Revision as of 07:53, 30 April 2014

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.

Description of the water channel

See description of UFR 2-13

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	$\nu ~\operatorname {=} ~0.48$

Measuring Technique

Particle-image velocimetry

See description for UFR 2-13

Laser distance sensor

See description for UFR 2-13

Numerical Simulation Methodology

Computational fluid dynamics (CFD)

See description for UFR 2-13

Computational structural dynamics (CSD)

See description for UFR 2-13

Coupling algorithm

See description for UFR 2-13

Numerical CFD Setup

Preliminary tests were conducted for the geometrically similar FSI-PfS-1a test case to identify the appropriate CFD domain size (see FSI-PfS-1). The outcome was that a consideration of a part of the whole test section combined with periodic boundary conditions on the lateral boundaries is sufficient to get reliable flow results. This configuration is denoted subset case. The geometry of the current FSI-PfS-2a test case is similar to FSI-PfS-1a. Owing to the rear mass in the present case avoiding three-dimensional deformations of the flexible structure, the restriction to the subset case is even stronger justified than for FSI-PfS-1a. Therefore, the subset case is directly applied here: The depth of the computational domain is not the entire test section width $W/D\approx 8.18$ , but a subset of the total width equal to the entire length of the splitter plate $\left(l_{1}+l_{2}\right)/D\approx 2.73$ yielding a quadratic plate. A block-structured grid is generated consisting of 91 blocks involving about 14 million control volumes (CVs). Note that this resolution is chosen based on the experiences made for the FSI-PfS-1a test case (de Nayer et al., 2014). Figure 4 shows the x-y cross-section of the grid. Since only every fourth grid line of the mesh is shown in Fig. 4, the angles between the grid lines and the transitions between the blocks appear to be worse than they are in the full grid. In order to fully resolve the viscous sublayer on the elastic structure, the first cell center is located at a distance of $\Delta y/D\approx 6.8\times 10^{-4}$ from the flexible structure, which leads to average $y^{+}$ values below 0.8, mostly even below 0.5 near the body. The geometrical stretching factors are kept below 1.1. In spanwise direction the grid consists of 72 equidistant cells. The inflow side is rounded in order to use a C-grid. Consequently, the computational domain in front of the cylinder is slightly larger than in the test section depicted in Fig. 1. On the right side a convective outlet boundary condition is applied. On the cylinder and on the flexible structure no-slip walls are defined. The top and the bottom of the domain are relatively far away from the flexible structure and are thus taken into account via slip walls. The lateral sides are assumed to be periodic as mentioned above.