Development of immersed boundary methods for isothermal and thermal flows 8

  Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Chapter Applications of Developed IBM Solvers to Simulate Three Dimensional Moving Boundary Flows In this chapter, the new boundary condition-enforced IBM proposed in the framework of NS solver is utilized to study the three-dimensional moving boundary flows where the immersed objects are undertaking complex and prescribed motions. Two biomimic problems are considered. The first one discusses the flow behaviors around a heaving and pitching finite span foil and the second one investigates the hydrodynamic performances around a fish-like swimming body. 8.1 Incompressible flow over a heaving and pitching finite span foil The three-dimensional incompressible flow around a finite span foil under a heaving and pitching motion has been investigated by several researchers. Von Ellenrieder et al. (2003) conducted a visualization study on the three-dimensional flow behaviors behind a rectangular heaving-and-pitching foil of aspect ratio (span-to-chord) 3.0 at a Reynolds number of 164 and found 276      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows sets of rings- and loops-like structures. A range of Strouhal numbers (between 0.2 and 0.4), pitch amplitudes (between 0D and 20D ) and heave/pitch phase angles (between 60D and 120D ) were tested which showed that the variation of these parameters had visible effects on the wake structures. Blondeaux et al. (2005), by employing a panel method, numerically investigated the vortex topologies behind the same foil as in the experiment of von Ellenrieder et al. (2003). Based on the numerical calculations, they identified that the vortex rings produced by the foils were shed every half a cycle. However, the vortex topologies observed in their numerical work were not completely consistent with those in the visualization of von Ellenrieder et al. (2003). Buchholz et al. (2006) performed flow visualization for the interrogation of wake structures generated by a rigid flat pitching panel of aspect ratio 0.54, from which they observed horseshoe vortices of alternating sign shed twice per flapping cycle. They also identified the robustness of the wake patterns with respect to changes in Reynolds number, aspect ratio and amplitude. Dong et al. (2006) carried out a detailed numerical analysis on the hydrodynamic mechanisms associated with the thrust generation of thin ellipsoidal flapping foils. The thrust and propulsive efficiency of these foils were also examined over a range of span-chord-ratios and Strouhal numbers as well as pitch-bias angles. They indicated a monotonic increase of thrust coefficient with aspect-ratio and Strouhal number for all foils. Shao et al. (2010) numerically predicted the wake structures and propulsion performance of a finite-span flapping foil with 277      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows five different aspect ratios at Re = 200 and reported that both the thrust coefficient and propulsion efficiency increase with increasing aspect ratio. In this subsection, the wake structure and propulsion performance of rigid finite-span foils heaving and pitching in a free stream is studied. The geometry of the foil is shown in Fig. 8.1. It has a rectangular platform and a symmetric cross section whose profile is similar to a NACA0012 profile. The chord and span width are denoted by c and W , while the area of the platform is denoted by S . The aspect ratio of the foil is defined as: AR = area of the rectangular platform S W = = (length of the chord) c c (8.1) The foil is placed in a free stream with uniform incoming velocity U ∞ , and oriented with x -axis along the streamwise direction and y -axis along the spanwise direction (Fig. 8.1). In the current simulations, the foil undergoes a combination of harmonic pitching (angular oscillation) and heaving (vertical oscillation) motion where the foil pitches about its mass center according to ϑ = ϑ0 sin(2π fct +ψ ) , (8.2) while the foil center, at the same time, heaves in the z -direction according to h = h0 sin(2π fct ) . (8.3) In Eqs. (8.2) and (8.3), ϑ0 and h0 are pitch and heave amplitudes, and ψ is the phase difference between the pitching and heaving motion. f c is the oscillating frequency. To better illustrate the combined “pitching-and-heaving” 278      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows motion, a schematic diagram is appended in Fig. 8.2 in its side view. By taking the chord length c and free stream velocity U ∞ as the reference length and velocity scales respectively, the problem can be characterized by several other key dimensionless parameters in addition to AR , ϑ0 and ψ : the Reynolds number Re = U∞c ν , the Strouhal number St = 2h0 f c U∞ and normalized heave amplitude h0 / c . The present study focuses on the variations of the wake structures and propulsion performance with respect to the aspect ratio AR and Strouhal number St at Reynolds number Re = 200 . The variation range of AR and St is summarized in Table 8.1. Other involved D parameters are set as h0 / c = 0.5 , ϑ0 = 30D and ψ = 90 . A computational domain of size 15c × 10c × 10c is chosen for the simulations, where the mass center of the foil is initially located at ( 5c , 5c , 5c ). As always, non-uniform meshes are employed for domain discretization, with a fine resolution of Δx = Δy = Δz = h = c / 40 covering the region swept by the flapping motion of the foil. For the time integration, a step size of Δt = × 10−4 is used. To have an idea of the wake evolution and vortex shedding process of this rectangular flapping foil, the vortex structures in their λ2 -definition at four phases in one flapping cycle is plotted in Fig. 8.3 for AR = and St = 0.6 . The left column gives the perspective views and the right column gives the 279      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows side view. In the first half cycle, the foil is first moving upward and then back to the equilibrium position. During its upward motion, it is observed that the flow separates from the leading edge along the lower foil surface (Fig. 8.3(a)). The free shear layer along the upper foil surface, on the other hand, rolls up and forms a negative vortex structure in the clockwise rotation (Fig. 8.3(b)). When the foil reverses its motion, this negative vortex structure is shedding from the trailing edge and a new negative vorticity is generated along the upper surface (Fig. 8.3(c)). At the same time, the shed vortex connects with the two tip vortices from the span ends, forming a vortex ring or vortex loop. In the second half cycle, the foil first heaves downward to the lowest position (Fig. 8.3 (d)) and then back to the central position again (Fig. 8.3(e)). The evolution of vortex structures in this half cycle follows a mirror image of those in the first half cycle. Thereafter, the downstream wake of this foil consists of two sets of vortex rings. These observations are consistent with the findings reported by Dong et al. (2006) for a thin ellipsoidal flapping foil and Shao et al. (2010) for the same rectangular foil. A full visualization is shown in Fig. 8.4 by plotting a top view of the vortex structure at the instant corresponding to Fig. 8.3(d). As the vortex rings are convected downstream, a narrowing of their spanwise lengths is observed, a phenomenon which has also been noted in the simulations of Blondeaux et al. (2005) for a similar rectangular flapping foil and in the experiments of 280      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Buchholz & Smits (2006) for a pitching plate. Dong et al. (2006) has also noticed this feature for their ellipsoidal foil. The general fluid dynamic behaviors for this rectangular foil (taking AR = and St = 0.6 as a representative) are also viewed by examining contours of the spanwise vorticity and contours of the mean streamwise velocity along the spanwise symmetry plane, which are respectively presented in Fig. 8.5 and Fig. 8.6. In Fig. 8.5, the formation of an inverse Karman vortex street is clearly shown, which, as discussed earlier, is always associated with the thrust-production. Contours of the mean streamwise velocity in Fig. 8.6, where the mean value was calculated over several consecutive flapping cycles after the steady force behaviors have been reached, show the existence of a single streamwise directed jet immediately behind the trailing edge of the foil. This high-intensity jet develops into a bifurcated one about three chord-lengths downstream of the trailing edge in the wake. The most significant wake behavior produced by the finite-span rectangular flapping foil has been discussed. Based on this knowledge, the effect of aspect ratio and Strouhal number is examined in the following. The aspect ratio analysis is limited to St = 0.6 . The vortex structures at different aspect ratios ( AR = 1, 2, and ) are given in Fig. 8.7 in 281      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows perspective and top visualizations. By comparing them with the case of AR = in Fig. 8.3, it is noted that at small AR , i.e. AR = , and 3, the wakes are dominated by two sets of vortex rings. The two sets of vortex rings separate from each other in the wake and convect downstream at an oblique angle to the wake centerline. Furthermore, the smaller the aspect ratio, the more circular the vortex rings and the larger the oblique angle. At large AR , i.e. AR = and 5, due to the large span size, the tip vortices are not strengthened enough to merge together and the wake is dominated by two sets of vortex loops. These vortex loops, as can be clearly observed in Fig. 8.7(c) and 8.7d), wrap end-to-end with their neighbors and keep twined together in the wake. Despite of the differences, the wake structures share a number of interesting features. For all the cases, the vortex rings/loops contract themselves in the spanwise direction as they move downstream. However, the streamwise length of the vortex rings/loops for different AR remains nearly the same. Moreover, the vortices seem to shed from these flapping foils almost at the same frequency, by noticing the same number of vortex rings in each case. Fig. 8.8 presents contours of the mean streamwise velocity in the spanwise symmetry plane ( y = ) for AR = 1, 2, and . Just as the corresponding plot for AR = , the wake in each case induces a jet behind the flapping foil. However, the topologies of the wake are quite different from each other. For 282      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows cases of small span size of AR = and 2, the jet bifurcates itself downstream of the trailing edge so that the wakes in Figs. 8.8(a) and 8.8(b) show a bifurcated shape. A closer examination shows that the bifurcation location moves downwards as the span size of the flapping foil increases. For example, at AR = , the jet bifurcates at about one chord-length downstream of the trailing edge while at AR = the bifurcation happens around two chord-lengths away. Compared with the case of AR = shown in Fig. 8.6, the bifurcations in the previous two low- AR cases appear earlier and their bifurcated patterns are clearer as well. In Figs. 8.8(c) and 8.8(d), no clear bifurcation is observed and a single jet maintains up to about two or three chord-lengths behind the trailing edge, showing that a quite distinct wake pattern is produced for high aspect ratios of AR = and 5. A roughly quantitative analysis reveals that the strength of the jet increases with an increase in the span size, which could be fairly consistent with the above qualitative discussions. To evaluate the effect of Strouhal number, simulations at three other values of St = 0.35, 0.8 and 1.0 are carried out for AR = . The top views of the vortex structures for these cases are presented in Fig. 8.9. In combination with the plot in Fig. 8.4 for St = 0.6 , it is noted that the wake at the low-flapping-frequency case of St = 0.3 is rather different from the other three. At St = 0.3 , as shown in Fig. 8.9(a), no linkage exists between the tip 283      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows vortices so that the vortices appear as loops rather than rings, which could be attributed to the lower strength of the tip vortices at this low flapping frequency. With low strength, the interaction between the tip vortices is weakened and they are able to be combined together and form a ring. The three larger- St cases in Fig. 8.4 and Figs. 8.9(b)-(c) share similar vortex structures. The two tip vortices are linked together and the vortex rings are clearly displayed. Moreover, it is observed that as the Strouhal number increases, the spanwise length of the vortices is reduced (for clarity, consider comparison of the spanwise spacing between the second pair tip vortices in each figure). This is because the tip vortices have different vortex strengths at different flapping frequencies. At higher ones like St = 1.0 , they are capable of moving closer together under their mutual induction. As previously, contours of the mean streamwise velocity in plane y = are plotted. At the low frequency case of St = 0.3 in Fig. 8.10(a), a single jet region with relatively low streamwise velocity appears immediately behind the trailing edge, which does not seem to bifurcate in the downstream wake. At higher frequency cases of St = 0.8 and 1.0 in Figs. 8.10(b) and 8.10(c), a diamond-shaped region of high velocity exists immediately behind the flapping foil and there is no evidence of clear bifurcation for this high-velocity jet. However, unlike the low frequency case, a large region of high streamwise velocity reappears around the centerline in the downstream wake region. 284      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Finally, the hydrodynamic performance of this rectangular flapping foil is assessed in terms of thrust coefficient CT and propulsion efficiency η , which are, respectively, defined as CT = η= FT (8.4) ρU ∞2 S FT U P (8.5) where FT and FT are the instantaneous and time-mean thrust, respectively, while P is the time-mean input power. Fig. 8.11 shows the variation of time-mean thrust coefficient versus foil aspect-ratio for St = 0.6 . It is found that the thrust increases monotonically with the aspect ratio and might approach a constant with a further increase in AR . Such a behavior is recognized in the simulation of thin ellipsoidal flapping foils by Dong et al. (2006) and well documented for the two-dimensional flapping foils by Anderson et al. (1998) and Jones et al. (1998). The numerical result of Shao et al. (2010) is also included for a direct comparison, from which a good agreement is obtained except the present thrust coefficient is a litter larger than the result of Shao et al. (2010) at AR = . Similar trend also holds for the propulsive efficiency in Fig. 8.12, where η increases monotonically with respect to AR and asymptotically approaches a constant value. 285      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows swimming direction, thus contributing positively to the fish swimming. Thereafter, the net force under this situation is referred to as being of thrust type. On the other hand, the net force with negative value C F < basically goes against the fish swimming and thus referred to as being of drag type. The time-evolution curves in Figs. 8.18(a) and 8.18(b) show several remarkable features of the hydrodynamic force caused by the fish undulation: (1) For all the St under consideration, the force coefficients exhibit two peaks in each tail-stroke cycle: one during forward stroke and the other during backward stroke, which is consistent with the experimental discoveries of Hess & Videler (1984) and numerical observations of Borazjani & Sotiropoulos (2008); (2) As St increases from the rigid case of St = , the force coefficient appears to be of drag-type throughout the entire stroke cycle until a critical St ( St ≈ 1.0 for Re = 300 and St ≈ 0.3 for Re = 4000 ), where the thrust-type regime begins to appear. With St increasing further beyond the critical St , the thrust-type regime in each stroke cycle becomes wider and larger. (3) At small St , approximately St < 0.3 , the net force is of drag-type and with its magnitude greater than the corresponding value of the rigid case of St = at the same Re . At moderate St ( 0.3 < St < 0.5 for Re = 300 and St ≈ 0.5 for Re = 4000 ), although the force still remains of drag-type, its magnitude lowers down as compared to its rigid counterpart. 292      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows These hydrodynamic features hold essentially in both transitional and inertial flow regimes. The critical St , however, depends on the Re . The time-mean force coefficient C F , which is calculated by averaging the force coefficient CF over several consecutive swimming cycles, is presented in Fig. 8.19 as a function of St . It is observed that for Re = 300 , our results match perfectly well with those of Borazjani & Sotiropoulos (2008) and Zhou & Shu (2012), and for Re = 4000 , our results fall within the regions enveloped by the curves of Borazjani & Sotiropoulos (2008) and Zhou & Shu (2012) and thus are basically good. From the two force variation curves, we can see that for both Re = 300 and 4000, the fish experiences a drag-type mean net force at low St and a thrust-type mean net force at high St . The mean net force variation versus St is almost monotonic except in the low St region, where the magnitude of the drag-type force generated on the fish first increases above that of the rigid case, then gradually diminishes, peaking at around St ≈ 0.1 for both Re considered. The critical Strouhal number Stcritical at which the mean net force transits from the drag-type to thrust-type decreases with the increase in Re . 8.3 Conclusions The applicability of the new boundary condition-enforced immersed boundary method to three dimensional incompressible viscous flows with moving 293      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows boundaries has been demonstrated in this chapter. Two interesting biofluid problems involving geometrically complex rigid or flexible bodies are simulated. The first is a three-dimensional rigid finite-span foil which undergoes a harmonic heaving and pitching motion in a uniform oncoming flow. The second concerns a flexible-body fish which commits a straight-swimming in ocean through an undulatory locomotion along its flexible body. The aerodynamic or hydrodynamic performances of the foil and fish and their dependence on various kinematic parameters have been carefully examined and compared with the established results in the literature, which show a very satisfactory agreement. The flow fields around these moving objects, in terms of three-dimensional wake structures, have also been visualized and matched well with those reported previously. Therefore, the boundary condition-enforced immersed boundary method offers a powerful tool for the complex moving boundary problems. 294      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Table 8.1 The variation range of AR and St AR St 1, 2, 3, 4, 0.6 0.35, 0.6, 0.8, 1.0 295      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows U∞ W c Fig. 8.1 Schematic view of a rigid finite-span foil heaving and pitching in a free stream c ψ h Fig. 8.2 Schematic diagram for the definition of parameters 296      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) t / T flap = (b) t / T flap = 1/ (c) t / T flap = / (d) t / T flap = / 297      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (e) t / T flap = / Fig. 8.3 Perspective (left column) and side (right column) views of vortex structures in their λ2 -definitions in one flapping cycle Fig. 8.4 Top view of the vortex structure in its λ2 -definition corresponding to Fig. 8.3(d) Fig. 8.5 Contours of the spanwise vorticity along the spanwise symmetry plane for AR = and St = 0.6 298      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Fig. 8.6 Contours of the mean streamwise velocity along the spanwise symmetry plane for AR = and St = 0.6 (a) AR = (b) AR = 299      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (c) AR = (d) AR = Fig. 8.7 Perspective (left column) and side (right column) views of vortex structures for different aspect ratios at St = 0.6 (a) AR = (b) AR = 300      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (c) AR = (d) AR = Fig. 8.8 Contours of the mean streamwise velocity in the spanwise symmetry plane (a) St = 0.35 (b) St = 0.8 (c) St = 1.0 Fig. 8.9 Top view of vortex sturctures for different Strouhal number at AR = 301      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) St = 0.35 (b) St = 0.8 (c) St = 1.0 Fig. 8.10 Contours of the mean streamwise velocity in plane y = for different Strouhal number at AR = Fig. 8.11 Time-averaged thrust coefficient versus foil aspect-ratio for St = 0.6 302      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Fig. 8.12 Propulsive efficiency versus foil aspect-ratio for St = 0.6 Fig. 8.13 Variations of the propulsive efficiency and time-averaged thrust coefficient with respect to Strouhal number at AR = Fig. 8.14 Schematic view of the fish geometry 303      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) St = 0.3 (b) St = 1.2 Fig. 8.15 The three-dimensional flow structures induced by the swimming fish at Re=300 for some representative Strouhal numbers      (a) St = 0.2   304        Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (b) St = 0.7 Fig. 8.16 The three-dimensional flow structures induced by the swimming fish at Re=4000 for some representative Strouhal numbers  (a) Re=300 (b) Re=4000 Fig. 8.17 The instantaneous in-plane streamlines and vorticity field in the mid plane of y = at St = 0.3 for Re=300 and 4000 305      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) Re = 300 (b) Re = 4000 Fig. 8.18 The time-dependent hydrodynamic force coefficients for Re = 300 and 4000 306      Chapter Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) Re = 300 (b) Re = 4000 Fig. 8.19 The mean force coefficient for Re = 300 and 4000 307    [...]... objects, in terms of three-dimensional wake structures, have also been visualized and matched well with those reported previously Therefore, the boundary condition-enforced immersed boundary method offers a powerful tool for the complex moving boundary problems 294      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Table 8. 1 The variation range of AR and St AR St 1,... AR = 3 and St = 0.6 2 98     Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Fig 8. 6 Contours of the mean streamwise velocity along the spanwise symmetry plane for AR = 3 and St = 0.6 (a) AR = 1 (b) AR = 2 299      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (c) AR = 4 (d) AR = 5 Fig 8. 7 Perspective (left column) and side... 3, 4, 5 0.6 3 0.35, 0.6, 0 .8, 1.0 295      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows U∞ W c Fig 8. 1 Schematic view of a rigid finite-span foil heaving and pitching in a free stream c ψ h Fig 8. 2 Schematic diagram for the definition of parameters 296      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) t / T flap = 0 (b)... 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) St = 0.35 (b) St = 0 .8 (c) St = 1.0 Fig 8. 10 Contours of the mean streamwise velocity in plane y = 0 for different Strouhal number at AR = 3 Fig 8. 11 Time-averaged thrust coefficient versus foil aspect-ratio for St = 0.6 302      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows Fig 8. 12... Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) Re = 300 (b) Re = 4000 Fig 8. 18 The time-dependent hydrodynamic force coefficients for Re = 300 and 4000 306      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (a) Re = 300 (b) Re = 4000 Fig 8. 19 The mean force coefficient for Re = 300 and 4000 307    ... streamlines and vorticity field in the mid plane of y = 0 at St = 0.3 are depicted for both Re in Fig 8. 17 As expected, at a lower Re where viscous effects play a dominant role, the viscous region around the fish body is thicker and the overall width of the wake is larger The time-dependent hydrodynamic forces experienced by the fish for Re = 300 and 4000 are plotted in Figs 8. 18( a) and 8. 18( b), respectively,... anguiliform mode On the 286       Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows other hand, if the large amplitude of the propulsive wave is mostly confined to the posterior part of the body and increases significantly in the caudal area, the swimming mode is called carangiform Recent experiments and computations have shed valuable insights into the performance and. .. basically goes against the fish swimming and thus referred to as being of drag type The time-evolution curves in Figs 8. 18( a) and 8. 18( b) show several remarkable features of the hydrodynamic force caused by the fish undulation: (1) For all the St under consideration, the force coefficients exhibit two peaks in each tail-stroke cycle: one during forward stroke and the other during backward stroke, which... side (right column) views of vortex structures for different aspect ratios at St = 0.6 (a) AR = 1 (b) AR = 2 300      Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows (c) AR = 4 (d) AR = 5 Fig 8. 8 Contours of the mean streamwise velocity in the spanwise symmetry plane (a) St = 0.35 (b) St = 0 .8 (c) St = 1.0 Fig 8. 9 Top view of vortex sturctures for different Strouhal... St It is observed that for Re = 300 , our results match perfectly well with those of Borazjani & Sotiropoulos (20 08) and Zhou & Shu (2012), and for Re = 4000 , our results fall within the regions enveloped by the curves of Borazjani & Sotiropoulos (20 08) and Zhou & Shu (2012) and thus are basically good From the two force variation curves, we can see that for both Re = 300 and 4000, the fish experiences . width of the wake is larger. The time-dependent hydrodynamic forces experienced by the fish for Re 300= and 4000 are plotted in Figs. 8. 18( a) and 8. 18( b), respectively, in forms of force. cases of 0.8St = and 1.0 in Figs. 8. 10(b) and 8. 10(c), a diamond-shaped region of high velocity exists immediately behind the flapping foil and there is no evidence of clear bifurcation for. topologies of the wake are quite different from each other. For  Chapter 8 Applications of Developed IBM Solvers to Simulate 3D Moving Boundary Flows 283   cases of small span size of 1 A R = and