Computational Fluid Dynamics Multiple Choice Questions on “High Resolution Schemes – TVD Framework”.
1. What is the total variation of a flow variable (Φ) at a particular time step t?
a) TVt=∏iΦi+1-Φi
b) TVt=∫n Φndn
c) TVt=∑iΦ(i+1)Φi
d) TVt=∑iΦ(i+1)Φi
Answer: d
Clarification: Total variation is the summation of variations of the flow variable between two consecutive nodes. This is mathematically given as
TVt=∑iΦi+1-Φi.
2. A numerical method is total variation diminishing if __________
a) the total variation remains constant with increasing time
b) the total variation increases with increasing time
c) the total variation does not increase with increasing time
d) the total variation decreases with increasing time
Answer: c
Clarification: Any numerical method is said to be total variation diminishing if the total variation diminishes with time. This means that the value of total variation should not increase with time. It can either decrease or remain the same.
3. A Total Variation Diminishing (TVD) scheme is always __________
a) continuous
b) monotonic
c) stable
d) bounded
Answer: b
Clarification: A TVD scheme is monotonic and any monotonically preserving scheme is TVD. This means that the value of a local minimum is non-decreasing and the value of a local maximum is non-increasing.
4. Consider the discretized form of an equation given by (frac{partial(rho uphi)}{partial x}=-a(phi_c-phi_u)+b(phi_d-phi_c).) For this numerical scheme to be TVD, what is the condition?
(Note: Φu, Φc and Φd are the flow variables at the far upwind, upwind and downwind schemes).
a) a≥0;b≥0;0≤a+b≤1
b) a≥0;b≤0;0≤a+b≤1
c) a≥0;b≥0;0≤a-b≤1
d) a≥0;≤0;0≤a-b≤1
Answer: a
Clarification: This condition is given by Sweby and Harten. The sufficient condition for a system having the discretized equation -a(Φc-Φu)+b(Φd-Φc) to be TVD is given by
a≥0;b≥0;0≤a+b≤1.
5. Developing a TVD scheme relies upon _________
a) the flux limiter
b) the coefficients
c) the PDE
d) the convection terms
Answer: a
Clarification: A TVD scheme should not be completely upwind or downwind. So, to develop a TVD scheme, an approach is used in which a portion of the anti-diffusive flux is added to the upwind scheme. This flux is limited by a flux limiter function. To find the best flux limiter is the work in developing a TVD scheme.
6. The flux limiter is a function of __________
a) the gradient at that central node
b) the ratio of two consecutive gradients
c) the product of two consecutive gradients
d) the difference between two consecutive gradients
Answer: b
Clarification: Flux limiter prevents the excessive use of flux in regions where oscillations might occur and maximizes the contribution in smooth areas. The flux limiter is denoted by Ψ(r), where r is usually taken as the ratio of two consecutive gradients.
7. The Sweby’s diagram is drawn in __________ plane.
a) (Ψ,r)
b) (Ψ,(tilde{phi_c}))
c) (Ψ,(tilde{phi_f}))
d) (Ψ,(tilde{phi_d}))
Answer: a
Clarification: A Sweby’s diagram is used to represent the TVD. This diagram is drawn with the flux limiter (Ψr) in the y-direction and the variable r in the x-direction. A high-resolution scheme should lie in a particular region of this diagram to be TVD and monotonic.
8. The condition that the flux limiter of a scheme should satisfy to be TVD is __________
a) Ψr=min(0.5r,r) & if r>0; Ψr=0 & if r<0
b) Ψr=min(r,1) & if r>0; Ψr=0 & if r≤0
c) Ψr=min(2r,r) & if r>0; Ψr=0 & if r≤0
d) Ψr=min(2r,2) & if r>0; Ψr=0 & if r<0
Answer: c
Clarification: Similar to the Convection Boundedness Criterion, a flux limiter should satisfy the following criterion to be a TVD. There is a list of conditions which has to be satisfied. Simplifying and combining all of them, we get
(Psi = left{begin{matrix}
min(2r,r) & r>0 \
0 & rleq 0
end{matrix}right}.)
9. What are the flux limiters for upwind and downwind schemes respectively?
a) 0 and 2
b) 0 and 1
c) 0 and ∞
d) 1 and ∞
Answer: a
Clarification: The TVD schemes are developed starting from the upwind scheme. If flux limiters are used, the upwind scheme will change its nature. So, no flux limiter is required in this case and hence flux limiter for an upwind scheme is 0. For downwind scheme, the whole profile in the Sweby’s diagram should be in the line Ψ=2. So, the flux limiter here is 2.
10. Give the relationship between NVF and TVD.
(tilde{phi_c}) → Normalized flow variable at the upwind node
rf → Variable of flux limiter
a) (tilde{phi_c}=frac{1}{1-r_f})
b) (tilde{phi_c}=frac{1}{1+r_f})
c) (tilde{phi_c}=frac{r_f}{1-r_f})
d) (tilde{phi_c}=frac{r_f}{1+r_f})
Answer: d
Clarification: The variable of flux limiter is given by
(r_f=frac{phi_c-phi_u}{phi_d-phi_c})
(r_f=frac{phi_c-phi_u}{phi_d-phi_u+phi_u-phi_c})
(r_f=frac{frac{phi_c-phi_u}{phi_d-phi_u}}{frac{phi_d-phi_u+phi_u-phi_c}{phi_d-phi_u}})
(r_f=frac{frac{phi_c-phi_u}{phi_d-phi_u}}{frac{phi_d-phi_u}{phi_d-phi_u}-frac{phi_c-phi_u}{phi_d-phi_u}})
(r_f=frac{tilde{phi_c}}{1-tilde{phi_c}})
(tilde{phi_c}=frac{r_f}{1+r_f})
This is the relationship between TVD and NVF.