UFR 2-13 Test Case

A fluid-structure interaction benchmark in turbulent flow (FSI-PfS-1a)

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The following part is divided into two different sections: in the first one numerical phasedresolved results obtained for the two configurations (full and subset case) are compared. Based on this evaluation one case is chosen for a parameter study. Then, in the second subsection the numerical phased-averaged results chosen are juxtaposed to the experimental ones in order to verify their quality. In both simulations (subset and full case) the flow is initialized by assuming the entire structure to be undeformable. In this case the shell attached to the backside of the cylinder acts like a splitter plate attenuating the generation of a von Karman vortex street behind the cylinder. Nevertheless, quasi-periodic vortex shedding is still observed with a Strouhal number of St fixed to 0.175. Owing to different loads on both sides the structure starts to deflect as soon as it is released. After a short initial phase, in which the amplitudes of the deflections successively increase, a new quasi-periodic mode of oscillation is reached. In accordance with the experiment in the numerical simulations the shell deforms in the first swiveling mode as visible in Fig. 1.

Comparison of numerical results

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 $\rho _{\text{rubber plate}}$=1425 kg m$^{-3}$ 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 shell in the full case deforms more strongly in z-direction than in the subset case. 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 $\pm 60mm$ in the spanwise direction. All three points are not located directly on the shell extremity but at a distance of 9 mm from the extremity. 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 structure extremity and thus points at a certain distance from the tail are chosen. The dimensionless y-displacements $U_{y}^{*}=U_{y}/D$ at these three points are monitored as shown in Fig. 2(a). The following observation can be made: 1. The displacements are in phase. 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 ($f_{{\text{FSI}}_{\text{num}}}=6.96$ Hz and ${\text{St}}_{{\text{FSI}}_{\text{num}}}=0.11$). 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 $R^{2}=1-\sum _{i}\left(U_{{y}_{i}}^{*}-{\hat {U_{{y}_{i}}^{*}}}\right)^{2}/\sum _{i}\left(U_{{y}_{i}}^{*}-{\overline {U_{y}^{*}}}\right)^{2}$ of the calculated mean phase is close to unity (0.9869 for the full case and 0.9782 for the subset case). ${\hat {U_{{y}_{i}}^{*}}}$ denotes the estimated mean value of $U_{y}^{*}$ for the point i. ${\overline {U_{y}^{*}}}$ 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 shell 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 $u_{\text{inflow}}$ 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 shell 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 $\mu _{f}$, the fluid density $\rho _{f}$, the inlet velocity $u_{\text{inflow}}$ for the fluid; the cylinder diameter $D$, the dimensions of the rubber plate $l$, $w$ and $h$; the Young's modulus $E$, the Poisson's ratio $\nu$ and the density of the rubber plate $\rho _{\text{rubber plate}}$; To describe the FSI phenomenon the FSI frequency $f_{\text{FSI}}$, the displacement extrema $\left.U_{y}\right|_{max}$ and $\left.U_{y}\right|_{min}$ are chosen. These 13 physical quantities lead to 10 dimensionless parameters: The Reynolds number ${\text{Re}}=\rho _{f}u_{\text{inflow}}D/\mu _{f}$ for the fluid; the length ratios $w/l$, $h/l$, $D/l$ for the geometry; $\nu$ for the material of the rubber plate; The density ratio $\rho _{f}/\rho _{\text{rubber plate}}$, the Cauchy number ${\text{Cy}}=\rho _{f}u_{\text{inflow}}^{2}/E$ (as defined in~\cite{delangre2002}), the extrema of the dimensionless y-displacements $\left.U_{y}^{*}\right|_{max}=\left.U_{y}\right|_{max}/D$ and $\left.U_{y}^{*}\right|_{min}=\left.U_{y}\right|_{min}/D$ and the Strouhal number ${\text{St}}_{\text{FSI}}=f_{\text{FSI}}\,D/u_{\text{inflow}}$ 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 ${\text{Re}}$, the geometrical ratios $w/l$ and $D/l$ 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 $h/l$ will be taken into account in the sensitivity study.

• The density of the rubber plate $\rho _{\text{rubber plate}}$ 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 $\rho _{f}\,/\,\rho _{\text{rubber plate}}$ 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 ${\text{Cy}}$ is tested to evaluate this influence.

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

All the tests were carried out without structural damping and are summarized in Tab. 1. The full case used in 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 $(240{\times }10^{-6}\leq {\text{Cy}}\leq 96{\times }10^{-6})$ 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 $({\text{Cy}}\geq 213{\times }10^{-6})$, the system oscillates in the second swiveling mode. With E larger than 12 MPa $(Cy\leq 160{\times }10^{-6})$ the structure deflection is dominated by the first bending mode of the structure. For a Young's modulus between 9 and 12 MPa $(213{\times }10^{-6}\leq Cy\leq 160{\times }10^{-6})$ 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 $(1320kgm^{-3}\leq \rho _{\text{rubber plate}}\leq 1725kgm^{-3})$ $(0.757\geq \rho _{f}/\rho _{\text{rubber plate}}\geq 0.580)$ 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 $f_{\text{FSI}}$ slightly decreases with the increase of the density.

• Comparing the results for both thicknesses for the range $14{\text{MPa}}\leq E\leq 16{\text{MPa}}$, 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 $f_{\text{FSI}}$ 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 $\rho _{\text{rubber plate}}=1360kgm^{-3}$ in the range $14{\text{MPa}}\leq E\leq 20{\text{MPa}}$, we observe that the FSI frequency $f_{\text{FSI}}$ 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 shell 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.

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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)

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