Three-dimensional numerical simulation of a vortex ring impacting a bump

Transcription

1 HEOREICAL & APPLIED MECHANICS LEERS 4, (2014) hree-dimensional numerical simulation of a vorte ring impacting a bump Heng Ren, 1, a) 2, Changue Xu b) 1) China Electronics echnolog Group Corporation No.38 Research Institute, Hefei , China 2) College of Aerospace Engineering, Nanjing Universit of Aeronautics and Astronautics, Nanjing , China (Received 15 Januar 2014; accepted 15 March 2014) Abstract A vorte ring impinging on a three-dimensional bump is studied using large edd simulation for a Renolds number Re = based on the initial translation speed and diameter of the vorte ring. he effects of bump height on the vortical flow phenomena and the underling phsical mechanisms are investigated. Based on the analsis of the evolution of vortical structures, two tpical kinds of vortical structures, i.e., the wrapping vortices and the hair-pin vortices, are identified and pla an important role in the flow state evolution. he circulation of the primar vorte ring reasonabl elucidates some tpical phases of flow evolution. Furthermore, the mechanism of flow transition from laminar to turbulent state has been revealed based on analsis of turbulent kinetic energ. c 2014 he Chinese Societ of heoretical and Applied Mechanics. [doi: / ] Kewords large edd simulation, vorte ring, bump, vortical structure, turbulent state As one of the tpical forms of vorte motion, vorte rings widel eist in nature and engineering. he interaction of vorte rings with solid or fluid boundaries is a fundamental problem in fluid dnamics and has received considerable attention. his subject is also associated with a variet of practical applications, such as cavitated rings used for underwater drilling 1 and the downburst and aircraft interaction. 2 Moreover, the underling flow phenomena and phsical mechanisms are still unclear and are worth of detailed studies. Vorte ring interacting with a flat wall has been etensivel studied. 3 8 hese studies showed that as the primar vorte ring moves graduall toward the wall, its rate of approach slows down and its radius continues to increase. When the Renolds number is larger than 500 based on the initial diameter and translational speed of the vorte ring, the formation of the secondar vorte ring occurs and then it impacts the primar one. Eperimental stud 3 has revealed that, beond Re = 3000, the primar vorte ring will no longer remain stable as it approaches the wall. hus, the instabilit and transition to turbulence for the vorte ring evolution should be considered when the Renolds number becomes large enough. For the instabilit of vorte rings, etensive investigations have been carried out. Krutsch 9 first studied this subject and found that the vorte ring becomes unstable with some stationar waves distributed around its aimuthal direction. hen Maworth 10 verified eperimentall that a) Corresponding author. renheng@mail.ustc.edu.cn. b) cu@nuaa.edu.cn.

2 H. Ren, C. Y. Xu heor. Appl. Mech. Lett. 4, (2014) the stationar aimuthal waves grow at 45 relative to the propagation direction of vorte ring, and the wave number depends on the slenderness ratio of core radius to ring radius. Widnall and sai 11 presented the theoretical eplanation of the instabilit and indicated that a straining field in the neighbourhood of the vorte core leads to the amplification of small perturbation. hen Shariff et al. 12 established a viscous correction to the growth rate proposed b Widnall and sai 11 based on their direct numerical simulation (DNS) results. Comparing with the numerous studies of vorte ring interacting with a flat surface, the investigation relevant to a vorte ring impacting a curved wall is scarce. Orlandi and Vericco 13 numericall studied vorte pairs interacting with a two-dimensional circular clinder with no-slip and free-slip boundar conditions. For the free-slip condition, the dipole is observed to split into two vortices and then to rejoin on the clinder. While for the no-slip interaction, the generation of dipolar and tripolar structures occurs on the clinder surface. Vericco et al. 14 further studied this problem. he found that the induced vortices become more apparent as the diameter of the clinder increases. Recentl, Sousa 15 studied a vorte ring impacting a stationar sphere for Re = 1000 using DNS. After the secondar vorte ring is formed, the found its interaction with the primar ring results in the fast deca of circulation for the secondar ring. In this paper, a large edd simulation (LES) technique is utilied to investigate the effects of bump height on the dnamics of vortical structures and the turbulent behaviors when a vorte ring impacts a three-dimensional bump at Renolds number Re = o our knowledge, the relevant work has never been performed before. he purpose is to investigate the comple flow phenomena and the underling mechanisms. he compressible Navier Stokes equations in generalied coordinates are solved using the LES coupled with dnamic subgrid-scale (SGS) models. As emploed in our previous work, 16,17 the viscous terms and convective terms are discretied b a fourth-order and a second-order centered scheme, respectivel. Moreover, the present numerical methods have alread been emploed successfull to a variet of turbulent flows and have been verified the reliable calculations. According to the schematic as depicted in Fig. 1, a Gaussian vorte ring 13 with radius R 0 is initiall placed at c =(0,0,H v ), where H v is the distance between the vorte ring center and the bottom surface. he bump has a circular base with a cosine-squared cross section which is defined as (,)=H b cos 2 (π /6), where H b is the bump height. he initial translational speed of the vorte ring can be represented as 21 u v =[Γ /(4πR 0 )](ln(8r 0 /σ 0 ) 1/4), where σ 0 is the initial core radius and Γ represents the circulation of the vorte ring. o deal with the instabilit of the vorte ring, an aimuthal disturbance with an amplitude of is introduced b imposing a radial displacement on the ais of the ring. 12 In the computation, the effect of bump height is investigated. he parameters of three cases are given as follows. For all the cases, the slenderness ratio and initial height of the vorte ring are σ 0 /R 0 = 0.2 7,17 and H v /R 0 = 6R 0, respectivel, and the Renolds number is Re = he bump height H b /R 0 are 1.8, 2.4, and 3.0 for case 1, case 2, and case 3, respectivel. he computational domain etends for 16R 0 in the and directions and 12R 0 in the or vertical direction, i.e., L /R 0 = L /R 0 = 16, L /R 0 = 12. Based on our careful eaminations, a mesh of sie N N N = with a resolution R 0 = 40Δ is used in the computation. he grid-spacing is uniform in the plane, and a grid stretching in the vertical direction is used to

3 hree-dimensional numerical simulation of a vorte ring impacting a bump θ = 0 L /2 r Γ R 0 σ 0 H v Bump H b Fig. 1. Schematic diagram of a vorte ring approaching a bump. increase the grid resolution near the surface. Periodic boundar conditions are used in the and directions. No-slip boundar condition is emploed on the bump surface and a far-field boundar condition is applied in the = L plane. We first investigate the evolution of vortical structures depicted in Fig. 2. For comparabilit of the vortical structures for cases 1 3 as shown in Fig. 2, it is indicated that the distances between the vorte ring center and the bump top are 0.32R 0 approimatel for case 1 at = 15.0, case 2 at = 12.5, and case 3 at = From Fig. 2(a) for case 1, when the primar vorte ring moves close to the bump, a vorticit laer is obviousl generated on the core surface of bump at = hen the separation of boundar laer occurs and the secondar vorte ring is generated at = Due to the growth of the aimuthal perturbation, the primar ring develops into a wavlike structure at = 17.5 and Furthermore, b means of Fourier analsis of the aimuthal perturbation, it is identified that the dominant mode of the instabilit for the primar ring is k = 11, consistent with the theoretical estimate of the dominant mode k = 2.26/σ 0 approimatel b Maworth 10 and the number of the wav-like structures observed at = 17.5 and After the primar ring collides with the bump surface, the secondar ring generated lifts up from the surface and then moves over the primar vorte ring. At = 22.5, it is seen that the secondar ring has alread moved up the primar ring. Subsequentl, a variet of loop-like vortices wrapping around both the primar and secondar vorte rings (briefl called wrapping vortices ) are formed at = 25.0 and he generation of these wrapping vortices is associated with the aimuthal instabilit of the vorte rings. 22 Finall, the complicated interactions of the wrapping vortices and vorte rings over the bump surface result in the breakdown of the vortical structures into smallscale vortices at = 30.0, and further lead to the vortical flow transition to turbulent state which will be analed below. o investigate the effect of the bump height on the flow structures, Figs. 2(b) and 2(c) show the vortical structures for cases 2 and 3, respectivel. he evolution of vortical structures for the generation of secondar vorte ring is similar to case 1. When the wav-like secondar ring locates over the primar vorte ring, the secondar ring stretches significantl and causes severe distortion of the secondar ring at = 20.0 for case 2 in Fig. 2(b) and = 17.5 for case 3 in Fig. 2(c). hen, the intense stretching effect results in the disconnection of the secondar ring and the generation of hair-pin vortices at = 22.5 for case 2 and = 20.0 for case 3. hese hair-pin

4 H. Ren, C. Y. Xu heor. Appl. Mech. Lett. 4, (2014) (a) = 15.0 = 17.5 = 20.0 = 22.5 = 25.0 = 27.5 = 30.0 (b) = 12.5 = 15.0 = 17.5 = 20.0 = 22.5 = 25.0 = 27.5 (c) = 10.0 = 12.5 = 15.0 = 17.5 = 20.0 = 22.5 = 25.0 Fig. 2. Evolution of vortical structures depicted b iso-surface of the Q criterion with Q = 0.5: (a) case 1, (b) case 2, (c) case 3. vortices evolve over the bump surface because of the induction of the primar vorte ring. When the hair-pin vortices collide with the surface, the vortices break into small-scale ones and move upwards over the bump surface at = 25.0 for case 2 and = 22.5 for case 3. Subsequentl, the interactions of the hair-pin vortices and vorte rings over the bump surface result in the flow transition. Further, to quantitativel anale the development of aimuthal instabilities for the vorte rings, we perform the Fourier decomposition of the vertical vorticit ω and get Ā k, 15 which denotes the aimuthal perturbation in the vortical structures. he evolution of Ā k for the primar vorte ring in case 1 is plotted in Fig. 3. At = 15.0, the vorticit component in the wall-normal direction ω appears and it is obvious that the dominant mode is k = 11, consistent with the vortical structures shown in Fig. 2(a). With the evolution of the vorte ring, the amplitude of ω increases rapidl impling the fast growth of the instabilit. At = 20.0, the second harmonic k = 22 is apparent and the amplitude of Ā k increases considerabl, as shown in Fig. 3(b). hen with the vertical vorticit ω breaking into small-scale vorticit, the dominant and second harmonic modes deca rapidl, as depicted in Figs. 3(c) and 3(d). (a) 10 1 (b) 10 1 (c) 10 1 (d) 10 1 = 15.0 = 20.0 = 25.0 = Fig. 3. Aimuthal perturbation of primar vorte ring for case 1. We then stud the circulation of the primar ring and the total kinetic energ in the flow field. Here, the circulation is calculated b Γ = ω θ dr d, 22 where the integration domain for the primar vorte ring is chosen as the region with ω θ < 0, 12 the smbol represents the average in the aimuthal direction after transforming the data from the Cartesian coordinate to

5 hree-dimensional numerical simulation of a vorte ring impacting a bump the clindrical coordinate sstem. Furthermore, the total kinetic energ can be written as E = (1/2) (ūu ūu)dv, where the integral domain is the whole flow field and ūu represents the resolved velocit. he circulation is shown in Fig. 4(a). Based on the profiles for all the cases, the evolution of circulation can be divided into three phases. Firstl, as the vorte ring is awa from the bump, the circulation is nearl constant. hen the interaction of the vorte ring with the bump occurs and the circulation decreases quickl. hirdl, after the vortices break up into small-scale ones, the strength of vorte ring becomes relativel weak and the circulation decreases slowl. As the bump height increases, the deca rate of circulation decreases. his behavior is attributed to a weaker interaction between the primar vorte ring and the surface when H b increases. (a) (b) Γ /Γ Case 1 Case 2 Case 3 Ε /Ε Case 1 Case 2 Case Fig. 4. (a) Circulation of the primar ring and (b) total kinetic energ in the flow field. Figure 4(b) depicts the evolution of total kinetic energ, where E 0 represents the initial kinetic energ. It can be seen that the kinetic energ decreases slowl due to viscous dissipation before the vorte ring collides with the bump surface. hen the kinetic energ deceases quickl as the vorte-surface interaction. With the increase of the bump height, the deca rate of the kinetic energ becomes smaller, consistent with the evolution of circulation shown in Fig. 4(a). Finall, we investigate the flow transition from laminar to turbulent state and the relevant turbulent behavior. he turbulent kinetic energ (KE) is defined as E KE =(1/2) (u u )dv, where u represents the velocit fluctuations and is defined as u = ūu ūu, and the integral domain is the whole flow field. Figure 5 shows the evolution of E KE. Before the vorte ring interacts with the bump, the turbulent kinetic energ vanishes, corresponding to laminar flow state. As a tpical eample, we mainl discuss the behavior for case 1. It is identified that the generation of secondar vorte ring at = 17.5 approimatel is an indication of the growth of E KE. hen with the development of the aimuthal instabilit in the vortical structures and the breakdown of these vortices, the E KE grows rapidl and reaches its maimum at approimatel = 28, representing the flow transition to turbulence. 23,24 After = 28, the E KE decas quickl due to the vortical evolution and viscous deca. Moreover, compared with the three cases, as the bump height increases, the maimum of E KE decreases graduall in Fig. 5.

6 H. Ren, C. Y. Xu heor. Appl. Mech. Lett. 4, (2014) According to the investigation of an isolated vorte ring b Archer et al., 22 the shedding of hair-pin vortices along the aimuthal direction of the ring indicates turbulent flow state. In this stud, two tpical vortical structures are identified. As both the wrapping and hair-pin vortices are mainl distributed b the vorticit components in the radial and vertical directions, we can reasonabl measure the strength of these vortices b integrating the enstroph in the whole flow field, Ω r =(1/2) (ω 2 r + ω 2 )dv, where ω r and ω represent the vorticit components in the radial and vertical directions, respectivel. he evolution of Ω r is shown in Fig. 6. Compared with the profiles of E KE in Fig. 5, itis interesting to notice that the time-dependent characters of both Ω r and E KE ehibit the similar manner. he generation of Ω r (or the wrapping vortices and hair-pin vortices) corresponds to the instant of the growth of E KE. his behavior reasonabl indicates that the formation of the wrapping and hair-pin vortices plas an important role in the flow transition Case1 Case2 Case3 EKE Ωr Case 1 Case 2 Case Fig. 5. Evolution of turbulent kinetic energ integrated over the whole domain Fig. 6. Evolution of the enstroph of wrapping and hair-pin vortices integrated over the whole domain. In summar, the interaction between a vorte ring and a three-dimensional bump has been studied b means of an LES technique. he effects of bump height on the vortical flow phenomena and the underling phsical mechanisms were investigated. wo tpical kinds of vortical structures are identified and briefl represented as the wrapping vortices and the hair-pin vortices. he circulation of the vorte ring and the total kinetic energ have been investigated to reveal the different stages of flow evolution. he evolution of circulation can be divided into three phases. Firstl, as the vorte ring is far awa from the bump, the circulation is nearl constant. hen the interaction of the vorte ring with the bump occurs and the circulation decreases quickl. hirdl, after the vortices break up into small-scale ones, the circulation decreases slowl. Moreover, the evolution of KE and enstroph has been analed. It is found that the formation of the wrapping and hair-pin vortices plas an important role in the flow transition from laminar to turbulent state. his work was supported b the National Natural Science Foundation of China ( ) and the Natural Science Fund in Jiangsu Province (BK ). 1. G. L. Chahine, P. F. Genou. Collapse of a cavitating vorte ring. Journal of Fluids Engineering 105, (1983). 2.. S. Lundgren, N. N. Mansour. Vorte ring bubbles. Journal of Fluid Mechanics 224, (1991).

7 hree-dimensional numerical simulation of a vorte ring impacting a bump 3. J. D. A. Walker, C. R. Smith, A. W. Cerra, et al. he impact of a vorte ring on a wall. Journal of Fluid Mechanics 181, (1987). 4. C. C. Chu, C.. Wang, C. C. Chang. A vorte ring impinging on a solid plane surface-vorte structure and surface force. Phsics of Fluids 7, (1995). 5. P. Orlandi. Vorte dipole rebound from a wall. Phsics of Fluids A 2, (1990). 6. H. J. H. Clerc, C. H. Bruneau. he normal and oblique collision of a dipole with a no-slip boundar. Computers & Fluids 35, (2006). 7. M. Cheng, J. Lou, L. S. Luo. Numerical stud of a vorte ring impacting a flat wall. Journal of Fluid Mechanics 660, (2010). 8. L. D. Couch, P. S. Krueger. Eperimental investigation of vorte rings impinging on inclined surfaces. Eperiments in Fluids 51, (2011). 9. C. Krutsch. Über eine eperimentell beobachtete erscheining an werbelringen bei ehrer translatorischen beivegung in weklechin, flussigheiter. Annln Phsics 5, (1939) (in German) Maworth. he structure and stabilit of vorte rings. Journal of Fluid Mechanics 51, (1972). 11. S. E. Widnall, C. Y. sai. he instabilit of the thin vorte ring of constant vorticit. Philosophical ransactions of the Roal Societ of London A 287, (1977). 12. K. Shariff, R. Vericco, P. Orlandi. A numerical stud of three-dimensional vorte ring instabilities: viscous corrections and earl nonlinear stage. Journal of Fluid Mechanics 279, (1994). 13. P. Orlandi, R. Vericco. Vorte rings impinging on walls: aismmetric and three-dimensional simulations. Journal of Fluid Mechanics 256, (1993). 14. R. Vericco, J. B. Flor, G. J. F. Van Heijst, et al. Numerical and eperimental stud of the interaction between a vorte dipole and a circular clinder. Eperiments in Fluids 18, (1995). 15. P. J. F. de Sousa. hree-dimensional instabilit on the interaction between a vorte and a stationar sphere. heoretical and Computational Fluid Dnamics 26, (2012). 16. C. Y. Xu, L. W. Chen, X. Y. Lu. Large edd simulation of the compressible flow past a wav clinder. Journal of Fluid Mechanics 665, (2010). 17. H. Ren, X. Y. Lu. Large edd simulation of a vorte ring impinging on a three-dimensional circular clinder. heoretical & Applied Mechanics Letters 3, (2013). 18. X. Y. Lu, S. W. Wang, H. G. Sung, et al. Large-edd simulations of turbulent swirling flows injected into a dump chamber. Journal of Fluid Mechanics 527, (2005). 19. L. W. Chen, C. Y. Xu, X. Y. Lu. Numerical investigation of the compressible flow past an aerofoil. Journal of Fluid Mechanics 643, (2010). 20. L. W. Chen, G. L. Wang, X. Y. Lu. Numerical investigation of a jet from a blunt bod opposing a supersonic flow. Journal of Fluid Mechanics 684, (2011). 21. P. G. Saffman. he number of waves on unstable vorte rings. Journal of Fluid Mechanics 84, (1978). 22. P. J. Archer,. G. homas, G. N. Coleman. Direct numerical simulation of vorte ring evolution from the laminar to the earl turbulent regime. Journal of Fluid Mechanics 598, (2008). 23. M. Sreedhar, S. Ragab. Large edd simulation of longitudinal stationar vortices. Phsics of Fluids 6, (1994). 24. S. Ragab, M. Sreedhar. Numerical simulation of vortices with aial velocit deficits. Phsics of Fluids 7, (1995).