UFR 2-13 Test Case
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.
%-------------------------------------------------------------------- \subsubsection{Deflection of the structure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.9\linewidth,draft=\draftmode]
{FSI-PfS-1a_structure_phase_averaged_timephase}
\caption{\label{fig:swiveling_mode_FSI-PfS-1a}
Experimental structural results: Structure contour
for the reference period.}
\end{figure}
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. % \begin{table}[!htbp]
\centering
\begin{tabular}{|l||c|c|c|c|c|c|c|}
\hline
Case & \multicolumn{7}{|c|}{Results} \\
\hline
& St (Hz) & $f_{FSI}$ (\%) & Error & \mbox{$\left.U_{y}^*\right|_{max}$} & Error (\%) & \mbox{$\left.U_{y}^*\right|_{min}$} & Error \\
\hline
\hline
Sim. (no damping) & 0.1125 & 7.08 & -0.25 & 0.456 & 9.1 & -0.464 & -10.6 \\
\hline
Sim. (damping) & 0.1140 & 7.18 & 1.15 & 0.396 & -5.32 & -0.395 & 6.02 \\
\hline
\hline
Experiments & 0.1128 & 7.10 & - & 0.418 & - & -0.420 & - \\
\hline
\end{tabular}
\caption{\label{tab:comparison_num_exp_damping}
Comparison between numerical results with and without structural
damping and the experiment.}
\end{table}
\begin{figure}[!htbp]
\centering
\begin{minipage}{\linewidth}
\centering
\subfigure[\label{fig:comp_mean_period:a}
Experimental raw signal: Dimensionless displacement.]{
\includegraphics[width=0.45\linewidth,draft=\draftmode]{raw_signal_dispy_exp_real2}
}\hspace{0.049\linewidth}
\subfigure[\label{fig:comp_mean_period:b}Numerical raw signal (without damping):
Dimensionless displacement.]{
\includegraphics[width=0.45\linewidth,draft=\draftmode]{raw_signal_dispy_num}
}
\end{minipage}
\begin{minipage}{\linewidth}
\centering
\subfigure[\label{fig:comp_mean_period:c}Numerical raw signal (with damping):
Dimensionless displacement.]{
\includegraphics[width=0.45\linewidth,draft=\draftmode]{raw_signal_dispy_num_damping}
}\hspace{0.049\linewidth}%
\subfigure[\label{fig:comp_mean_period:d}Comparison of the averaged phase of the
experimental and numerical data.]{
\includegraphics[width=0.45\linewidth,draft=\draftmode]{comp_mean_period_damping}
}
\end{minipage}
\caption{\label{fig:comp_mean_period}Comparison of experimental
and numerical results; raw signals and averaged phases of a
point located at 9 mm distance from the shell extremity.}
\end{figure}
%-------------------------------------------------------------------- \subsubsection{Phase-resolved flow field}
Owing to improved results in case of the structural damping, this case is chosen for the direct comparison with the measurements. The phase-averaging process delivers the phase-resolved flow fields. Four phase-averaged positions, which describe the most important phases of the FSI phenomenon, are chosen for the comparison: Fig.~\ref{fig:comparison_rubber_plate:1} shows the flexible structure reaching a maximal upward deflection at \mbox{$t \approx T /
4$}. Then, it deforms in the opposite direction and moves
downwards. At \mbox{$t \approx T / 2$} the shell is almost in its undeformed state (see Fig.~\ref{fig:comparison_rubber_plate:2}). Afterwards, the flexible structure reaches a maximal downward deformation at \mbox{$t \approx 3
T / 4$} as seen in Fig.~\ref{fig:comparison_rubber_plate:3}. At
\mbox{$t \approx T$} the period cycle is completed and the shell is near its initial state presented in Fig.~\ref{fig:comparison_rubber_plate:4}.
For each of the given phase-averaged positions, the experimental and numerical results (dimensionless streamwise and transverse velocity component) are plotted for comparison. Note that the shell in the experimental figures is shorter than in the simulation plots. Indeed, in the experiment it is not possible to get exactly the whole experimental structure, around 1 mm at the end of the structure is missing. As in Section~\ref{sec:Comparison_of_numerical_results} an additional figure shows the error between the simulation and the experiment for the velocity magnitude.
At \mbox{$t \approx T / 4$} (see Fig.~\ref{fig:comparison_rubber_plate:1}), when the structure is in its maximal upward deflection, the acceleration zone above the structure has reached its maximum. The acceleration area below the plate is growing. Both phenomena are correctly predicted in the simulations. The computed acceleration area above the structure is slightly overestimated. However the local error is mostly under 20 \%.
The separation points at the cylinder are found to be in close agreement between measurements and predictions. Accordingsly, also the location of the shear layers shows a good agreement between simulations and experiments. The shedding phenomenon behind the structure generates a turbulent wake, which is correctly reproduced by the computations. Owing to the phase-averaging procedure, as expected all small-scale structures are averaged out.
At \mbox{$t \approx T / 2$} (see Fig.~\ref{fig:comparison_rubber_plate:2}), the plate is near its undeformed state. The acceleration zone above the structure has shrunk in favor of the area below the plate. Regarding these areas the predictions show a very good agreement with the measurements (marginal local errors). The predicted wake directly behind the structure matches the measured one.
At \mbox{$t \approx 3 T / 4$} (see Fig.~\ref{fig:comparison_rubber_plate:3}), the downward deformation of the plate is maximal, the flow is the symmetrical to the flow observed at \mbox{$t \approx T / 4$} with respect to \mbox{y/D = 0}. Again the acceleration areas around the structure show a very good agreement with the measurements. Once more the wake is correctly predicted in the near-field of the structure.
At \mbox{$t \approx T$} (see Fig.~\ref{fig:comparison_rubber_plate:4}) the flow is symmetrical to the flow observed at \mbox{$t \approx T /
2$} with respect to \mbox{y/D = 0}. The computed acceleration area
above the structure is slightly overestimated, but the local error is under 20 \%. The wake is again correctly predicted except directly after the flexible structure.
For every position the local error is mostly under $20\,\%$. In the error plot the areas with a bigger local error are near the structure and in the shear layers. This can be explained by the fact that near the structure and in the shear layers the gradients of the flow quantities are large. Since the mesh used for the simulation is much finer than the PIV measurement grid, the accuracy of the numerical solution is much higher than the precision of the PIV measurements in these regions. Another reason is that the error expected by the PIV method is more important for low flow velocities. Close to the flexible structure and directly after its tail the flow velocity is small, which at least partially explains the deviations observed between the experimental and numerical results.
In summary, for every position the computed flow is in good agreement with the measured one. The shedding phenomenon behind the cylinder and the positions of the vortices convected downstream are correctly predicted.
Contributed by: Michael Breuer — Helmut-Schmidt Universität Hamburg
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