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 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 $l_{2}$ yield 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 have 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.

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

Fig. 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 2.2 m/s. At an inflow velocity of uinflow = 1.385 m/s the displacement are symmetrical, reasonably large and well reproducible. Based on the inflow velocity chosen and the cylinder diameter the Reynolds number of the experiment is equal to Re = 30,470.

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

Although the material shows a strong non-linear elastic behavior for large strains, the application of a linear elastic constitutive law would be favored, to enable the reproduction of this FSI benchmark by a variety of different computational analysis codes without the need of complex material laws. This assumption can be justified by the observation that in the FSI test case, a formulation for large deformations but small strains is applicable. Hence, the identification of the material parameters is done on the basis of the moderate strain expected and the St. Venant-Kirchhoff constitutive law is chosen as the simplest hyper-elastic material model.

The density of the rubber material can be determined to be $\rho _{rubberplate}=1090kg/m^{3}$ for a thickness of the plate h = 0.002 m. This permits the accurate modeling of inertia effects of the structure and thus dynamic test cases can be used to calibrate the material constants. For the chosen material model, there are only two parameters to be defined: the Young's modulus E and the Poisson's ratio ν. In order to avoid complications in the needed element technology due to incompressibility, the material was realized to have a Poisson's ratio which reasonably differs from 0.5. Material tests of the manufacturer and complementary experimental/numerical structure test studies (dynamic and decay test scenarios) indicate that the Young's modulus is E=16 MPa and the Poisson's ratio is ν=0.48 (a detailed description of the structure tests is available in De Nayer et al., 2013).

Structure parameters

Density \rho_{\text{rubber plate}}=1360\,kg/m^3

Young's modulus E=16\,MPa

Poisson's ratio \nu=0.48\,

Test Case Experiments

Provide a brief description of the test facility, together with the measurement techniques used. Indicate what quantities were measured and where.

Discuss the quality of the data and the accuracy of the measurements. It is recognized that the depth and extent of this discussion is dependent upon the amount and quality of information provided in the source documents. However, it should seek to address:

How close is the flow to the target/design flow (e.g. if the flow is supposed to be two-dimensional, how well is this condition satisfied)?

Estimation of the accuracy of measured quantities arising from given measurement technique

Checks on global conservation of physically conserved quantities, momentum, energy etc.

Consistency in the measurements of different quantities.

Discuss how well conditions at boundaries of the flow such as inflow, outflow, walls, far fields, free surface are provided or could be reasonably estimated in order to facilitate CFD calculations

CFD Methods

Provide an overview of the methods used to analyze the test case. This should describe the codes employed together with the turbulence/physical models examined; the models need not be described in detail if good references are available but the treatment used at the walls should explained. Comment on how well the boundary conditions used replicate the conditions in the test rig, e.g. inflow conditions based on measured data at the rig measurement station or reconstructed based on well-defined estimates and assumptions.

Discuss the quality and accuracy of the CFD calculations. As before, it is recognized that the depth and extent of this discussion is dependent upon the amount and quality of information provided in the source documents. However the following points should be addressed: