Engineering Questions with Answers - Multiple Choice Questions

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

1 - Question

Consider the following stencil. What is Φe as given by the central difference scheme?
(Note: Φ represents the flow variable).
a) ϕE=ϕc(ϕE+ϕc)(xExC)(xexC)
b) ϕE=ϕc+(ϕE+ϕc)(xExC)(xexC)
c) ϕE=ϕc+(ϕEϕc)(xExC)(xexC)
d) ϕE=ϕc(ϕEϕc)(xExC)(xexC)

Explanation: The central difference scheme incorporates values both behind and ahead of the central node in the equation. Therefore,
ϕE=ϕc+(ϕEϕc)(xExC)(xexC).

2 - Question

Consider the following stencil. Assume that the grid is a uniform Cartesian grid. What is φw as given by the central difference scheme?
(Note: Φ represents the flow variable).
a) Φc
b) ϕc+ϕw2
c) ϕcϕw2
d) ϕwϕc2

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.

3 - Question

What is the central differencing scheme similar to?
a) Interpolation profile
b) Linear interpolation profile
c) Weighted average method
d) Geometric mean
Explanation: The central difference scheme matches the linear interpolation profile. The general form (given below) of both are the same. Φ=k0+k1(x-xC).

4 - Question

What is the relationship between ϕcϕwϕEϕw and the Peclet number (Pe) when the grid is uniform?
a) ϕcϕwϕEϕw=12(1Pe2)
b) ϕcϕwϕEϕw=12(1+Pe2)
c) ϕcϕwϕEϕw=12(Pe21)
d) ϕcϕwϕEϕw=(Pe4)

Explanation: For the central difference scheme applied to the convection-diffusion problems
ϕcϕwϕEϕw=aEaW+aE
Applying this to
ϕcϕwϕEϕw=12(1Pe2).

5 - Question

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
Explanation: 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.

6 - Question

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
Explanation: 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.

7 - Question

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
Explanation: 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.

8 - Question

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
Explanation: 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.

9 - Question

The order of accuracy of the central differencing scheme is _____________
a) fourth-order
b) third-order
c) second-order
d) first-order
Explanation: 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.

10 - Question

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