_{u}

The term (1-l-k)Φ_{u} in this equation is the deferred correction source term.

The high-resolution schemes formulated using the NWF method with the equation ϕf~=lϕc~+k are stable without any alteration when __________<br/>

a) k>2<br/>

b) l>2<br/>

c) k>l<br/>

d) l>k

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Answer: dExplanation: The NWF formulation of the high-resolution schemes, when the value of l is greater than the value of k, the diagonal coefficients are all positive and hence the solution is highly stable. This is the case everywhere except a narrow region in NVD.

What is DWF_{f} for the second-order upwind scheme?

a) ϕc~2(1−ϕc~)

b) 12(1−ϕc~)

c) ϕc~4(1−ϕc~)

d) 14(1−ϕc~)

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Explanation: For the second order upwind scheme,

ϕf~=32ϕc~

Therefore,

DWF

_{f}=32ϕc~−ϕc~1−ϕc~

DWF

_{f}=ϕc~2(1−ϕc~).

Along the downwind line of the NVD, the values of _____________ are changed to make the system stable.

a) a_{c}

b) (l,k)

c) Φ_{c}

d) Φ_{f}

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Answer: b

Explanation: Along the downwind line of NVD, the values of (l,k)=(0,1), a value of zero is obtained for the diagonal coefficient and the system becomes unstable. To overcome this problem, the values of (l,k) are set equal to (L,1-LΦ_{f}). The value of L can be chosen which is usually set to l in the previous interval.