Computational Fluid Dynamics Questions and Answers – FVM for Multi-dimensional Steady State Diffusion

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This set of Computational Fluid Dynamics Questions and Answers for Entrance exams focuses on “FVM for Multi-dimensional Steady State Diffusion”.

1. Which of these equations represent the semi-discretized equation of a 2-D steady-state diffusion problem?
a) \(\int_A(\Gamma A\frac{\partial \phi}{\partial x})dA+\int_A(\Gamma A \frac{\partial\phi}{\partial y}) dA+\int_{\Delta V} S\,dV=0\)
b) \(\int_A\frac{\partial}{\partial x}(\Gamma A \frac{\partial\phi}{\partial x})dA+\int_A\frac{\partial}{\partial y}(\Gamma A\frac{\partial\phi}{\partial y})dA+\int_{\Delta V}S\, dV=0\)
c) \(\int_A(\Gamma A\frac{d\phi}{dx})dA+\int_A(\Gamma A \frac{d\phi}{dy})dA+\int_{\Delta V}S\, dV=0\)
d) \(\frac{\partial \phi}{\partial t}+\int_A\frac{\partial}{\partial x}(\Gamma A \frac{\partial \phi}{\partial x}) dA+\int_A\frac{\partial}{\partial y}(\Gamma A \frac{\partial \phi}{\partial y})dA+\int_{\Delta V}S\, dV=0\)
View Answer

Answer: a
Explanation: The general governing equation for a 2-D steady-state diffusion problem is given by
\(\frac{\partial}{\partial x}(\Gamma\frac{\partial \phi}{\partial x})+\frac{\partial}{\partial y}(\Gamma\frac{\partial \phi}{\partial y})+S=0\)
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,
\(\int_{\delta V}\frac{\partial}{\partial x}(\Gamma\frac{\partial\phi}{\partial x})dV+\int_{\delta V}\frac{\partial}{\partial y}(\Gamma\frac{\partial \phi}{\partial y})dV+\int_{\Delta V} S \,dV=0\)
Applying Gauss Divergence theorem,
\(\int_A(\Gamma A\frac{\partial\phi}{\partial x})dA+\int_A(\Gamma A\frac{\partial\phi}{\partial y})dA+\int_{\Delta V}S \,dV=0\)
This is the semi-discretized form of the equation.
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2. 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

Answer: b
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. Consider the following stencil.
computational-fluid-dynamics-questions-answers-entrance-exams-q4
What is the flux across the northern face?
a) \(\Gamma_Na_N\frac{(\phi _N-\phi _P)}{\delta x_{PN}}\)
b) \(\Gamma_Na_N\frac{(\phi _N-\phi _P)}{\delta y_{PN}}\)
c) \(\Gamma_Na_N\frac{(\phi _E-\phi _P)}{\delta y_{PN}}\)
d) \(\Gamma_Na_N\frac{(\phi _E-\phi _P)}{\delta x_{PN}}\)
View Answer

Answer: b
Explanation: Flux across the northern face is \(\Gamma_N a_N\frac{\partial\phi}{\partial y}\Big|_n\). Expanding this using the central difference scheme, we get
\(\Gamma_N a_N\frac{\partial\phi}{\partial y}\Big|_n = \Gamma_Na_N \frac{(\phi _N-\phi _P)}{\delta y_PN}\).

4. Consider the following stencil.
computational-fluid-dynamics-questions-answers-entrance-exams-q4
For a source-less 2-D steady-state diffusion problem, the coefficient of the flow variable ΦP is ____
a) \(\frac{\Gamma_W A_W}{\delta x_{WP}}+\frac{\Gamma_E A_E}{\delta x_{PE}}+\frac{\Gamma_S A_S}{\delta y_SP}+\frac{\Gamma_N A_N}{\delta y_{PN}}\)
b) \(\frac{\Gamma_W A_W}{\delta y_{WP}}+\frac{\Gamma_E A_E}{\delta y_{PE}}+\frac{\Gamma_S A_S}{\delta x_SP}+\frac{\Gamma_N A_N}{\delta x_{PN}}\)
c) \(\frac{\Gamma_W A_W}{\delta y_{WP}}+\frac{\Gamma_S A_S}{\delta y_{SP}}+\frac{\Gamma_E A_E}{\delta x_{PE}}+\frac{\Gamma_N A_N}{\delta x_{PN}}\)
d) \(\frac{\Gamma_W A_W}{\delta x_{WP}}+\frac{\Gamma_S A_S}{\delta x_{SP}}+\frac{\Gamma_E A_E}{\delta y_{PE}}+\frac{\Gamma_N A_N}{\delta y_{PN}}\)
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
\(\frac{\Gamma_W A_W}{\delta x_{WP}}+\frac{\Gamma_E A_E}{\delta x_{PE}}+\frac{\Gamma_S A_S}{\delta y_SP}+\frac{\Gamma_N A_N}{\delta y_{PN}}\).

5. 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?
computational-fluid-dynamics-questions-answers-entrance-exams-q4
a) \(\frac{\Gamma_E A_E}{\delta y_{PE}}\)
b) \(\frac{\Gamma_E A_E}{\delta y_{PE}}\)
c) \(\frac{\Gamma_E A_E}{\delta x_{PE}}\)
d) \(\frac{\Gamma_E A_E}{\delta x_{WP}}\)
View Answer

Answer: c
Explanation: Flux in the eastern direction is given by
\(\Gamma_E A_E\frac{\partial\phi}{\partial x}\Big|_e=\Gamma_E A_E\frac{(\phi _E-\phi _P)}{\delta x_{PE}}\)
\(\Gamma_E A_E\frac{\partial\phi}{\partial x}\Big|_e=\Gamma _e A_E\frac{\phi_E}{\delta x_{PE}}-\Gamma_E a_E\frac{\phi_P}{\delta x_{PE}}\)
Expanding this while forming the general equation, we will get
\(a_E=\frac{\Gamma_E A_E}{\delta x_{PE}}\).
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6. 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

Answer: d
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. 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?
a) ∑anb =aT+aB
b) ∑anb =aS+ aN
c) ∑anb =aW+aE
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.

8. 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

Answer: c
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.
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9. 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

Answer: a
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. Consider the stencil.
computational-fluid-dynamics-questions-answers-entrance-exams-q4
The values of \(\vec{A_w}\, and\, \vec{A_s}\) are _____________
a) \(\vec{A_w}=\Delta x; \vec{A_s}=\Delta y\)
b) \(\vec{A_w}=-\Delta x; \vec{A_s}=-\Delta y\)
c) \(\vec{A_w}=-\Delta y; \vec{A_s}=-\Delta x\)
d) \(\vec{A_w}=\Delta y; \vec{A_s}=\Delta x\)
View Answer

Answer: b
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, \(\vec{A_w}=-\Delta x; \vec{A_s}=-\Delta y\).

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn