Computational Fluid Dynamics Multiple Choice Questions on “Convection-Diffusion Problems – FROMM Scheme”.
1. Which of these profiles is used by the FROMM scheme?
a) Φ(x)=k0+k1 (x-xc)+k2 (x-xc)2
b) Φ(x)=k1 (x-xc )+k2 (x-xc)2
c) Φ(x)=k0+k1 (x-xc)
d) Φ(x)=k1 (x-xc)
Answer: c
Clarification: The FROMM scheme uses a linear interpolation method to approximate the cell face values. So, Φ(x)=k0+k1 (x-xc) is the profile used by the FROMM scheme. But, the approach is different from the other profiles using a linear profile.
2. What is the order of accuracy of the FROMM scheme?
a) First-order
b) Second-order
c) Third-order
d) Fourth-order
Answer: b
Clarification: The first term of the truncation error while implementing the Taylor series in the FROMM scheme is of order two. Therefore, the FROMM scheme is second-order accurate using a linear profile.
3. FROMM scheme ____________
a) gives weighted importance to the upwind and downwind schemes
b) gives equal importance to upwind and downwind scheme
c) is downwind biased
d) is upwind biased
Answer: d
Clarification: The FROMM scheme is upwind biased. It gives more importance to the upwind nodes than the downwind nodes. IT uses two upwind nodes and one downwind node (totally three nodes).
4. Which of these is correct about the FROMM scheme?
a) A linear profile is obtained between the immediate upwind and the far downwind nodes
b) A linear profile is obtained between the far upwind and the immediate downwind nodes
c) A linear profile is obtained between the far upwind and the immediate upwind nodes
d) A linear profile is obtained between the far upwind and the far downwind nodes
Answer: b
Clarification: A linear profile is obtained by connecting the values of the far upwind node and the immediate downwind node. A profile with the same slope obtained here is created between the immediate upwind and the current node to get the required value.
5. Consider the following stencil.
What is Φe according to the QUICK scheme?
a) Φe=(phi_P+frac{x_e-x_P}{x_E-x_W})(ΦE-ΦW)
b) Φe=(phi_P+frac{x_e-x_P}{x_E-x_W})(ΦE+ΦW)
c) Φe=(phi_P-frac{x_e-x_P}{x_E-x_W})(ΦE-ΦW)
d) Φe=(phi_P-frac{x_e-x_P}{x_E-x_W})(ΦE+ΦW)
Answer: a
Clarification: To find Φe, the FROMM scheme first finds ΦP using the profile between ΦW and ΦE given by
ΦP=ΦW+(frac{x_P-x_W}{x_E-x_W})(ΦE-ΦW)
ΦW=ΦP–(frac{x_P-x_W}{x_E-x_W}) (ΦE-ΦW)
Now, Φe is given by,
Φe=ΦW+(frac{x_e-x_W}{x_E-x_W})(ΦE-ΦW)
Which becomes
Φe=ΦP–(frac{x_P-x_W}{x_E-x_W})(ΦE-ΦW)+(frac{x_e-x_W}{x_E-x_W})(ΦE-ΦW)
Therefore,
Φe=ΦP+(frac{x_e-x_P}{x_E-x_W})(ΦE-ΦW).
6. Consider the following stencil.
Assume a uniform grid. What is Φe according to the QUICK scheme?
a) ΦP+(frac{2}{3})(ΦE-ΦW)
b) ΦP+(frac{1}{2})(ΦE-ΦW)
c) ΦP+(frac{1}{4})(ΦE-ΦW)
d) ΦP+(frac{3}{4})(ΦE-ΦW)
Answer: c
Clarification: In general,
Φe=ΦP+(frac{x_e-x_P}{x_E-x_W}) (ΦE-ΦW)
For a uniform grid,
(frac{x_e-x_P}{x_E-x_W})=1/4
So,
Φe=ΦP+(frac{1}{4})(ΦE-ΦW).
7. Which of these is correct about the FROMM scheme?
a) Stable and bounded for a variable velocity system
b) Stable but not bounded for a variable velocity system
c) Stable but not bounded for a constant velocity system
d) Stable and bounded for a constant velocity system
Answer: c
Clarification: The FROMM scheme is numerically stable when the velocity field is constant. But, when the velocity is varying, the scheme is unstable. It includes undershoots and overshoots and hence not bounded.
8. What is the normalized relationship between Φf and Φc for the FROMM scheme?
a) (tilde{phi_f}=tilde{phi_c}+frac{1}{4})
b) (tilde{phi_f}=tilde{phi_c}-frac{1}{4})
c) (tilde{phi_f}=frac{1}{4}-tilde{phi_c})
d) (tilde{phi_f}=frac{1}{4} tilde{phi_c})
Answer: a
Clarification: The relationship between Φf and Φc is
(phi_f=phi_c+frac{1}{4}(phi_D-phi_U))
The normalized forms of Φf, Φc, ΦD and ΦU are (tilde{phi_f}, tilde{phi_c},) 1 and 0 respectively. Therefore,
(tilde{phi_f}=tilde{phi_c}+frac{1}{4}.)
9. For the FROMM scheme, what is the flux limiter ψ(r) equal to?
a) 1-(frac{r}{2})
b) 1+(frac{r}{2})
c) (frac{1-r}{2})
d) (frac{1+r}{2})
Answer: d
Clarification: To find the flux limiter,
Φf=Φc+(frac{1}{2}) Ψ(r)(ΦD-Φc )
For the FROMM scheme,
Φf=Φc+(frac{1}{4})(ΦD-ΦU)
Comparing both,
Ψ(r)(ΦD-Φc)=(frac{1}{2})(ΦD-ΦU)
Ψ(r)=(frac{1}{2}frac{(phi_D-phi_U)}{(phi_D-phi_c)})
Ψ(r)=(frac{1}{2}frac{(phi_D-phi_c+phi_c-phi_U)}{(phi_D-phi_c)})
But,
(frac{(phi_c-phi_U)}{(phi_D-phi_c)}=r)
Therefore,
(psi(r)=frac{1}{2}(1+r)).
10. Consider the following stencil.
Assume a uniform grid. What is the convective flux at the western face ((dot{m_w}phi_w)) using the FROMM scheme?
a) ((phi_P-frac{phi_E}{4}+frac{phi_W}{4}) max(dot{m_w},0)-(phi_Wfrac{-phi_{WW}}{4}+frac{phi_C}{4}) max(-dot{m_w},0))
b) ((phi_P-frac{phi_W}{4}+frac{phi_E}{4}) max(dot{m_w},0)-(phi_Wfrac{-phi_{WW}}{4}+frac{phi_C}{4}) max(-dot{m_w},0))
c) ((phi_Pfrac{-phi_E}{4}+frac{phi_W}{4}) max(dot{m_w},0)-(phi_Wfrac{-phi_C}{4}+frac{phi_{WW}}{4}) max(-dot{m_w},0))
d) ((phi_Pfrac{-phi_W}{4}+frac{phi_E}{4}) max(dot{m_w},0)-(phi_Wfrac{-phi_C}{4}+frac{phi_{WW}}{4})max(-dot{m_w},0))
Answer: a
Clarification: When the flow direction is positive,
(phi_w=phi_P-frac{phi_E}{4}+frac{phi_W}{4})
When the flow direction is negative,
(phi_w=phi_W-frac{phi_{WW}}{4}+frac{phi_C}{4})
Therefore,
(dot{m_w}phi_w=(phi_P-frac{phi_E}{4}+frac{phi_W}{4})max(dot{m_w},0)-(phi_W-frac{phi_{WW}}{4}+frac{phi_C}{4})max(-dot{m_w},0)).