A fluid-structure interaction benchmark in turbulent flow (FSI-PfS-1a)

Comparison between numerical and experimental results

The investigations presented in Section~\ref{sec:Full_case_vs_Subset_case} based on slightly different material characteristics than defined in Section~\ref{sec:Material_parameters} have shown that the subset case permits a gain in CPU-time but nevertheless nearly identical results as the full case. Therefore, the numerical computation with the structural parameters defined in Section~\ref{sec:Material_parameters} ($E$ = 16 MPa, $h$ = 0.0021~m, $\rho_\text{rubber plate}$ = 1360 kg m$^{-3}$) is carried out for the subset case.

Two simulations are considered: one with the structural damping defined in Section~\ref{sec:validation_structure_model}, the other one without damping. These results are compared with the experimental data to check their accuracy. In order to quantitatively compare the experimental and numerical data, both are phase-averaged as explained in Section~\ref{sec:Generation_of_phase-resolved_data}. Similar to the numerical comparison presented in Section~\ref{sec:Full_case_vs_Subset_case} the displacement of the structure will be first analyzed and then the phase-resolved flow field is considered.

Fig. xx Experimental structural results: Structure contour for the reference period.

The structure contour of the phased-averaged experimental results for the reference period is depicted in Fig.~\ref{fig:swiveling_mode_FSI-PfS-1a}. Obviously, the diagram represents the first swiveling mode of the FSI phenomenon showing only one wave mode at the clamping. Figure~\ref{fig:comp_mean_period:a} depicts the experimental dimensionless raw signal obtained at a point located in the midplane at a distance of 9~mm from the shell extremity (see Fig.~\ref{fig:comp_full_subset:c}). Figure~\ref{fig:comp_mean_period:b} shows the numerical signal predicted without structural damping and Fig.~\ref{fig:comp_mean_period:c} the one computed with damping. Applying the phase-averaging process the mean phase of the FSI phenomenon for the experiment and for the simulations is generated. The outcome is presented in Fig.~\ref{fig:comp_mean_period:d} with the phase as the abscissa and the dimensionless displacement \mbox{$U_y^* = U_y / D$} as the ordinate. The amplitudes of the experimental signal varies more than in the predictions. Therefore, the maximal standard deviation of each point of the averaged phase is for the experiment bigger (0.083) than for the simulation (0.072 with and without damping). In order to check the reliability of the computed mean phase the coefficient of determination $R^2$ is computed: it is smaller for the mean experimental phase (0.9640) than for the mean simulation ones (0.9770 without damping and 0.9664 with damping). However, the values are close to unity, which is an indication that the averaged phases are representative for the signals. In Fig.~\ref{fig:comp_mean_period:d} the mean period calculated from the simulation without damping is quasi-antisymmetric with respect to \mbox{$U_y^* = 0$}. On the contrary the period derived from the experiment is not exactly antisymmetric with respect to the midpoint of the phase \mbox{$\phi = \pi$}: the cross-over is not at the midpoint of the phase but slightly deviates to the right. However, the absolute values of the minimum and maximum are nearly identical. As for the experimental phase, the simulation with damping generates a phase signal, which is not completely antisymmetric. In the experiment this weak asymmetry can be attributed to minor asymmetries in the setup or in the rubber material. The comparison in Fig.~\ref{fig:comp_mean_period:d} shows some differences in the extrema and a summary is presented in Table~\ref{tab:comparison_num_exp_damping}. Without structural damping the simulations produce extrema which are too large by about 10~\%. With structural damping the extrema are smaller, even smaller than in the experiment by about 6~\%. Thus, the structural damping also has a significant influence on the FSI predictions and can not be overlooked.

The frequency of the FSI phenomenon, i.e., the frequency of the y-displacements, is about \mbox{$f_{{FSI}_{\text{exp}}}=7.10\,$Hz} in the experimental investigations, which corresponds to a Strouhal number \mbox{$\text{St} \approx 0.11$}. In the numerical predictions without damping this frequency is \mbox{$f_{{FSI}_{\text{num}}}^{\text{no damping}}=7.08\,$Hz} and with damping \mbox{$f_{{FSI}_{\text{num}}}^{\text{damping}}=7.18\,$Hz}. This comparison shows an error of \mbox{$\epsilon_{f}=-0.25\,\%$} for the results without damping and \mbox{$\epsilon_{f}=1.15\,\%$} for the cases with damping. Nevertheless, the FSI frequency is also very well predicted in both cases. One can notice that the frequency of the coupled system slightly increases due to the structural damping.

Tab. xx Comparison between numerical results with and without structural damping and the experiment.

Fig. xx Comparison of experimental and numerical results; raw signals and averaged phases of a point located at 9 mm distance from the shell extremity.

Conclusion

A new FSI benchmark case denoted FSI-PfS-1a is proposed. The definition of the test case is driven by the idea to setup a well-defined but nevertheless challenging benchmark for fluid-structure interaction in the turbulent flow regime. A rigid front cylinder and a flexible membranous rubber tail attached to the backside of the cylinder form the structure which is exposed to a uniform inflow at a low turbulence level. Thus three critical issues of precursor benchmarks are circumvented, i.e., an additional degree of freedom of a rotating front cylinder, an extremely thin flexible structure and an additional weight at the end of the membranous structure. The investigations comprise three parts.

First, two dynamic structural tests were carried out experimentally and numerically in order to evaluate an appropriate material model and to check and evaluate the material parameters of the rubber (Young's modulus, damping). This preliminary work has shown that the St.\ Venant-Kirchhoff material model is sufficient to describe the deflection of the flexible structure.

Second, detailed experimental investigations in a water tunnel using optical measurement techniques for both, the fluid flow and the structure deformation, were carried out. A quasi-periodic oscillating flexible structure in the first swiveling mode with a corresponding Strouhal number of about St = 0.11 is found. A post-processing of the extensive data sets delivered the phase-averaged flow field and the structural deformations.

Third, various simulations relying on a newly developed FSI simulation tool combining a partitioned solution strategy with an eddy-resolving scheme (LES) were performed. A subset case and full case are taken into account. Owing to the wider structure and less constraints of the lateral nodes the deformations in the spanwise direction were found to be larger in the full case reflecting some kind of mild waves in the structure. Nevertheless, in relation to the deformation of the structure in cross-flow direction the spanwise deflections are insignificant, especially for the comparison of the phase-averaged signals.

A study on three parameters for the subset case without structural damping yields that the Young's modulus has a very important influence on the system. It controls in which swiveling mode the flexible structure oscillates. The thickness of the plate h plays a role in the results, too, but not so significant as the Young's modulus. The parameter with the least effect on the FSI simulations is the density of the rubber plate: large variations of the density do not have major influence on the predictions.

As usual for rubber material, a certain level of structural damping has to be expected. To model this phenomenon in a simple and straightforward way, classical Rayleigh damping is used and adjusted based on one of the pure structural test presented. The FSI simulations with and without structural damping are compared with the experiment. It turns out that the structural damping can not be ignored in the present case and significantly affects the deflection of the structure. Without taken the damping into account the structural deflections are overpredicted. Including the simple damping model improves the results. The eddy-resolving FSI simulations are found to be in close agreement with the experiment for every position of the flexible structure. Solely the amplitudes of the deflections are slightly underpredicted with damping. Nevertheless, the shedding phenomenon behind the cylinder/structure and the positions of the vortices convected downstream are correctly predicted. Furthermore, the FSI frequency found in the simulations matches particularly well the measured one.

Evaluation

Comparison of CFD Calculations with Experiments

Discuss how well the CFD calculations of the assessment quantities compare with experiment and with one another. Present some key comparisons in the form of tables or graphical plots and, where possible, provide hyperlinks to the appropriate results database. Results with different turbulence models covering as wide a range as possible should be included in the discussion. However, if too many different calculation results are available (e.g. from workshops) do not present all the comparisons here. A selection should be made showing results only for the most typical and practically important models. Comprehensive comparisons can be made available via a link to the associated databases. Finally, draw conclusions on the ability of the models used to simulate the test case flow.

Contributed by: Michael Breuer — Helmut-Schmidt Universität Hamburg