Numerical realization of helical vortices: application to vortex instability
Tóm tắt
The need to numerically represent a free vortex system arises frequently in fundamental and applied research. Many possible techniques for realizing this vortex system exist but most tend to prioritize accuracy either inside or outside of the vortex core, which therefore makes them unsuitable for a stability analysis considering the entire flow field. In this article, a simple method is presented that is shown to yield an accurate representation of the flow inside and outside of the vortex core. The method is readily implemented in any incompressible Navier–Stokes solver using primitive variables and Cartesian coordinates. It can potentially be used to model a wide range of vortices but is here applied to the case of two helices, which is of renewed interest due to its relevance for wind turbines and helicopters. Three-dimensional stability analysis is performed in both a rotating and a translating frame of reference, which yield eigenvalue spectra that feature both mutual inductance and elliptic instabilities. Comparison of these spectra with available theoretical predictions is used to validate the proposed baseflow model, and new insights into the elliptic instability of curved Batchelor vortices are presented. Furthermore, it is shown that the instabilities in the rotating and the translating reference frames have the same structure and growth rate, but different frequency. A relation between these frequencies is provided.
Tài liệu tham khảo
Levy, H., Forsdyke, A.G.: The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120(786), 670 (1928). http://www.jstor.org/stable/95006
Kida, S.: A vortex filament moving without change of form. J. Fluid Mech. 112, 397 (1981). https://doi.org/10.1017/S0022112081000475
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)
Crow, S.C.: Stability theory for a pair of trailing vortices. AIAA J. 8(12), 2172 (1970). https://doi.org/10.2514/3.6083
Rosenhead, L.: The spread of vorticity in the wake behind a cylinder. Proc. R. Soc. Lond. A 127(806), 590 (1930). https://doi.org/10.1098/rspa.1930.0078
Moore, D.W.: Finite amplitude waves on aircraft trailing vortices. Aeronaut. Q. 23(4), 307 (1972). https://doi.org/10.1017/S000192590000620X
Saffman, P.G.: The velocity of viscous vortex rings. Stud. Appl. Math. 49(4), 371 (1970). https://doi.org/10.1002/sapm1970494371
Widnall, S.E., Bliss, D., Zalay, A.: Theoretical and experimental study of the stability of a vortex pair, pp. 339–354. Springer, Boston (1971). https://doi.org/10.1007/978-1-4684-8346-8_19
Hardin, J.C.: The velocity field induced by a helical vortex filament. Phys. Fluids 25(11), 1949 (1982). https://doi.org/10.1063/1.863684
Okulov, V.L.: On the stability of multiple helical vortices. J. Fluid Mech. 521, 319 (2004). https://doi.org/10.1017/S0022112004001934
Callegari, A.J., Ting, L.: Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Math. 35(1), 148 (1978). https://doi.org/10.1137/0135013
Blanco-Rodríguez, F.J., Le Dizès, S., Selçuk, C., Delbende, I., Rossi, M.: Internal structure of vortex rings and helical vortices. J. Fluid Mech. 785, 219 (2015). https://doi.org/10.1017/jfm.2015.631
Segalini, A., Alfredsson, P.H.: A simplified vortex model of propeller and wind-turbine wakes. J. Fluid Mech. 725, 91 (2013). https://doi.org/10.1017/jfm.2013.182
Sørensen, J.N., Shen, W.Z.: Numerical modeling of wind turbine wakes. J. Fluids Eng. 124(2), 393 (2002). https://doi.org/10.1115/1.1471361
Kleusberg, E., Schlatter, P., Henningson, D.S.: Parametric study of the actuator line method in high-order codes. Technical report, Department of Mechanics, KTH Royal Institute of Technology (2017)
Selçuk, S., Delbende, I., Rossi, M.: Helical vortices: quasiequilibrium states and their time evolution. Phys. Rev. Fluids 2, 084701 (2017). https://doi.org/10.1103/PhysRevFluids.2.084701
Quaranta, H.U., Brynjell-Rahkola, M., Leweke, T., Henningson, D.S.: Local and global pairing instabilities of two interlaced helical vortices. J. Fluid Mech. 863, 927 (2019). https://doi.org/10.1017/jfm.2018.904
Saffman, P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1992)
Batchelor, G.K.: Axial flow in trailing line vortices. J. Fluid Mech. 20(4), 645 (1964). https://doi.org/10.1017/S0022112064001446
Quaranta, H.U., Bolnot, H., Leweke, T.: Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687 (2015). https://doi.org/10.1017/jfm.2015.479
Ivanell, S., Mikkelsen, R., Sørensen, J.N., Henningson, D.: Stability analysis of the tip vortices of a wind turbine. Wind Energy 13(8), 705 (2010). https://doi.org/10.1002/we.391
Sarmast, S., Dadfar, R., Mikkelsen, R.F., Schlatter, P., Ivanell, S., Sørensen, J.N., Henningson, D.S.: Mutual inductance instability of the tip vortices behind a wind turbine. J. Fluid Mech. 755, 705 (2014). https://doi.org/10.1017/jfm.2014.326
Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69 (1995). https://doi.org/10.1017/S0022112095000462
Fischer, P.F., Lottes, J.W., Kerkemeier, S.G.: Nek5000 web page (2008). https://nek5000.mcs.anl.gov
Maday, Y., Patera, A.T.: Spectral element methods for the Navier–Stokes equations. In: Noor, A.K. (ed.) State of the Art Surveys in Computational Mechanics, pp. 71–143. ASME, New York (1989)
Fischer, P.F., Lottes, J.W.: Hybrid Schwarz–Multigrid Methods for the Spectral Element Method: Extensions to Navier–Stokes, pp. 35–49. Springer, Berlin (2005). https://doi.org/10.1007/3-540-26825-1_3
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856 (1986). https://doi.org/10.1137/0907058
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in FORTRAN 77: Volume 1, Volume 1 of Fortran Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (1992)
Okulov, V.L., Sørensen, J.N.: Maximum efficiency of wind turbine rotors using Joukowsky and Betz approaches. J. Fluid Mech. 649, 497 (2010). https://doi.org/10.1017/S0022112010000509
Malm, J., Schlatter, P., Fischer, P.F., Henningson, D.S.: Stabilization of the spectral element method in convection dominated flows by recovery of skew-symmetry. J. Sci. Comput. 57(2), 254 (2013). https://doi.org/10.1007/s10915-013-9704-1
Negi, P., Schlatter, P., Henningson, D.S.: A re-examination of filter-based stabilization for spectral element methods. Techical report, Department of Mechanics, KTH Royal Institute of Technology (2017)
Sipp, D., Jacquin, L., Cossu, C.: Self-adaptation and viscous selection in concentrated two-dimensional vortex dipoles. Phys. Fluids 12(2), 245 (2000). https://doi.org/10.1063/1.870325
Le Dizès, S., Verga, A.: Viscous interactions of two co-rotating vortices before merging. J. Fluid Mech. 467, 389 (2002). https://doi.org/10.1017/S0022112002001532
Leweke, T., Le Dizès, S., Williamson, C.H.K.: Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48(1), 507 (2016). https://doi.org/10.1146/annurev-fluid-122414-034558
Moore, D.W., Saffman, P.G.: Structure of a Line Vortex in an Imposed Strain, pp. 339–354. Springer, Boston (1971). https://doi.org/10.1007/978-1-4684-8346-8_20
Widnall, S.E.: The stability of a helical vortex filament. J. Fluid Mech. 54, 641 (1972). https://doi.org/10.1017/S0022112072000928
Gupta, B.P., Loewy, R.G.: Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12(10), 1381 (1974). https://doi.org/10.2514/3.49493
Hill, D.C.: Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183 (1995). https://doi.org/10.1017/S0022112095001480
Bagheri, S., Åkervik, E., Brandt, L., Henningson, D.S.: Matrix-free methods for the stability and control of boundary layers. AIAA J. 47(5), 1057 (2009). https://doi.org/10.2514/1.41365
Maschhoff, K.J., Sorensen, D.C.: P\_ARPACK: an efficient portable large scale eigenvalue package for distributed memory parallel architectures. In: Waśniewski, J., Dongarra, J., Madsen, K., Olesen, D. (eds.) Applied Parallel Computing Industrial Computation and Optimization, pp. 478–486. Springer, Berlin (1996)
Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK User’s Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. Society for Industrial and Applied Mathematics, Philadelphia (1998). https://doi.org/10.1137/1.9780898719628
Lacaze, L., Ryan, K., Le Dizès, S.: Elliptic instability in a strained Batchelor vortex. J. Fluid Mech. 577, 341 (2007). https://doi.org/10.1017/S0022112007004879
Roy, C., Schaeffer, N., Le Dizès, S., Thompson, M.: Stability of a pair of co-rotating vortices with axial flow. Phys. Fluids 20(9), 094101 (2008). https://doi.org/10.1063/1.2967935
Selçuk, S.C., Delbende, I., Rossi, M.: Helical vortices: linear stability analysis and nonlinear dynamics. Fluid Dyn. Res. 50(1), 011411 (2017). https://doi.org/10.1088/1873-7005/aa73e3
Hattori, Y., Blanco-Rodríguez, F.J., Le Dizès, S.: Numerical stability analysis of a vortex ring with swirl. J. Fluid Mech. 878, 5 (2019). https://doi.org/10.1017/jfm.2019.621
Robinson, A.C., Saffman, P.G.: Three-dimensional stability of vortex arrays. J. Fluid Mech. 125, 411 (1982). https://doi.org/10.1017/S0022112082003413
Blanco-Rodríguez, F.J., Le Dizès, S.: Curvature instability of a curved Batchelor vortex. J. Fluid Mech. 814, 397 (2017). https://doi.org/10.1017/jfm.2017.34
Moore, D.W., Saffman, P.G.: The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346(1646), 413 (1975). https://doi.org/10.1098/rspa.1975.0183
Eloy, C., Le Dizès, S.: Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13(3), 660 (2001). https://doi.org/10.1063/1.1345716
Sipp, D., Jacquin, L.: Widnall instabilities in vortex pairs. Phys. Fluids 15(7), 1861 (2003). https://doi.org/10.1063/1.1575752
Blanco-Rodríguez, F.J., Le Dizès, S.: Elliptic instability of a curved Batchelor vortex. J. Fluid Mech. 804, 224 (2016). https://doi.org/10.1017/jfm.2016.533
Lacaze, L., Birbaud, A.L., Le Dizès, S.: Elliptic instability in a Rankine vortex with axial flow. Phys. Fluids 17(1), 017101 (2005). https://doi.org/10.1063/1.1814987
Kleusberg, E., Schlatter, P., Henningson, D.S.: Wind Energy (2019, accepted for publication)
Brocklehurst, A., Barakos, G.: A review of helicopter rotor blade tip shapes. Prog. Aerosp. Sci. 56, 35 (2013). https://doi.org/10.1016/j.paerosci.2012.06.003