250+ TOP MCQs on High Resolution Schemes – Downwind and Normalized Weighing Factor and Answers

Computational Fluid Dynamics Multiple Choice Questions 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}})
Answer: a
Clarification: The Downwind Weighing Factor is given by
DWF_f=(frac{phi_f-phi_c}{phi_d-phi_c})
Normalizing this, we get
DWF_f=(frac{tilde{phi_f}-tilde{phi_c}}{tilde{phi_d}-tilde{phi_c}})
But, the value of (tilde{phi_d}) is 1. So,
DWF_f=(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
Answer: c
Clarification: 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 Φ_f computed using Φ_c and Φ_c. 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
Answer: b
Clarification: The relation between the DWF formulation and the TVD formulation is given by
DWF_f=(frac{1}{2}) ψ(r_f)
The ψ(r_f) value for downwind scheme is 2. Therefore, the DWF_f value is 1.

4. DWF_f 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})
Answer: b
Clarification: For FROMM scheme,
(tilde{phi_f}=tilde{phi_c}+frac{1}{4})
Therefore,
DWF_f=(frac{(tilde{phi_c}+1/4)-tilde{phi_c}}{1-tilde{phi_c}})
DWF_f=(frac{1}{4(1-tilde{phi_c})}.)

5. For a scheme modelled using the DWF method, the diagonal coefficient becomes zero when ___________
a) DWF_f > 0
b) DWF_f > 1
c) DWF_f > 0.5
d) DWF_f > 2
Answer: c
Clarification: For values of DWF_f larger than 0.5, results in a system with negative diagonal coefficients. So, the system becomes unsolvable by iterative methods. This happens whenever Φ_f > 0.5(Φ_c+Φ_d).

6. The value of DWF_f for the central difference scheme is __________
a) 1
b) (frac{1}{3})
c) (frac{1}{4})
d) (frac{1}{2})
Answer: d
Clarification: For the central difference scheme,
ψ(r_f)=1
So, the value of DWF_f for this scheme is ½.

7. The deferred correction source term of the NWF method using he normalized interpolation profile (tilde{phi_f}=ltilde{phi_c}+k) is _________
a) (1-l-k)Φ_u
b) (k)Φ_u
c) (-l)Φ_u
d) (l-k)Φ_u
Answer: a
Clarification: We have the equation
(tilde{phi_f}=ltilde{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}+kfrac{phi_d-phi_u}{phi_d-phi_u})
Φ_f=l(Φ_c-Φ_u)+k(Φ_d-Φ_u)+Φ_u
Φ_f=l(Φ_c))+k(Φ_d))+(1-l-k)Φ_u
The term (1-l-k)Φ_u 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} = ltilde{phi_c}+k) are stable without any alteration when __________
a) k>2
b) l>2
c) k>l
d) l>k
Answer: d
Clarification: 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 DWF_f 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})})
Answer: a
Clarification: For the second order upwind scheme,
(tilde{phi_f}=frac{3}{2} tilde{phi_c})
Therefore,
DWF_f=(frac{frac{3}{2}tilde{phi_c}-tilde{phi_c}}{1-tilde{phi_c}})
DWF_f=(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) a_c
b) (l,k)
c) Φ_c
d) Φ_f
Answer: b
Clarification: 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-LΦ_f). The value of L can be chosen which is usually set to l in the previous interval.