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# Computational Fluid Dynamics – Convection-Diffusion Problems – Central Difference Schemes

Consider the following stencil.

What is Φ_{e} as given by the central difference scheme?

(Note: Φ represents the flow variable).

a) ϕE=ϕc−(ϕE+ϕc)(xE−xC)(xe−xC)

b) ϕE=ϕc+(ϕE+ϕc)(xE−xC)(xe−xC)

c) ϕE=ϕc+(ϕE−ϕc)(xE−xC)(xe−xC)

d) ϕE=ϕc−(ϕE−ϕc)(xE−xC)(xe−xC)

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Answer: c

Explanation: The central difference scheme incorporates values both behind and ahead of the central node in the equation. Therefore,

ϕE=ϕc+(ϕE−ϕc)(xE−xC)(xe−xC).

Consider the following stencil.

Assume that the grid is a uniform Cartesian grid. What is φ_{w} as given by the central difference scheme?

(Note: Φ represents the flow variable).

a) Φ_{c}

b) ϕc+ϕw2

c) ϕc−ϕw2

d) ϕw−ϕc2

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Answer: b

Explanation: The formula for central differencing is

ϕE=ϕw+(ϕc−ϕw)(xC−xW)(xC−xw)

For uniform grids,

ϕE=ϕw+(ϕc−ϕw)2(1)

ϕE=(ϕc+ϕw)2.

What is the central differencing scheme similar to?

a) Interpolation profile

b) Linear interpolation profile

c) Weighted average method

d) Geometric mean

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Answer: bExplanation: The central difference scheme matches the linear interpolation profile. The general form (given below) of both are the same. Φ=k0+k1(x-xC).

What is the relationship between ϕc−ϕwϕE−ϕw and the Peclet number (Pe) when the grid is uniform?

a) ϕc−ϕwϕE−ϕw=12(1−Pe2)

b) ϕc−ϕwϕE−ϕw=12(1+Pe2)

c) ϕc−ϕwϕE−ϕw=12(Pe2−1)

d) ϕc−ϕwϕE−ϕw=(Pe4)

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Answer: a

Explanation: For the central difference scheme applied to the convection-diffusion problems

ϕc−ϕwϕE−ϕw=aEaW+aE

Applying this to

ϕc−ϕwϕE−ϕw=12(1−Pe2).

The central difference approximation goes wrong when _____________

a) Peclet number is negative

b) Peclet number is positive

c) Peclet number is low

d) Peclet number is high

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Answer: dExplanation: When the Peclet number is low in the positive or negative direction, the central differencing scheme is valid. If the Peclet number goes beyond a certain value both in the positive and negative direction, this approximation gives unphysical answers.

Which of these is correct about the central differencing scheme?

a) The importance of upwind and downwind nodes depends on the problem

b) It gives more importance to the downwind nodes

c) It gives equal importance to upwind and downwind nodes

d) It gives more importance to the upwind nodes

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Answer: cExplanation: The central differencing scheme gives equal importance to the upwind and the downwind nodes. The contribution of all the neighbouring nodes is considered for this approximation.

The central differencing scheme becomes inconsistent when the Peclet number _____________

a) is higher than 2

b) is less than 2

c) is higher than 5

d) is less than 5

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Answer: aExplanation: When the Peclet number goes beyond 2, the central difference approximation fails. The discretization process becomes inconsistent here. In this case, an increase in the neighbouring value will lead to a decrease in the value at the central node.

The central difference scheme gives unphysical results when the problem is _____________

a) depends on the boundary conditions

b) equally dominated by diffusion and convection

c) diffusive dominant

d) convective dominant

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Answer: dExplanation: In diffusion problems, both the upwind and the downwind neighbours will be equally important. As this is the case in the central differencing scheme, the scheme fits diffusive dominant problems.

The order of accuracy of the central differencing scheme is _____________

a) fourth-order

b) third-order

c) second-order

d) first-order

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Answer: cExplanation: The central differencing scheme is second-order accurate. This can be proved by using the Taylor series expansion. This is more accurate when compared to the upwind or the downwind schemes.

The central differencing scheme gives good results when _____________

a) the grid is coarse

b) the grid is very fine

c) the grid is Cartesian and uniform

d) the gird is on-Cartesian

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Answer: bExplanation: The central differencing scheme is good when the cell Peclet number is less than 2. For this, the grid should be very fine. So, the central differencing scheme is good to use when the grid is fine.