Fluid-structure interaction II

Flows Around Bodies

Underlying Flow Regime 2-14

Test Case Study

Description of the geometrical model and the test section

FSI-PfS-2a consists of a flexible rubber structure with an attached steel weight clamped behind a fixed rigid non-rotating cylinder installed in a water channel (see Fig. 1). The experiments use the same set-up as used in FSI-PfS-1a. As a consequence all channel related parameters suchs as test section geometries, blocking ratio and the working conditions including the inflow profile remain the same as described in FSI-PfS-1a. The difference in the investigations are the changed structure definitions of FSI-PfS-2a. The deformable structure used in the experiment behind the cylinder has a slightly shorter length with $l_{1}\operatorname {=} 0.050\,m~(l_{1}/D\approx 2.27)$ . The attached steel weight has a length of $l_{2}\operatorname {=} 0.010\,m~(l_{2}/D\approx 0.46)$ and the width w that the addition of $l_{1}$ and $l_{2}$ yields the length $l\operatorname {=} 0.060\,m$ identical to the plate of FSI-PfS-1a. The whole structure including the rigid cylinder, the flexible plate and the steel weight has a width $w\operatorname {=} 0.177\,m~(w/D\approx 8.05)$ . Again the small gap of about $1.5\times 10^{-3}\,m$ between the side of the structure and both lateral channel walls is present. In contrast to the rubber material applied in FSI-PfS-1a the rubber used in FSI-PfS-2a has an almost constant thickness $h\operatorname {=} 0.002\,m~(h/D\approx 0.09)$ . All parameters of the geometrical configuration of the FSI-PfS-2a benchmark are summarized as follows:

Fig. 1: Geometrical configuration of the FSI-PfS-2a Benchmark.

Flow parameters

Several preliminary tests were performed to find the best working conditions in terms of maximum structure displacement, good reproducibility and measurable structure frequencies within the turbulent flow regime.

Figure 3 depicts the measured extrema of the structure displacement versus the inlet velocity and Fig. 4 gives the frequency and Strouhal number as a function of the inlet velocity. These data were achieved by measurements with the laser distance sensor explained in Section Laser distance sensor. The entire diagrams are the result of a measurement campaign in which the inflow velocity was consecutively increased from 0 to 2.4 m/s. At an inflow velocity of $u_{\text{inflow}}=1.385m/s$ (same inflow velocity as in FSI-PfS-1a) the displacements are symmetrical, reasonably large and well reproducible. Based on the inflow velocity chosen and the cylinder diameter, the Reynolds number is equal to Re = 30,470.

Fig. 3: Experimental displacements of the structure extremity versus the inlet velocity.

Fig. 4: Experimental measurements of the frequency and the corresponding Strouhal number of the FSI phenomenon versus the inlet velocity.

Regarding the flow around the front cylinder, at this inflow velocity the flow is again in the sub-critical regime. That means the boundary layers are still laminar, but transition to turbulence takes place in the free shear layers evolving from the separated boundary layers behind the apex of the cylinder.

Material Parameters

The density of the rubber material is found to be $\rho _{rubberplate}=1090~kg/m^{3}$ for a thickness of the plate h = 0.002 m. This permits the accurate modeling of inertia effects of the structure and thus static and dynamic test cases can be used to calibrate the material constants (see Kalmbach and Breuer, 2013). Again the St. Venant-Kirchhoff constitutive law is chosen as the simplest hyper-elastic material model. Similar to FSI-PfS-1a, there are only two parameters to be defined: the Young's modulus E and the Poisson's ratio ν. Complementary experimental/numerical structure test studies (static, dynamic and decay test scenarios) indicate that the Young's modulus is E=3.15 MPa and the Poisson's ratio is ν=0.48 (a detailed description of the structure tests is available in Kalmbach and Breuer, 2013). The density of the steel weight is given by $\rho _{steel}=7850~kg/m^{3}$ .

Structure Parameters:

Density	$\rho _{rubberplate}~\operatorname {=} ~1090~kg/m^{3}$
Young's modulus	$E~\operatorname {=} ~3.15~MPa$
Poisson's ratio	$\nu ~\operatorname {=} ~0.48$