Engineering Questions with Answers - Multiple Choice Questions
Computational Fluid Dynamics – FVM for Multi-dimensional Steady State Diffusion
1 - Question
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
View Answer
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.
2 - Question
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
View Answer
Explanation: 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.
3 - Question
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
View Answer
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
4 - Question
Consider the following stencil.
For a source-less 2-D steady-state diffusion problem, the coefficient of the flow variable ΦP is ____
a) ΓWAWδxWP+ΓEAEδxPE+ΓSASδySP+ΓNANδyPN
b) ΓWAWδyWP+ΓEAEδyPE+ΓSASδxSP+ΓNANδxPN
c) ΓWAWδyWP+ΓSASδySP+ΓEAEδxPE+ΓNANδxPN
d) ΓWAWδxWP+ΓSASδxSP+ΓEAEδyPE+ΓNANδyPN
View Answer
Answer: a
Explanation: The general form is given by aPΦP=aEΦP+aWΦW+aNΦN+aSΦS
Here, for source-less problem, aP is the addition of all fluxes given by
ΓWAWδxWP+ΓEAEδxPE+ΓSASδySP+ΓNANδyPN.
5 - Question
If aPΦP=aEΦP+aWΦW+aNΦN+aSΦS+S is the general form of a 2-D steady-state diffusion problem, what is aE by considering the following stencil?
a) ΓEAEδyPE
b) ΓEAEδyPE
c) ΓEAEδxPE
d) ΓEAEδxWP
View Answer
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.
6 - Question
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
View Answer
Explanation: 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).
7 - Question
I general, for all the steady-state diffusion problems, the discretized equation can be given as aPΦ P = ∑anbΦnb-S. For a one-dimensional problem, which of these is wrong?<br/>
a) ∑anb =aT+aB<br/>
b) ∑anb =aS+ aN<br/>
c) ∑anb =aW+aE<br/>
d) ∑anb =aP+aE
View Answer
Answer: d
Explanation: For a one-dimensional problem is x-direction, ∑anb =aW+aE. For a one-dimensional problem is y-direction, ∑anb =aS+ aN. For a one-dimensional problem is z-direction, ∑anb =aT+aB.
Explanation: For a one-dimensional problem is x-direction, ∑anb =aW+aE. For a one-dimensional problem is y-direction, ∑anb =aS+ aN. For a one-dimensional problem is z-direction, ∑anb =aT+aB.
8 - Question
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
View Answer
Explanation: 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.
9 - Question
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
View Answer
Explanation: 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.
10 - Question
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
View Answer
Explanation: 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.