# Computational Fluid Dynamics Questions and Answers – High Resolution Schemes – Downwind and Normalized Weighing Factor

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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

Answer: a
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=$$\frac{\tilde{\phi_f}-\tilde{\phi_c}}{1-\tilde{\phi_c}}$$

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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

Answer: c
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

Answer: b
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

Answer: b
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

Answer: c
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(Φcd).
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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

Answer: d
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

Answer: a
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(Φcu)+k(Φdu)+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

Answer: d
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

Answer: a
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=$$\frac{\tilde{\phi_c}}{2(1-\tilde{\phi_c})}$$.
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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

Answer: b
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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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