This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “High Resolution Schemes – Downwind and Normalized Weighing Factor”.
1. The Downwind Weighing Factor in the normalized form is given by __________
a) \(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{1-\tilde{\phi_c}}\)
b) \(\frac{\tilde{\phi_c}-\tilde{\phi_f}}{1-\tilde{\phi_c}}\)
c) \(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{1-\tilde{\phi_f}}\)
d) \(\frac{\tilde{\phi_c}-\tilde{\phi_f}}{1-\tilde{\phi_f}}\)
View Answer
Explanation: The Downwind Weighing Factor is given by
DWFf=\(\frac{\phi_f-\phi_c}{\phi_d-\phi_c}\)
Normalizing this, we get
DWFf=\(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{\tilde{\phi_d}-\tilde{\phi_c}}\)
But, the value of \(\tilde{\phi_d}\) is 1. So,
DWFf=\(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{1-\tilde{\phi_c}}\)
2. The value of the Downwind Weighing Factor (DWF) lies between ___________
a) 0≤DWF≤∞
b) DWF≥0
c) 0≤DWF≤1
d) DWF≤1
View Answer
Explanation: By using DWF, the high-resolution estimate of \(\tilde{\phi_f}\, or\, \phi_f\) is redistributed between the upwind and the downwind nodes. As the value of Φf computed using Φc and Φc. The value of DWF always lies between 0 and 1.
3. The value of DWF for the downwind scheme is __________
a) 0
b) 1
c) 2
d) 3
View Answer
Explanation: The relation between the DWF formulation and the TVD formulation is given by
DWFf=\(\frac{1}{2}\) ψ(rf)
The ψ(rf) value for downwind scheme is 2. Therefore, the DWFf value is 1.
4. DWFf for the FROMM scheme is ___________
a) \(\frac{1}{2(1-\tilde{\phi_c})}\)
b) \(\frac{1}{4(1-\tilde{\phi_c})}\)
c) \(\frac{1}{2}\)
d) \(\frac{1}{4}\)
View Answer
Explanation: For FROMM scheme,
\(\tilde{\phi_f}=\tilde{\phi_c}+\frac{1}{4}\)
Therefore,
DWFf=\(\frac{(\tilde{\phi_c}+1/4)-\tilde{\phi_c}}{1-\tilde{\phi_c}}\)
DWFf=\(\frac{1}{4(1-\tilde{\phi_c})}.\)
5. For a scheme modelled using the DWF method, the diagonal coefficient becomes zero when ___________
a) DWFf > 0
b) DWFf > 1
c) DWFf > 0.5
d) DWFf > 2
View Answer
Explanation: For values of DWFf larger than 0.5, results in a system with negative diagonal coefficients. So, the system becomes unsolvable by iterative methods. This happens whenever Φf > 0.5(Φc+Φd).
6. The value of DWFf for the central difference scheme is __________
a) 1
b) \(\frac{1}{3}\)
c) \(\frac{1}{4}\)
d) \(\frac{1}{2}\)
View Answer
Explanation: For the central difference scheme,
ψ(rf)=1
So, the value of DWFf for this scheme is ½.
7. The deferred correction source term of the NWF method using he normalized interpolation profile \(\tilde{\phi_f}=l\tilde{\phi_c}+k\) is _________
a) (1-l-k)Φu
b) (k)Φu
c) (-l)Φu
d) (l-k)Φu
View Answer
Explanation: We have the equation
\(\tilde{\phi_f}=l\tilde{\phi_c}+k\)
This can be expanded as
\(\frac{\phi_f-\phi_u}{\phi_d-\phi_u}=l \frac{\phi_c-\phi_u}{\phi_d-\phi_u}+k\)
\(\frac{\phi_f-\phi_u}{\phi_d-\phi_u}=l \frac{\phi_c-\phi_u}{\phi_d-\phi_u}+k\frac{\phi_d-\phi_u}{\phi_d-\phi_u}\)
Φf=l(Φc-Φu)+k(Φd-Φu)+Φu
Φf=l(Φc))+k(Φd))+(1-l-k)Φu
The term (1-l-k)Φu in this equation is the deferred correction source term.
8. The high-resolution schemes formulated using the NWF method with the equation \(\tilde{\phi_f} = l\tilde{\phi_c}+k\) are stable without any alteration when __________
a) k>2
b) l>2
c) k>l
d) l>k
View Answer
Explanation: The NWF formulation of the high-resolution schemes, when the value of l is greater than the value of k, the diagonal coefficients are all positive and hence the solution is highly stable. This is the case everywhere except a narrow region in NVD.
9. What is DWFf for the second-order upwind scheme?
a) \(\frac{\tilde{\phi_c}}{2(1-\tilde{\phi_c})}\)
b) \(\frac{1}{2(1-\tilde{\phi_c})}\)
c) \(\frac{\tilde{\phi_c}}{4(1-\tilde{\phi_c})}\)
d) \(\frac{1}{4(1-\tilde{\phi_c})}\)
View Answer
Explanation: For the second order upwind scheme,
\(\tilde{\phi_f}=\frac{3}{2} \tilde{\phi_c}\)
Therefore,
DWFf=\(\frac{\frac{3}{2}\tilde{\phi_c}-\tilde{\phi_c}}{1-\tilde{\phi_c}}\)
DWFf=\(\frac{\tilde{\phi_c}}{2(1-\tilde{\phi_c})}\).
10. Along the downwind line of the NVD, the values of _____________ are changed to make the system stable.
a) ac
b) (l,k)
c) Φc
d) Φf
View Answer
Explanation: Along the downwind line of NVD, the values of (l,k)=(0,1), a value of zero is obtained for the diagonal coefficient and the system becomes unstable. To overcome this problem, the values of (l,k) are set equal to (L,1-LΦf). The value of L can be chosen which is usually set to l in the previous interval.
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