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# Computational Fluid Dynamics – FVM for Multi-dimensional Steady State Diffusion

Which of these equations represent the semi-discretized equation of a 2-D steady-state diffusion problem?

a) ∫A(ΓA∂ϕ∂x)dA+∫A(ΓA∂ϕ∂y)dA+∫ΔVSdV=0

b) ∫A∂∂x(ΓA∂ϕ∂x)dA+∫A∂∂y(ΓA∂ϕ∂y)dA+∫ΔVSdV=0

c) ∫A(ΓAdϕdx)dA+∫A(ΓAdϕdy)dA+∫ΔVSdV=0

d) ∂ϕ∂t+∫A∂∂x(ΓA∂ϕ∂x)dA+∫A∂∂y(ΓA∂ϕ∂y)dA+∫ΔVSdV=0

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Answer: a

Explanation: The general governing equation for a 2-D steady-state diffusion problem is given by

∂∂x(Γ∂ϕ∂x)+∂∂y(Γ∂ϕ∂y)+S=0

Here, partial differentiation is used as the variable φ varies in both x and y directions, but the differentiation is only in the required direction.

Integrating the equation with respect to the control volume,

∫δV∂∂x(Γ∂ϕ∂x)dV+∫δV∂∂y(Γ∂ϕ∂y)dV+∫ΔVSdV=0

Applying Gauss Divergence theorem,

∫A(ΓA∂ϕ∂x)dA+∫A(ΓA∂ϕ∂y)dA+∫ΔVSdV=0

This is the semi-discretized form of the equation.

The area in the western face of a 2-D steady-state diffusion stencil (uniform) is _______________

a) grid size in the x-direction

b) grid size in the y-direction

c) product of the grid sizes in the x and y-directions

d) ratio of the grid sizes in the x and y-directions

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Answer: bExplanation: In the one-dimensional case, the area is taken to be unity. In the two-dimensional case, the area is the grid size in the perpendicular direction multiplied by unity. So, for area Ae=Aw=Δy and An=As=Δx.

Consider the following stencil.

What is the flux across the northern face?

a) ΓNaN(ϕN−ϕP)δxPN

b) ΓNaN(ϕN−ϕP)δyPN

c) ΓNaN(ϕE−ϕP)δyPN

d) ΓNaN(ϕE−ϕP)δxPN

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Answer: b

Explanation: Flux across the northern face is ΓNaN∂ϕ∂y∣∣n. Expanding this using the central difference scheme, we get

ΓNaN∂ϕ∂y∣∣n=ΓNaN(ϕN−ϕP)δyPN

Consider the following stencil.

For a source-less 2-D steady-state diffusion problem, the coefficient of the flow variable Φ_{P} is ____

a) ΓWAWδxWP+ΓEAEδxPE+ΓSASδySP+ΓNANδyPN

b) ΓWAWδyWP+ΓEAEδyPE+ΓSASδxSP+ΓNANδxPN

c) ΓWAWδyWP+ΓSASδySP+ΓEAEδxPE+ΓNANδxPN

d) ΓWAWδxWP+ΓSASδxSP+ΓEAEδyPE+ΓNANδyPN

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Answer: a

Explanation: The general form is given by a_{P}Φ_{P}=a_{E}Φ_{P}+a_{W}Φ_{W}+a_{N}Φ_{N}+a_{S}Φ_{S}

Here, for source-less problem, a_{P} is the addition of all fluxes given by

ΓWAWδxWP+ΓEAEδxPE+ΓSASδySP+ΓNANδyPN.

If a_{P}Φ_{P}=a_{E}Φ_{P}+a_{W}Φ_{W}+a_{N}Φ_{N}+a_{S}Φ_{S}+S is the general form of a 2-D steady-state diffusion problem, what is a_{E} by considering the following stencil?

a) ΓEAEδyPE

b) ΓEAEδyPE

c) ΓEAEδxPE

d) ΓEAEδxWP

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Answer: c

Explanation: Flux in the eastern direction is given by

ΓEAE∂ϕ∂x∣∣e=ΓEAE(ϕE−ϕP)δxPE

ΓEAE∂ϕ∂x∣∣e=ΓeAEϕEδxPE−ΓEaEϕPδxPE

Expanding this while forming the general equation, we will get

aE=ΓEAEδxPE.

Consider the following 2-D surface with the numbers inside as the global indices of their cells.

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

The general discretized equation is of the form aPΦP=aEΦP+aWφW+aNΦN+aSΦS+S. Which of the following is correct regarding the cell numbered “13”?

a) aE=0; aW=0

b) aW=0; aN=0

c) aN=0; aS=0

d) aS=0; aW=0

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Answer: dExplanation: For the control volumes adjacent to the boundary of the global domain, the boundary-side coefficient is set to zero. Therefore, for the cell numbered “13”, the southern and the western coefficients are zero (aS=0; aW=0).

I general, for all the steady-state diffusion problems, the discretized equation can be given as a_{P}Φ _{P} = ∑a_{nb}Φ_{nb}-S. For a one-dimensional problem, which of these is wrong?<br/>

a) ∑a_{nb} =a_{T}+a_{B<br/>}

b) ∑a_{nb} =a_{S}+ a_{N<br/>}

c) ∑a_{nb} =a_{W}+a_{E<br/>}

d) ∑a_{nb} =a_{P}+a_{E}

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Explanation: For a one-dimensional problem is x-direction, ∑a

_{nb}=a

_{W}+a

_{E}. For a one-dimensional problem is y-direction, ∑a

_{nb}=a

_{S}+ a

_{N}. For a one-dimensional problem is z-direction, ∑a

_{nb}=a

_{T}+a

_{B}.

In a control volume adjacent to the boundary, the flux crossing the boundary is _______________ in the discretized equation.

a) set to some arbitrary constant

b) set to zero

c) introduced as a source term

d) introduced as a convective flux

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Answer: cExplanation: As the boundary-side coefficients are set to zero in the discretized equations of the boundary-based control volumes, the information in the boundary may be lost. To avoid this, the flux crossing the boundary is introduced as a source term in the equation.

Consider a source-less 3-D steady-state diffusion problem. The general discretized equation is aP ΦP = ∑anb Φnb. What is aP?

a) aP=aW+aE+aS+aN+aT+aB

b) aP=aW+aE+aS+aN

c) aP=aW+aE+aS+aN+aT

d) aP=0

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Answer: aExplanation: For all steady-state diffusion problems, in the absence of source term, aP=∑anb. Therefore, for the three-dimensional case, aP=aW+aE+aS+aN+aT+aB which includes the coefficients of all the neighbouring flow variables.

Consider the stencil.

The values of Aw→andAs→ are _____________

a) Aw→=Δx;As→=Δy

b) Aw→=−Δx;As→=−Δy

c) Aw→=−Δy;As→=−Δx

d) Aw→=Δy;As→=Δx

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Answer: bExplanation: The values of Aw and As are Δ x and Δ y respectively. The signs of the area vectors depend on their directions. Therefore, Aw→=−Δx;As→=−Δy.