Page 334 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Simulation of flow-induced vibrations in tube bundles using URANS 303

Table 6.2.2.2 Nondimensional parameters of the simulations
β Γ e u v TI (%) NTLS h Re D Π 0

Rigid 0 0 27.88 0 0.4 0.1–10 0.0023– 0.0768 15,600– 9124.2
0.023 156,000
Flexible 0.57 1.21 27.88 3.0– 0.4 0.05 0.0023 0.0768 19,800– 9124.2
11.25 74,400

The parameters that are used throughout the simulations are listed in
Table 6.2.2.2. The nondimensionalization used is the same one as in Modarres-
Sadeghi et al. (2008).
1=2
ρA f TL 2 L ρA f D o EA s L 2
β ¼ , Γ ¼ , E ¼ , u ¼ vL, h ¼ , Π 0 ¼
ρA f + m EI D o EI D h EI
in which ρ is the fluid density, ρA f the added mass of the fluid per unit length, A f the
2
cross section of the tube in the fluid (¼πD o /4), m the tube’s mass per unit length, T the
external tension applied on the tube, E Young’s modulus, I the area moment of inertia,
v the mean axial flow speed, L the tube’s length, D o the outer diameter of the tube,
D h the hydraulic channel diameter, and A s the cross-sectional area of the tube
2
2
(¼π(D o D i )/4). Three additional parameters are included: the inlet turbulence inten-
sity (TI), a nondimensional characteristic inlet turbulence length scale (NTLS), and the
Reynolds number of the fluid (Re D ).
TLS ρvD o
NTLS ¼ , TI, Re D ¼
L μ
with TLS the turbulent length scale and μ the fluid’s viscosity. Those parameters
were not specified in Modarres-Sadeghi et al. (2008), but they are necessary to char-
acterize the simulations.
In the following graphs, changes of fluid velocity will be studied, which are
reported as changes in nondimensional flow velocity (u). It should however be taken
into account that the Reynolds number is changing concurrently.
6.2.2.3.3 Dynamics in stable regime

In this and the following sections, the results of the coupled simulations are presented.
The resulting vibration is analyzed by fitting modal expressions to it. The frequency
(f ), damping (c), and deflection (w) are reported nondimensionally ( ND ) as a function
of nondimensional flow velocity. Following Modarres-Sadeghi et al. (2008), they are
given by the following formulas:
1=2 1=2

f ND ¼ L 2 m + ρA f f , c ND ¼ L 2 m + ρA f c, w ND ¼ w
L
EI EI
The computed modal characteristics are reported in Fig. 6.2.2.9. Consistent with
the linear angle dependency of viscous forces, the damping increases linearly with
flow velocity. The frequency decreases with increasing flow velocity. There is