Page 204 - Introduction to Computational Fluid Dynamics

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6.3 UNSTRUCTURED MESHES
This discretised equation, however, is of little use because the values of variables 11:10 183
at fictitious points P 2 and E 2 and at vertices a and b are not known. We must therefore
relate values at these fictitious points to the values at nodes P and E. This matter is
developed in the next subsection.
6.3.7 Interpolation of Φ at P 2 ,E 2 , a, and b

If it is assumed that the variation between P and P 2 is linear then to first-order
accuracy

·∇ P , (6.77)
P 2 = P + P = P + l PP 2
where
− x 2,P ), (6.78)
l PP 2 = i (x 1,P 2 − x 1,P ) + j (x 2,P 2
and

∂ ∂
∇ P = i + j . (6.79)

∂x 1 P ∂x 2 P
Taking the dot product in Equation 6.77 therefore gives

2

P = (x i,P 2 − x i,P ) ∂ , (6.80)
∂x i
i=1 P
− x i,P mustbeevaluatedintermsofpointswhosecoordinatesareknown.
where x i,P 2
Thus
− x i,c + x i,c − x i,P . (6.81)
x i,P 2 − x i,P = x i,P 2
However, from the construction shown in Figure 6.11,
− x i,e . (6.82)
x i,P 2 − x i,c = x i,P 1
Therefore, Equation 6.81 is further reformulated as
− x i,e + x i,e − x i,P ] + x i,c − x i,e . (6.83)
x i,P 2 − x i,P = [x i,P 1
Now, the equation to the face-normal passing through e is given by

1

) i β + j β 2
i (x 1,e − x 1,P 1 ) + j (x 2,e − x 2,P 1 1 1
n = = , (6.84)
l P 1 e A c
therefore
l P 1 e i
− x i,e =− β (6.85)
1
x i,P 1
A c
and Equation 6.83 can be written as
− x i,P = l xi + d xi , (6.86)
x i,P 2