UFR 2-13 Test Case

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Test Case Study

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. 1). The cylinder has a diameter D=0.022m. It is positioned in the middle of the experimental test section with ${\displaystyle H_{c}\operatorname {=} H/2\operatorname {=} 0.120\,m}$ ${\displaystyle (H_{c}/D\approx 5.45)}$, whereas the test section denotes a central part of the entire water channel (see Fig. 2). Its center is located at ${\displaystyle L_{c}\operatorname {=} 0.077\,m}$ ${\displaystyle (L_{c}/D\operatorname {=} 3.5)}$ downstream of the inflow section. The test section has a length of ${\displaystyle L\operatorname {=} 0.338\,m}$ ${\displaystyle (L/D\approx 15.36)}$, a height of ${\displaystyle H\operatorname {=} 0.240\,m}$ ${\displaystyle (H/D\approx 10.91)}$ and a width ${\displaystyle W\operatorname {=} 0.180\,m}$ ${\displaystyle (W/D\approx 8.18)}$. The blocking ratio of the channel is about ${\displaystyle 9.2\%}$. The gravitational acceleration ${\displaystyle g}$ 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 ${\displaystyle l\operatorname {=} 0.060\,m}$ ${\displaystyle (l/D\approx 2.72)}$ and a width ${\displaystyle w\operatorname {=} 0.177\,m}$ ${\displaystyle (w/D\approx 8.05)}$. Therefore, in the experiment there is a small gap of about ${\displaystyle 1.5\times 10^{-3}\,m}$ between the side of the deformable structure and both lateral channel walls. The thickness of the plate is ${\displaystyle h\operatorname {=} 0.0021\,m}$ ${\displaystyle (h/D\approx 0.09)}$. 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 as follows:

Fig. 1: Geometrical configuration of the FSI-PfS-1a Benchmark.

Description of the water channel

In order to validate numerical FSI simulations based on reliable experimental data, the special research unit on FSI (Bungartz et al. (2006, 2010)) designed and constructed a water channel (Göttingen type, see Fig. 2) for corresponding experiments with a special concern regarding controllable and precise boundary and working conditions Gomes et al. (2006, 2010, 2011, 2013). The channel (2.8 m x 1.3 m x 0.5 m, fluid volume of 0.9 m³) 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 Fig. 1. 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 ${\displaystyle u_{\text{max}}=6}$ m/s is possible. To prevent asymmetries the gravity force is aligned with the x-axis in main flow direction.

Fig. 2: Sketch of the flow channel (dimensions given in mm).

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.

Fig. 3: Experimental displacements of the structure extremity versus the inlet velocity.

Fig. 3 depicts the measured extrema of the structure displacement versus the inlet velocity and Fig. 4 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 Laser distance sensor. The entire diagrams are the result of a measurement campaign in which the inflow velocity was consecutively increased from 0 to ${\displaystyle 2.2}$ m/s. At an inflow velocity of ${\displaystyle u_{\text{inflow}}=1.385}$ m/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 ${\displaystyle {\text{Re}}=30,470}$.

Fig. 4: Experimental measurements of the frequency and the corresponding Strouhal number of the FSI phenomenon versus the inlet velocity.

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. 5). The velocity components ${\displaystyle {\overline {u}}}$ and ${\displaystyle {\overline {v}}}$ are measured with two-component laser-Doppler velocimetry (LDV) along the y-axis in the middle of the measuring section at ${\displaystyle x/D=4.18}$ and ${\displaystyle z/D=0}$. It can be assumed that the velocity component ${\displaystyle {\overline {w}}}$ shows a similar velocity profile as ${\displaystyle {\overline {v}}}$. Furthermore, a low inflow turbulence level of ${\displaystyle {\text{Tu}}_{\text{inflow}}={\sqrt {{\frac {1}{3}}~\left({\overline {u'^{2}}}+{\overline {v'^{2}}}+{\overline {w'^{2}}}\right)}}/u_{\text{inflow}}=0.02}$ is measured. All experiments were performed with water under standard conditions at ${\displaystyle T=20^{\circ }\,C}$. The flow parameters are summarized in the following table:

Inflow velocity ${\displaystyle u_{\text{inflow}}=1.385\,m/s}$ ${\displaystyle \rho _{f}=1000\,kg/m^{3}}$ ${\displaystyle \mu _{f}=1.0\times 10^{-3}\,Pa\,s}$

Fig. 5 Profiles of the mean streamwise and normal velocity as well as the turbulence level at the inflow section of the water channel.

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 model.

The density of the rubber material can be determined to be ${\displaystyle \rho _{\text{rubber plate}}}$=1360 kg/m${\displaystyle ^{3}}$ 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 ${\displaystyle \nu }$. 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 ${\displaystyle 0.5}$. Material tests of the manufacturer and complementary experimental/numerical structure test studies (dynamic and decay test scenarios) indicate that the Young's modulus is E=16 MPa and the Poisson's ratio is ${\displaystyle \nu }$=0.48 (a detailed description of the structure tests is available in De Nayer et al., 2014).

Density ${\displaystyle \rho _{\text{rubber plate}}=1360\,kg/m^{3}}$ ${\displaystyle E=16\,MPa}$ ${\displaystyle \nu =0.48\,}$

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 Gomes et al. (2006, 2010, 2013) 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 Gomes et al. (2006, 2010, 2013) 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 (Adrian, 1991) setup 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 ${\displaystyle \Delta }$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, with the displacements obtained and the time interval ${\displaystyle \Delta }$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 phase-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 ${\displaystyle d_{\text{avg,SHGS}}}$=10~µm and a density of ${\displaystyle \rho _{SHGS}}$ = 1400 kg m${\displaystyle ^{-3}}$ as tracer particles. To prove the following behavior of these particles a Stokes number Sk=1.08 and a particle sedimentation velocity ${\displaystyle u_{\text{SHGS}}=2.18\times 10^{-5}~{\text{m/s}}}$ is calculated. With this Stokes number and a particle sedimentation velocity which is much lower than the expected velocities in the experiments, it is ensured that the tracers closely follow the fluid flow. 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 (The processing of the phase-resolved fluid velocity fields involving the structure deflections is described in Section 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 ${\displaystyle \mu }$ 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 ${\displaystyle \Delta }$ t=200 ${\displaystyle \mu }$s. Laser and camera were controlled by a TSI synchronizer (TSI 610035) with 1 ns resolution.

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. 6). 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 800 profiles per second. The laser line was positioned in a horizontal (x/D = 3.2, see Fig. 6(a)) and in a vertical alignment(z/D = 0, see Fig. 6(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.

Fig. 6: Setup and alignment of multiple-point laser sensor on the flexible structure in a) z-direction and b) x-direction.

Numerical Simulation Methodology

The applied numerical method relies on an efficient partitioned coupling scheme developed for dynamic fluid-structure interaction problems in turbulent flows (Breuer et al, 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 (Breuer, 2002). 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 Breuer et al, 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 space conservation law (SCL) (Demirdzic 1988 and 1990, Lesoinne, 1996). 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 (Durst et al, 1996a, b) 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 Rhie and Chow (1983) 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 Stone (1968). The corrector step is repeated (about 3 to 8 iterations) until a predefined convergence criterion (${\displaystyle \Delta {m}<{O}(10^{-9})}$) is reached and the final velocities and the pressure of the new time step are obtained. In Breuer et al (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 Ssmagorinsky (1963) 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 Germano et al (1991) was omitted and instead a well established standard constant ${\displaystyle C_{s}=0.1}$ is used.

The CFD prediction determines the forces on the structure and delivers them to the CSD calculation. In the other direction the CSD prediction determines displacements at the moving boundaries of the computational domain for the fluid flow. The task is to adapt the grid of the inner computational domain based on these displacements at the interface. For moderate deformations algebraic methods are found to be a good compromise since they are extremely fast and deliver reasonable grid point distributions maintaining the required high grid quality. Thus, the grid adjustment is performed based on a transfinite interpolation (Thompson et al., 1985). It consists of three shear transformations plus a tensor-product transformation.

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 (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-${\displaystyle \alpha }$ 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. Furthermore, special care is given to prevent locking phenomena by applying the well-known Assumed Natural Strain (ANS) (Hughes and Tezduyar, 1981; Park and Stanley, 1986) and Enhanced Assumed Strain (EAS) methods (Bischoff et al., 2004).

Both, shell and membrane elements reflect geometrically reduced structural models with a two-dimensional representation of the mid-surface which can describe the three-dimensional physical properties by introducing mechanical assumptions for the thickness direction. Due to this reduced model additional operations are required to transfer information between the two-dimensional structure and the three-dimensional fluid model. Thus in the case of shells, the surface of the interface is found by moving the two-dimensional surface of the structure half of the thickness normal to the surface on both sides and the closing of the volume (Bletzinger et al., 2006).. On these two moved surfaces the exchange of data is performed consistently with respect to the shell theory (Hojjat et al., 2010).

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 ${\displaystyle L_{2}}$ norm of the displacement differences between two FSI-subiterations normalized by the ${\displaystyle L_{2}}$ norm of the changes in the displacements between the current and the last time step drops below a predefined limit, e.g. ${\displaystyle \varepsilon _{FSI}=10^{-4}}$ 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 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 ${\displaystyle \Delta }$z/D=1.7 x 10${\displaystyle ^{-2}}$.

Fig. 7: 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 ${\displaystyle \Delta }$y/D=9 x 10${\displaystyle ^{-4}}$ from the flexible structure. Based on the wall shear stresses on the flexible structure the average y${\displaystyle ^{+}}$ 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. Ther

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. 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 plate 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 ${\displaystyle \Delta t_{f}=2\times 10^{-5}s~(\Delta t_{f}^{\ast }=1.26\times 10^{-3})}$ in dimensionless form using ${\displaystyle u_{\text{inflow}}}$ and D as reference quantities) is chosen and the same time-step size is applied for the structural solver based on the generalized-${\displaystyle \alpha }$ method with the spectral radius ${\displaystyle \varrho _{\infty }}$=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 ${\displaystyle \omega }$=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 ${\displaystyle \varepsilon _{FSI}=10^{-4}}$ for the ${\displaystyle L_{2}}$ 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 ${\displaystyle f={\bar {f}}+{\tilde {f}}+{f}\prime }$, where ${\displaystyle {\bar {f}}}$ describes the global mean part, ${\displaystyle {\tilde {f}}}$ the quasi-periodic part and ${\displaystyle {f}\prime }$ a random turbulence-related part (Reynolds et al., 1972; Cantwell et al., 1983). This splitting can also be written in the form ${\displaystyle f=+f\prime }$, where ${\displaystyle }$ 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 ${\displaystyle f\prime }$ 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 ${\displaystyle }$ 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 Gomes et al. (2006). 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 (see Section Full case vs. subset case for a discussion on the deviation from 2D) 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 plate 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 trailing edge of the rubber plate. 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 (${\displaystyle U_{y}}$=0) are looked for and used to limit the periods. Third, each period ${\displaystyle T_{i}}$ 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 ${\displaystyle T_{i}}$ corresponds to the sub-part j of the period ${\displaystyle T_{i+1}}$ 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. 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 Laser distance 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 Gomes et al. (2006), 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. A more detailed description of the phase-averaging method is available in De Nayer et al. (2014).

Contributed by: G. De Nayer, A. Kalmbach, M. Breuer — Helmut-Schmidt Universität Hamburg (with support by S. Sicklinger and R. Wüchner from Technische Universität München)

© copyright ERCOFTAC 2021 ${\displaystyle Insertformulahere}$