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Key Physics

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.

At an inflow velocity of $u_{\text{inflow}}=1.385$ m/s the displacements are quasi two-dimensional, symmetrical, reasonably large and well reproducible. The rubber plate deforms in the first swiveling mode. This FSI phenomenon is dominated by an instability-induced excitation (IIE) (Naudascher and Rockwell (1994)). IIE is provoked by flow instability which gives rise to flow fluctuations if a specific flow velocity is reached. These fluctuations and the resulting forces become well correlated and their frequency is close to a natural frequency of the flexible structure ("lock-in" phenomenon).

Based on the inflow velocity chosen ($u_{\text{inflow}}=1.385$ m/s) 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. Transition to turbulence means that from that point onwards the flow is three-dimensional and chaotic, and consists of a variety of different length and time scales. The low-frequency components of the turbulent flow dominate the coupled FSI problem, whereas the high-frequency contributions are visible in the fluid forces but are filtered out by the flexible structure. That is the reason why the signals for the deflections show the quasi-periodic signals without high-frequency fluctuations.

Numerical Modelling

CFD

• Discretization accuracy: In order to perform LES predictions it is required that spatial and temporal discretization are both at least of second-order accuracy. It is also important that the numerical schemes applied possesses low numerical diffusion (and dispersion) properties in order to resolve all the scales and not to dampen them out. A predictor-corrector scheme (projection method) of second-order accuracy forms the kernel of the fluid solver. In the predictor step an explicit Runge-Kutta scheme advances the momentum equation in time. This explicit method is chosen because of that accuracy, speed and low memory consumption. The discretization in space is done with second order central discretization scheme without any flux blending.

• Grid resolution: The second critical issue to perform LES is the grid resolution. The mesh near the wall, in the free-shear layers and also in the interior flow domain has to be enough fine. For wall-resolved LES the recommendations given by Piomelli and Chasnov (1996) should be followed or outperformed, e.g. y+ < 2. In the present investigation two different block-structured grids either for the subset and for the full case are used. In the first case the entire grid consists of about 13.5 million control volumes (CVs). For the full geometry the grid possesses about 22.5 million CVs. The first cell center is positioned at a distance of $\Delta$y/D=1.7 x 10$^{-2}$. For both setups it was found to be sufficient to resolve the flow accurately at walls as well as in the internal flow domain.

• Grid quality: The third point is the quality of the grid. Smoothness and orthogonality is a very important issue for LES computations. In order to capture separations and reattachments at the cylinder and on the plate reliably, the orthogonality of the curvilinear grid in the vicinity of the walls has to be high.

• Inlet boundary condition: 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.

• Outlet boundary condition: 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 $u_{\text{inflow}}$.

• Boundary conditions at the lateral sides: In the subset 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 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.

CSD

• Grid resolution: In the present investigation shell elements were used. A grid study on a simple structure case has shown that a mesh with $10\times 10$ shell elements for the subset case and with $10\times 30$ shell elements for the full case were sufficient.

• Non-linear deformation: In the present test case the deformations of the flexible structure are moderate. However, in the realized computations possible geometrical non-linearities are taken into account with the CSD solver Carat++.

• Time discretization: In the present study the standard Newmark algorithm was sufficient.

• Boundary conditions at the cylinder: At the rigid cylinder a clamped support is realized and all degrees of freedom are equal to zero.

• Boundary conditions at the extremity: On the downstream trailing-edge side, the rubber plate is free to move and all nodes have the full set of six degrees of freedom.

• Boundary conditions at the lateral sides: 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 as explained in details in Section Numerical CFD Setup.

FSI

• FSI subiterations: When the FSI phenomenon is fully developed, 5 FSI subiterations are sufficient to reach a FSI convergence criterion set to $\varepsilon _{\text{FSI}}=10^{-4}$ for the ${\text{L}}_{2}$ norm of the displacement differences.

• Unterrelaxation: To stabilize and speed up the FSI convergence an static unterrelaxation on the displacement is used. A constant underrelaxation factor of $\omega =0.5$ is considered for the displacements. The loads are transferred without underrelaxation.

Physical Modelling

CFD

• Wall-resolved LES: As mentioned above the flow in the present test case is in the sub-critical regime. Since in LES a large number of scales are resolved by the numerical method, this methodology is well suited. The near-wall regions are resolved too in order to obtain a reference LES solution. Later, wall functions can be used and compared.

CSD

• Shell: The flexible structure of the test case is a simple rubber plate. It can be modelized with different FEM elements (plate, shell, 3D-rigid elements...). In the present study shell elements were used because we would like to carry out more complex test cases with exactly the same software environment in the future and hence, will also deal with curved surfaces.

• St. Venant-Kirchhoff material model: Although the rubber material used for the test case shows a strong non-linear elastic behavior for large strains, the application of a linear elastic constitutive law is 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.

• Rayleigh damping: Due to the usual dynamic behavior of rubber, a certain level of damping has to be expected. To represent this adequately in the numerical simulation, various approaches are possible. However, damping in a vibrating structure is a complex phenomenon with various sources (Petersen, 2000). The goal within this benchmark description is to provide a simple modeling for the damping effects to enable the computation with standard finite-element codes without adapting them. Moreover, based on the experimental observations it can be observed, that within the proposed FSI benchmark only the lowest eigenmodes (nearly exclusively the first bending mode) of the structure are excited. A proper choice for the damping model is therefore the Rayleigh damping. More details can be found in De Nayer et al., 2013.

FSI

• Partionned approach with strong implicit coupling: 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. For 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 avoids instabilities due to the added-mass effect known from loose coupling schemes and maintains the explicit character of the time-stepping scheme beneficial for LES. For more details about this semi-implicit coupling scheme, we refer to Breuer et al., 2012.

Application Uncertainties

Application uncertainties can arise due to:

• CFD Boundary conditions in spanwise direction: The CFD boundary conditions at the lateral sides used in the full case setup are slip walls. In the reality there are of cource boundary layers here which are not resolved. The CFD boundary conditions at the lateral sides used in the subset case setup are periodic. It has been shown that the model is a good compromise between low CPU consumtion and good physical representation. Nevertheless, it still induces a certain level of uncertainty.

• CSD Boundary conditions in spanwise direction: The CSD boundary conditions used, particularly that the nodes have to stay in same z-plane, are not optimal. Because of these the material is numerically more stiff. These boundary conditions provoke uncertainties.

• Top and bottom slip wall boundary conditions: In the water channel the top and bottom sides are walls. In the simulations slip walls are used to reduce CPU-time consumption. However, this is an approximation which can lead to uncertainties.

• Phase-averaging method: To reduce statistical errors due to insufficient sampling to a reasonable minimum, the flow field was averaged in spanwise direction and for each phase over 27 cycles. Nevertheless, the results are not completely free of uncertainties arising from the averaging process.

Recommendations for Future Work

• The numerical computations were done with wall-resolved LES. It implies big meshes and also bis CPU-time consumption. Wall functions should be tested to reduced that.

• 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.

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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)

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