1
Let us introduce a and b two vectors. The dyadic product of these two
vectors produces a second rank symmetric tensor S :
S = a⊗b
(0.1)
S has the following property: for all vector u:
S · u = (a ⊗ b)· u
= a (b · u)
(0.2)
This means that S maps the vector u onto a vector parallel to a with magnitude |a|(b · u).
Weller’s expression for the interpolated velocity vector reads:
! −1
!
1
1
u= ∑
S f ⊗S f
Sf φf
· ∑
f Sf
f Sf
(0.3)
Where u it the velocity vector, S f is the area vector and φ f the volumetric flow
rate through the face S f . φ f represents S f · u f .
Let us introduce the symmetric tensor D :
D=∑
f
1
S f ⊗S f
Sf
Multiplying on either side of (0.3) by D leads to:
!
1
1
∑ S S f ⊗S f ·u = ∑ S S f φ f
f
f
f
f
(0.4)
(0.5)
Introducing the unit normal vector to the face S f : n f , applying the distributivity of the inner product of a Tensor and vector to the left hand side of (0.5) and
replacing φ f by S f · u f on the right hand side of (0.5), leads to the following
expression:
(0.6)
n
⊗
S
∑ | f {z f } ·u = ∑ n f S f · u f
f
f
Df
Introducing the property of the dyadic product (0.2) into the left hand side of
the above equation, leads to:
(0.7)
∑n f S f ·u = ∑n f S f ·u f
f
f
The dyad D f = n f ⊗ S f maps the
velocity vector u onto a vector parallel
to n f with a magnitude n f S f · u ≈ φ f , i.e. onto a vector ≈ φ f n f . In his reconstruction from the volumetric flow rates, Weller produces an approximate
velocity vector u assuming that the equation (0.7) is true.