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# Computational Fluid Dynamics – Convection-Diffusion Problems – FROMM Scheme

Which of these profiles is used by the FROMM scheme?

a) Φ(x)=k_{0}+k_{1} (x-x_{c})+k_{2} (x-x_{c})^{2}

b) Φ(x)=k_{1} (x-x_{c} )+k_{2} (x-x_{c})^{2}

c) Φ(x)=k_{0}+k_{1} (x-x_{c})

d) Φ(x)=k_{1} (x-x_{c})

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Answer: c

Explanation: The FROMM scheme uses a linear interpolation method to approximate the cell face values. So, Φ(x)=k_{0}+k_{1} (x-x_{c}) is the profile used by the FROMM scheme. But, the approach is different from the other profiles using a linear profile.

What is the order of accuracy of the FROMM scheme?

a) First-order

b) Second-order

c) Third-order

d) Fourth-order

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Answer: bExplanation: The first term of the truncation error while implementing the Taylor series in the FROMM scheme is of order two. Therefore, the FROMM scheme is second-order accurate using a linear profile.

FROMM scheme ____________

a) gives weighted importance to the upwind and downwind schemes

b) gives equal importance to upwind and downwind scheme

c) is downwind biased

d) is upwind biased

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Answer: dExplanation: The FROMM scheme is upwind biased. It gives more importance to the upwind nodes than the downwind nodes. IT uses two upwind nodes and one downwind node (totally three nodes).

Which of these is correct about the FROMM scheme?

a) A linear profile is obtained between the immediate upwind and the far downwind nodes

b) A linear profile is obtained between the far upwind and the immediate downwind nodes

c) A linear profile is obtained between the far upwind and the immediate upwind nodes

d) A linear profile is obtained between the far upwind and the far downwind nodes

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Answer: bExplanation: A linear profile is obtained by connecting the values of the far upwind node and the immediate downwind node. A profile with the same slope obtained here is created between the immediate upwind and the current node to get the required value.

Consider the following stencil

What is Φ_{e} according to the QUICK scheme?

a) Φ_{e}=ϕP+xe−xPxE−xW(Φ_{E}-Φ_{W})

b) Φ_{e}=ϕP+xe−xPxE−xW(Φ_{E}+Φ_{W})

c) Φ_{e}=ϕP−xe−xPxE−xW(Φ_{E}-Φ_{W})

d) Φ_{e}=ϕP−xe−xPxE−xW(Φ_{E}+Φ_{W})

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Answer: a

Explanation: To find Φ_{e}, the FROMM scheme first finds Φ_{P} using the profile between Φ_{W} and Φ_{E} given by

Φ_{P}=Φ_{W}+xP−xWxE−xW(Φ_{E}-Φ_{W})

Φ_{W}=Φ_{P}–xP−xWxE−xW (Φ_{E}-Φ_{W})

Now, Φ_{e} is given by,

Φ_{e}=Φ_{W}+xe−xWxE−xW(Φ_{E}-Φ_{W})

Which becomes

Φ_{e}=Φ_{P}–xP−xWxE−xW(Φ_{E}-Φ_{W})+xe−xWxE−xW(Φ_{E}-Φ_{W})

Therefore,

Φ_{e}=Φ_{P}+xe−xPxE−xW(Φ_{E}-Φ_{W}).

Consider the following stencil.

Assume a uniform grid. What is Φ_{e} according to the QUICK scheme?

a) Φ_{P}+23(Φ_{E}-Φ_{W})

b) Φ_{P}+12(Φ_{E}-Φ_{W})

c) Φ_{P}+14(Φ_{E}-Φ_{W})

d) Φ_{P}+34(Φ_{E}-Φ_{W})

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Answer: c

Explanation: In general,

Φ_{e}=Φ_{P}+xe−xPxE−xW (Φ_{E}-Φ_{W})

For a uniform grid,

xe−xPxE−xW=1/4

So,

Φ_{e}=Φ_{P}+14(Φ_{E}-Φ_{W}).

Which of these is correct about the FROMM scheme?

a) Stable and bounded for a variable velocity system

b) Stable but not bounded for a variable velocity system

c) Stable but not bounded for a constant velocity system

d) Stable and bounded for a constant velocity system

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Answer: cExplanation: The FROMM scheme is numerically stable when the velocity field is constant. But, when the velocity is varying, the scheme is unstable. It includes undershoots and overshoots and hence not bounded.

What is the normalized relationship between Φ_{f} and Φ_{c} for the FROMM scheme?

a) ϕf~=ϕc~+14

b) ϕf~=ϕc~−14

c) ϕf~=14−ϕc~

d) ϕf~=14ϕc~

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Answer: a

Explanation: The relationship between Φ_{f} and Φ_{c} is

ϕf=ϕc+14(ϕD−ϕU)

The normalized forms of Φ_{f}, Φ_{c}, Φ_{D} and Φ_{U} are ϕf~,ϕc~, 1 and 0 respectively. Therefore,

ϕf~=ϕc~+14.

For the FROMM scheme, what is the flux limiter ψ(r) equal to?

a) 1-r2

b) 1+r2

c) 1−r2

d) 1+r2

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Answer: d

Explanation: To find the flux limiter,

Φ_{f}=Φ_{c}+12 Ψ(r)(Φ_{D}-Φ_{c} )

For the FROMM scheme,

Φ_{f}=Φ_{c}+14(Φ_{D}-Φ_{U})

Comparing both,

Ψ(r)(Φ_{D}-Φ_{c})=12(Φ_{D}-Φ_{U})

Ψ(r)=12(ϕD−ϕU)(ϕD−ϕc)

Ψ(r)=12(ϕD−ϕc+ϕc−ϕU)(ϕD−ϕc)

But,

(ϕc−ϕU)(ϕD−ϕc)=r

Therefore,

ψ(r)=12(1+r).

Consider the following stencil.

Assume a uniform grid. What is the convective flux at the western face (mw˙ϕw) using the FROMM scheme?

a) (ϕP−ϕE4+ϕW4)max(mw˙,0)−(ϕW−ϕWW4+ϕC4)max(−mw˙,0)

b) (ϕP−ϕW4+ϕE4)max(mw˙,0)−(ϕW−ϕWW4+ϕC4)max(−mw˙,0)

c) (ϕP−ϕE4+ϕW4)max(mw˙,0)−(ϕW−ϕC4+ϕWW4)max(−mw˙,0)

d) (ϕP−ϕW4+ϕE4)max(mw˙,0)−(ϕW−ϕC4+ϕWW4)max(−mw˙,0)

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Answer: a

Explanation: When the flow direction is positive,

ϕw=ϕP−ϕE4+ϕW4

When the flow direction is negative,

ϕw=ϕW−ϕWW4+ϕC4

Therefore,

mw˙ϕw=(ϕP−ϕE4+ϕW4)max(mw˙,0)−(ϕW−ϕWW4+ϕC4)max(−mw˙,0)