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# Computational Fluid Dynamics – Convection-Diffusion Problems – Upwind and Downwind Schemes

The upwind scheme is suitable for _____________

a) convection term

b) diffusion term

c) both convection and diffusion terms

d) either convection or diffusion term

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Answer: aExplanation: The upwind scheme is not suitable for non-directional phenomena. The diffusion scheme is a non-directional phenomenon. The convection scheme is a directional phenomenon. So, the scheme is suitable for the convection term.

The upwind scheme is dependent on the _____________

a) Convection term

b) Peclet number

c) Flow direction

d) Gradient

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Answer: cExplanation: The upwind scheme reflects the physics of advection. The cell-face value is dependent on the upwind nodal value. So, we can say it is dependent on the flow direction and it is suitable for directional flows.

Consider the stencil.

Give the advection flux at face e (me˙ϕE) using the upwind scheme.

(Note: m˙ and Φ are the mass flow rate and the flow variable respectively).

a) me˙ϕE=max(me˙,0)ϕC−max(−me˙,0)ϕE

b) me˙ϕE=min(me˙,0)ϕC−max(−me˙,0)ϕE

c) me˙ϕE=max(me˙,0)ϕC−min(−me˙,0)ϕE

d) me˙ϕE=min(me˙,0)ϕC−min(−me˙,0)ϕE

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Answer: a

Explanation: According to the upwind scheme,

ϕE={ϕCϕEifme˙>0ifme˙<0

Where, me˙>0 means that the flow is from C to e and me˙<0 means that the flow direction is opposite. Therefore,

me˙ϕE=max(me˙,0)ϕC−max(−me˙,0)ϕE.

The neighbour coefficients yielded by the upwind scheme for convection is _____________

a) zero

b) cannot predict

c) negative

d) positive

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Answer: cExplanation: The upwind scheme leads to negative neighbouring coefficients. If continuity is ensured, the diagonal coefficients (coefficients of the central nodes) are the addition of the neighbouring coefficients.

Consider the following stencil.

Give the relationship between ϕC−ϕWϕE−ϕW and Peclet number (Pe) for the upwind scheme while used for the convection term.

(Note: Φ is the flow variable).

a) ϕC−ϕWϕE−ϕW=2−max(Pe,0)4+∣∣Pe∣∣

b) ϕC−ϕWϕE−ϕW=2−max(−Pe,0)4+∣∣Pe∣∣

c) ϕC−ϕWϕE−ϕW=2+max(Pe,0)4+∣∣Pe∣∣

d) ϕC−ϕWϕE−ϕW=2+max(−Pe,0)4+∣∣Pe∣∣

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Answer: d

Explanation: For the upwind scheme,

Φ_{C}-Φ_{W} = 2+max(-Pe,0) and Φ_{E}-Φ_{C} = 2+max(Pe,0)

Therefore,

ϕC−ϕWϕE−ϕC+ϕC−ϕW=2+max(−Pe,0)2+max(Pe,0)+2+max(−Pe,0)

ϕC−ϕWϕE−ϕW=2+max(−Pe,0)4+∣∣Pe∣∣.

The order of accuracy of the upwind scheme is _____________

a) first-order

b) second-order

c) third-order

d) fourth-order

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Answer: aExplanation: The upwind scheme is first-order accurate. This is why, even though the fashion of the upwind scheme matches with that of the exact solution, it varies much. The downwind scheme is also first-accurate.

Consider the stencil.

Give the advection flux at face w (mw˙ϕW) using the downwind scheme.

(Note: m˙ and Φ are the mass flow rate and the flow variable respectively).

a) mw˙ϕW=−max(mw˙,0)ϕC+max(mw˙,0)ϕW

b) mw˙ϕW=max(mw˙,0)ϕC+max(mw˙,0)ϕW

c) mw˙ϕW=−max(−mw˙,0)ϕC+max(mw˙,0)ϕW

d) mw˙ϕW=max(−mw˙,0)ϕC+max(mw˙,0)ϕW

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Answer: c

Explanation: According to the downwind scheme,

ϕE={ϕCϕEifme˙>0ifme˙<0

Where, me˙>0 means that the flow is from C to e and me˙<0 means that the flow direction is opposite. Therefore,

mw˙ϕW=−max(−mw˙,0)ϕC+max(mw˙,0)ϕW.

The advantage of the upwind scheme is over the central-difference scheme is _____________

a) accuracy

b) stability

c) high convergence rate

d) consistency

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Answer: aExplanation: The upwind scheme is less accurate than the central difference schemes. But the central difference schemes are oscillatory. They do not give answers which are physically correct. This is the advantage of the upwind scheme over the central-difference scheme.

The upwind scheme is _____________

a) conservative but wiggles

b) bounded and conservative

c) bounded but not conservative

d) neither conservative nor bounded

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Answer: bExplanation: The upwind scheme does not produce results which wiggle. So, it is bounded. It uses consistent expressions to calculate fluxes through cells. Therefore, it is sure that the formulation is conservative.

When the flow is not aligned with the grid lines, the diffusion produced by the upwind scheme is ____________

a) false advection

b) false convection

c) anti-diffusion

d) false diffusion

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Answer: cExplanation: A major drawback of the scheme is that it does not produce correct results when the flow is not aligned with the grid lines. The error has a diffusion-like appearance and is referred to as false diffusion.