Computational Fluid Dynamics Questions and Answers for Entrance exams 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 Afrac{partial phi}{partial x})dA+int_A(Gamma A frac{partialphi}{partial y}) dA+int_{Delta V} S,dV=0)
b) (int_Afrac{partial}{partial x}(Gamma A frac{partialphi}{partial x})dA+int_Afrac{partial}{partial y}(Gamma Afrac{partialphi}{partial y})dA+int_{Delta V}S, dV=0)
c) (int_A(Gamma Afrac{dphi}{dx})dA+int_A(Gamma A frac{dphi}{dy})dA+int_{Delta V}S, dV=0)
d) (frac{partial phi}{partial t}+int_Afrac{partial}{partial x}(Gamma A frac{partial phi}{partial x}) dA+int_Afrac{partial}{partial y}(Gamma A frac{partial phi}{partial y})dA+int_{Delta V}S, dV=0)
Answer: a
Clarification: The general governing equation for a 2-D steady-state diffusion problem is given by
(frac{partial}{partial x}(Gammafrac{partial phi}{partial x})+frac{partial}{partial y}(Gammafrac{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}(Gammafrac{partialphi}{partial x})dV+int_{delta V}frac{partial}{partial y}(Gammafrac{partial phi}{partial y})dV+int_{Delta V} S ,dV=0)
Applying Gauss Divergence theorem,
(int_A(Gamma Afrac{partialphi}{partial x})dA+int_A(Gamma Afrac{partialphi}{partial y})dA+int_{Delta V}S ,dV=0)
This is the semi-discretized form of the equation.
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
Answer: b
Clarification: 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.
What is the flux across the northern face?
a) (Gamma_Na_Nfrac{(phi _N-phi _P)}{delta x_{PN}})
b) (Gamma_Na_Nfrac{(phi _N-phi _P)}{delta y_{PN}})
c) (Gamma_Na_Nfrac{(phi _E-phi _P)}{delta y_{PN}})
d) (Gamma_Na_Nfrac{(phi _E-phi _P)}{delta x_{PN}})
Answer: b
Clarification: Flux across the northern face is (Gamma_N a_Nfrac{partialphi}{partial y}Big|_n). Expanding this using the central difference scheme, we get
(Gamma_N a_Nfrac{partialphi}{partial y}Big|_n = Gamma_Na_N frac{(phi _N-phi _P)}{delta y_PN}).
4. Consider the following stencil.
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}})
Answer: a
Clarification: 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?
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}})
Answer: c
Clarification: Flux in the eastern direction is given by
(Gamma_E A_Efrac{partialphi}{partial x}Big|_e=Gamma_E A_Efrac{(phi _E-phi _P)}{delta x_{PE}})
(Gamma_E A_Efrac{partialphi}{partial x}Big|_e=Gamma _e A_Efrac{phi_E}{delta x_{PE}}-Gamma_E a_Efrac{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}}).
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