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| = A fluid-structure interaction benchmark in turbulent flow (FSI-PfS-1a) = | | =Fluid-structure interaction in turbulent flow past cylinder/plate configuration I (First swiveling mode)= |
| {{UFRHeader | | {{UFRHeader |
| |area=2 | | |area=2 |
| |number=13 | | |number=13 |
| }} | | }} |
| | |
| | |
| __TOC__ | | __TOC__ |
| = Introduction = | | |
| | == Flows around bodies == |
| | === Underlying Flow Regime 2-13 === |
| | |
| | = Description = |
| | |
| | == Introduction == |
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| A flexible structure exposed to a fluid flow is deformed and deflected | | A flexible structure exposed to a fluid flow is deformed and deflected |
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| simulation tools are required. | | simulation tools are required. |
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| The long-term objective of the present research project is the coupled | | The long-term objective of the research reported here is the coupled |
| simulation of big lightweight structures such as thin membranes | | simulation of big lightweight structures such as thin membranes |
| exposed to turbulent flows (outdoor tents, awnings...). To study these | | exposed to turbulent flows (outdoor tents, awnings...). To study these |
| complex FSI problems, a multi-physics code framework was recently | | complex FSI problems, a multi-physics code framework was recently |
| developed~\citep{fsi-les-2012}. In order to assure reliable numerical | | developed (Breuer et al., 2012) combining Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) solvers . In order to assure reliable numerical |
| simulations of complex configurations, the whole FSI code needs to be | | simulations of complex configurations, the whole FSI code needs to be |
| validated at first on simpler test cases with trusted reference | | validated at first on simpler test cases with trusted reference |
| data. In~\cite{fsi-les-2012} the verification process of the code | | data. In Breuer et al. (2012) the verification process of the code |
| developed is detailed. The CFD and CSD solvers were at first checked | | developed is detailed. The CFD and CSD solvers were at first checked |
| separately and then, the coupling algorithm was considered in detail | | separately and then, the coupling algorithm was considered in detail |
| based on a laminar benchmark. A 3D turbulent test case was also taken | | based on a laminar benchmark. A 3D turbulent test case was also calculated to prove the applicability of the newly developed |
| into account to prove the applicability of the newly developed
| |
| coupling scheme in the context of large-eddy simulations | | coupling scheme in the context of large-eddy simulations |
| (LES). However, owing to missing reference data a full validation was | | (LES). However, owing to missing reference data a full validation was |
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| interacting with flexible structures. | | interacting with flexible structures. |
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| 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. | | == Review of previous work == |
| The well-known 2D purely numerical laminar benchmarks of Turek and Hron (2006,2010) developed in a collaborative research effort on | | |
| | The present study is mainly related to two former investigations |
| | of Turek and Hron (2006, 2010) and Gomes et al. (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 | | FSI (DFG Forschergruppe 493) consists of an elastic cantilever |
| plate which is clamped behind a rigid circular cylinder. Three different | | plate which is clamped behind a rigid circular cylinder. Three different |
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| additional self-contained CFD and CSD test cases to check both solvers | | additional self-contained CFD and CSD test cases to check both solvers |
| independently. These test cases were also used to validate the solvers | | independently. These test cases were also used to validate the solvers |
| applied in the present study (Breuer et al, 2012). | | 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 | | 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 | | was provided by Gomes et al. (2006, 2013) and Gomes (2011). Here, a |
| very thin metal sheet with an additional weight at the end is attached | | very thin metal sheet with an additional weight at the end is attached |
| behind a rotating circular cylinder and mounted inside a channel | | behind a rotating circular cylinder and mounted inside a channel |
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| demanding. | | demanding. |
| There are also turbulent FSI benchmarks involving 2D structures: | | There are also turbulent FSI benchmarks involving 2D structures: |
| in~\cite{stab-fsi-2008} a rigid plate with a single rotational degree | | in Gomes et al. (2010) a rigid plate with a single rotational degree |
| of freedom was mounted into a water channel and experimentally studied | | of freedom was mounted into a water channel and experimentally studied |
| by particle-image velocimetry (PIV). This study also presents the | | by particle-image velocimetry (PIV). This study also presents the |
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| achieved by the present code for a turbulent FSI problem. As another | | achieved by the present code for a turbulent FSI problem. As another |
| turbulent experimental benchmark, the investigations | | turbulent experimental benchmark, the investigations |
| of~\cite{gomes2010,gomes2013} and \cite{gomes2011b} have to be | | of Gomes et al. (2010, 2013) and Gomes (2010) have to be |
| cited: the same geometry as in~\cite{gomes2006} was used, but this | | cited: the same geometry as in Gomes et al. (2006) was used, but this |
| time with water as the working fluid leading to much higher Reynolds | | time with water as the working fluid leading to much higher Reynolds |
| numbers within the turbulent regime. The resulting FSI test case was | | numbers within the turbulent regime. The resulting FSI test case was |
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| structural simulation and the grid adaptation of the flow | | structural simulation and the grid adaptation of the flow |
| prediction. | | prediction. |
| | |
| | == Choice of test case == |
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| Thus, in the present study a slightly different configuration is | | Thus, in the present study a slightly different configuration is |
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| for the coupled problem) and the processing of the respective data to | | for the coupled problem) and the processing of the respective data to |
| guarantee a reliable reproduction of the proposed test case with | | guarantee a reliable reproduction of the proposed test case with |
| various suitable methods. | | various suitable methods. A detailed description of the present test case is published in De Nayer et al. (2014). |
| | |
| = Test case description =
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| == Description of the geometrical model and the test section ==
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| FSI-PfS-1a consists of a flexible thin structure with a distinct
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| thickness clamped behind a fixed rigid non-rotating cylinder installed
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| in a water channel (see Fig.~\ref{fig:rubber_plate_geom}). The
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| cylinder has a diameter D=0.022m. It is positioned in the
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| middle of the experimental test section with <math>H_c = H/2 \operatorname{=} 0.120m</math> <math>H_c/D \approx 5.45</math>, whereas the test section denotes a
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| central part of the entire water channel (see
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| Fig.~\ref{fig:water_channel}). Its center is located at <math>L_c \operatorname{=}
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| 0.077m</math> <math>(L_c/D \operatorname{=} 3.5)</math> downstream of the inflow
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| section. The test section has a length of <math>L \operatorname{=} 0.338m</math>
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| <math>(L/D \approx 15.36)</math>, a height of <math>H \operatorname{=} 0.240m</math>
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| <math>(H/D \approx 10.91)</math> and a width <math>W \operatorname{=} 0.180m</math>
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| <math>(W/D \approx 8.18)</math>. The blocking ratio of the channel is
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| about <math>9.2\%</math>. The gravitational acceleration <math>g</math> points in
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| x-direction (see Fig.~\ref{fig:rubber_plate_geom}), i.e. in the
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| experimental setup this section of the water channel is turned 90
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| degrees. The deformable structure used in the experiment behind the
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| cylinder has a length <math>l \operatorname{=} 0.060m</math> <math>(l/D \approx 2.72)</math>
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| and a width <math>w \operatorname{=} 0.177m</math> <math>(w/D \approx 8.05)</math>.
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| Therefore, in the experiment there is a small gap of about <math>1.5
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| \times 10^{-3}m</math> between the side of the deformable structure and
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| both lateral channel walls.
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| The thickness of the plate is <math>h \operatorname{=} 0.0021m</math> <math>(h/D
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| \approx 0.09)</math>. This thickness is an averaged value. The material
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| is natural rubber and thus it is difficult to produce a perfectly
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| homogeneous 2 mm plate. The measurements show that the thickness of
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| the plate is between 0.002 and 0.0022 m. All parameters of the
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| geometrical configuration of the FSI-PfS-1a benchmark are summarized
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| in Table~\ref{tab:geom_conf_bench}.
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| [[File:FSI-PfS-1a_Benchmark_Rubberplate_geometry0001_new.png]]
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| Fig. 1: Geometrical configuration of the FSI-PfS-1a Benchmark.
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| == Description of the water channel ==
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| In order to validate numerical FSI simulations based on reliable
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| experimental data, the special research unit on FSI~\citep{for493}
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| designed and constructed a water channel (G\"ottingen type, see
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| Fig.~\ref{fig:water_channel}) for corresponding experiments with a
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| special concern regarding controllable and precise boundary and
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| working conditions \citep{gomes2006, gomes2010, gomes2011b}. The
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| channel (\mbox{$2.8~$m$~ \times~1.3~$m$~\times~0.5~$m}, fluid volume
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| of $0.9~$m$^3$) has a rectangular flow path and includes several
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| rectifiers and straighteners to guarantee a uniform inflow into the
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| test section. To allow optical flow measurement systems like
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| Particle-Image Velocimetry, the test section is optically accessible
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| on three sides. It possesses the same geometry as the test section
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| described in Section~\ref{sec:Description_model}. The structure is
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| fixed on the backplate of the test section and additionally fixed in
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| the front glass plate. With a 24~kW axial pump a water inflow of up to
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| \mbox{$u_{\text{max}}=6$ m/s} is possible. To prevent asymmetries the
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| gravity force is aligned with the x-axis in main flow direction.
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| [[File:waterchannel_new.png]]
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| Fig. 2: Sketch of the flow channel (dimensions given in mm).
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| == Flow parameters ==
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| Several preliminary tests were performed to find the best working
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| conditions in terms of maximum structure displacement, good
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| reproducibility and measurable structure frequencies within the
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| turbulent flow regime. Figure~\ref{fig:structure_lastpoint_peaks_ramp}
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| depicts the measured extrema of the structure displacement versus the
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| inlet velocity and
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| Figure~\ref{fig:structure_lastpoint_frequency_St_ramp} gives the
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| frequency and Strouhal number as a function of the inlet
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| velocity. These data were achieved by measurements with the laser
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| distance sensor explained in Section~\ref{sec:Laser_Sensor}. The
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| entire diagrams are the result of a measurement campaign in which the
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| inflow velocity was consecutively increased from 0 to <math>2.2 m/s</math>. At an inflow velocity of <math>u_{\text{inflow}}=1.385 m/s </math>
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| the displacement are symmetrical, reasonably large and well
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| reproducible. Based on the inflow velocity chosen and the cylinder
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| diameter the Reynolds number of the experiment is equal to
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| <math>\text{Re}=30,470</math>. Regarding the flow around the front
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| cylinder, at this inflow velocity the flow is in the sub-critical
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| regime. That means the boundary layers are still laminar, but
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| transition to turbulence takes place in the free shear layers evolving
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| from the separated boundary layers behind the apex of the
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| cylinder. Except the boundary layers at the section walls the inflow
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| was found to be nearly uniform (see
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| Fig.~\ref{fig:water_channel_inflow}). The velocity components
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| <math>\overline{u} </math> and <math>\overline{v}</math> are measured with two-component
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| laser-Doppler velocimetry (LDV) along the y-axis in the middle of the
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| measuring section at <math>x/D=4.18 </math> and <math>z/D=0 </math>. It can be assumed that
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| the velocity component $\overline{w}$ shows a similar velocity profile
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| as <math>\overline{v} </math>. Furthermore, a low inflow turbulence level of
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| <math>\text{Tu}_{\text{inflow}}=\sqrt{0.5~\left(\overline{u'^2}+\overline{v'^2} \right)}/u_{\text{inflow}}=0.02 </math> is measured. All
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| experiments were performed with water under standard conditions at
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| <math>T=20^\circ~C </math>. The flow parameters are summarized to
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| Inflow velocity <math>u_{\text{inflow}}=1.385 m/s </math>
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| Flow density <math> \rho_f=1000 kg/m^3</math>
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| Flow dynamic viscosity <math> \mu_f=1.0 \times 10^{-3}Pa s </math>
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| [[File:flow_conditions.png]]
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| Fig. 3 Profiles of the mean streamwise and normal velocity as well as the turbulence level at the inflow section of the water channel.
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| == Material Parameters ==
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| Although the material shows a strong non-linear elastic behavior for
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| large strains, the application of a linear elastic constitutive law
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| would be favored, to enable the reproduction of this FSI benchmark by
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| a variety of different computational analysis codes without the need
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| of complex material laws. This assumption can be justified by the
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| observation that in the FSI test case, a formulation for large
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| deformations but small strains is applicable. Hence, the
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| identification of the material parameters is done on the basis of the
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| moderate strain expected and the St. Venant-Kirchhoff constitutive law
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| is chosen as the simplest hyper-elastic material.
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| The density of the rubber material can be determined to be
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| <math>\rho_{\text{rubber plate}}</math>=1360 kg/m<math>^3</math> for a thickness
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| of the plate h = 0.0021 m. This permits the accurate modeling
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| of inertia effects of the structure and thus dynamic test cases can be
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| used to calibrate the material constants. For the chosen material
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| model, there are only two parameters to be defined: the Young's
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| modulus E and the Poisson's ratio <math>\nu</math>. In order to avoid
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| complications in the needed element technology due to
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| incompressibility, the material was realized to have a Poisson's ratio
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| which reasonably differs from <math>0.5</math>. Material tests of the
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| manufacturer indicate that the Young's modulus is E=16 MPa and
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| the Poisson's ratio is <math>\nu</math>=0.48.
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| density <math>\rho_{\text{rubber plate}}</math>=1360 kg/m<math>^3</math>, Young's modulus E=16 MPa, Poisson's ratio <math>\nu</math>=0.48
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| = Measuring Techniques =
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| Experimental FSI investigations need to contain fluid and structure
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| measurements for a full description of the coupling process. Under
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| certain conditions, the same technique for both disciplines can be
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| used. The measurements performed by
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| Gomes et. al. (2006, 2010, 2013) used the same camera system for
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| the simultaneous acquisition of the velocity fields and the structural
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| deflections. This procedure works well for FSI cases involving laminar
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| flows and 2D structure deflections. In the present case the structure
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| deforms slightly three-dimensional with increased cycle-to-cycle
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| variations caused by turbulent variations in the flow. The applied
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| measuring techniques, especially the structural side, have to deal
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| with those changed conditions especially the formation of
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| shades. Furthermore, certain spatial and temporal resolutions as well
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| as low measurement errors are requested. Due to the different
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| deformation behavior a single camera setup for the structural
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| measurements like in Gomes et. al. (2006, 2010, 2013) used was not
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| practicable. Therefore, the velocity fields were captured by a 2D
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| Particle-Image Velocimetry (PIV) setup and the structural deflections
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| were measured with a laser triangulation technique. Both devices are
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| presented in the next sections.
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| === Particle-image velocimetry ===
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| A classic Particle-Image Velocimetry (Adrian, 1991) setup consists of a single camera obtaining two
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| components of the fluid velocity on a planar surface illuminated by a
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| laser light sheet. Particles introduced into the fluid are following the
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| flow and reflecting the light during the passage of the light sheet.
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| By taking two reflection fields in a short time interval <math>\Delta</math>t, the most-likely displacements of several particle groups on an
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| equidistant grid are estimated by a cross-correlation technique or a
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| particle-tracking algorithm. Based on a precise preliminary
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| calibration, the displacements obtained and the time interval
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| <math>\Delta</math>t chosen the velocity field can be calculated. To prevent
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| shadows behind the flexible structure a second light sheet was used to
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| illuminate the opposite side of the test section.
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| The phased-resolved PIV-measurements (PR-PIV) were carried out with a
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| 4 Mega-pixel camera (TSI Powerview 4MP, charge-coupled device (CCD)
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| chip) and a pulsed dual-head Neodym:YAG laser (Litron NanoPIV 200)
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| with an energy of 200 mJ per laser pulse. The high energy of
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| the laser allowed to use silver-coated hollow glass spheres (SHGS)
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| with an average diameter of <math>d_{\text{avg,SHGS}}</math>=10~µm and
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| a density of <math>\rho_{SHGS}</math> = 1400 kg m<math>^{-3}</math> as tracer
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| particles. To prove the following behavior of these particles a
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| Stokes number Sk=1.08 and a particle sedimentation velocity
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| <math>u_{\text{SHGS}}=2.18 \times 10^{-5}~\text{m/s}</math> is calculated
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| With this Stokes number and a particle sedimentation velocity which is
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| much lower than the expected velocities in the experiments, an eminent
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| following behavior is approved. The camera takes 12 bit pictures with
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| a frequency of about 7.0 Hz and a resolution of 1695 x 1211 px with respect to the rectangular size of the test
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| section. For one phase-resolved position (described in
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| Section~\ref{sec:Generation_of_phase-resolved_data}) 60 to 80
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| measurements are taken. Preliminary studies with more and fewer
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| measurements showed that this number of measurements represent a good
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| compromise between accuracy and effort. The grid has a size of 150 x 138 cells and was calibrated with an average
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| factor of 126 <math>\mu</math> m/px}, covering a planar flow field of
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| x/D = -2.36 to 7.26 and y/D = -3.47 to ~3.47 in the middle of the test section at
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| z/D = 0. The time between the frame-straddled laser
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| pulses was set to <math>\Delta</math> t=200 <math>\mu</math>s. Laser and camera were
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| controlled by a TSI synchronizer (TSI 610035) with 1 ns
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| resolution. The processing of the phase-resolved fluid velocity fields
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| involving the structure deflections is described in
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| Section "Generation of phase-resolved data".
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| === Laser distance sensor ===
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| Non-contact structural measurements are often based on laser distance
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| techniques. In the present benchmark case the flexible structure shows
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| an oscillating frequency of about 7.1 Hz. With the
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| requirement to perform more than 100 measurements per period, a
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| time-resolved system was needed. Therefore, a laser triangulation was
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| chosen because of the known geometric dependencies, the high data
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| rates, the small measurement range and the resulting higher accuracy
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| in comparison with other techniques such as laser phase-shifting or
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| laser interferometry. The laser triangulation uses a laser beam which
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| 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
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| the object from the sensor changes, also the angle changes and thus
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| the position of its image on the CCD-chip. From this change in
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| position the distance to the object is calculated by simple
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| trigonometric functions and an internal length calibration adjusted to
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| the applied measurement range. To study simultaneously more than one
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| point on the structure, a multiple-point triangulation sensor was
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| applied (Micro-Epsilon scanControl 2750, see
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| Fig. 4). This sensor uses a matrix of CCD
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| chips to detect the displacements on up to 640 points along a laser
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| line reflected on the surface of the structure with a data rate of
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| \mbox 800 profiles per second. The laser line was positioned in a
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| horizontal (x/D = 3.2, see
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| Fig. 4(a)) and in a vertical
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| alignment(z/D = 0, see
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| Fig. 4(b)) and has an accuracy of
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| 40 µm}. Due to the different refraction indices of air,
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| glass and water a custom calibration was performed to take the
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| modified optical behavior of the emitted laser beams into account.
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| [[File:structure_sensors_scancontrolonly0001_new.png]]
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| Fig. 4: Setup and alignment of multiple-point laser sensor on the flexible structure in a) z-direction and b) x-direction.
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| = Numerical Simulation Methodology =
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| The applied numerical method relies on an efficient partitioned
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| coupling scheme developed for dynamic fluid-structure interaction
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| problems in turbulent flows (Breuer et al, 2012). The fluid flow is
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| predicted by an eddy-resolving scheme, i.e., the large-eddy simulation
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| technique. FSI problems very often encounter instantaneous
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| non-equilibrium flows with large-scale flow structures such as
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| separation, reattachment and vortex shedding. For this kind of flows
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| the LES technique is obviously the best
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| choice (Breuer, 2002). Based on a semi-implicit scheme the LES code
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| is coupled with a code especially suited for the prediction of shells
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| and membranes. Thus an appropriate tool for the time-resolved
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| prediction of instantaneous turbulent flows around light, thin-walled
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| structures results. Since all details of this methodology were
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| recently published in Breuer et al, 2012, in the following only a
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| brief description is provided.
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| == Computational fluid dynamics (CFD) ==
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| Within a FSI application the computational domain is no longer fixed
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| but changes in time due to the fluid forces acting on the structure.
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| This temporally varying domain is taken into account by the Arbitrary
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| Lagrangian-Eulerian (ALE) formulation expressing the conservation
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| equations for time-dependent volumes and surfaces. Here the filtered
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| Navier-Stokes equations for an incompressible fluid are solved. Owing
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| to the deformation of the grid, extra fluxes appear in the governing
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| equations which are consistently determined considering the
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| \emph{space conservation law (SCL)}(Demirdzic 1988 and 1990, Lesoinne, 1996). The SCL is expressed
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| by the swept volumes of the corresponding cell faces and assures that
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| no space is lost during the movement of the grid lines. For this
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| purpose the in-house code FASTEST-3D (Durst et al, 1996a, b)
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| relying on a three-dimensional finite-volume scheme is used. The
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| discretization is done on a curvilinear, block-structured body-fitted
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| grid with collocated variable arrangement. A midpoint rule
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| approximation of second-order accuracy is used for the discretization
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| of the surface and volume integrals. Furthermore, the flow variables
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| are linearly interpolated to the cell faces leading to a second-order
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| accurate central scheme. In order to ensure the coupling of pressure
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| and velocity fields on non-staggered grids, the momentum interpolation
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| technique of Rhie (1983) is used.
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| A predictor-corrector scheme (projection method) of second-order
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| accuracy forms the kernel of the fluid solver. In the predictor step
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| an explicit three substep low-storage Runge-Kutta scheme advances
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| the momentum equation in time leading to intermediate
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| velocities. These velocities do not satisfy mass conservation. Thus,
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| in the following corrector step the mass conservation equation has to
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| be fulfilled by solving a Poisson equation for the
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| pressure-correction based on the incomplete LU decomposition method
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| of Stone (1968). The corrector step is repeated (about 3 to 8
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| iterations) until a predefined convergence criterion (<math>\Delta\dot{m} <
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| {\cal O}(10^{-9})</math>) is reached and the final velocities and the
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| pressure of the new time step are obtained. In Breuer et al (2012) it
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| is explained that the original pressure-correction scheme applied
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| for fixed grids has not to be changed concerning the mass conservation
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| equation in the context of moving grids. Solely in the momentum
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| equation the grid fluxes have to be taken into account as described
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| above.
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| In LES the large scales in the turbulent flow field are resolved
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| directly, whereas the non-resolvable small scales have to be taken
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| into account by a subgrid-scale model. Here the well-known and most
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| often used eddy-viscosity model, i.e., the Ssmagorinsky (1963) model
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| is applied. The filter width is directly coupled to the volume of the
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| computational cell and a Van Driest damping function ensures a
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| reduction of the subgrid length near solid walls. Owing to minor
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| influences of the subgrid-scale model at the moderate Reynolds number
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| considered in this study, a dynamic procedure to determine the
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| Smagorinsky parameter as suggested by Germano et al (1991) was omitted and
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| instead a well established standard constant <math>C_s = 0.1</math> is used.
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| | |
| == Computational structural dynamics (CSD) ==
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| The dynamic equilibrium of the structure is described by the momentum
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| equation given in a Lagrangian frame of reference. Large deformations,
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| where geometrical non-linearities are not negligible, are allowed
| |
| (Hojjat et al, 2010). According to preliminary structure considerations, a total Lagrangian
| |
| formulation in terms of the second Piola-Kirchhoff stress tensor and
| |
| the Green-Lagrange strain tensor which are linked by the
| |
| St.\ Venant-Kirchhoff material law is used in the present study.
| |
| For the solution of the governing equation the finite-element solver
| |
| Carat++, which was developed with an emphasis on the prediction of
| |
| shell or membrane behavior, is applied. Carat++ is based on several
| |
| finite-element types and advanced solution strategies for form finding
| |
| and non-linear dynamic
| |
| Problems (Wüchner et al, 2005; Bletzinger et al, 2005; Dieringer et al, 2012). For the
| |
| dynamic analysis, different time-integration schemes are available,
| |
| e.g., the implicit generalized-$\alpha$ method (Chung et al, 1993). In the
| |
| modeling of thin-walled structures, membrane or shell elements are
| |
| applied for the discretization within the finite-element model. In the
| |
| current case, the deformable solid is modeled with a 7-parameter shell
| |
| element.
| |
| | |
| == Coupling algorithm ==
| |
| | |
| To preserve the advantages of the highly adapted CSD and CFD codes and
| |
| to realize an effective coupling algorithm, a partitioned but
| |
| nevertheless strong coupling approach is chosen. Since LES
| |
| typically requires small time steps to resolve the turbulent flow
| |
| field, the coupling scheme relies on the explicit
| |
| predictor-corrector scheme forming the kernel of the fluid solver.
| |
| | |
| Based on the velocity and pressure fields from the corrector step, the
| |
| fluid forces resulting from the pressure and the viscous shear
| |
| stresses at the interface between the fluid and the structure are
| |
| computed. These forces are transferred by a grid-to-grid data
| |
| interpolation to the CSD code Carat++ using a conservative
| |
| interpolation scheme (Farhat et al, 1998) implemented in the coupling
| |
| interface CoMA (Gallinger et al, 2009). Using the fluid forces provided via
| |
| CoMA, the finite-element code Carat++ determines the stresses in the
| |
| structure and the resulting displacements of the structure. This
| |
| response of the structure is transferred back to the fluid solver via
| |
| CoMA applying a bilinear interpolation which is a consistent scheme
| |
| for four-node elements with bilinear shape functions.
| |
| | |
| For moderate and high density ratios between the fluid and the
| |
| structure, e.g., a flexible structure in water, the added-mass effect
| |
| by the surrounding fluid plays a dominant role. In this situation a
| |
| strong coupling scheme taking the tight interaction between the fluid
| |
| and the structure into account, is indispensable. In the coupling
| |
| scheme developed in Breuer et al (2012) this issue is taken into
| |
| account by a FSI-subiteration loop which works as follows:
| |
| | |
| A new time step begins with an estimation of the displacement of the
| |
| structure. For the estimation a linear extrapolation is applied taking
| |
| the displacement values of two former time steps into account.
| |
| According to these estimated boundary values, the entire computational
| |
| grid has to be adapted as it is done in each FSI-subiteration
| |
| loop. Then the predictor-corrector scheme of the next time step is
| |
| carried out and the cycle of the FSI-subiteration loop is
| |
| entered. After each FSI-subiteration first the FSI convergence is
| |
| checked. Convergence is reached if the <math>L_2</math> norm of the displacement
| |
| differences between two FSI-subiterations normalized by the <math>L_2</math> norm
| |
| of the changes in the displacements between the current and the last
| |
| time step drops below a predefined limit, e.g. <math>\varepsilon_{FSI} =
| |
| 10^{-4}</math> for the present study. Typically, convergence is not reached
| |
| within the first step but requires a few FSI-subiterations (5 to
| |
| 10). Therefore, the procedure has to be continued on the fluid
| |
| side. Based on the displacements on the fluid-structure interface,
| |
| which are underrelaxated by a constant factor $\omega$ during the
| |
| transfer from the CFD to the CSD solver, the inner computational CFD
| |
| grid is adjusted. The key point of the coupling procedure suggested in
| |
| Breuer et al (2012) is that subsequently only the corrector step of
| |
| the predictor-corrector scheme is carried out again to obtain a new
| |
| velocity and pressure field. Thus the clue is that the pressure is
| |
| determined in such a manner that the mass conservation is finally
| |
| satisfied. Furthermore, this extension of the predictor-corrector
| |
| scheme assures that the pressure forces as the most relevant
| |
| contribution to the added-mass effect, are successively updated until
| |
| dynamic equilibrium is achieved. In conclusion, instabilities due to
| |
| the added-mass effect known from loose coupling schemes are avoided
| |
| and the explicit character of the time-stepping scheme beneficial for
| |
| LES is still maintained.
| |
| | |
| The code coupling tool CoMA is based on the
| |
| Message-Passing-Interface (MPI) and thus runs in parallel to the
| |
| fluid and structure solver. The communication in-between the codes is
| |
| performed via standard MPI commands. Since the parallelization in
| |
| FASTEST-3D and Carat++ also relies on MPI, a hierarchical
| |
| parallelization strategy with different levels of parallelism is
| |
| achieved. According to the CPU-time requirements of the different
| |
| subtasks, an appropriate number of processors can be assigned to the
| |
| fluid and the structure part. Owing to the reduced structural models
| |
| on the one side and the fully three-dimensional highly resolved fluid
| |
| prediction on the other side, the predominant portion of the CPU-time
| |
| is presently required for the CFD part. Additionally, the communication
| |
| time between the codes via CoMA and within the CFD solver takes a
| |
| non-negligible part of the computational resources.
| |
| | |
| == Numerical CFD Setup ==
| |
| | |
| For the CFD prediction of the flow two different block-structured
| |
| grids either for a subset of the entire channel (w'/l=1) or for
| |
| the full channel but without the gap between the flexible structure
| |
| and the side walls (w/l=2.95) are used. In the first
| |
| case the entire grid consists of about 13.5 million control volumes
| |
| (CVs), whereas 72 equidistant CVs are applied in the spanwise
| |
| direction. For the full geometry the grid possesses about 22.5 million
| |
| CVs. In this case starting close to both channel walls the grid is
| |
| stretched geometrically with a stretching factor $1.1$ applying in
| |
| total 120 CVs with the first cell center positioned at a distance of
| |
| <math>\Delta</math>y/D=1.7 x 10<math>^{-2}</math>.
| |
| | |
| | |
| [[File:Benchmark_FSI-PfS-1_full_and_subset_case.jpg]]
| |
| | |
| Fig. xx X-Y cross-section of the grid used for the simulation (Note that only every fourth grid line in each direction is displayed here).
| |
| | |
| | |
| The gap between the elastic structure and the walls is not taken into
| |
| account in the numerical model and thus the width of the channel is
| |
| set to w instead of W. The stretching factors are kept below 1.1 with the first cell
| |
| center located at a distance of <math>\Delta</math>y/D=9 x 10<math>^{-4}</math> from
| |
| the flexible structure. Based on the wall shear stresses on the
| |
| flexible structure the average y<math>^+</math> values are predicted to be below
| |
| 0.8, mostly even below 0.5. Thus, the viscous sublayer on the
| |
| elastic structure and the cylinder is adequately resolved. Since the
| |
| boundary layers at the upper and lower channel walls are not
| |
| considered, no grid clustering is required here.
| |
| | |
| On the CFD side no-slip boundary conditions are applied at the rigid
| |
| front cylinder and at the flexible structure. Since the resolution of
| |
| the boundary layers at the channel walls would require the bulk of the
| |
| CPU-time, the upper and lower channel walls are assumed to be slip
| |
| walls. Thus the blocking effect of the walls is maintained without
| |
| taking the boundary layers into account. At the inlet a constant
| |
| streamwise velocity is set as inflow condition without adding any
| |
| perturbations. The choice of zero turbulence level is based on the
| |
| consideration that such small perturbations imposed at the inlet will
| |
| generally not reach the cylinder due to the coarseness of the grid at
| |
| the outer boundaries. Therefore, all inflow fluctuations will be
| |
| highly damped. However, since the flow is assumed to be sub-critical,
| |
| this disregard is insignificant. At the outlet a convective outflow
| |
| boundary condition is favored allowing vortices to leave the
| |
| integration domain without significant disturbances
| |
| (Breuer, 2002). The convection velocity is set to
| |
| <math>u_\text{inflow}</math>.
| |
| | |
| As mentioned above two different cases are considered. In order to
| |
| save CPU-time in the first case only a subset of the entire spanwise
| |
| extension of the channel is taken into account. Thus the computational
| |
| domain has a width of w'/l=1 in z-direction and the
| |
| flexible structure is a square in the x-z-plane. In this case a
| |
| reasonable approximation already applied in Breuer et al (2012) is to
| |
| apply periodic boundary conditions in spanwise direction for both
| |
| disciplines. For LES predictions periodic boundary conditions
| |
| represent an often used measure in order to avoid the formulation of
| |
| appropriate inflow and outflow boundary conditions. The approximation
| |
| is valid as long as the turbulent flow is homogeneous in the specific
| |
| direction and the width of the domain is sufficiently large. The
| |
| latter can be proven by predicting two-point correlations, which have
| |
| to drop towards zero within the half-width of the domain. The impact
| |
| of periodic boundary conditions on the CSD predictions are discussed
| |
| below.
| |
| | |
| For the full case with w/l=2.95 periodic boundary conditions
| |
| can no longer be used. Instead, for the fluid flow similar to the
| |
| upper and lower walls also for the lateral boundaries slip walls are
| |
| assumed since the full resolution of the boundary layers would be
| |
| again too costly. Furthermore, the assumption of the slip wall is
| |
| consistent with the disregard of the small gap between the flexible
| |
| structure and the side walls discussed above.
| |
| | |
| == Numerical CSD Setup ==
| |
| | |
| Motivated by the fact that in the case of LES frequently a domain
| |
| modeling based on periodic boundary conditions at the lateral walls is
| |
| used to reduce the CPU-time requirements, this special approach was
| |
| also investigated for the FSI test case. The detailed discussion of
| |
| this specific boundary modeling for the spanwise direction is given in
| |
| Section~\ref{section:bc}. As a consequence, there are two different
| |
| structure meshes used: For the CSD prediction of the case with a
| |
| subset of the full channel the elastic structure is resolved by the
| |
| use of 10 x 10 quadrilateral four-node 7-parameter shell elements. For the
| |
| case discretizing the entire channel, 10 quadrilateral four-node shell
| |
| elements are used in the main flow direction and 30 in the spanwise
| |
| direction.
| |
| | |
| On the CSD side, the flexible shell is loaded on the top and bottom
| |
| surface by the fluid forces, which are transferred from the fluid mesh
| |
| to the structure mesh. These Neumann boundary conditions for the
| |
| structure reflect the coupling conditions. Concerning the Dirichlet
| |
| boundary conditions, the four edges need appropriate support modeling:
| |
| on the upstream side at the rigid cylinder a clamped support is
| |
| realized and all degrees of freedom are equal to zero. On the opposite
| |
| downstream trailing-edge side, the rubber plate is free to move and
| |
| all nodes have the full set of six degrees of freedom. The edges which
| |
| are aligned to the main flow direction need different boundary
| |
| condition modeling, depending on whether the subset or the full case
| |
| is computed:
| |
| | |
| For the subset case due to the fluid-motivated periodic boundary
| |
| conditions, periodicity for the structure is correspondingly assumed
| |
| for consistency reasons. As it turns out, this assumption seems to
| |
| hold for this specific benchmark configuration and its deformation
| |
| pattern which has strong similarity with an oscillation in the first
| |
| eigenmode of the plate. Hence, this modeling approach may be used for
| |
| the efficient processing of parameter studies, e.g., to evaluate the
| |
| sensitivity of the FSI simulations with respect to slight variations
| |
| in model parameters shown in a sensitivity study. For
| |
| this special type of support modeling, there are always two structure
| |
| nodes on the lateral sides (one in a plane z=-w/2 and its twin in
| |
| the other plane z=+w/2) which have the same load. These two nodes
| |
| must have the same displacements in x- and y-direction and their
| |
| rotations have to be identical. Moreover, the periodic boundary
| |
| conditions imply that the z-displacement of the nodes on the sides are
| |
| forced to be zero.
| |
| | |
| For the full case the presence of the walls in connection with the
| |
| small gap implies that there is in fact no constraining effect on the
| |
| structure, as long as no contact between the plate and the wall takes
| |
| place. Out of precise observations in the lab, the possibility of
| |
| contact may be disregarded. In principle, this configuration would
| |
| lead to free-edge conditions like at the trailing edge. However, the
| |
| simulation of the fluid with a moving mesh needs a well-defined mesh
| |
| situation at the side walls which made it necessary to tightly connect
| |
| the structure mesh to the walls (the detailed representation of the
| |
| side edges within the fluid mesh is discarded due to computational
| |
| costs and the resulting deformation sensitivity of the mesh in these
| |
| regions). Also the displacement in z-direction of the structure nodes
| |
| at the lateral boundaries is forced to be zero.
| |
| | |
| == Coupling conditions ==
| |
| | |
| For the turbulent flow a time-step size of <math>\Delta t_{f} = 2 \times
| |
| 10^{-5}s (\Delta t_{f}^{\ast} = 1.26 \times 10^{-3}</math> in
| |
| dimensionless form using <math>u_\text{inflow}</math> and D as reference
| |
| quantities) is chosen and the same time-step size is applied for the
| |
| structural solver based on the generalized-<math>\alpha</math> method with the
| |
| spectral radius <math>\varrho_\infty</math>=1.0, i.e, the Newmark standard
| |
| method. For the CFD part this time-step size corresponds to a CFL
| |
| number in the order of unity. Furthermore, a constant underrelaxation
| |
| factor of <math>\omega</math>=0.5 is considered for the displacements and the
| |
| loads are transferred without underrelaxation. In accordance with
| |
| previous laminar and turbulent cases in Breuer et. al. (2012) the FSI
| |
| convergence criterion is set to <math>\varepsilon_{FSI} = 10^{-4}</math> for the
| |
| <math>L_2</math> norm of the displacement differences. As estimated from previous
| |
| cases Breuer et. al. (2012) 5 to 10 FSI-subiterations are required to
| |
| reach the convergence criterion.
| |
| | |
| After an initial phase in which the coupled system reaches a
| |
| statistically steady state, each simulation is carried out for about
| |
| 4 s real-time corresponding to about 27 swiveling cycles of the
| |
| flexible structure.
| |
| | |
| For the coupled LES predictions the national supercomputer
| |
| SuperMIG/SuperMUC was used applying either 82 or 140 processors for
| |
| the CFD part of the reduced and full geometry,
| |
| respectively. Additionally, one processor is required for the coupling
| |
| code and one processor for the CSD code, respectively.
| |
| | |
| = Generation of Phase-resolved Data =
| |
| | |
| Each flow characteristic of a quasi-periodic FSI problem can be
| |
| written as a function $f=\bar{f}+\tilde{f}+f'$, where $\bar{f}$
| |
| describes the global mean part, $\tilde{f}$ the quasi-periodic part
| |
| and $f'$ a random turbulence-related
| |
| part~\citep{reynolds72,cantwell83}. This splitting can also be written
| |
| in the form $f = \left<f\right> + f'$, where $\left<f\right>$ is the
| |
| phase-averaged part, i.e., the mean at constant phase. In order to be
| |
| able to compare numerical results and experimental measurements, the
| |
| irregular turbulent part $f'$ has to be averaged out. This measure is
| |
| indispensable owing to the nature of turbulence which solely allows
| |
| reasonable comparisons based on statistical data. Therefore, the
| |
| present data are phase-averaged to obtain only the phase-resolved
| |
| contribution $\left<f\right>$ of the problem, which can be seen as a
| |
| representative and thus characteristic signal of the underlying FSI
| |
| phenomenon.
| |
| | |
| The procedure to generate phase-resolved results is the same for the
| |
| experiments and the simulations and is also similar to the one
| |
| presented in \cite{gomes2006}. The technique can be split up into
| |
| three steps:
| |
| | |
| * '''Reduce the 3D-problem to a 2D-problem''' - Due to the facts that in the present benchmark the structure deformation in spanwise direction is negligible and that the delivered experimental PIV-results are solely available in one x-y-plane, first the 3D-problem is reduced to a 2D-problem. For this purpose the flow field and the shell position in the CFD predictions are averaged in spanwise direction.
| |
| | |
| * '''Determine n reference positions for the FSI Problem''' - A representative signal of the FSI phenomenon is the history of the y-displacements of the shell extremity. Therefore, it is used as the trigger signal for this averaging method leading to phase-resolved data. Note that the averaged period of this signal is denoted $T$. At first, it has to be defined in how many sub-parts the main period of the FSI problem will be divided and so, how many reference positions have to be calculated (for example in the present work $n = 23$). Then, the margins of each period of the y-displacement curve are determined. In order to do that the intersections between the y-displacement curve and the zero crossings ($U_y = 0$) are looked for and used to limit the periods. Third, each period $T_i$ found is divided into $n$ equidistant sub-parts denoted $j$.
| |
| | |
| * '''Sort and average the data corresponding to each reference Position''' - The sub-part $j$ of the period $T_i$ corresponds to the sub-part $j$ of the period $T_{i+1}$ and so on. Each data set found in a sub-part $j$ will be averaged with the other sets found in the sub-parts $j$ of all other periods (see Fig.~\ref{fig:num_phase-resolved_method2_step2}). Finally, data sets of $n$ phase-averaged positions for the representative reference period are achieved.
| |
| | |
| The simulation data containing structure positions, pressure and
| |
| velocity fields, are generated every 150 time steps. According to the
| |
| frequency observed for the structure and the time-step size chosen
| |
| about 50 data sets are obtained per swiveling period. With respect to
| |
| the time interval predicted and the number of subparts chosen, the
| |
| data for each subpart are averaged from about 50 data sets. A
| |
| post-processing program is implemented based on the method described
| |
| above. It does not require any special treatment and thus the
| |
| aforementioned method to get the phase-resolved results is
| |
| straightforward.
| |
| | |
| The current experimental setup consists of the multiple-point triangulation
| |
| sensor described in Section~\ref{sec:Laser_Sensor} and the
| |
| synchronizer of the PIV system. Each measurement pulse of the PIV
| |
| system is detected in the data acquisition of the laser distance
| |
| sensor, which measures the structure deflection continuously with 800
| |
| profiles per second. With this setup, contrary to \cite{gomes2006},
| |
| the periods are not detected during the acquisition but in the
| |
| post-processing phase. After the run a specific software based on the
| |
| described method mentioned above computes the reference structure
| |
| motion period and sorts the PIV data to get the phase-averaged
| |
| results.
| |
|
| |
|
| | The described test case FSI-PfS-1a is a part of a series of reference |
| | test cases designed to improve numerical FSI codes. A second test case |
| | FSI-PfS-2a is described in Kalmbach and Breuer (2013). The geometry is |
| | similar to the first one: A fixed rigid cylinder with a plate clamped |
| | behind it. However, this time a rear mass is added at the extremity of |
| | the flexible structure and the material (para-rubber) is less |
| | stiff. The flexible structure deforms in the second swiveling mode and |
| | the structure deflections are completely two-dimensional and |
| | larger. |
| | |
| | ---- |
|
| |
|
| {{ACContribs | | {{ACContribs |
| | authors=Michael Breuer | | |authors=G. De Nayer, A. Kalmbach, M. Breuer |
| | organisation=Helmut-Schmidt Universität Hamburg | | |organisation= Helmut-Schmidt Universität Hamburg (with support by S. Sicklinger and R. Wüchner from Technische Universität München) |
| }} | | }} |
| | |
| {{UFRHeader | | {{UFRHeader |
| |area=2 | | |area=2 |