Evaluation

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:

Figure 9 shows the experimental raw signal of dimensionless y-displacements from a point located at a distance of ~2 mm (x/D=3.13) from the trailing edge of the steel weight in the midplane of the test section (see Fig. 8). Note that only a small extract of the entire data containing several thousand cycles is shown for the sake of visibility. The signal shows only slight variations in the extrema: The maxima of Uy/D (full data set, not the extract depicted in Fig. 9) vary between 0.573 and 0.727 and the minima between -0.584 and -0.701. The standard deviations on the extrema are about ${\displaystyle \pm 0.03~(\pm 5\%)}$ of the mean value of the extrema). Minor variations are observed regarding the period in Figure 9. The monitoring point trajectory plotted in the phase plane describes a '8', which is typical for the second swiveling mode for this kind of configurations. The cycle-to-cycle variations in these plots are small. Therefore, the FSI phenomenon can be characterized as quasi-periodic.

Figure 10 shows the numerical raw signal of dimensionless y-displacements for the same monitoring point. The phase plane is also plotted. The numerical raw results are very similar to those obtained in the experiment.

Fig. 8: Position of the monitoring point (in red).

Fig. 9: Dimensionless experimental raw y-displacements and phase plane at the monitoring point sketched in Fig 8.

Fig. 10: Dimensionless numerical raw y-displacements and phase plane at the monitoring point sketched in Fig 8.

Figure 11 is composed of four 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 second swiveling mode. Thus, there are two wave nodes: one is located at the clamping of the flexible structure as in the first swiveling mode; the second one is found close to the bond of the rubber and the steel weight. At the beginning of the period (t = 0) the structure is in its undeformed state (not shown here). 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. 11 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. 11. 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. 11 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 in Generation of Phase-resolved Data is ideal for this purpose and the results obtained are presented in the next section.

Fig. 11: Experimental unsteady flow field, magnitude of the flow velocity shown by contours (x-y plane located in the middle of the rubber plate).

The rubber plate mounted behind the cylinder acts as a splitter plate (Anderson and Szewczyk, 1997). Nevertheless, quasi-periodic vortex shedding occurs. The shed vortices visualized by iso-surfaces of the velocity magnitude with a value of u = 1.1 m/s (${\displaystyle u/u_{\text{inflow}}=0.79}$) move downstream and start to interact with the flexible structure leading to an oscillating quasi-periodic motion. The extra steel weight at the end of the tail additionally supports the deflection by the higher inertia of the swiveling system. Fig. 12 shows two different views of a single coupled measurement (without phase-averaging) which allows a detailed analysis of the flow behavior in the wake of the structure. In these snapshots two vortex rolls with a distance of about Δx/D = 2.5 and Δy/D = 1 to each other are visible. These vortex rolls illustrate several three-dimensional flow structures, for example the contraction in the middle of the upper vortex roll. Additionally, the contours of the instantaneous spanwise velocity component are mapped onto the iso-surfaces. As obvious from these figures the wake behind the nominally two-dimensional structure is strongly three-dimensional including vortical structures with vorticity components aligned to the main flow direction. Thus for a detailed FSI simulation eddy-resolving schemes such as large-eddy simulations (Breuer et al., 2012) are required.

Fig. 12: Experimental unsteady 3D flow field, structure and flow results showing a single coupled measurement at ${\displaystyle t=T/2}$. The iso-surfaces represent the dimensionless velocity magnitude ${\displaystyle u/u_{\text{inflow}}=0.79}$. The contours on the iso-surfaces depict the spanwise velocity component w. The subfigures (a) and (b) show the same configuration from two different perspectives.

Phase-averaged results

Deflection of the structure

Figure 13 depicts the x–y-cross-sections of the whole structure plotted for 12 characteristic moments of the reference period. Each profile consists of approximately 180 measurement points. This figure represents the entire period and clearly shows that the flexible structure oscillates in the second swiveling mode as mentioned above. Due to the steel weight attached to the rubber there is no measurable displacement in the z-direction, leading to a fully two-dimensional structure deformation behavior.

Fig. 13: Experimental structural results: Structure contour for the reference period.

The averaged phases of the numerical and experimental investigations are compared in Fig. 14. The maximum of the dimensionless y-displacement ${\displaystyle \left.U_{y}^{*}\right|_{max}=\left.U_{y}\right|_{max}/D}$ is very well predicted with a value of 0.670. Compared to the measurements (${\displaystyle \left.U_{y}^{*}\right|_{max}=0.667}$) a small error of 0.5 % is found. For the minimum of the dimensionless y-displacement ${\displaystyle \left.U_{y}^{*}\right|_{min}=\left.U_{y}\right|_{min}/D}$, the error is larger (7.2 %), but acceptable. The magnitude of the minimum found in the FSI simulation (${\displaystyle \left.U_{y}^{*}\right|_{min}=-0.674}$) is near to the maximum value (0.670). It shows the antisymmetry of the phase with respect to the midpoint ${\displaystyle \phi =\pi }$. The averaged phase of the measurements is not exactly antisymmetric with respect to ${\displaystyle \phi =\pi }$: The minimum is observed at -0.629, whereas the maximum is 0.667. The imbalance of the experimental phase can be attributed to minor asymmetries in the setup or in the rubber material.

As written in the Abstract the frequency of the FSI phenomenon, i.e., the frequency of the y-displacements, is about ${\displaystyle f_{{\text{FSI}}_{\text{exp}}}=11.25}$ Hz in the experimental investigations, which corresponds to a Strouhal number ${\displaystyle {\text{St}}_{{\text{FSI}}_{\text{exp}}}\approx 0.179}$. The predicted FSI frequency is ${\displaystyle f_{{\text{FSI}}_{\text{num}}}=11.53}$ Hz (${\displaystyle {\text{St}}_{{\text{FSI}}_{\text{num}}}\approx 0.183}$). The error is about ${\displaystyle \epsilon _{f}=2.49\%}$. The FSI frequency is also well predicted. A summary of the numerical and experimental values is presented in the table below.

Fig. 14: Comparison of the predicted and measured averaged phase.

 ${\displaystyle f_{\text{FSI}}}$ (Hz) ${\displaystyle {\text{St}}_{\text{FSI}}}$ Error (%) ${\displaystyle \left.U_{y}^{*}\right|_{max}}$ Error (%) ${\displaystyle \left.U_{y}^{*}\right|_{min}}$ Error (%) Simulation 11.53 0.183 2.49 0.670 0.5 -0.674 7.2 Experiment 11.25 0.179 - 0.667 - -0.629 -

Phase-resolved flow field

The flow and structure results of the phase-averaged reference period obtained in the experiment (by the planar PIV system) and in the FSI simulation are put side by side in order to be compared (see Fig. 15). Once again the symmetric vortex centers shedding alternatively from the upper and lower sides of the cylinder are now visible in the velocity contour plots. Looking at the temporal development of these flow structures over the entire period, the convective transport of the vortex centers in the main flow direction is noticeable.

For each of the given phase-averaged positions the predicted structure deformations and flow fields are in good agreement with the measurements. The size of the acceleration area and the position of vortices in the wake are very similar in experiment and simulation. The shedding phenomenon behind the cylinder is correctly predicted. The computed shear layers are in good agreement with the measured one except in the vicinity of the cylinder. Note that the reasons for the deviations can be explained as will be done below. The recirculation areas found in the present FSI simulation approximately correspond to the experimental observations.

Fig. 15: Comparison of experimental and numerical phase-averaged streamwise velocity component.

Figure 16 compares the experimental and numerical phase-averaged transverse velocity component only for the first three phases (${\displaystyle \phi \approx \{\pi /12,~5\pi /12,~9\pi /12\}}$). Since the FSI phenomenon is antisymmetric, the last three phases (${\displaystyle \phi \approx \{13\pi /12,~17\pi /12,~21\pi /12\}}$) are similar to the first three ones and thus can be omitted for the sake of brevity. For all phase-averaged positions the transverse velocity component is in very good agreement with the measured one. Again the predicted sizes and positions of the vortices visible in the contour plots of the transverse velocity component coincide with the measurements.

Fig. 16: Comparison of experimental and numerical phase-averaged transverse velocity component.

In order to investigate the predicted results more deeply, the dimensionless absolute error between the simulation and the experiment for two representative positions of the whole FSI phenomenon is visualized in Fig. 17. For both positions the dimensionless absolute local error on the velocity magnitude is mostly below ${\displaystyle 20\%}$. The areas with high local errors are located near the structure, in the shear layers and in the zones of maximum velocity of the wake. Three reasons can be found for these differences:

• In the vicinity of the structure, in the shear layers and in the zones of maximum velocity the gradients of the flow quantities are large. Since the grid used for the simulation is much finer than the PIV measurement mesh, the accuracy of the numerical solution is much higher than the precision of the PIV measurements in these regions.
• Reflections of the laser light at the surface of the cylinder lead to inaccuracies of the PIV measurements in the proximity of the structure.
• The measurement error is more important for low flow velocities since the uncertainties for the velocity expected by the PIV method is calculated to about 0.076 m/s. The areas exhibiting higher errors are surrounding the cylinder and the rubber plate, where the flow velocity is small.

Fig. 17: Dimensionless absolute local error (velocity magnitude) in %.

For the present rubber test case with the steel tail, 550 single V3V measurements were taken. Divided into the 23 intervals of the phase-averaging procedure, each structure position was assigned on average to 24 single V3V measurements. To permit the validation of numerical data, these measurements were phase-averaged as explained above. Figure 18 depicts the volumetric phase-averaged data at the same time-phase angle ${\displaystyle \phi =\pi }$ as shown in Fig. 12. In contrast to the instantaneous measurements at a specific instant in time, only small variations in the spanwise direction of the vortex rolls are visible. By using more V3V measurements for the phase averaging (not possible here due to storage limitations of the V3V system), the phase-averaged flow field should converge towards a two-dimensional problem. For this reason also the spanwise velocity component diminishes for a longer averaging period and finally vanishes completely.

Fig. 18: Experimental phase-averaged 3D flow field, structure and flow results showing a phase-averaged coupled measurement at ${\displaystyle \phi =\pi }$. The iso-surfaces represent the dimensionless velocity magnitude ${\displaystyle u/u_{\text{inflow}}=0.79}$. The contours on the iso-surfaces depict the spanwise velocity component w. The subfigures (a) and (b) show the same configuration from two different perspectives.

Movie (Comparison Experiment vs. Simulation)

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

Sensitivity study

In order to better understand the current test case a sensitivity study based on a dimensional analysis was carried out. The dimensional analysis presented in De Nayer and Breuer (2014) leads to eleven dimensionless numbers. Due to the experimental setup a variety of physical quantities are fixed. Thus, the final analysis includes only four dimensionless numbers:

• the Cauchy number ${\displaystyle {\text{Cy}}_{\text{rubber}}}$,
• the dimensionless extrema ${\displaystyle \left.U_{y}^{*}\right|_{max}}$ and ${\displaystyle \left.U_{y}^{*}\right|_{min}}$ of the y-displacement and
• the Strouhal number ${\displaystyle {\text{St}}_{\text{FSI}}}$.

However, for a simulation, modeling aspects are of course significant and have to be taken into account in the sensitivity study.

The present test case is within the turbulent regime. Hence, turbulence plays an important role. The modeled part of the LES method is based on the subgrid-scale (SGS) model. For the parameter of the chosen model, there is a range of possible values: For example for the Smagorinsky model parameters in the range ${\displaystyle C_{s}\in [0.065,0.2]}$ are found in the literature. Therefore, simulations with different SGS models and different parameter values were carried out. Three simulations using the Smagorinsky model are presented: the Smagorinsky constant is adjusted to test the whole range (${\displaystyle C_{s}=0.065,\ 0.1{\text{ and }}0.18}$). Furthermore, a simulation with the dynamic procedure of Germano et al. (1991) still relying on the Smagorinsky model is carried out. Finally, the WALE model is tested with a ${\displaystyle C_{W}=0.33}$, corresponding to a Smagorinsky constant of ${\displaystyle C_{s}=0.1}$.

As already explained, the modeling of the material has a strong influence on the coupled simulation. For rubber it is usual to expect a certain level of damping. This effect is modeled by a Rayleigh damping. The damping factor proportional to the stiffness is approximated to a value of ${\displaystyle \beta =0.006}$, whereas the quantity proportional to the mass ${\displaystyle \alpha }$ is assumed to be zero (De Nayer and Breuer, 2014). Thus, a simulation with Rayleigh damping is also carried out and completes the sensitivity study.

All tests are summarized in the Table 1. The experimental results are also added as the reference. Each simulation was done for a time interval of 2 s physical time and comprises about 15 swiveling periods without the initial starting phase. Relative errors between the numerical and experimental values are given.

Table 1: Parameter study and corresponding results. Bold values define the numerical reference case used for comparison with the experiment. The errors are defined according to the deviations to the measurements.

The following results and trends can be seen concerning the influence of the Rayleigh damping:

• The frequency ${\displaystyle f_{\text{FSI}}}$ and consequently the Strouhal number ${\displaystyle {\text{St}}_{\text{FSI}}}$ is slightly lower for the simulation with structural damping than for the simulation without.
• The extrema of the y-displacement stay the same with or without structural damping.

With respect to the influence of the SGS model, the following observations can be made:

• The SGS model has no significant influence on the frequency of the FSI phenomenon.
• Variations of the parameter ${\displaystyle C_{s}}$ of the Smagorinsky model lead to small changes of the maxima of the y-displacements. When ${\displaystyle C_{s}}$ increases, ${\displaystyle \left.U_{y}^{*}\right|_{max}}$ and ${\displaystyle \left.U_{y}^{*}\right|_{min}}$ decreases. However, the influence of ${\displaystyle C_{s}}$ on the FSI results is very limited: Large modifications of ${\displaystyle C_{s}}$ around the standard value ${\displaystyle C_{s}=0.1}$ alter the maximal values less than 3 %.
• The dynamic Germano model produces extrema of the y-displacement equivalent to those obtained by the Smagorinsky model with ${\displaystyle C_{s}=0.065}$. The frequency is slightly lower than with the Smagorinsky model with ${\displaystyle C_{s}=0.065}$ and is near the value reached by the Smagorinsky model with ${\displaystyle C_{s}=0.18}$.
• The WALE model with ${\displaystyle C_{W}=0.33}$ generates similar extrema of the y-displacement to those obtained by the Smagorinsky model with ${\displaystyle C_{s}=0.1}$. The frequency is slightly lower than for the Smagorinsky model with ${\displaystyle C_{s}=0.1}$. Similar to the dynamic model, the frequency is close to the value achieved by the Smagorinsky model with ${\displaystyle C_{s}=0.18}$.

Finally, the influence of Young's modulus is the following:

• As expected for a problem involving a plate, when ${\displaystyle E_{\text{rubber}}}$ increases, ${\displaystyle f_{\text{FSI}}}$ slowly increases.
• Small changes of ${\displaystyle E_{\text{rubber}}}$ result in substantial variations of the maxima of the y-displacements.

In summary, the present parameter study shows that Young's modulus ${\displaystyle E_{\text{rubber}}}$ (or the dimensionless Cauchy number ${\displaystyle {\text{Cy}}_{\text{rubber}}}$) has the largest influence on the FSI phenomenon for the setup used in FSI-PfS-2a. The structural damping has a minor effect and can be neglected. The choice of the SGS model and parameter does not play an important role for FSI-PfS-2a. The classical Smagorinsky model with a standard parameter set to ${\displaystyle C_{s}=0.1}$ can be used.

Conclusions

A new FSI benchmark case denoted FSI-PfS-2a was proposed. It consists of the turbulent flow past a cylinder with a flexible splitter plate and a rear mass. The flow is in the so-called sub-critical regime (Re = 30,470), which means that due to the transition in the free shear layers the wake is 3D and chaotic. The flexible structure deforms in the second swiveling mode with a corresponding Strouhal number of about ${\displaystyle {\text{St}}_{\text{FSI}}=0.179}$. However, the large deformations observed are almost 2D due to the steel weight attached behind the flexible rubber plate. This test case is supported by detailed experimental (Kalmbach and Breuer, 2013) and numerical (De Nayer and Breuer, 2014) investigations.

The simulation shows a similar unsteady behavior of the flow and the structure as the experiment. However, due to the chaotic nature of turbulence the experimental and numerical results have to be phase-averaged prior to a detailed comparison. The simulated phase-averaged signal fits pretty well to the experimental one concerning form and extrema. Moreover, the frequency of the FSI problem is well predicted with an error of only 2.49 %. Six characteristic positions of the structure representing the FSI phenomenon are chosen to compare the phase-averaged numerical flow field with the phase-averaged PIV measurements. For each of these positions the streamwise and transverse velocity components are in very good agreement between the simulation and the experiment. The shedding phenomenon behind the cylinder with the flexible structure and the positions of the vortices convected downstream are correctly predicted.

To better comprehend the current test case a sensitivity study including physical and numerical parameters was carried out leading to the following conclusions:

• Young's modulus ${\displaystyle E_{\text{rubber}}}$ has the major influence on the FSI phenomenon for the geometry used in FSI-PfS-2a. Compared to the value provided by Kalmbach and Breuer (2013) based on a static structural test, Young's modulus was newly determined based on a more reasonable dynamic structural test. Another parameter study (De Nayer et al., 2014) conducted with the geometry of the test case FSI-PfS-1a (similar geometry but other material parameters and no rear mass) has led to the same conclusion.
• Furthermore, in FSI-PfS-2a the structural damping has a minor effect and can be omitted. This outcome is exactly the contrary to the one found with FSI-PfS-1a. This observation can be explained by the fact that FSI-PfS-1a is an instability-induced excitation and thus more sensitive to structural damping. FSI-PfS-2a, however, is a movement-induced excitation. Here, the damping generated by the viscous fluid exceeds the structural damping by orders of magnitude.
• The sensitivity study also permits to verify that the SGS model and its parameter does not strongly affect the simulation. Indeed, the Smagorinsky model, the dynamic model of Germano and the WALE model yield similar results for the test case FSI-PfS-2a. This is due to the moderate Reynolds number and the fine grid applied.

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.

Experimental data

The experimental data files below contains the phase-resolved flow results obtained with the PIV setup presented before. Each file has 5 columns: The 2 first ones contain the x- and y-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 dimensionless results obtained with the LES computation presented before (as presented in Section Phase-averaged results). Each file has 6 columns: The 2 first ones contain the dimensionless x- and y-positions of each cell center. The 4 next columns contain the dimensionless 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: Andreas Kalmbach, Guillaume De Nayer, Michael Breuer — Helmut-Schmidt Universität Hamburg