# UFR 2-13 Evaluation

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# Evaluation

The following part is divided into five different sections: in the first numerical phase-resolved results obtained for the two configurations (full and subset case) are compared. Based on this evaluation one case is chosen for a parameter study presented in the second section. The third section presents some unsteady results before in Section 4 a detailed comparison between the measured and the predicted data is carried out. Finally, an overview on the data files is provided.

# Comparison of numerical results for the 2 setups

Two numerical setups are used to run the FSI-PfS-1a simulation: the full case and the subset case. These configurations differ regarding the geometry and the boundary conditions as described in Section Numerical CFD Setup. The subset case represents a simpler model than the full case requiring less CPU-time (one second real-time is predicted in about 170 hours wall-clock with the subset case on 84 processors and in about 310 hours wall-clock with the full case on 142 processors) and thus is worth to be considered. The question, however, is which influence these modeling assumptions have on the numerical results?

## Full case vs. subset case

Both setups are performed with slightly different material characteristics than defined in Section Material Parameters: The Young's modulus is set to E=14 MPa, the thickness of the plate is equal to h = 0.002 m, the solid density is ${\displaystyle \rho _{\text{rubber plate}}}$=1425 kg m${\displaystyle ^{-3}}$ and no structural damping is used. The reason is that this comparison was a preliminary study carried out prior to the final definition of the test case. Because of the similitudes of the values used here and those defined in Section Material Parameters and because of the large CPU-time requested, the comparison of the numerical results is not repeated with the parameters defined in Section Material Parameters.

### Deflection of the structure

At first the predicted deformation of the structure is analyzed. For this purpose Fig. 1 depicts an arbitrarily chosen snapshot of the deformed structure for both cases taken from the quasi-periodic oscillation mode. It is observed that the rubber plate in the full case shows a stronger variation of the y-deformations in z-direction than in the subset case, i.e., a stronger three-dimensional deformation. This observation can be explained as follows: the full setup has a wider structure and the lateral nodes are exposed to less constraints than in the subset case.

Fig. 1: Comparison of the structure deformations in y- and z-direction between the full and subset case

In order to quantify these displacement variations along the z-axis in the full case, three characteristic points on the structure in three parallel planes depicted in Fig. 2(c) are chosen: one plane is set in the middle of the structure, the others are shifted ${\displaystyle \pm 60mm}$ in the spanwise direction. All three points are not located directly on the trailing edge of the rubber plate but at a distance of 9 mm from the trailing edge. This choice is motivated by the planned comparison with the measured data (Section Comparison between numerical and experimental results) and the limitation in the experiment. The laser distance sensor does not allow to follow the trailing edge of the structure and thus points at a certain distance from the tail are chosen. The dimensionless y-displacements ${\displaystyle U_{y}^{*}=U_{y}/D}$ at these three points are monitored and displayed by three different colors as shown in Fig. 2(a). The following observation can be made:

1. The displacements are in phase and more or less the three curves lie upon each other.

2. Local differences between the curves are observed in the extrema.

3. These variations are, however, not constant in time. In other words the displacement in one plane is not always bigger than another. The variations reflect some kind of waves in the structure that move in the spanwise direction.

Comparing those three raw signals with the z-averaged displacements depicted in Fig. 2(b), a maximal difference of 5% regarding the extrema is noticed. Hence the variations are small. The corresponding z-variations of the subset case are even smaller (<0.5%). Therefore, it was decided to continue the analysis by averaging both cases in z-direction.

The next step is to compare the structure deformations obtained with the full and the subset case. Fig. 2(b) shows the dimensionless y-displacements of both cases. Notice that by the averaging procedure in z-direction the 3D-problem is reduced to a 2D-problem. The frequencies are identically predicted in both cases (${\displaystyle f_{{\text{FSI}}_{\text{num}}}=6.96}$ Hz and ${\displaystyle {\text{St}}_{{\text{FSI}}_{\text{num}}}=0.11}$). Minor differences appear in the extrema of the raw signals presented in Fig. 2(b). As before these variations are not constant in time and thus the maximal values are found irregularly for either the full or the subset case. As a consequence the comparison of the phase-averaged displacement signal (see Fig. 2(d)) shows no significant changes between both cases and the coefficient of determination ${\displaystyle R^{2}=1-\sum _{i}\left(U_{{y}_{i}}^{*}-{\hat {U_{{y}_{i}}^{*}}}\right)^{2}/\sum _{i}\left(U_{{y}_{i}}^{*}-{\overline {U_{y}^{*}}}\right)^{2}}$ of the calculated mean phase is close to unity (0.9869 for the full case and 0.9782 for the subset case). ${\displaystyle {\hat {U_{{y}_{i}}^{*}}}}$ denotes the estimated mean value of ${\displaystyle U_{y}^{*}}$ for the point i. ${\displaystyle {\overline {U_{y}^{*}}}}$ is the mean value of all the displacements. The standard deviation for each point of the averaged phase is also computed: the maximum for the full case is 0.055 (dimensionless) and for the subset case 0.065 (dimensionless). These values are small compared to the signal, which is another indication for the reliability of the averaged phase. The subset case predicts structure deformations very similar to the full case. In order to check if the FSI results are quasi identical for the full and the subset case, the phase-resolved flow field has to be additionally taken into account.

Fig. 2 Comparison of the structure deformations in y- and z-direction between the full and subset case

### Phase-resolved flow field

The phase-averaging process described in Section Generation of Phase-resolved Data delivers the phase-resolved flow fields for the full and the subset case. In order to compare them just two representative phase-averaged positions of the FSI problem are chosen to limit this subsection. Figure 3 shows the flow field in the vicinity of the plate during its maximal deformation at t=T/4 and Fig. 4 depicts it close to its undeformed position at t=T, where T denotes the period time of the phase-averaged signal. The figures display the contours of the phase-averaged streamwise and transverse velocity components. Furthermore, the local error of the velocity magnitude defined by the deviation between the absolute values of the velocity vector of both cases normalized by the inflow velocity ${\displaystyle u_{\text{inflow}}}$ is depicted. For both positions the results obtained for the subset and full case are nearly identical. Figures 3(e) and 4(e) underline that the local error of the velocity magnitude between both cases is about zero everywhere except in the region near the structure. For the position t=T/4 (Fig. 3(e)) small local errors are located behind the structure in the vortex shedding region. For the position t=T (Fig. 4(e)) the phase-averaged position of the rubber plate for the subset case differs slightly from the one of the full case. Since the flow field is rapidly changing during the vortex shedding process, this minor deviation in the phase-angle explains the small local errors observed near the structure and in the shear layer.

Fig. 3 Comparison of the results for the full and subset case; phase-averaged data at t=T/4.

Fig. 4 Comparison of the results for the full and subset case; phase-averaged data at t=T.

The comparison of the phase-averaged flow fields shows no significant changes between both cases. The subset case predicts the phase-averaged flow field very similar to the full case. As said before, the subset setup is simpler and less expensive in CPU-time. Therefore, the subset case is very interesting in order to simulate the present test case using LES.

# Sensitivity study for the subset case - Dimensional analysis

In order to better understand the test case a dimensional analysis was carried out. The physical quantities of the present FSI problem are: The dynamic viscosity ${\displaystyle \mu _{f}}$, the fluid density ${\displaystyle \rho _{f}}$, the inlet velocity ${\displaystyle u_{\text{inflow}}}$ for the fluid; the cylinder diameter ${\displaystyle D}$, the dimensions of the rubber plate ${\displaystyle l}$, ${\displaystyle w}$ and ${\displaystyle h}$; the Young's modulus ${\displaystyle E}$, the Poisson's ratio ${\displaystyle \nu }$ and the density of the rubber plate ${\displaystyle \rho _{\text{rubber plate}}}$; To describe the FSI phenomenon the FSI frequency ${\displaystyle f_{\text{FSI}}}$, the displacement extrema ${\displaystyle \left.U_{y}\right|_{max}}$ and ${\displaystyle \left.U_{y}\right|_{min}}$ are chosen. These 13 physical quantities lead to 10 dimensionless parameters: The Reynolds number ${\displaystyle {\text{Re}}=\rho _{f}\,u_{\text{inflow}}\,D/\,\mu _{f}}$ for the fluid; the length ratios ${\displaystyle w/l}$, ${\displaystyle h/l}$, ${\displaystyle D/l}$ for the geometry; ${\displaystyle \nu }$ for the material of the rubber plate; The density ratio ${\displaystyle \rho _{f}/\rho _{\text{rubber plate}}}$, the Cauchy number ${\displaystyle {\text{Cy}}=\rho _{f}\,u_{\text{inflow}}^{2}/E}$ (as defined in~\cite{delangre2002}), the extrema of the dimensionless y-displacements ${\displaystyle \left.U_{y}^{*}\right|_{max}=\left.U_{y}\right|_{max}/D}$ and ${\displaystyle \left.U_{y}^{*}\right|_{min}=\left.U_{y}\right|_{min}/D}$ and the Strouhal number ${\displaystyle {\text{St}}_{\text{FSI}}=f_{\text{FSI}}\,D/u_{\text{inflow}}}$ for the FSI coupling.

In the present experimental investigation the operating conditions for the fluid are well-known. The length and the width of the rubber plate are well defined, too. Therefore, the Reynolds number ${\displaystyle {\text{Re}}}$, the geometrical ratios ${\displaystyle w/l}$ and ${\displaystyle D/l}$ are fixed in the sensitivity study.

• On the contrary, the material is natural rubber and to manufacture a perfectly homogeneous 2 mm plate is not easy. The experimental measurements show that the thickness varies between 0.002 and 0.0022 m. Two values of h are tested: the theoretical value of 0.002 m and the average value 0.0021 m. Consequently, the geometrical ratio ${\displaystyle h/l}$ will be taken into account in the sensitivity study.
• The density of the rubber plate ${\displaystyle \rho _{\text{rubber plate}}}$ is determined by a scale and the volume of the structure. Due to the dependency of this volume on the plate thickness the density determination can be inaccurate. As an additional dimensionless parameter the density ratio ${\displaystyle \rho _{f}\,/\,\rho _{\text{rubber plate}}}$ is a part of the sensitivity study.
• The last parameter of the structure is the Young's modulus, because it has an important influence on the modeling of the material. A large spectrum of values for E and consequently of ${\displaystyle {\text{Cy}}}$ is tested to evaluate this influence.

The dimensional analysis presented here will also be reduced to the six following dimensionless numbers: the density ratio ${\displaystyle \rho _{f}\,/\,\rho _{\text{rubber plate}}}$, the geometrical ratio ${\displaystyle h/l}$, the Cauchy number ${\displaystyle {\text{Cy}}}$, the dimensionless y-displacement extrema ${\displaystyle \left.U_{y}^{*}\right|_{max}}$ and ${\displaystyle \left.U_{y}^{*}\right|_{min}}$ and the Strouhal number ${\displaystyle {\text{St}}_{\text{FSI}}}$.

All the tests were carried out without structural damping and are summarized in Tab. 1. The full case used in the Section above and the experimental results are also added as references. Each simulation was done for a time interval of 4 s physical time and comprises about 27 swiveling periods. Relative errors between the numerical and experimental values are given.

Tab. 1: Parameter study for the subset case of the FSI test case (without structural damping).

The following results and trends can be seen:

• By varying the Young's modulus E between 8 and 16 MPa ${\displaystyle (240{\times }10^{-6}\leq {\text{Cy}}\leq 96{\times }10^{-6})}$ it is possible to control the mode of the FSI phenomenon. Thus E (or the Cauchy number) turns out to be the most crucial material parameter. With E smaller than 9 MPa ${\displaystyle ({\text{Cy}}\geq 213{\times }10^{-6})}$, the system oscillates in the second swiveling mode. With E larger than 12 MPa ${\displaystyle (Cy\leq 160{\times }10^{-6})}$ the structure deflection is dominated by the first bending mode of the structure. For a Young's modulus between 9 and 12 MPa ${\displaystyle (213{\times }10^{-6}\leq Cy\leq 160{\times }10^{-6})}$ a mode transition phase appears in which both swiveling modes are apparent. In this situation the y-displacements of the plate are no longer quasi-periodic and can not be described by a unique frequency.
• Non-negligible variations in the density ${\displaystyle (1320kgm^{-3}\leq \rho _{\text{rubber plate}}\leq 1725kgm^{-3})}$ ${\displaystyle (0.757\geq \rho _{f}/\rho _{\text{rubber plate}}\geq 0.580)}$ for a fixed thickness (h = 0.002 m) and Young's modulus (E = 14 MPa) do not drastically change the results of the frequency and of the mean period extrema. The FSI frequency ${\displaystyle f_{\text{FSI}}}$ slightly decreases with the increase of the density.
• Comparing the results for both thicknesses for the range ${\displaystyle 14{\text{MPa}}\leq E\leq 16{\text{MPa}}}$, it is obvious that a mild variation of the thickness of the plate (0.1 mm, equivalent to 5%) has a non-negligible influence on the extrema of the mean period and no significant influence on the frequency.
• Overall the frequency of the FSI phenomenon ${\displaystyle f_{\text{FSI}}}$ is very well predicted (relative error under 2.22%) for all tested parameters leading to the first swiveling mode.
• Comparing the results for the density ${\displaystyle \rho _{\text{rubber plate}}=1360{\text{ kg m}}^{-3}}$ in the range ${\displaystyle 14{\text{MPa}}\leq E\leq 20{\text{MPa}}}$, we observe that the FSI frequency ${\displaystyle f_{\text{FSI}}}$ slightly increases with the Young's modulus and that the displacement extrema decrease.

In summary, the parameter study shows that the Young's modulus (or the dimensionless number of Cauchy Cy) is the most important parameter: It controls the swiveling mode of the flexible structure. Furthermore, it can be observed that mild modifications of the plate thickness (or of the geometrical ratio h/l) have a certain effect on the predicted FSI phenomenon. Contrarily, this parameter study shows that variations of the density ratio do not have major influence on the predictions. Therefore, errors in the density measurement do not play an important role. With the support of these extensive preliminary numerical investigations we can now compare the final numerical results with the experiment.

In order to comprehend the real structure deformation and the turbulent flow field found in the present test case, experimentally and numerically obtained unsteady results are presented in this section.

A high-speed camera movie of the structure deflection illustrates the deflection of the rubber plate over several periods:

Download movie or view online at http://vimeo.com/59130974


Figure 8 shows experimental raw signals of dimensionless displacements from a point located at a distance of 9 mm from the trailing edge of the rubber plate in the midplane of the test section. Note that only a small extract of the entire data containing several thousand cycles is shown for the sake of visibility. In Figure 8a) the history of the y-displacement ${\displaystyle U_{y}^{*}=U_{y}/D}$ obtained in the experiment is plotted. The signal shows significant variations in the extrema: The maxima of ${\displaystyle U_{y}^{*}}$ (full data set, not the extract depicted in Fig. 8) vary between 0.298 and 0.523 and the minima between -0.234 and -0.542. The standard deviations on the extrema are about ${\displaystyle \pm 0.05~(\pm 12\%)}$ of the mean value of the extrema). Minor variations are observed regarding the period in Figure 8a). Figure 8b) and 8c) show the corresponding experimental phase portrait and phase plane, respectively. The phase portrait has a quasi-ellipsoidal form. The monitoring point trajectory plotted in the phase plane describes an inversed 'C', which is typical for the first swiveling mode. The cycle-to-cycle variations in these plots are small. Therefore, the FSI phenomenon can be characterized as quasi-periodic.

Fig. 8: Experimental raw signals of dimensionless displacements from a point in the midplane of the test section located at a distance of 9 mm from the trailing edge of the rubber plate.

Figure 9 is composed of eight images of the instantaneous flow field (streamwise velocity component) experimentally measured in the x-y plane located in the middle of the rubber plate. These pictures constitute a full period T of the FSI phenomenon arbitrarily chosen. As mentioned before, the rubber plate deforms in the first swiveling mode. Thus, there is only one wave node located at the clamping of the flexible structure. At the beginning of the period (t = 0) the structure is in its undeformed state. Then, it starts to deform upwards and reaches a maximal deflection at t = T / 4. Afterwards, the plate deflects downwards until its maximal deformation at t =3T/4. Finally the plate deforms back to its original undeformed state and the end of the period is reached.

As visible in Fig. 9 the flow is highly turbulent, particularly near the cylinder, the flexible structure and in the wake. The strong shear layers originating from the separated boundary layers are clearly visible. This is the region where for the sub-critical flow the transition to turbulence takes place as visible in Fig. 9. Consequently, the flow in the wake region behind the cylinder is obviously turbulent and shows cycle-to-cycle variations. That means the flow field in the next periods succeeding the interval depicted in Fig. 9 will definitely look slightly different due to the irregular chaotic character of turbulence. Therefore, in order to be able to compare these results an averaging method is needed leading to a statistically averaged representation of the flow field. Since the FSI phenomenon is quasi-periodic the phase-averaging procedure presented above is ideal for this purpose and the results obtained are presented in the next section.

Fig. 9: Experimental unsteady flow field, magnitude of the flow velocity shown by contours (x-y plane located in the middle of the rubber plate).

Prior to this, however, it should be pointed out that very similar figures as depicted in Fig. 9 could also be shown from the numerical predictions based on LES. Exemplary and for the sake of brevity, Fig. 10 displays the streamwise velocity component of the flow field in a x-y-plane solely at t=3T/4. As expected the LES prediction is capable to resolve small-scale flow structures in the wake region and in the shear layers. Furthermore, the figure visualizes the deformed structure showing nearly no variation in spanwise direction.

Fig. 10: Numerical unsteady flow field (movie, x-y plane located in the middle of the rubber plate).

# Comparison between numerical and experimental results

The investigations presented in Section Full case vs. subset case based on slightly different material characteristics than defined in Section 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 Material Parameters (E = 16 MPa, h = 0.0021 m, ${\displaystyle \rho _{\text{rubber plate}}}$ = 1360 kg m${\displaystyle ^{-3}}$) is carried out for the subset case.

Two simulations are considered: one with the structural damping, 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 Generation of Phase-resolved Data. Similar to the numerical comparison presented in Section Full case vs. subset case the displacement of the structure will be first analyzed and then the phase-resolved flow field is considered.

Fig. 1 Experimental structural results: Structure contour for the reference period.

## Structure results

The structure contour of the phased-averaged experimental results for the reference period is depicted in Fig. 1. Obviously, the diagram represents the first swiveling mode of the FSI phenomenon showing only one wave mode at the clamping. Figure 2(a) depicts the experimental dimensionless raw signal obtained at a point located in the midplane at a distance of 9 mm from the trailing edge of the plate (see Fig. 2(c)). Figure 2(b) shows the numerical signal predicted without structural damping and Fig. 2(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. 2(d) with the phase as the abscissa and the dimensionless displacement ${\displaystyle U_{y}^{*}=U_{y}/D}$ 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 R² 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. 2(d) the mean period calculated from the simulation without damping is quasi-antisymmetric with respect to ${\displaystyle U_{y}^{*}=0}$. On the contrary the period derived from the experiment is not exactly antisymmetric with respect to the midpoint of the phase ${\displaystyle \phi =\pi }$: 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. 2(d) shows some differences in the extrema and a summary is presented in Table 1. 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 ${\displaystyle f_{{FSI}_{\text{exp}}}=7.10Hz}$ in the experimental investigations, which corresponds to a Strouhal number St=0.11. In the numerical predictions without damping this frequency is ${\displaystyle f_{{\text{FSI}}_{\text{num}}}^{\text{no damping}}=7.08Hz}$ and with damping ${\displaystyle f_{{\text{FSI}}_{\text{num}}}^{\text{damping}}=7.15Hz}$. This comparison shows an error of ${\displaystyle \epsilon _{f}=-0.25}$ for the results without damping and ${\displaystyle \epsilon _{f}=0.65}$ 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. 1 Comparison between numerical results with and without structural damping and the experiment.

Fig. 2 Comparison of experimental and numerical results; raw signals and averaged phases of a point located at 9 mm distance from the trailing edge of the plate.

## 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. 3 shows the flexible structure reaching a maximal upward deflection at t=T/4. Then, it deforms in the opposite direction and moves downwards. At t=T/2 the plate is almost in its undeformed state (see Fig. 4). Afterwards, the flexible structure reaches a maximal downward deformation at t=3T/4 as seen in Fig. 5. At t=T the period cycle is completed and the rubber plate is near its initial state presented in Fig. 6.

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 rubber plate in the experimental figures is shorter than in the simulation plots, because it was not possible to measure the structure deformation by the laser distance sensor (Section Laser distance sensor) up to the trailing edge. Thus, in the experiment about 1 mm at the end of the structure is missing. As in Section Comparison of numerical results an additional figure shows the error between the simulation and the experiment for the velocity magnitude.

At t=T/4 (see Fig. 3), 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 t=T/2 (see Fig. 4), 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 t=3T/4 (see Fig. 5), the downward deformation of the plate is maximal, the flow is the symmetrical to the flow observed at t=T/4 with respect to 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 t=T (see Fig. 6) the flow is symmetrical to the flow observed at t=T/2 with respect to 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.

Fig. 3 Comparison of experimental and numerical results (subset case with damping, see Table 2, phase-averaged data at t=T/4.

Fig. 4 Comparison of experimental and numerical results (subset case with damping, see Table 2, phase-averaged data at t=T/2.

Fig. 5 Comparison of experimental and numerical results (subset case with damping, see Table 2, phase-averaged data at t=3T/4.

Fig. 6 Comparison of experimental and numerical results (subset case with damping, see Table 2, phase-averaged data at t=T.

## Movie (Comparison Experiment vs. Simulation)

Movie: Comparison of phase-averaged 2D flow measured by PIV (left) and numerical LES computation (right) (streamwise velocity)

    Download movie    or     view online at https://vimeo.com/78057152


## Conclusions

A new FSI benchmark case denoted FSI-PfS-1a was proposed. Detailed experimental investigations in a water tunnel and various simulations relying on a newly developed FSI simulation tool were performed. A quasi-periodic oscillating flexible structure in the first swiveling mode with a corresponding Strouhal number of about ${\displaystyle {\text{St}}_{\text{FSI}}=0.11}$ was found. Post-processing of the extensive data sets delivered the phase-averaged flow field and the structural deformations.

Calculations were carried our for a subset case and for a full case. 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, relative 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 taking 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.

# Data files

As explained in Section Generation of Phase-resolved Data 23 reference positions were calculated with the phase-resolved post-processing algorithm. 23 phase-averaged data are enough to precisely describe the period of the FSI phenomenon. All quantities present in the data sets are dimensionless.

## Experimental data

The experimental data files below contains the phase-resolved flow results obtained with the PIV setup presented before. Each file has 6 columns: The 3 first ones contain the x-, y- and z-positions of each cell center. The 3 next columns contain the x-, y-velocity and the velocity magnitude at the point.

Phase-averaged 2D flow fields:

The experimental data files below contains the phase-resolved structural results obtained with the laser distance sensor presented before. Each file has 3 columns with the x-, y- and z-position of the flexible structure.

Phase-averaged structure:

## Numerical data

The numerical data files contains the phase-resolved results obtained with the LES computation presented before: subset case with Rayleigh damping (as presented in Section Comparison between numerical and experimental results). Each file has 6 columns: The 2 first ones contain the x- and y-positions of each cell center. The 4 next columns contain the x-, y- and z-velocity and the pressure at the point.

Phase-averaged 2D flow fields (The structure is included and thus not provided here separately):

Contributed by: G. De Nayer, A. Kalmbach, M. Breuer — Helmut-Schmidt Universität Hamburg (with support by S. Sicklinger and R. Wüchner from Technische Universität München)