UFR 2-13 Test Case

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

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Test Case Studies

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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.

Numerical Simulation Methodology

The applied numerical method relies on an efficient partitioned coupling scheme developed for dynamic fluid-structure interaction problems in turbulent flows \citep{fsi-les-2012}. The fluid flow is predicted by an eddy-resolving scheme, i.e., the large-eddy simulation technique. FSI problems very often encounter instantaneous non-equilibrium flows with large-scale flow structures such as separation, reattachment and vortex shedding. For this kind of flows the LES technique is obviously the best choice~\citep{habil2002}. Based on a semi-implicit scheme the LES code is coupled with a code especially suited for the prediction of shells and membranes. Thus an appropriate tool for the time-resolved prediction of instantaneous turbulent flows around light, thin-walled structures results. Since all details of this methodology were recently published in \citet{fsi-les-2012}, in the following only a brief description is provided.

Computational fluid dynamics (CFD)

Within a FSI application the computational domain is no longer fixed but changes in time due to the fluid forces acting on the structure. This temporally varying domain is taken into account by the Arbitrary Lagrangian-Eulerian (ALE) formulation expressing the conservation equations for time-dependent volumes and surfaces. Here the filtered Navier-Stokes equations for an incompressible fluid are solved. Owing to the deformation of the grid, extra fluxes appear in the governing equations which are consistently determined considering the \emph{space conservation law (SCL)} \citep{demirdzic1988,demirdzic1990,lesoinne96}. The SCL is expressed by the swept volumes of the corresponding cell faces and assures that no space is lost during the movement of the grid lines. For this purpose the in-house code FASTEST-3D \citep{durstsch96,duscwe961} relying on a three-dimensional finite-volume scheme is used. The discretization is done on a curvilinear, block-structured body-fitted grid with collocated variable arrangement. A midpoint rule approximation of second-order accuracy is used for the discretization of the surface and volume integrals. Furthermore, the flow variables are linearly interpolated to the cell faces leading to a second-order accurate central scheme. In order to ensure the coupling of pressure and velocity fields on non-staggered grids, the momentum interpolation technique of \citet{rhie83} is used.

A predictor-corrector scheme (projection method) of second-order accuracy forms the kernel of the fluid solver. In the predictor step an explicit three substep low-storage Runge-Kutta scheme advances the momentum equation in time leading to intermediate velocities. These velocities do not satisfy mass conservation. Thus, in the following corrector step the mass conservation equation has to be fulfilled by solving a Poisson equation for the pressure-correction based on the incomplete LU decomposition method of \citet{stone68}. The corrector step is repeated (about 3 to 8 iterations) until a predefined convergence criterion ($\Delta\dot{m} < {\cal O}(10^{-9})$) is reached and the final velocities and the pressure of the new time step are obtained. In \cite{fsi-les-2012} it is explained that the original pressure-correction scheme applied for fixed grids has not to be changed concerning the mass conservation equation in the context of moving grids. Solely in the momentum equation the grid fluxes have to be taken into account as described above.

In LES the large scales in the turbulent flow field are resolved directly, whereas the non-resolvable small scales have to be taken into account by a subgrid-scale model. Here the well-known and most often used eddy-viscosity model, i.e., the \citet{smagorinsky} model is applied. The filter width is directly coupled to the volume of the computational cell and a Van Driest damping function ensures a reduction of the subgrid length near solid walls. Owing to minor influences of the subgrid-scale model at the moderate Reynolds number considered in this study, a dynamic procedure to determine the Smagorinsky parameter as suggested by \citet{germano} was omitted and instead a well established standard constant $C_s = 0.1$ is used.

Computational structural dynamics (CSD)

The dynamic equilibrium of the structure is described by the momentum equation given in a Lagrangian frame of reference. Large deformations, where geometrical non-linearities are not negligible, are allowed \citep{hojjat2010}. According to the preliminary considerations described in Section~\ref{sec:Structural_Tests}, 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~\citep{wuechner2005,bletzinger2005,dieringer2012}. For the dynamic analysis, different time-integration schemes are available, e.g., the implicit generalized-$\alpha$ method \citep{chung93}. 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 \citep{farhat98} implemented in the coupling interface CoMA \citep{coma}. 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 \citet{fsi-les-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 L$_2$ norm of the displacement differences between two FSI-subiterations normalized by the L$_2$ norm of the changes in the displacements between the current and the last time step drops below a predefined limit, e.g.\ $\varepsilon_{FSI} = 10^{-4}$ 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 \citet{fsi-les-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 (see Fig.~\ref{fig:Benchmark_FSI-PfS-1_full_and_subset_case}). 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 $\Delta y / D = 1.7 \times 10^{-2}$.

Fig. xx CFD Grid

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 \mbox{$w$} instead of \mbox{$W$}. Two main reasons are responsible for this simplification. If the gap would be considered in the simulation, the boundary layers of the channel walls had to be fully resolved, which is too costly. Moreover, the cells in this gap would be subjected to heavy distortions during the FSI simulation, which would massively complicate the purpose of grid adaptation during the movement of the flexible structure close to the side walls and may even lead to convergence problems of the coupled solver.

\begin{figure}[!htbp]

 \centering
 \includegraphics[width=0.7\linewidth,draft=\draftmode]
                 {Benchmark_Rubberplate_mesh_coarser_4}
 \caption{\label{fig:rubber_plate_mesh}x-y cross-section of the
   grids used for the simulation (Note that only every fourth grid
   line in each direction is displayed here).}

\end{figure}

In the x-y cross-section both grids are identical (see Fig.~\ref{fig:rubber_plate_mesh}). Since only every fourth grid line of the mesh is shown in Fig.~\ref{fig:rubber_plate_mesh}, the angles between grid lines and the transitions between the blocks appear to be worse than in the original grid. The numerical domain has a length of $L$. Since the inflow side is rounded in order to use a C-grid, the computational domain in front of the cylinder is slightly larger than in the test section depicted in Fig.~\ref{fig:rubber_plate_geom}. The grid points are clustered towards the rigid cylinder and the flexible structure using a stretching function according to a geometric series. The stretching factors are kept below 1.1 with the first cell center located at a distance of $\Delta y / D = 9 \times 10^{-4}$ from the flexible structure. Based on the wall shear stresses on the flexible structure the average $y^+$ 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.\\

\paragraph{CSD prediction}~

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 $10 \times 10$ quadrilateral four-node 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. These choices are derived from a grid independency study explained below.

The finite elements for the structure are 7-parameter shell elements with 6 degrees of freedom per node \citep{buechter1992,buechter1994,bischoff2000}. The specific degrees of freedom are the three deformations of the mid-surface and the three components of the difference vector of the shell director. Special treatments for the thickness stretch are included to avoid the undesirable effect which is called 'thickness locking' \citep{bischoff1997}. For the present test cases the ANS method is not activated and the EAS method is used according to the recommendations in~\cite{bischoff2004} and~\cite{bischoff1999}, to have an effective counter-measure against transverse shear-, membrane-, in-plane shear and thickness locking. For a detailed derivation of the element, an in-depth discussion of valid shell formulations and locking phenomena in shell element analysis, the reader is referred to these two studies. The shell elements used are formulated for large deformations, i.e., geometrically non-linear analysis. In the given benchmark scenario, the ratio of the thickness and the length of the thin structure, $h/l = 0.035 $, is smaller than $1/10$. So even a 5-parameter shell using the Reissner-Mindlin kinematics~\citep{reissner45,mindlin51} would be valid. The 7-parameter theory yields higher accuracy for the representation of through-the-thickness effects and is for the structure considered fully comparable to a solid model.

Based on the experimentally observed deformation pattern, a representative pure structure simulation test case with a very similar deformation pattern was chosen for grid independency studies of the structural model. This measure allows to avoid the high computational overhead of fully coupled FSI simulations. Thus, as a preliminary investigation for the choice of a proper structure mesh, a small test case with a constant pressure distribution on one plate surface is carried out using the material parameters defined in Section~\ref{sec:Material_parameters}. The magnitude of the pressure is chosen such that the final tip deformation is in the range of the maximally expected tip deflection of the flexible structure in the FSI case. This load scenario is computed with a sequence of systematically refined meshes (using quadrilateral four-node shell elements) for the structure used in the subset case (quadratic plate) and the full case (rectangular plate). The tip deformations obtained are summarized in Table~\ref{tab:grid_study}. Moreover, a cross-check by a simulation using a very fine mesh of solid elements shows a deviation in the tip displacement of less than 1~\%. As a result of this grid independency study, a mesh with 30 shell elements in streamwise direction can be considered as converged for both cases. Astonishingly, even a mesh with only 10 shell elements of the used type in streamwise direction could be seen as reasonable for a deflection comparable to the one observed in the FSI experiments. Thus in order to save CPU-time 10 $\times$ 10 and 10 $\times$ 30 shell elements are applied in the FSI prediction of the subset and the full case, respectively.

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:

• What numerical procedures were used (discretisation scheme and solver)?

• What grid resolution was used? Were grid sensitivity studies carried out?

• Did any of the analyses check or demonstrate numerical accuracy?

• Were sensitivity tests carried out to explore the effect of uncertainties in boundary conditions?

• If separate calculations of the assessment parameters using the same physical model have been performed and reported, do they agree with one another?

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Contributed by: Michael Breuer — Helmut-Schmidt Universität Hamburg

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