Revision as of 12:50, 4 October 2013

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

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.

Test Case Study

Brief Description of the Study Test Case

This should:

Convey the general set up of the test-case configuration( e.g. airflow over a bump on the floor of a wind tunnel)
Describe the geometry, illustrated with a sketch
Specify the flow parameters which define the flow regime (e.g. Reynolds number, Rayleigh number, angle of incidence etc.)
Give the principal measured quantities (i.e. assessment quantities) by which the success or failure of CFD calculations are to be judged. These quantities should include global parameters but also the distributions of mean and turbulence quantities.

The description can be kept fairly short if a link can be made to a data base where details are given. For other cases a more detailed, fully self-contained description should be provided.

Test Case Experiments

Provide a brief description of the test facility, together with the measurement techniques used. Indicate what quantities were measured and where.

Discuss the quality of the data and the accuracy of the measurements. It is recognized that the depth and extent of this discussion is dependent upon the amount and quality of information provided in the source documents. However, it should seek to address:

How close is the flow to the target/design flow (e.g. if the flow is supposed to be two-dimensional, how well is this condition satisfied)?

Estimation of the accuracy of measured quantities arising from given measurement technique

Checks on global conservation of physically conserved quantities, momentum, energy etc.

Consistency in the measurements of different quantities.

Discuss how well conditions at boundaries of the flow such as inflow, outflow, walls, far fields, free surface are provided or could be reasonably estimated in order to facilitate CFD calculations

CFD Methods

Provide an overview of the methods used to analyze the test case. This should describe the codes employed together with the turbulence/physical models examined; the models need not be described in detail if good references are available but the treatment used at the walls should explained. Comment on how well the boundary conditions used replicate the conditions in the test rig, e.g. inflow conditions based on measured data at the rig measurement station or reconstructed based on well-defined estimates and assumptions.

Discuss the quality and accuracy of the CFD calculations. As before, it is recognized that the depth and extent of this discussion is dependent upon the amount and quality of information provided in the source documents. However the following points should be addressed: