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# Computational Fluid Dynamics – Convection-Diffusion Problems – Hybrid Differencing Scheme

The hybrid differencing scheme is a combination of ____________ and ___________

a) upwind and downwind schemes

b) downwind and central difference schemes

c) central difference and upwind schemes

d) two types of central difference schemes

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Answer: cExplanation: Hybrid differencing scheme was introduced by Spalding in 1970s. It is the hybrid between upwind and central differencing schemes so that the advantages of both of these schemes is utilized.

The difference scheme to be used in the hybrid system is chosen by evaluating the ____________

a) Local Peclet number

b) Global Peclet number

c) Reynolds number

d) Nusselt number

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Answer: aExplanation: The central differencing scheme works well with low Peclet numbers and the upwind scheme works well for high Peclet numbers. So, the differencing scheme to be used in this method is chosen using the Peclet number.

The hybrid differencing scheme is ____________

a) never bounded

b) bounded unconditionally

c) bounded in the low Peclet number

d) bounded in the high Peclet number

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Answer: bExplanation: The coefficients of the hybrid differencing scheme is always positive. So, it is unconditionally bounded. There is no particular region for the boundedness of the hybrid differencing scheme.

What is the order of accuracy of the hybrid differencing scheme?

a) Fourth-order

b) Third-order

c) Second-order

d) First-order

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Answer: dExplanation: The major disadvantage of the hybrid difference scheme is its low order of accuracy based on the Taylor series truncation term. It is first-order accurate. Yet, it is useful for solving practical flow problems.

Which of these is correct about the hybrid differencing scheme?

a) It is conservative but not transportive

b) It is conservative and transportive

c) It is transportive but not conservative

d) It is neither transportive nor conservative

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Answer: bExplanation: The hybrid differencing scheme is fully conservative. It satisfies the transportiveness condition by using upwind scheme for high Peclet numbers. So, the scheme is conservative and transportive as well.

What is the advantage of the hybrid differencing scheme compared to the QUICK scheme?

a) Transportiveness

b) Accuracy

c) Stability

d) Conservativeness

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Answer: cExplanation: The QUICK scheme also possesses conservativeness and transportiveness. QUICK scheme has a higher order of accuracy. But, it is not stable. Stability is the advantage of the hybrid scheme over this scheme.

The Peclet number is calculated at the ____________

a) control volume

b) cell centres

c) vertices

d) faces

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Answer: dExplanation: The hybrid differencing scheme uses piecewise formulae based on the Peclet number evaluated at the faces of each control volume. Based on this Peclet number, a scheme is chosen.

In which of these ranges is the central differencing scheme used?

a) -2≤Pe≤2

b) -1≤Pe≤1

c) -0.5≤Pe≤0.5

d) -5≤Pe≤5

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Answer: aExplanation: The central differencing scheme is valid until the Peclet number reaches a value of two. So, in the hybrid difference scheme, in the range -2≤Pe≤2, the central differencing scheme is used.

Consider the following stencil.

Which is correct about the hybrid differencing scheme?

(Note: q, φ and F are the net flux per unit area, the flow variable and the convective mass flux per unit area respectively).

a) q_{w}=F_{w} φ_{C} if Pe≥2; q_{w}=F_{w} φ_{C} if Pe≤-2

b) q_{w}=F_{w} φ_{W} if Pe≥2; q_{w}=F_{w} φ_{C} if Pe≤-2

c) q_{w}=F_{w} φ_{W} if Pe≥2; q_{w}=F_{w} φ_{W} if Pe≤-2

d) q_{w}=F_{w} φ_{C} if Pe≥2; q_{w}=F_{w} φ_{W} if Pe≤-2

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Answer: b

Explanation: When the Peclet number goes more than positive two or less than negative two, the upwind scheme is employed. Therefore,

qw={FwϕWifPe≥2FwϕCifPe≤−2

Consider the following stencil.

For steady two-dimensional convection-diffusion problem, if the general discretized equation is a_{P} φ_{P}=a_{W} φ_{W}+a_{E} φ_{E}+a_{S} φ_{S}+a_{N} φ_{N}, what is a_{N} using the hybrid differencing scheme?

(Note: ϕ, a, F and D are the flow variable, coefficients, convective mass flux per unit area and diffusion conductance respectively).

a) max(F_{n},(D_{n}–Fn2),0)

b) max(-F_{n},(D_{n}+Fn2),0)

c) max(-F_{n},(D_{n}–Fn2),0)

d) max(F_{n},(D_{n}+Fn2),0)

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Answer: c

Explanation: The coefficients are decided by the Peclet numbers. Put into a simple form, they are given as the maximum among the three as

a_{N}=max(-F_{n},(D_{n}–Fn2),0).