UFR 2-14 Best Practice Advice

Fluid-structure interaction in turbulent flow past cylinder/plate configuration II

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Key Physics

FSI-PfS-2a consists of the turbulent flow past a fixed rigid non-rotating cylinder with a flexible splitter plate and a rear mass. Compared to FSI-PfS-1a the setup is similar. However, the less stiff rubber and the addition of the rear mass change completely the governing mechanism responsible for the deformations of the structure.

At an inflow velocity of $u_{\text{inflow}}$ = 1.385 m/s the displacements are two-dimensional, symmetrical, large and well reproducible. The flexible structure deforms in the second swiveling mode (there are two wave nodes) with a frequency of $f_{\text{FSI}}=11.25$ Hz. The corresponding Strouhal number in the experiment is ${\text{St}}_{\text{FSI}}=f_{\text{FSI}}\,D/u_{\text{inflow}}=0.179$. Contrary to FSI-PfS-1a this FSI phenomenon is dominated by a movement-induced excitation (MIE) (Naudascher and Rockwell, 1994). MIE is directly linked to body movements and disappears if the body comes to rest. MIE represents a self-excitation: If a body is accelerated in a flow, fluid forces acting on this body are modified by the unsteady flow induced.

Based on the given inflow velocity and the cylinder diameter the Reynolds number is equal to Re = 30,470. The flow around the cylinder is in the so-called sub-critical regime: The boundary layers are still laminar and transition to turbulence occurs in the free shear layers evolving from the separated boundary layers behind the apex of the cylinder. From that point onwards transition to turbulence causes the flow to become three-dimensional and chaotic and to consist of a variety of different length and time scales. All these different scales lead to a wide range of frequencies, which are partially visible for example in the fluid forces acting on the structure. The highest frequencies are filtered out by the structure causing the signals for the deflections to show 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 possess low numerical diffusion (and dispersion) properties in order to resolve all the scales and not to damp 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 its accuracy, speed and low memory consumption. For discretization in space a second-order central discretization scheme without any flux blending is used.

• Grid resolution: The second critical issue in 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 fine enough. For wall-resolving LES the recommendations given by Piomelli and Chasnov (1996) should be followed or outperformed, e.g., $y^{+}<2,\;\Delta x^{+}<50,\;\Delta z^{+}<50-150$. In the present investigation a block-structured grid for the subset case is used. The entire grid consists of about 14 million control volumes (CVs). The first cell center is located at a distance of $\Delta y/D\approx 6.8\times 10^{-4}$. It was found to be sufficient to resolve the flow accurately at walls as well as in the free shear layers. Similar to the classical flow around a cylinder also in the present configuration it is important to resolve the region close to the separation point and the evolving shear layer region adequately.

• 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. For deforming grids such as in the present FSI case, it is furthermore crucial to keep a high quality grid after strong grid movements and deformations.

• 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, in general, small perturbations imposed at the inlet will not reach the cylinder anyway due to the coarseness of the grid at the outer boundaries. Therefore, all inflow fluctuations will be highly damped. However, since the flow is sub-critical and the inflow turbulence level measured in the experimental setup found to be rather small, the neglect of inflow perturbations is of no relevance.

• 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: A reasonable approximation already applied in Breuer et al. (2012) is to use periodic boundary conditions in spanwise direction for both the fluid and the structure. For LES predictions periodic boundary conditions represent an often used approach 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.

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 $30\times 10$ shell elements for the subset case was sufficient.

• Non-linear deformation: In the present test case the deformations of the flexible structure are large. Therefore, in the realized computations 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 trailing edge: 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: At the edges which are aligned to the main flow direction, special boundary conditions are used, as explained in details in Section Numerical CSD 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 L2 norm of the displacement differences.

• Unterrelaxation: To stabilize and speed up the FSI convergence a static unterrelaxation on the displacement is used. A constant underrelaxation factor of ω = 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 due to a moderate Reynolds number. Since in LES a wide range of scales is directly resolved by the numerical method, the influence of the SGS model is expected to be small (see below). Thus the LES methodology is well suited to generate reliable data (comparable to DNS). The near-wall regions are also well resolved allowing the application of the no-slip boundary condition. All these issues are decisive for the objective to obtain a reference LES solution. In future studies, it is also planned to investigate the influence of wall functions on the coupled FSI results.

• SGS model: A sensitivity study presented in De Nayer and Breuer (2014) permits to verify that the SGS model and its parameter does not strongly affect the simulation. Indeed, the Smagorinsky model (with $C_{s}=0.065,\ 0.1{\text{ and }}0.18$), the dynamic model of Germano and the WALE model (with $C_{W}=0.33$) deliver similar results for the test case FSI-PfS-2a. This is due to the moderate Reynolds number and the fine grid applied. The classical Smagorinsky model with a standard parameter set to $C_{s}=0.1$ can be used.

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. A simple choice is the Rayleigh damping. A sensitivity study carried out in De Nayer and Breuer (2014) shows a limited influence of the structural damping on the FSI results. Indeed, the damping slightly modifies the FSI frequency, but the displacement extrema are not affected. This can be explained by the nature of the FSI problem: As written before, FSI-PfS-2a is a movement-induced excitation (MIE). The damping generated by the viscous fluid also exceeds the structural damping by orders of magnitude.

FSI

• Partitioned 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 are periodic. The subset model induces a certain level of uncertainty. However, due to the steel rear mass the deformation in spanwise direction are quasi-nonexistent. The deformation of the flexible structure are quasi-2D. Therefore, the use of periodic boundaries is recommended.

• 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 rubber material is numerically more stiff. However, due to the steel rear mass the z-deformation are quasi-nonexistent. Therefore, for FSI-PfS-2a this special treatment at lateral sides is usable.

• 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 numerical flow field was averaged in spanwise direction and for each phase over 15 cycles. Nevertheless, the numerical results are not completely free of uncertainties arising from the averaging process.

• PIV method: The uncertainties for the velocity expected by the PIV method is calculated to about 0.076 m/s (Kalmbach, 2014).

Recommendations for Future Work

• The numerical computations were conducted based on wall-resolved LES. It implies very fine meshes and also large CPU-time consumption. Wall functions should be tested to reduce this effort. Corresponding studies are in progress.

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Contributed by: Andreas Kalmbach, Guillaume De Nayer, Michael Breuer — Helmut-Schmidt Universität Hamburg

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