Page 465 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 437

where
-
h = h(L/2Di), (6.105)
L and Di being the length and internal diameter of the curved pipe, while h is the frictional
resistance coefficient for turbulent flow in a curved pipe. The resistance coefficient h for
a curved pipe is somewhat larger than that for a straight pipe (Lo), and according to
Schlichting (1960) is given by
h = h,[l + 0.075 Re0.25(Di/2Ro)0.5], (6.106)

where Re = UDi/v is the Reynolds number, and R, is the radius of curvature of the pipe
segment.
Equation (6.103) corresponds to a single finite element. Such equations for all the
elements are assembled to form the global equation of static equilibrium, which is then
solved numerically.

(b) Analysis of motion around the static equilibrium
Similarly to the analysis of the static equilibrium equations, the variational statement used
for the finite element model of the in-plane perturbations is

(6.107)

where Sylr and S$ are the variations in the dimensionless in-plane displacement pertur-
bations, while A:l (q?, r;) and At(v;, $) represent the left-hand sides of equations (6.71)
and (6.73), respectively; n and <j have the same meaning as in the foregoing.
Proceeding as before, one obtains the matrix differential equation governing the motion
of a typical element; the associated matrices are given in Appendix K. Again, the equations
for all the finite elements are assembled to form the global equation of motion, which is
then converted into an eigenvalue problem that is solved numerically.

6.4 CURVED PIPES WITH SUPPORTED ENDS
Solutions of the global equation of motion yields the system eigenfrequencies, on the basis
of which stability also is decided. For convenience of comparison with other results, two
forms of nondimensionalization are used for the circular frequency Q:

(6.108a)

Similarly, either U, defined in equations (6.46), or

(6.108b)

is uscd for thc dimensionless flow velocity. In the results to be presented, unless otherwise
specified, the values of y, Ba, pa, A and 2 are zero. Dissipation in the material of the
pipe is ignored for simplicity; hence, by the same reasoning as for straight pipes with
supported ends, the system is conservative (Section 3.2). Therefore, the eigenfrequencies