sci.math.numanalysis
SUNA39, Reduced quad for internal flow
======================================
For the problem at hand, incompressible and irrotational flow at the shell side
of tube bundles (SUNA38), we will restrict ourselves to rectangular cylindrical
symmetric elements. While considering the incompressibility condition, fluxes
through cylinder surfaces are important. Accompanying surfaces are defined as
follows:
_________ < r2,z2 If the cylinder axis is on the left, then:
  O(3) 
a  ^  O(1) = pi.(r2.r2  r1.r1)
x O(4)   O(2) O(2) = 2.pi.r2.(z2  z1)
i ^   ^ O(3) = O(1)
s _________ O(4) = 2.pi.r1.(z2  z1)
 r1,z1 > O(1)
^
Conservation of mass now gives rise to the following equation:
 O(1).v1 + O(2).u2 + O(3).v3  O(4).u4 = 0
Here the numbering of area's O and velocities (u,v) have been taken equal.
Or, apart from the constants (pi):
2.(r2.u2  r1.u4).(z2  z1) + (r2.r2  r1.r1).(v3  v1) = 0
Divide the whole expression by 2.(r2  r1).(z2  z1) :
r2.u2  r1.u4 v3  v1
 + (r2 + r1)/2 .  = 0
r2  r1 z2  z1
In the limit for r2  r1 > 0 and z2  z1 > 0 this is consistent with the
partial differential equation for incompressibility in cylinder coordinates:
d(r.u)/dr + r.dv/dz = 0
Much easier to handle is the irrotational condition. In fact, it is completely
equivalent with the condition in flat 2D cartesian coordinates:
(r2  r1).u1 + (z2  z1).v2 + (r1  r2).u3 + (z1  z2).v4 = 0
In the limit for r2  r1 > 0 and z2  z1 > 0 this is consistent with the
partial differential equation for irrotationality (?) in cylinder coordinates:
du/dz  dv/dr = 0
In the above, fluxes have been defined as if the velocities are "midside" ones.
But in the first place we are stucked with the Reduced Quadrilateral, which has
its velocities defined at the vertices. The relationship between the former and
the latter kind of velocities is simply found by taking mean values. This means
that, in the above discretizations, substitutions like the following have to be
made:
f1 := (f1 + f2)/2
midside vertex
Such a procedure results in schemes like below:
A(8) A(6) A(7) A(5) v v
________ ________ 4 ________ 3
A(7)   A(5) A(8)   A(6) u   u
_ _ _ _ _ _
     
A(1) ____ ____ A(3) A(2) ____ ____ A(4) u ____ ____ u
  1  2
A(2) A(4) A(1) A(3) v v
Incompressible Irrotational Vertex velocities,
each distributed
A(2*k1) = coefficient of Approximation for u(k) over two halves of
A( 2*k ) = coefficient of Approximation for v(k) the quad's sides.
Remember that the reduced quadrilateral is a "wrong" element. Despite of this,
alas (:), it frequently gives _exellent_ results.
To be continued ...

* Han de Bruijn; Applications&Graphics  "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354  would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands.  Mathematics" (HdB). * for
* Email: Han.deBruijn@RC.TUDelft.NL  Fax: +31 15 78 37 87 * Blood