UFR 2-14 Evaluation

Fluid-structure interaction in turbulent flow past cylinder/plate configuration II

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Unsteady results

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/59130975

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 $\pm 0.03~(\pm 5\%)$ 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 ($u/u_{\text{inflow}}=0.79$) 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 $t=T/2$, the iso-surfaces represent the dimensionless velocity magnitude $u/u_{\text{inflow}}=0.79$, the contours on the iso-surfaces depict the spanwise velocity component w.).

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 $\left.U_{y}^{*}\right|_{max}=\left.U_{y}\right|_{max}/D$ is very well predicted with a value of 0.670. Compared to the measurements ($\left.U_{y}^{*}\right|_{max}=0.667$) a small error of 0.5 % is found. For the minimum of the dimensionless y-displacement $\left.U_{y}^{*}\right|_{min}=\left.U_{y}\right|_{min}/D$, the error is larger (7.2 %), but acceptable. The magnitude of the minimum found in the FSI simulation ($\left.U_{y}^{*}\right|_{min}=-0.674$) is near to the maximum value (0.670). It shows the antisymmetry of the phase with respect to the midpoint $\phi =\pi$. The averaged phase of the measurements is not exactly antisymmetric with respect to $\phi =\pi$: 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 $f_{{\text{FSI}}_{\text{exp}}}=11.25$ Hz in the experimental investigations, which corresponds to a Strouhal number ${\text{St}}_{{\text{FSI}}_{\text{exp}}}\approx 0.179$. The predicted FSI frequency is $f_{{\text{FSI}}_{\text{num}}}=11.53$ Hz (${\text{St}}_{{\text{FSI}}_{\text{num}}}\approx 0.183$). The error is about $\epsilon _{f}=2.49\%$. 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.

	$f_{\text{FSI}}$ (Hz)	${\text{St}}_{\text{FSI}}$	Error (%)	$\left.U_{y}^{*}\right|_{max}$	Error (%)	$\left.U_{y}^{*}\right|_{min}$	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 fit well between the experiment and the 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.

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Figure 16 compares the experimental and numerical phase-averaged transverse velocity component only for the first three phases ($\phi \approx \{\pi /12,~5\pi /12,~9\pi /12\}$). Since the FSI phenomenon is antisymmetric, the last three phases ($\phi \approx \{13\pi /12,~17\pi /12,~21\pi /12\}$) 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 $20\%$. 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 are taken. Divided into the 23 intervals of the phase-averaging procedure, each structure position is assigned on average to 24 single V3V measurements. To permit the validation of numerical data, these measurements are phase-averaged as explained above. Fig. 18 depicts the volumetric phase-averaged data at the same time-phase angle $\phi =\pi$ as shown in Fig. 12. In contrast to the single measurement 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 $\phi =\pi$, the iso-surfaces represent the dimensionless velocity magnitude $u/u_{\text{inflow}}=0.79$, the contours on the iso-surfaces depict the spanwise velocity component w.).

Data files

As explained in Section Generation of Phase-resolved Data in FSI-PfS-1a 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:

Media:FSI-PfS-2a_exp_2Dflow_01.dat Media:FSI-PfS-2a_exp_2Dflow_02.dat Media:FSI-PfS-2a_exp_2Dflow_03.dat Media:FSI-PfS-2a_exp_2Dflow_04.dat

Media:FSI-PfS-2a_exp_2Dflow_05.dat Media:FSI-PfS-2a_exp_2Dflow_06.dat Media:FSI-PfS-2a_exp_2Dflow_07.dat Media:FSI-PfS-2a_exp_2Dflow_08.dat

Media:FSI-PfS-2a_exp_2Dflow_09.dat Media:FSI-PfS-2a_exp_2Dflow_10.dat Media:FSI-PfS-2a_exp_2Dflow_11.dat Media:FSI-PfS-2a_exp_2Dflow_12.dat

Media:FSI-PfS-2a_exp_2Dflow_13.dat Media:FSI-PfS-2a_exp_2Dflow_14.dat Media:FSI-PfS-2a_exp_2Dflow_15.dat Media:FSI-PfS-2a_exp_2Dflow_16.dat

Media:FSI-PfS-2a_exp_2Dflow_17.dat Media:FSI-PfS-2a_exp_2Dflow_18.dat Media:FSI-PfS-2a_exp_2Dflow_19.dat Media:FSI-PfS-2a_exp_2Dflow_20.dat

Media:FSI-PfS-2a_exp_2Dflow_21.dat Media:FSI-PfS-2a_exp_2Dflow_22.dat Media:FSI-PfS-2a_exp_2Dflow_23.dat

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:

Media:FSI-PfS-2a_exp_2Dstructure_01.dat Media:FSI-PfS-2a_exp_2Dstructure_02.dat Media:FSI-PfS-2a_exp_2Dstructure_03.dat Media:FSI-PfS-2a_exp_2Dstructure_04.dat

Media:FSI-PfS-2a_exp_2Dstructure_05.dat Media:FSI-PfS-2a_exp_2Dstructure_06.dat Media:FSI-PfS-2a_exp_2Dstructure_07.dat Media:FSI-PfS-2a_exp_2Dstructure_08.dat

Media:FSI-PfS-2a_exp_2Dstructure_09.dat Media:FSI-PfS-2a_exp_2Dstructure_10.dat Media:FSI-PfS-2a_exp_2Dstructure_11.dat Media:FSI-PfS-2a_exp_2Dstructure_12.dat

Media:FSI-PfS-2a_exp_2Dstructure_13.dat Media:FSI-PfS-2a_exp_2Dstructure_14.dat Media:FSI-PfS-2a_exp_2Dstructure_15.dat Media:FSI-PfS-2a_exp_2Dstructure_16.dat

Media:FSI-PfS-2a_exp_2Dstructure_17.dat Media:FSI-PfS-2a_exp_2Dstructure_18.dat Media:FSI-PfS-2a_exp_2Dstructure_19.dat Media:FSI-PfS-2a_exp_2Dstructure_20.dat

Media:FSI-PfS-2a_exp_2Dstructure_21.dat Media:FSI-PfS-2a_exp_2Dstructure_22.dat Media:FSI-PfS-2a_exp_2Dstructure_23.dat

Numerical data

to be added !

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Contributed by: Andreas Kalmbach, Guillaume De Nayer, Michael Breuer — Helmut-Schmidt Universität Hamburg

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