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

Introduction

A flexible structure exposed to a fluid flow is deformed and deflected owing to the fluid forces acting on its surface. These displacements influence the flow field resulting in a coupling process between the fluid and the structure shortly denoted fluid-structure interaction (FSI). Due to its manifold forms of appearance it is a topic of major interest in many fields of engineering. Based on enhanced numerical algorithms and increased computational resources numerical simulations have become an important and valuable tool for solving this kind of problem within the last decade. Today FSI simulations complement additional experimental investigations. A long-lasting vision of the computational engineer is to completely replace or at least strongly reduce expensive experimental investigations in the foreseeable future. However, to attain this goal validated and thus reliable simulation tools are required.

The long-term objective of the present research project is the coupled simulation of big lightweight structures such as thin membranes exposed to turbulent flows (outdoor tents, awnings...). To study these complex FSI problems, a multi-physics code framework was recently developed~\citep{fsi-les-2012}. In order to assure reliable numerical simulations of complex configurations, the whole FSI code needs to be validated at first on simpler test cases with trusted reference data. In~\cite{fsi-les-2012} the verification process of the code developed is detailed. The CFD and CSD solvers were at first checked separately and then, the coupling algorithm was considered in detail based on a laminar benchmark. A 3D turbulent test case was also taken into account to prove the applicability of the newly developed coupling scheme in the context of large-eddy simulations (LES). However, owing to missing reference data a full validation was not possible. The overall goal of the present paper is to present a turbulent FSI test case supported by experimental data and numerical predictions based on the multi-physics code developed. Thus, on the one hand the current FSI methodology involving LES and shell structures undergoing large deformations is validated. On the other hand, a new turbulent FSI benchmark configuration is defined, based on the specific insights into numerical flow simulation, computational structural analysis as well as coupling issues. Hence, the present study should provide a precisely described test case to the FSI community for the technically relevant case of turbulent flows interacting with flexible structures.

The present study is mainly related to two former investigation of Turek and Hron (2006,2010) and Gomes and Lienhart (2006, 2012) on vortex-induced fluid-structure interactions. The well-known 2D purely numerical laminar benchmarks of Turek and Hron (2006,2010) developed in a collaborative research effort on FSI (DFG Forschergruppe 493) consists of an elastic cantilever plate which is clamped behind a rigid circular cylinder. Three different test cases, named FSI1, FSI2 and FSI3 are provided, complemented by additional self-contained CFD and CSD test cases to check both solvers independently. These test cases were also used to validate the solvers applied in the present study (Breuer et al, 2012). In order to close the gap of complementary experimantel and numerical data, a nominally 2D laminar experimental case was provided by~\cite{gomes2006,gomes2013} and \cite{gomes2011b}. Here, a very thin metal sheet with an additional weight at the end is attached behind a rotating circular cylinder and mounted inside a channel filled with a mixture of polyglycol and water to reach a low Reynolds number in the laminar regime. Experimental data are provided for several inflow velocities and two different swiveling motions could be identified depending on the inflow velocity. Owing to the thin metal sheet and the rear mass the accurate prediction of this case is demanding. There are also turbulent FSI benchmarks involving 2D structures: in~\cite{stab-fsi-2008} a rigid plate with a single rotational degree of freedom was mounted into a water channel and experimentally studied by particle-image velocimetry (PIV). This study also presents the first comparison between experimental data and predicted results achieved by the present code for a turbulent FSI problem. As another turbulent experimental benchmark, the investigations of~\cite{gomes2010,gomes2013} and \cite{gomes2011b} have to be cited: the same geometry as in~\cite{gomes2006} was used, but this time with water as the working fluid leading to much higher Reynolds numbers within the turbulent regime. The resulting FSI test case was found to be very challenging from the numerical point of view. Indeed, the prediction of the deformation and motion of the very thin flexible structure requires two-dimensional finite-elements. On the other hand the discretization of the extra weight mounted at the end of the thin metal sheet calls for three-dimensional volume elements. Thus for a reasonable prediction of this test case both element types have to be used concurrently and have to be coupled adequately. Additionally, the rotational degree of freedom of the front cylinder complicates the structural simulation and the grid adaptation of the flow prediction.

Thus, in the present study a slightly different configuration is considered to provide in a first step a less ambitious test case for the comparison between predictions and measurements focusing the investigations more to the turbulent flow regime and its coupling to a less problematic structural model. For this purpose, a fixed cylinder with a thicker rubber tail and without a rear mass is used. This should open the computation of the proposed benchmark case to a broader spectrum of codes and facilitates its adoption in the community. Strong emphasis is put on a precise description of the experimental measurements, a comprehensive discussion of the modeling in the numerical simulation (for the single field solutions as well as for the coupled problem) and the processing of the respective data to guarantee a reliable reproduction of the proposed test case with various suitable methods.

Test case description

Description of the geometrical model and the test section

FSI-PfS-1a consists of a flexible thin structure with a distinct thickness clamped behind a fixed rigid non-rotating cylinder installed in a water channel (see Fig.~\ref{fig:rubber_plate_geom}). The cylinder has a diameter $D\operatorname {=} 0.022m$ . It is positioned in the middle of the experimental test section with $H_{c}=H/2\operatorname {=} 0.120m$ $H_{c}/D\approx 5.45$ , whereas the test section denotes a central part of the entire water channel (see Fig.~\ref{fig:water_channel}). Its center is located at $L_{c}\operatorname {=} 0.077m$ $(L_{c}/D\operatorname {=} 3.5)$ downstream of the inflow section. The test section has a length of $L\operatorname {=} 0.338m$ $(L/D\approx 15.36)$ , a height of $H\operatorname {=} 0.240m$ $(H/D\approx 10.91)$ and a width $W\operatorname {=} 0.180m$ $(W/D\approx 8.18)$ . The blocking ratio of the channel is about $9.2\%$ . The gravitational acceleration $g$ points in x-direction (see Fig.~\ref{fig:rubber_plate_geom}), i.e. in the experimental setup this section of the water channel is turned 90 degrees. The deformable structure used in the experiment behind the cylinder has a length $l\operatorname {=} 0.060m$ $(l/D\approx 2.72)$ and a width $w\operatorname {=} 0.177m$ $(w/D\approx 8.05)$ . Therefore, in the experiment there is a small gap of about $1.5\times 10^{-3}m$ between the side of the deformable structure and both lateral channel walls. The thickness of the plate is $h\operatorname {=} 0.0021m$ $(h/D\approx 0.09)$ . This thickness is an averaged value. The material is natural rubber and thus it is difficult to produce a perfectly homogeneous 2 mm plate. The measurements show that the thickness of the plate is between 0.002 and 0.0022 m. All parameters of the geometrical configuration of the FSI-PfS-1a benchmark are summarized in Table~\ref{tab:geom_conf_bench}.

Description of the water channel

In order to validate numerical FSI simulations based on reliable experimental data, the special research unit on FSI~\citep{for493} designed and constructed a water channel (G\"ottingen type, see Fig.~\ref{fig:water_channel}) for corresponding experiments with a special concern regarding controllable and precise boundary and working conditions \citep{gomes2006, gomes2010, gomes2011b}. The channel (\mbox{$2.8~$m$~ \times~1.3~$m$~\times~0.5~$m}, fluid volume of $0.9~$m$^3$) has a rectangular flow path and includes several rectifiers and straighteners to guarantee a uniform inflow into the test section. To allow optical flow measurement systems like Particle-Image Velocimetry, the test section is optically accessible on three sides. It possesses the same geometry as the test section described in Section~\ref{sec:Description_model}. The structure is fixed on the backplate of the test section and additionally fixed in the front glass plate. With a 24~kW axial pump a water inflow of up to \mbox{$u_{\text{max}}=6$ m/s} is possible. To prevent asymmetries the gravity force is aligned with the x-axis in main flow direction.

Flow parameters

Several preliminary tests were performed to find the best working conditions in terms of maximum structure displacement, good reproducibility and measurable structure frequencies within the turbulent flow regime. Figure~\ref{fig:structure_lastpoint_peaks_ramp} depicts the measured extrema of the structure displacement versus the inlet velocity and Figure~\ref{fig:structure_lastpoint_frequency_St_ramp} gives the frequency and Strouhal number as a function of the inlet velocity. These data were achieved by measurements with the laser distance sensor explained in Section~\ref{sec:Laser_Sensor}. The entire diagrams are the result of a measurement campaign in which the inflow velocity was consecutively increased from 0 to $2.2m/s$ . At an inflow velocity of $u_{\text{inflow}}=1.385m/s$ the displacement are symmetrical, reasonably large and well reproducible. Based on the inflow velocity chosen and the cylinder diameter the Reynolds number of the experiment is equal to ${\text{Re}}=30,470$ . Regarding the flow around the front cylinder, at this inflow velocity the flow is in the sub-critical regime. That means the boundary layers are still laminar, but transition to turbulence takes place in the free shear layers evolving from the separated boundary layers behind the apex of the cylinder. Except the boundary layers at the section walls the inflow was found to be nearly uniform (see Fig.~\ref{fig:water_channel_inflow}). The velocity components ${\overline {u}}$ and ${\overline {v}}$ are measured with two-component laser-Doppler velocimetry (LDV) along the y-axis in the middle of the measuring section at $x/D=4.18$ and $z/D=0$ . It can be assumed that the velocity component $\overline{w}$ shows a similar velocity profile as ${\overline {v}}$ . Furthermore, a low inflow turbulence level of ${\text{Tu}}_{\text{inflow}}={\sqrt {0.5~\left({\overline {u'^{2}}}+{\overline {v'^{2}}}\right)}}/u_{\text{inflow}}=0.02$ is measured. All experiments were performed with water under standard conditions at $T=20^{\circ }~C$ . The flow parameters are summarized to

 Inflow velocity  $u_{\text{inflow}}=1.385m/s$ 
 Flow density  $\rho _{f}=1000kg/m^{3}$ 
 Flow dynamic viscosity  $\mu _{f}=1.0\times 10^{-3}Pas$

Material Parameters

Although the material shows a strong non-linear elastic behavior for large strains, the application of a linear elastic constitutive law would be favored, to enable the reproduction of this FSI benchmark by a variety of different computational analysis codes without the need of complex material laws. This assumption can be justified by the observation that in the FSI test case, a formulation for large deformations but small strains is applicable. Hence, the identification of the material parameters is done on the basis of the moderate strain expected and the St. Venant-Kirchhoff constitutive law is chosen as the simplest hyper-elastic material.

The density of the rubber material can be determined to be $\rho _{\text{rubber plate}}$ =1360 kg/m $^{3}$ for a thickness of the plate h = 0.0021 m. This permits the accurate modeling of inertia effects of the structure and thus dynamic test cases can be used to calibrate the material constants. For the chosen material model, there are only two parameters to be defined: the Young's modulus E and the Poisson's ratio $\nu$ . In order to avoid complications in the needed element technology due to incompressibility, the material was realized to have a Poisson's ratio which reasonably differs from $0.5$ . Material tests of the manufacturer indicate that the Young's modulus is E=16 MPa and the Poisson's ratio is $\nu$ =0.48.

density  $\rho _{\text{rubber plate}}$ =1360 kg/m $^{3}$ , Young's modulus E=16 MPa, Poisson's ratio  $\nu$ =0.48

Measuring Techniques

Experimental FSI investigations need to contain fluid and structure measurements for a full description of the coupling process. Under certain conditions, the same technique for both disciplines can be used. The measurements performed by \cite{gomes2006,gomes2010,gomes2013} used the same camera system for the simultaneous acquisition of the velocity fields and the structural deflections. This procedure works well for FSI cases involving laminar flows and 2D structure deflections. In the present case the structure deforms slightly three-dimensional with increased cycle-to-cycle variations caused by turbulent variations in the flow. The applied measuring techniques, especially the structural side, have to deal with those changed conditions especially the formation of shades. Furthermore, certain spatial and temporal resolutions as well as low measurement errors are requested. Due to the different deformation behavior a single camera setup for the structural measurements like in \cite{gomes2006,gomes2010,gomes2013} used was not practicable. Therefore, the velocity fields were captured by a 2D Particle-Image Velocimetry (PIV) setup and the structural deflections were measured with a laser triangulation technique. Both devices are presented in the next sections.

Particle-image velocimetry

A classic Particle-Image Velocimetry~\citep{adrian1991} setup depicted in Fig.~\ref{fig:piv} consists of a single camera obtaining two components of the fluid velocity on a planar surface illuminated by a laser light sheet. Particles introduced into the fluid are following the flow and reflecting the light during the passage of the light sheet. By taking two reflection fields in a short time interval $\Delta$ t, the most-likely displacements of several particle groups on an equidistant grid are estimated by a cross-correlation technique or a particle-tracking algorithm. Based on a precise preliminary calibration, the displacements obtained and the time interval $\Delta$ t chosen the velocity field can be calculated. To prevent shadows behind the flexible structure a second light sheet was used to illuminate the opposite side of the test section.

The phased-resolved PIV-measurements (PR-PIV) were carried out with a 4 Mega-pixel camera (TSI Powerview 4MP, charge-coupled device (CCD) chip) and a pulsed dual-head Neodym:YAG laser (Litron NanoPIV 200) with an energy of 200 mJ per laser pulse. The high energy of the laser allowed to use silver-coated hollow glass spheres (SHGS) with an average diameter of $d_{\text{avg,SHGS}}$ =10~µm and a density of $\rho _{SHGS}$ = 1400 kg m $^{-3}$ as tracer particles. To prove the following behavior of these particles a Stokes number Sk=1.08 and a particle sedimentation velocity $u_{\text{SHGS}}=2.18\times 10^{-5}~{\text{m/s}}$ is calculated With this Stokes number and a particle sedimentation velocity which is much lower than the expected velocities in the experiments, an eminent following behavior is approved. The camera takes 12 bit pictures with a frequency of about 7.0 Hz and a resolution of 1695 x 1211 px with respect to the rectangular size of the test section. For one phase-resolved position (described in Section~\ref{sec:Generation_of_phase-resolved_data}) 60 to 80 measurements are taken. Preliminary studies with more and fewer measurements showed that this number of measurements represent a good compromise between accuracy and effort. The grid has a size of 150 x 138 cells and was calibrated with an average factor of 126 $\mu$ m/px}, covering a planar flow field of x/D = -2.36 to 7.26 and y/D = -3.47 to ~3.47 in the middle of the test section at z/D = 0. The time between the frame-straddled laser pulses was set to $\Delta$ t=200 $\mu$ s. Laser and camera were controlled by a TSI synchronizer (TSI 610035) with 1 ns resolution. The processing of the phase-resolved fluid velocity fields involving the structure deflections is described in Section~\ref{sec:Generation_of_phase-resolved_data}.

Laser distance sensor

Non-contact structural measurements are often based on laser distance techniques. In the present benchmark case the flexible structure shows an oscillating frequency of about 7.1 Hz. With the requirement to perform more than 100 measurements per period, a time-resolved system was needed. Therefore, a laser triangulation was chosen because of the known geometric dependencies, the high data rates, the small measurement range and the resulting higher accuracy in comparison with other techniques such as laser phase-shifting or laser interferometry. The laser triangulation uses a laser beam which is focused onto the object. A CCD-chip located near the laser output detects the reflected light on the object surface. If the distance of the object from the sensor changes, also the angle changes and thus the position of its image on the CCD-chip. From this change in position the distance to the object is calculated by simple trigonometric functions and an internal length calibration adjusted to the applied measurement range. To study simultaneously more than one point on the structure, a multiple-point triangulation sensor was applied (Micro-Epsilon scanControl 2750, see Fig.~\ref{fig:sensor_alignment}). This sensor uses a matrix of CCD chips to detect the displacements on up to 640 points along a laser line reflected on the surface of the structure with a data rate of \mbox 800 profiles per second. The laser line was positioned in a horizontal (x/D = 3.2, see Fig.~\ref{fig:sensor_alignment}(a)) and in a vertical alignment(z/D = 0, see Fig.~\ref{fig:sensor_alignment}(b)) and has an accuracy of 40 µm}. Due to the different refraction indices of air, glass and water a custom calibration was performed to take the modified optical behavior of the emitted laser beams into account.