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# Computational Fluid Dynamics – Diffusion Problem – Orthogonal and Non-Orthogonal Grids

Which of these statements is true?

a) The Cartesian and non-Cartesian orthogonal grids lead to the same discretized equation

b) A non-Cartesian orthogonal grid leads to an extra source term when compared to the Cartesian orthogonal grids

c) The equations of the Cartesian and non-Cartesian orthogonal grids differ by a trigonometric function

d) The equations obtained from a non-Cartesian grid has fewer terms when compared to that obtained from a Cartesian grid

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Answer: aExplanation: The discretized equation for the non-Cartesian grids should be exactly the same as obtained from the Cartesian grids. The solution should also be the same if the boundary conditions match.

Non-orthogonality creates a problem in _________ of the steady-state diffusion equation.

a) the neighbouring terms

b) the source term

c) the direction of the surface vector

d) the magnitude of the surface vector

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Answer: cExplanation: The surface vector and the vector joining the owner and the neighbouring elements are not collinear for non-orthogonal grids. Thus, non-orthogonal grids need special attention in the steady-state diffusion equation.

Non-orthogonality leads to ________ in diffusion problems.

a) cubic-diffusion

b) less-diffusion

c) additional-diffusion

d) cross-diffusion

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Answer: dExplanation: In general, the surface vector of the non-orthogonal grids is given as the sum of the vector connecting the owner and the neighbour elements and an additional vector. This leads to an extra term in the diffusion equation called the cross-diffusion or non-orthogonal diffusion.

Which of these statements is false?

a) Unstructured grids are always non-orthogonal

b) Structured grids are always orthogonal

c) Non-orthogonal grids can be structured or unstructured

d) Curvilinear structured grids are non-orthogonal

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Answer: bExplanation: Non-orthogonality can exist in structured grids also when it is curvilinear. Structured curvilinear grids and unstructured grids are non-orthogonal. So, the statement “Structured grids are always orthogonal” is wrong.

In the minimum correction approach of decomposing the surface vector of a non-orthogonal grid, the relationship between the vector connecting the owner and the neighbour node (Ef→) and the surface vector (Sf→) is given as _________

a) Sf→sinθ.e⃗

b) Sf→cosθ.e⃗

c) (Sfcosθ)e⃗

d) (Sfsinθ)e⃗

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Answer: c

Explanation: Here, a right-angled triangle is formed by the vectors Ef→,Sf→andTf→. The Tf→ vector is orthogonal to the Ef→ vector in this case. The relation is given by (e⃗ .Sf→)e⃗ =(Sfcosθ)e⃗ .

Which of these is correct regarding the minimum correction approach?

a) The non-orthogonal correction is kept as small as possible

b) The non-orthogonal correction is kept as large as possible

c) The surface vector is kept as small as possible

d) The surface vector is kept as large as possible

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Answer: a

Explanation: In the minimum correction approach, the decomposition of the surface vector is done in a way that the non-orthogonal correction is as small as possible, thus making the Ef→andtheTf→ orthogonal.

In the orthogonal correction approach, the relationship between Ef→andSf→ is ________

a) Ef→=Sf→×e⃗

b) Ef→=Sfcosθe⃗

c) Ef→=Sfe⃗

d) Ef→=Sf→.e⃗

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Answer: c

Explanation: In this approach, the contribution of the term involving Φ_{F} and Φ_{C} are kept the same as that of the orthogonal mesh. This is achieved by the relation

Ef→=Sfe⃗ .

In the over-relaxed approach, the importance of the term involving ΦF and ΦC _________ as the non-orthogonality _________

a) decreases, increases

b) remains the same, increases

c) increases, remains the same

d) increases, increases

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Answer: dExplanation: The importance of the term involving ΦF and ΦC decreases when the non-orthogonality decreases for the minimum correction approach. For the over-relaxed approach, the importance increases.

What is the relationship between Ef→andSf→ using the over-relaxed approach?

a) Ef→=(Sf→).e⃗

b) Ef→=(Sfcosθ)e⃗

c) Ef→=(Sf→)×e⃗

d) Ef→=((Sf→).e⃗ )e⃗

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Answer: b

Explanation: The relationship is given by (Sfcosθ)e⃗ . This is calculated in CFD packages as Ef→=Sf→.Sf→e⃗ .Sf→e⃗ . The derivation is given as

Ef→=(Sfcosθ)e⃗ =(S2fSfcosθ)e⃗ =Sf→.Sf→e⃗ .Sf→e⃗ .

Which of these methods is used to treat the non-orthogonal diffusion term?

a) Deferred correction

b) Predictor–corrector

c) Green-gauss

d) Trial and error method

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Answer: aExplanation: The cross-diffusion term cannot be expressed in terms of nodal values. So, deferred correction is used here. Its value is computed using the current gradient field and this is added as a source term in the right-hand side of the algebraic equation.