# 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) $$\frac{\tilde{\phi_c}-\tilde{\phi_f}}{1-\tilde{\phi_f}}$$

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

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

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) $$\frac{1}{4}$$

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

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).

6. The value of DWFf for the central difference scheme is __________
a) 1
b) $$\frac{1}{3}$$
c) $$\frac{1}{4}$$
d) $$\frac{1}{2}$$

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

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

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) $$\frac{1}{4(1-\tilde{\phi_c})}$$

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 