Development of immersed boundary methods for isothermal and thermal flows 6

  Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows Chapter Applications of Developed IBM Solvers to Simulate Two-Dimensional Fluid and Thermal Flows In this chapter, the fluid IBM solver proposed in Chapter and the two thermal IBM solvers proposed in Chapters and are numerically examined by studying a collection of two-dimensional fluid and thermal flow problems. The unsteady insect hovering flight which undertakes a harmonic translational and rotational motion is simulated in Section 6.1. Section 6.2 discusses particle sedimentations through vertical channels. In Section 6.3, the forced convective heat transfer of a transversely oscillating cylinder in the wake of an upstream cylinder is investigated. 6.1 Unsteady insect hovering flight at low Reynolds numbers Flying insects in nature share plenty of fantastic aerodynamic performances and maneuverabilities like taking off backward, flying sideways, and landing upside down (Nachtigall 1974; Collett & Land 1975a, 1975b; Dalton & Kings 1975), which even our state-of-the-art vehicles are not capable of achieving. Enchanted by the brilliant flight behaviors, researchers have endeavored to 153      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows explore the mechanisms behind it. One major effort is on how the flapping wings generate forces (e.g. to support insects hovering in the air). Early quasi-steady analysis, pioneered by Weis-Fogh & Jensen (1956) and Weis-Fogh (1973), assumed that the force generation relied solely on the instantaneous velocity and angle of attack while totally ignoring the past history influence of wing motion, thus failing to provide sufficient force required for hovering (Ellington, 1984). Recent studies recognized that three distinct but interactive unsteady mechanisms: dynamic stall, rotational circulation and wake capture, were responsible for the enhanced aerodynamic performance of insects (Ellington et al. 1996; Dickinson & Gätz 1993; Dickinson 1994; Dickinson et al. 1999; Sane & Dickinson 2001, 2002; Birch & Dickinson 2001). By measuring the time-dependent aerodynamic forces on an impulsively started aerofoil within the Reynolds number range for flies of the genus Drosophila and other small insects, Dickinson & Gätz (1993) indicated for the first time that, the lift was largely enhanced during the translational motion of the wing due to the presence of a leading edge vortex (or dynamic stall vortex). Ellington et al. (1996) conducted smoke-visualizations on wings of a tethered hawkmoth Manduca sexta and discovered that this leading edge vortex (LEV) remained attached to the wing surface during translational motions in both the up- and down-stroke and the high lift benefited from it maintains for the entire 154      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows stroke. By analyzing the momentum imparted to the fluid by the vortex, the researchers believed that this must be a new high lift mechanism and named it as delayed stall or dynamic stall mechanism (Ellington et al. 1996; Dickinson et al. 1999). The LEV attachment, together with the large lift preservation, was also observed by Birch and Dickinson (2001) for a fruit fly model Drosophila melanogaster whose Reynolds number is high. Dickinson et al. (1999) performed force measurements on wings of a dynamically scaled model of the fruit fly Drosophila melanogaster and noticed the existence of transient peaks in lift at each half-stroke reversal. Dickinson et al. (1999) suggested that two mechanisms were responsible for the force peaks: wake capture mechanism at the beginning of the stroke due to large effective fluid velocity and rotational circulation mechanism at the end of the stroke due to rapid pitching-up rotation. Apart from valuable studies using aerodynamic models, computational fluid dynamic analysis based on fluid-structure interaction has been developed as a complementary method for the unsteady aerodynamics analysis. By using the pseudo-compressibility method (Liu et al. 1995), Liu et al. (1998) confirmed the delayed stall mechanism during the hovering flight. Sun and Tang (2002a), by employing the artificial compressibility algorithm (Rogers and Kwak 1990), discovered the close association between the force peaks and the translational acceleration. After realizing that there is no evident spanwise flow at 155      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows low Re ∼ 102 (Birch & Dickinson 2001), Wang et al. (2004) compared the forces between two-dimensional computations and three-dimensional experiments for several qualitatively different kinematic patterns (at low Re ranging from 75 to 115). They found that the results could achieve a satisfactory match especially in advanced rotation and symmetrical cases. This indicates that at low Re , two-dimensional calculations could give reasonable results. In the literature, a number of works have employed the two-dimensional approach in the insect flight study (Gustafson & Leben 1991; Wang 2000a; Wang 2004). In this section, the two-dimensional unsteady flow around an elliptical flapping wing which undergoes a prescribed harmonic translational and rotational motion is simulated, to mimick insects’ hovering flight. Two most common hovering modes, normal hovering (Fig. 6.1) and dragonfly hovering (Fig. 6.2), are studied. Their governing equations can be written in a unified form as X = A cos(2π t / T flap ) cosψ (6.1) Y = A cos(2π t / Tflap )sinψ (6.2) α AOA = α AOA,0 + α AOA, m sin(2π t / T flap + φ ) (6.3) where X and Y are the coordinates of the center of the flapping wing, A is the amplitude of the translational oscillation motion, T flap is the flapping period, ψ is the inclined angle of stroke plane. α AOA is the angle of attack, 156      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows α AOA,0 and α AOA, m are the angles describing the mean value and amplitude of the sinusoidal oscillating rotation, respectively. φ is the phase difference of the rotation relative to translation. A schematic diagram is provided in Fig. 6.3 for a better illustration of these parameters. In normal hovering, the wing strokes in a horizontal plane with ψ = while in dragonfly hovering, the wing strokes in an inclined plane with a non-zero ψ . In all our simulations, the Reynolds number Re = U ref c ν the wing and the velocity U ref = 6.1.1 is defined based on the chord length c of 2π A . T flap Normal hovering mode Most hovering insects, such as fruit flies, bees and beetles, adopt symmetric back-and-forth strokes along a horizontal plane (Weis-Fogh 1973), which is frequently referred to as “normal hovering”. In this subsection, normal hovering flight is studied. 6.1.1.1 Normal hovering flight without ground effect The insect-hovering far above the ground is first simulated. Following the work of Wang et al. (2004) and Eldredge (2007), simulations are carried out at Re = 75 with characteristic parameters of A = 2.8 , α AOA,0 = π / and α AOA, m = π / . Three values of φ = π / , and −π / for phase difference are taken into consideration, corresponding to advanced, symmetric 157      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows and delayed rotation respectively. For the considered cases, a computational domain of size 30 × 30 is capable of providing domain-independent results. The non-uniform mesh is employed for the domain discretization, with a fine resolution of Δx = Δy = h = 0.02 in the region involving the insect movement. Meanwhile, a step size of Δt = 0.001 is used for the time integration. The drag and lift coefficients CH and CV , which are defined and normalized in the same way as Eldredge (2007) , are recorded for the first four flapping cycles in Figs. 6.4 and 6.5. In all the cases, a quasi-periodic state has been attained after two strokes. A comparison with the published results shows that for all the three cases, the time histories of CH have good agreements with both the experimental and numerical results of Wang et al. (2004). The lift evolution, although it does not match very well with the three-dimensional experiment of Wang et al. (2004), can capture and well predict the major features of the force profile, like the timing of the peaks and their values. Furthermore, it is noted that the agreement between all of our simulation results and those from Eldredge’s VVPM (2007) is favorably well. The resultant time-mean drag and lift coefficients C H and CV for the cases of advanced, symmetric and delayed rotation are (0.598, 0.593), (0.728, 0.464) and (0.674, 0.262), respectively, where C H and CV are defined as 158      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows CH = CV = T flap T flap ∫C ∫C dt (6.4) dt . (6.5) H V The data reveal that the mean lift drops monotonically with φ decreasing from π / to − π / , indicating that the mean lift is sensitive to the phase difference, as emphasized by several researchers (Dickinson et al. 1999; Wang 2000; Wang et al. 2004). This variation can be explained from the vorticity fields (say, in the first half-stroke during the third cycle), as shown in Figs. 6.6 to 6.8 for φ = π / , and − π / , respectively. After the wing begins a new cycle, a negative LEV and a positive trailing edge vortex (TEV) are gradually formed and enhanced on the downwind side of the wing. In the case of φ = π / (Fig. 6.6), the LEV remains attached to the wing surface throughout the translational process (dynamic stall). This LEV corresponds to a region of low pressure above the wing and thus will result in lift enhancement. At the same time, the positive LEV formed in the previous half-stroke is “recaptured” by the wing (Fig. 6.6(a)-(b)). It combines with the newly formed TEV and induces the augmentation of rotational circulation, which in turn contributes to a high lift. For the symmetric rotation case (Fig. 6.7), the LEV attachment is not very significant (comparing Figs. 6.6(a) and 6.7(a)). Although the LEV recapture is also observed (Fig. 6.7(a)), the strength of the LEV is not as large as that in advanced rotation. Therefore, the 159      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows generated lift could not be enhanced remarkably. In the case of delayed rotation (Fig. 6.8), the wing translates at an angle of attack greater than π / , and the flow separates quickly, thus no LEV attachment and recapturing occur. Meanwhile, a wing with attack angle larger than π / would result in a downward lift and all of these lead to a low lift. 6.1.1.2 Normal hovering flight with ground effect When insects are flying near the ground or perching on bodies, the ground effect would play an important role. In this subsection, the normal hovering flight near a surface is simulated. Another important parameter, ground clearance Gc as shown in Fig. 6.3, is introduced to distinguish different flight conditions. It is defined as the distance between the ground surface and the wing center (in the time-mean sense). During the assessment of the ground effect, simulations are carried out at Re = 100 . Several ground clearances are considered while the other parameters are kept fixed at A = 2.5 , φ = , α AOA,0 = π / and α AOA, m = π / . Computational domains of size 30 × (15 + Gc ) are used to simulate the considered cases. The non-uniform mesh is used for the domain discretization, with a fine resolution of Δx = Δy = h = 0.02 in the region swept by the inset movement. Meanwhile, Δt = 0.001 is selected for the time integration. The time courses of lift and drag coefficients in one flapping cycle after steady 160      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows or quasi-steady state has been achieved are depicted at typical clearances of Gc = , and in Fig. 6.9. Their comparisons with the corresponding curves reported by Gao & Lu (2008) demonstrates a satisfactory consistency. The lift coefficient at Gc = almost repeats itself in the up- and down-stroke, and the corresponding drag coefficient, on the other hand, shows anti-symmetry about each other in the two half-strokes. Furthermore, both the lift and drag peak around t / T flap = 0.1 and 0.6. This should be explained by the rotational circulation and wake capture mechanism. For Gc = and 5, the symmetric and anti-symmetric behaviors are lost and the lift generated in the downstroke is obviously larger than that in the upstroke, peaking around t / T flap = 0.8 . The drag variation due to the increase in Gc from to is not significant except a slight increase in the peak values. The histories of force coefficients for infinity ground clearance Gc = ∞ (corresponding to the case where the insect is hovering far above the ground) are also included in the figure. It is observed the two curves almost coincide with those for Gc = , which indicates that the ground has little effect on the hovering flight when the insect flies above the ground at a height no less than Gc = . The time-mean drag and lift coefficients C H and CV have been evaluated after a steay state of periodicity or quasi-periodicity is achieved and lasts for at least 10-15 cycles. Their variations versus ground clearance are plotted in Fig. 6.10. To get a clear view of the ground effect on the aerodynamic forces, C H 161      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows and CV for the flight far away from ground ( Gc = ∞ ), denoted by CH ,∞ and CV ,∞ are also included. It should be noted that the mean drag coefficients are calculated based on their absolute values, otherwise the drag forces in the fore- and backstroke would almost cancel out. Fig. 6.10 demonstrates that the ground has a very significant influence on the aerodynamic forces when the insect hovers very near the ground (say, at Gc = in present study), where both the lift and drag are greatly enhanced and reach their respective maxima. As Gc increases, CH and CV first decrease quickly to the minima around Gc = 2.5 , and then increase asymptotically towards CH ,∞ and CV ,∞ . In fact, as the insect hovers at a height around Gc = above the ground, the ground effect has become weak enough to be negligible. The developments of the vortex structure for the first half stroke (forward stroke) at some representative ground clearances of Gc = , and are depicted in Figs. 6.11-6.13. When the wing begins translation and rotation in the forward-stroke, a pair of negative LEV and positive TEV is produced. At Gc = , the positive LEV formed in the previous back-stroke, due to the close proximity between the wing and ground surface, flows over the wing to the downwind side and enhances the newly formed negative LEV (Fig. 6.11(a)-(b)). The LEV is attached to the wing during its translational motion (dynamic stall mechanism). The TEV, however, is stretched in the narrow gap and quickly dissipated (Fig. 6.11(b)-(c)). During the stroke reversal, the LEV 162      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) t / Tflap = 1/ (b) t / T flap = / (c) t / T flap = / (d) t / T flap = / Fig. 6.13 The development of vortex structure in the forth stroke at Gc = 199      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows Fig. 6.14 Comparison of time-dependent drag and lift coefficient for dragonfly hovering mode   (a) t / Tflap = 1/ (b) t / T flap = / (c) t / T flap = / (d) t / T flap = / 200      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (e) t / T flap = / (f) t / T flap = / (g) t / T flap = / (h) t / T flap = / Fig. 6.15 Vorticity field evolution during one stroke for dragonfly hovering   Fig. 6.16 Time-mean drag and lift coefficients versus inclined angle 201      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) Horizontal force coefficient (b) Vertical force coefficient Fig. 6.17 Time evolution of force coefficients during two strokes (a) horizontal force (b) vertical force   202      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows Fig. 6.18 Snapshots of particle sedimentation at blockage ratios: 12/13, 18/13, 20/13, 22/13, 32/13 (from left to right) Fig. 6.19 Trajectories of particle center at different blockage ratios 203      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows Fig. 6.20 Instantaneous vorticity field at different blockage ratios corresponding to Fig. 6.18 (a) Gr = 100 204      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (b) Gr = 564 (c) Gr = 1500 (d) Gr = 2500 205      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (e) Gr = 5000 Fig. 6.21 Streamlines (left), the vorticity (middle) and temperature contours (right) at different Gr Fig. 6.22 Time histories of the lateral particle positions at different Gr 206      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows Fig. 6.23 The terminal-settling-velocity based Reynolds number Retmn versus the Grashof number Gr y = A sin(2π f ct ) U∞ L Fig. 6.24 Configuration of tandem cylinder system         207      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows   (a) fc / f st = 1.3 and A = 0.15 (b) fc / f st = 1.5 and A = 0.15 (c) fc / f st = 1.3 and A = 0.35 (d) fc / f st = 1.5 and A = 0.35 Fig. 6.25 Instantaneous vorticity contours for G = at different vibration frequencies and amplitudes   (b) 2nd cycle (a) 1st cycle Fig. 6.26 Instantaneous vorticity contours for two consecutive cycles of excitation at fc / f st = 0.9 and A = 0.35   (a) G = 208      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (b) G = Fig. 6.27 Instantaneous vorticity contours of a stationary tandem cylinder system at G = and for Re = 100   (a) fc / f st = 0.4 and A = 0.15 (b) fc / f st = 1.7 and A = 0.35 Fig. 6.28 Instantaneous vorticity contours for G = at different vibration frequencies and amplitudes (b) 2nd cycle (a) 1st cycle Fig. 6.29 Instantaneous vorticity contours for two consecutive cycles of excitation at the lock-on frequency of fc / f st = 1.0 and A = 0.15 Fig. 6.30 Instantaneous vorticity contours at the lock-on frequency of fc / f st = 1.0 for G = and A = 0.35 209      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) fc / f st = 1.1 and A = 0.15 (b) fc / f st = 1.3 and A = 0.15 Fig. 6.31 Instantaneous vorticity contours for G = at different vibration frequencies and amplitudes (a) fc / f st = 0.4 (b) fc / f st = 0.9 (c) fc / f st = 1.3 (d) fc / f st = 1.5 Fig. 6.32 Instantaneous isotherms for G = and A = 0.15 at different excitation frequencies Fig. 6.33 Instantaneous isotherms for G = and A = 0.35 at the excitation frequency fc / f st = 0.9 210      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) fc / f st = 0.4 and A = 0.15 (b) fc / f st = 1.7 and A = 0.35 Fig. 6.34 Instantaneous isotherms for G = at different excitation conditions Fig. 6.35 Instantaneous isotherms for G = at the excitation condition of fc / f st = 1.7 and A = 0.35 211      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) G = (b) G = (c) G = Fig. 6.36 Time-mean drag coefficient versus vibration frequency 212      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) G = (b) G = (c) G = Fig. 6.37 Time-mean r.m.s of lift coefficient versus vibration frequency 213      Chapter Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows (a) G = (b) G = (c) G = Fig. 6.38 Time-mean Nusselt number versus vibration frequency 214    [...]... and Fig 6. 26( a)) There is no change in 178      Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows longitudinal vortex spacing (behind the downstream cylinder) at different vibration amplitudes (see Fig 6. 25(a) and Fig .6. 25(c) for comparison), but higher vibration frequencies play a role of progressively reducing the longitudinal vortex spacing (see Fig 6. 26( b) and. .. in the range of 0.9 − 1.3, the oscillating motion of the downstream cylinder takes control of the instability mechanism of the whole system and the lock-on state of vortex formation exists for both vibration amplitudes of A = 0.15 and 0.35 The flows behind the downstream cylinder are perfectly periodic This can be revealed from the vortical structures in Fig 6. 26 for f c / f st = 0.9 and A = 0.35... exhibit in the downstroke 6. 2 Particulate flow Particulate flows widely exist in natural and industrial processes, such as 165       Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows sedimentation of dust particles in aerial, dispersion of fuel particles in combustion chambers, reaction of catalyst particles in slurry reactors, and transport of sand particles in rivers,... reverses its stroke direction and flaps upward, the positive TEV sheds into the wake (Fig 6. 15(d)) The negative LEV (Fig 6. 15(e)), on the 164       Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows other hand, combines with a newly formed negative TEV, which is then coupled with the aforementioned positive TEV, forming a dipole vortex pair (Fig 6. 15(f)-(h)) This vortex... aspect ratio of 0.25 and undertakes asymmetric strokes along an inclined stroke plane For the convenience of comparison, the kinematic parameters closely follow those in Wang (2000) and Xu & Wang (20 06) , where Re = 157 , A = 1.25 , α AOA,0 = 135 , α AOA, m = 45 and φ = 0 A computational domain of size 163       Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows 30 ×... rear region of the upstream cylinder during the roll-up process of vortex development, leading to a thin boundary layer there (compared to G = 2 ) At larger spacing of G = 7 , the thermal fields in the gap region are similar to those for a single bluff body (Fig 6. 35) With sufficient room for the vortices 182      Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows to...  Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows formed in the previous half-stroke is separated from the wing (Fig 6. 11(d)) and then sweeps away along the horizontal direction When Gc is increased, at Gc = 3 and 5, as shown in Figs 6. 12 and 6. 13, the positive LEV formed in the previous back-stroke combines with the... 183      Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows significant influence on the drag, oscillation amplitude of lift as well as Nusselt number 6. 3.3.1 Average drag Variation of the time-mean drag coefficient CD at various excitation amplitude and spacing is plotted in Fig .6. 36 as a function of f c / f st It is noticed that the drag curve for the upstream... the frequency-axis in the case of G = 4 and 7 (Fig 6. 36( b)-(c)) In contrast, at small spacing (i.e G = 2 in Fig 6. 36 (a)), a clear increase in CD , as high as 6. 02% for A = 0.15 and 11.05% for A = 0.35 (compared to their stationary counterparts), can be identified in the lock-on region The shape of the drag curve for the downstream cylinder is more complicated Firstly, for a given vibration frequency,... positive values For both amplitudes, the mean drag is maximized around f c = 1.3 f st , approximately corresponding to the end of lock-on regime For G = 4 (Fig 6. 36( b)), the drag force experiences a gentle and nearly monotonic increase and maximizes itself around the synchronization point, f c = f st For G =7 (Fig 6. 36 (c)), in general, the drag force increases greatly below the natural frequency and then . 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows 153  Chapter 6 Applications of Developed IBM Solvers to Simulate Two-Dimensional Fluid and Thermal Flows. time-mean drag and lift coefficients H C and V C for the cases of advanced, symmetric and delayed rotation are (0.598, 0.593), (0.728, 0. 464 ) and (0 .67 4, 0. 262 ), respectively, where H C and V C . Fig. 6. 10. To get a clear view of the ground effect on the aerodynamic forces, H C  Chapter 6 Applications of Developed IBM Solvers to Simulate 2D Fluid and Thermal Flows 162   and