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Fig. 3. Influence of the thickness (right) and the penalization parameter (left). Both figures show the deflection line of the beam. The mean flow has an orientation of 45‘ Although the thickness varies by a factor of 10, the difference in deformation between the thinnest (red) and thickest (green) beam is less than 8%. The penalization paramete is decreased from ¢ = 10~? (black) to ¢=5 x 107° (red) and the difference is even smaller. We can conclude that both t and « do not have a strong influence on the results. (Fo interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Figure 3 Influence of the thickness (right) and the penalization parameter (left). Both figures show the deflection line of the beam. The mean flow has an orientation of 45‘ Although the thickness varies by a factor of 10, the difference in deformation between the thinnest (red) and thickest (green) beam is less than 8%. The penalization paramete is decreased from ¢ = 10~? (black) to ¢=5 x 107° (red) and the difference is even smaller. We can conclude that both t and « do not have a strong influence on the results. (Fo interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Related Figures (9)

Fig. 1. Fluid-structure configuration with constant uniform mean flow and a clamped flexible beam. The rectangular computational domain Q comprises the fluid domain Q,, the solid domain Q, and the sponge layer Qxp. respectively. Here, Q; is the solid density, E its Young’s modulus and h the (physical) height of the cross section, which is not the same as the numerical thickness t that is used to define the geometry of the problem, cf. Fig. 1. Due to the normalization, both uw and y charac- terize the solid material as much as the fluid flow.
Fig. 1. Fluid-structure configuration with constant uniform mean flow and a clamped flexible beam. The rectangular computational domain Q comprises the fluid domain Q,, the solid domain Q, and the sponge layer Qxp. respectively. Here, Q; is the solid density, E its Young’s modulus and h the (physical) height of the cross section, which is not the same as the numerical thickness t that is used to define the geometry of the problem, cf. Fig. 1. Due to the normalization, both uw and y charac- terize the solid material as much as the fluid flow.
3.1.3. Time discretization Fig. 2. Flow past an object without (left) and with (right) sponge technique. Shown is vorticity with superimposed white streamlines. Only part of the computational domai is shown. The technique used in the present work relies on using a smooth mask rather than a discontinuous one. Hence we choose a continuous approximation to the Heaviside function, with a cer- tain width of the transition from one to zero. The width of this smoothing layer decreases when the resolution increases, so the smooth approximation gets sharper and converges, in the limit of small Ax, to the Heaviside function. This smooth function can then be displaced by an increment as small as desired and not limited by the spatial resolution. Among the various possible approximations we choose We add a penalization term for the vorticity, i.e. 7¢/&p-(@ — Wo), to Eq. (5) in order to impose a Dirichlet condition on o, typically with wo = 0. The mask function for the sponge, 7%», is defined like a frame around the computational domain, as illustrated in Fig. 1. There is no need for the use of two distinct penalization parameters, since both have the same convergence properties and imply the same restrictions on the time step. Hence we set Epc = €. Using this technique is particularly suitable for free-flow configurations and therefore used for the simulations in Section 4. Fig. 2 illustrates the effect of the sponge. In the periodic case (left), the wake of the obstacle re-enters the domain. The penalization term for the vorticity cancels the wake and prevents it from an interaction with the obstacle.
3.1.3. Time discretization Fig. 2. Flow past an object without (left) and with (right) sponge technique. Shown is vorticity with superimposed white streamlines. Only part of the computational domai is shown. The technique used in the present work relies on using a smooth mask rather than a discontinuous one. Hence we choose a continuous approximation to the Heaviside function, with a cer- tain width of the transition from one to zero. The width of this smoothing layer decreases when the resolution increases, so the smooth approximation gets sharper and converges, in the limit of small Ax, to the Heaviside function. This smooth function can then be displaced by an increment as small as desired and not limited by the spatial resolution. Among the various possible approximations we choose We add a penalization term for the vorticity, i.e. 7¢/&p-(@ — Wo), to Eq. (5) in order to impose a Dirichlet condition on o, typically with wo = 0. The mask function for the sponge, 7%», is defined like a frame around the computational domain, as illustrated in Fig. 1. There is no need for the use of two distinct penalization parameters, since both have the same convergence properties and imply the same restrictions on the time step. Hence we set Epc = €. Using this technique is particularly suitable for free-flow configurations and therefore used for the simulations in Section 4. Fig. 2 illustrates the effect of the sponge. In the periodic case (left), the wake of the obstacle re-enters the domain. The penalization term for the vorticity cancels the wake and prevents it from an interaction with the obstacle.
Fig. 4. Stability map in the Re-u* plane for ju = 1/3. Inviscid [18] (--) and viscous linear stability analysis [3] (—) Symbols indicate present simulations, found to be stable (C1) or in an either periodic (O) or chaotic (*) fluttering state. Insets show corresponding snapshots of the absolute vorticity |j@| at Re = 750.
Fig. 4. Stability map in the Re-u* plane for ju = 1/3. Inviscid [18] (--) and viscous linear stability analysis [3] (—) Symbols indicate present simulations, found to be stable (C1) or in an either periodic (O) or chaotic (*) fluttering state. Insets show corresponding snapshots of the absolute vorticity |j@| at Re = 750.
Fig. 5. The influence of the Reynolds number. Phase-diagram of the trailing edge in the displacement-velocity plane (y,u,) (left column) and a vorticity snapshot (right column) with superimposed white streamlines. Only part of the computational domain is shown. The Reynolds number is Re = 25,100,200,1000 (from top to bottom). The point in phase space belonging to the snapshot is marked with a red star. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The influence of the Reynolds number. Phase-diagram of the trailing edge in the displacement-velocity plane (y,u,) (left column) and a vorticity snapshot (right column) with superimposed white streamlines. Only part of the computational domain is shown. The Reynolds number is Re = 25,100,200,1000 (from top to bottom). The point in phase space belonging to the snapshot is marked with a red star. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Amplitude and frequency of y-displacement and frequencies of drag and lift forces as a function of the Reynolds number. The weight ratio is 4. = 1/3 and the reduced inflow velocity is u* = 14.
Amplitude and frequency of y-displacement and frequencies of drag and lift forces as a function of the Reynolds number. The weight ratio is 4. = 1/3 and the reduced inflow velocity is u* = 14.
Fig. 6. Periodic state: Re = 750 and u®* = 14. Left side: (A-G) time history of a typical oscillation period with a snapshot every At = 0.1. Top: the whole vorticity field at T = 84.5 Bottom, middle: Phase diagram of the end point, colored with the absolute acceleration. Bottom, right: time history of the deflection line for one period of motion, witl At = 0.05, color ranging from dark to light blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.
Fig. 6. Periodic state: Re = 750 and u®* = 14. Left side: (A-G) time history of a typical oscillation period with a snapshot every At = 0.1. Top: the whole vorticity field at T = 84.5 Bottom, middle: Phase diagram of the end point, colored with the absolute acceleration. Bottom, right: time history of the deflection line for one period of motion, witl At = 0.05, color ranging from dark to light blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.
Fig. 7. Left: Strouhal number St = 2 zAs of the displacement, for simulations resulting in periodic behavior. Re = 200 (O), Re = 500 (LD), Re = 750 (v), Re = 1000 (+*) and values presented by Michelin et al. (x) [18]. “The dashed line corresponds to the average Strouhal number. Right: spectra of the integral lift force for Re = 750 in a periodic (u* = 14, light red) and chaotic (u* = 30, blue) state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Left: Strouhal number St = 2 zAs of the displacement, for simulations resulting in periodic behavior. Re = 200 (O), Re = 500 (LD), Re = 750 (v), Re = 1000 (+*) and values presented by Michelin et al. (x) [18]. “The dashed line corresponds to the average Strouhal number. Right: spectra of the integral lift force for Re = 750 in a periodic (u* = 14, light red) and chaotic (u* = 30, blue) state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Chaotic state: Re = 750 and u* = 30. Left side: (A-G) time history of a snapping event with a snapshot every At = 0.05. Top: the whole vorticity field at T= 11.70, exhibiting the typical vortex pairs of different sign (X and Y), and the strongly asymmetric vortex pair resulting from the snapping event (Z). Bottom, middle: Phase diagram of the end point, colored with absolute acceleration. Bottom, right: time history of the deflection line with At = 0.05, color ranging from dark to light blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Chaotic state: Re = 750 and u* = 30. Left side: (A-G) time history of a snapping event with a snapshot every At = 0.05. Top: the whole vorticity field at T= 11.70, exhibiting the typical vortex pairs of different sign (X and Y), and the strongly asymmetric vortex pair resulting from the snapping event (Z). Bottom, middle: Phase diagram of the end point, colored with absolute acceleration. Bottom, right: time history of the deflection line with At = 0.05, color ranging from dark to light blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Considering the u* =22 case, a different behavior can be ob- served. The drag force exhibits a periodic behavior until T ~ 22, For the u* = 20 case, one distinct dominant frequency can be found, corresponding to a horizontal band in wavelet-space. This frequency does not alter significantly, albeit some deviations from Fig. 9. Wavelet analyses of the drag force for case u* = 20 (left) and u* = 22 (right). The modulus of the complex valued Morlet wavelet coefficients are plotted as a function position and scale. The original signal is plotted on the top.
Considering the u* =22 case, a different behavior can be ob- served. The drag force exhibits a periodic behavior until T ~ 22, For the u* = 20 case, one distinct dominant frequency can be found, corresponding to a horizontal band in wavelet-space. This frequency does not alter significantly, albeit some deviations from Fig. 9. Wavelet analyses of the drag force for case u* = 20 (left) and u* = 22 (right). The modulus of the complex valued Morlet wavelet coefficients are plotted as a function position and scale. The original signal is plotted on the top.
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Parallel ComputingBiological Fluid DynamicsSpectral MethodsComputational Fluid MechanicsFluid structure interactionLarge Eddy SimulationDirect Numerical SimulationAerodynamicsMultiphase flowsTransport Phenomena in Porous Media
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