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# Computational Fluid Dynamics – Diffusion Problem – Green-Gauss Gradient for Cartesian Grids

To get the gradient of the flow variable using the Green-Gauss Theorem, which of these theorems is used?

a) Mean value theorem

b) Stolarsky mean

c) Racetrack principle

d) Newmark-beta method

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Answer: a

Explanation: The Green-Gauss theorem states that for a closed volume V with the surrounding surface ∂V and outward pointing incremental surface vector dS⃗ ,

∫_{V} ∇ΦdV=∮∂VΦdS⃗

Using the mean value theorem,

∫_{V} ∇ΦdV=∇Φ¯¯¯¯¯¯¯¯V

Where, ∇Φ¯¯¯¯¯¯¯¯V is the average gradient over the volume V.

What is the final form of the Green-Gauss gradient method for finding the gradient of Φ over element C?

a) ∇Φ_{C}=∑_{f~nb(c)}Φ_{f} Sf→

b) ∇Φ_{C}=1/V_{C} ∑_{f~nb(c)}Φ_{f} Sf→

c) ∇Φ_{C}=1/V_{C} ∑_{f~nb(c)}Φ_{f}

d) ∇Φ_{C}=1/V_{C} ∑_{f~nb(c)}a_{f} Φ_{f}

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Answer: b

Explanation: The final form of the Green-Gauss gradient method is given by

∇Φ_{C}=1/V_{C} ∑_{f~nb(c)}Φ_{f} Sf→

Where,

Sf→ is known from the geometry of the grids.

Φ_{f} for all the faces should be known to compute ∇Φ_{C}.

The gradient at the face of an element is obtained using ________

a) Linear interpolation

b) Geometric values

c) Green-Gauss theorem

d) Weighted average

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Answer: c

Explanation: Gradient at a face of an element is given by the weighted average of the gradients at the centroids of the element sharing that surface.

∇Φ_{f}=g_{C} ∇Φ_{C}+g_{f} ∇Φ_{f}.

The face-based stencil used for computing Φf in the Green-Gauss Gradient formula is ________

a) more accurate and needs a large stencil

b) less accurate and needs a large stencil

c) more accurate and needs a compact stencil

d) less accurate and needs a compact stencil

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Answer: dExplanation: The face-based computation of Φf uses a compact stencil involving face neighbours. This is less accurate than the vertex-based computation involving a large stencil of vertex neighbours.

It is easy to construct _________ in the face-based computation.

a) Grids

b) Stencil

c) Global matrix

d) Jacobian matrices

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Answer: dExplanation: Compact stencil uses implicit methods. So, it is easy to construct compact Jacobian matrices. But the large stencil brings more information into the reconstruction and therefore is more accurate.

The value of the flow variable at face centre (Φ_{f}) in terms of the flow variable at the owner cell’s centre (Φ_{C}) and the neighbouring cell’s centre (Φ_{f}) as given by the face-based stencil is ________

(Note: g is the weighted average).

a) Φ_{f}=g_{C} Φ_{C}+g_{C} Φ_{f}

b) Φ_{f}=g_{C} Φ_{C}+g_{f} Φ_{f}

c) Φ_{f}=g_{C} Φ_{C}+(1-g_{C})Φ_{f}

d) Φ_{f}=g_{C} Φ_{C}+(1+g_{C})Φ_{f}

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Answer: c

Explanation: In compact stencil, the value of Φ_{f} is calculated using the weighted average values of the two cells sharing the face. It is given by

Φ_{f}=g_{C} Φ_{C}+(1-g_{C}) Φ_{f}

Where,

g_{C}=distanceFfdistanceFC.

The face-centred stencil is second-order _________

a) always

b) only when the centroid of the face and the line connecting the two cells meet

c) only when the centroid of the face and the line connecting the two cells do not meet

d) never

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Answer: bExplanation: The face-centred stencil is second-order accurate when the centroid of the face is the same as the point of intersection of the face and the line connecting the two cells. This happens when the centroid and the line connecting the two cells meet.

In the vertex-based stencil, the flow variable at the face centroid is computed as ___________

a) the mean of the values at the vertices defining the surface

b) the mean of the values at the cell centres

c) the mean of the values at the vertices of the cells

d) the mean of the values at the centroids of the neighbouring faces

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Answer: aExplanation: In the enlarged stencil, the value Φf is calculated as the mean of the values at the vertices defining the surface. Mathematically representing it, Φf=ϕn1+ϕn22.

The flow variable at the vertex node is calculated using the weighted average of the values at the cells sharing it. What is the weight used here?

a) Inverse of the distance of the vertex from the cell centroid

b) Distance of the vertex from the cell

c) Centroids of the cells

d) Mass of the cells

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Answer: dExplanation: The impact decreases as the centroid of the cell moves away from the vertex. So, the inverse is used as the weight here. This is why the method needs an enlarged stencil. But it leads to more accurate approximation.

To overcome the disadvantage caused by the information from the wrong side of cells ____________ is used in the vertex-based method.

a) upwind biased scheme

b) weighted average

c) downwind biased scheme

d) central scheme

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Answer: aExplanation: The major disadvantage of using the enlarged stencil is that information from the wrong side of the face may also contribute to the weighted average while calculating the vertex values. To avoid this, the upwind biased method is used.