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.

Generation of Phase-resolved Data

Each flow characteristic of a quasi-periodic FSI problem can be written as a function $f=\bar{f}+\tilde{f}+f'$, where $\bar{f}$ describes the global mean part, $\tilde{f}$ the quasi-periodic part and $f'$ a random turbulence-related part~\citep{reynolds72,cantwell83}. This splitting can also be written in the form $f = \left<f\right> + f'$, where $\left<f\right>$ is the phase-averaged part, i.e., the mean at constant phase. In order to be able to compare numerical results and experimental measurements, the irregular turbulent part $f'$ has to be averaged out. This measure is indispensable owing to the nature of turbulence which solely allows reasonable comparisons based on statistical data. Therefore, the present data are phase-averaged to obtain only the phase-resolved contribution $\left<f\right>$ of the problem, which can be seen as a representative and thus characteristic signal of the underlying FSI phenomenon.

The procedure to generate phase-resolved results is the same for the experiments and the simulations and is also similar to the one presented in \cite{gomes2006}. The technique can be split up into three steps:

Reduce the 3D-problem to a 2D-problem - Due to the facts that in the present benchmark the structure deformation in spanwise direction is negligible and that the delivered experimental PIV-results are solely available in one x-y-plane, first the 3D-problem is reduced to a 2D-problem. For this purpose the flow field and the shell position in the CFD predictions are averaged in spanwise direction.

Determine n reference positions for the FSI Problem - A representative signal of the FSI phenomenon is the history of the y-displacements of the shell extremity. Therefore, it is used as the trigger signal for this averaging method leading to phase-resolved data. Note that the averaged period of this signal is denoted $T$. At first, it has to be defined in how many sub-parts the main period of the FSI problem will be divided and so, how many reference positions have to be calculated (for example in the present work $n = 23$). Then, the margins of each period of the y-displacement curve are determined. In order to do that the intersections between the y-displacement curve and the zero crossings ($U_y = 0$) are looked for and used to limit the periods. Third, each period $T_i$ found is divided into $n$ equidistant sub-parts denoted $j$.

Sort and average the data corresponding to each reference Position - The sub-part $j$ of the period $T_i$ corresponds to the sub-part $j$ of the period $T_{i+1}$ and so on. Each data set found in a sub-part $j$ will be averaged with the other sets found in the sub-parts $j$ of all other periods (see Fig.~\ref{fig:num_phase-resolved_method2_step2}). Finally, data sets of $n$ phase-averaged positions for the representative reference period are achieved.

The simulation data containing structure positions, pressure and velocity fields, are generated every 150 time steps. According to the frequency observed for the structure and the time-step size chosen about 50 data sets are obtained per swiveling period. With respect to the time interval predicted and the number of subparts chosen, the data for each subpart are averaged from about 50 data sets. A post-processing program is implemented based on the method described above. It does not require any special treatment and thus the aforementioned method to get the phase-resolved results is straightforward.

The current experimental setup consists of the multiple-point triangulation sensor described in Section~\ref{sec:Laser_Sensor} and the synchronizer of the PIV system. Each measurement pulse of the PIV system is detected in the data acquisition of the laser distance sensor, which measures the structure deflection continuously with 800 profiles per second. With this setup, contrary to \cite{gomes2006}, the periods are not detected during the acquisition but in the post-processing phase. After the run a specific software based on the described method mentioned above computes the reference structure motion period and sorts the PIV data to get the phase-averaged results.

Test Case Study

Brief Description of the Study Test Case

This should:

Convey the general set up of the test-case configuration( e.g. airflow over a bump on the floor of a wind tunnel)
Describe the geometry, illustrated with a sketch
Specify the flow parameters which define the flow regime (e.g. Reynolds number, Rayleigh number, angle of incidence etc.)
Give the principal measured quantities (i.e. assessment quantities) by which the success or failure of CFD calculations are to be judged. These quantities should include global parameters but also the distributions of mean and turbulence quantities.

The description can be kept fairly short if a link can be made to a data base where details are given. For other cases a more detailed, fully self-contained description should be provided.

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: