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Effect of Geometry on Direct and Adjoint Linear Global Modes of Low Reynolds Number Laminar Flow over Body


We study a two-dimensional flow over a cylinder and three ellipses with the aspect ratio \(a / b = 2.04\), \(3.65\), and \(5\) at a Reynolds number \(Re = 100\) based on the inflow velocity and hydraulic diameter using a direct numerical simulation (DNS) and linear stability analysis. The DNS shows that increase in \(a / b\) leads to stabilization of the flow due to decrease in the recirculation zone. The linear stability analysis based on the time-averaged velocity field shows that the modes of interest describe a Karman vortex street. The oscillation frequency reconstructed from the linear stability analysis is in excellent agreement with the DNS. Solving the linearized adjoint equations made it possible to identify the flow area where direct and adjoint modes overlap. These areas of the field \(S_w\), usually called “wavemaker,” change their shape with increase in \(a / b\). Non-zero values of \(S_w\) tend to approach the bottom part of the “tail” of the ellipse due to the asymmetry of the recirculation region, attached to the top side of the ellipse.

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  1. 1

    Mathis, C., Provansal, M., and Boyer, L., The Benard-von Karman Instability: An Experimental Study near the Threshold, J. Phys. Lett., 1984, vol. 45, pp. 483–491.

  2. 2

    Williamson, C.H., Vortex Dynamics in the Cylinder Wake, Ann. Rev. Fluid Mech., 1996, vol. 28, pp. 477–539.

  3. 3

    Schmid, P.J. and Henningso, D.S., Stability and Transition in Shear Flows, New York: Springer-Verlag, 2001.

  4. 4

    Hill, D.C., A Theoretical Approach for Analysing the Restabilization of Wakes, AIAA, 1992, pp. 92–106.

  5. 5

    Hill, D.C., Adjoint Systems and Their Role in the Receptivity Problem for Boundary Lyers, J. Fluid Mech., 1995, vol. 292, pp. 183–204.

  6. 6

    Bottaro, A., Corbett, P., and Luchini, P., The Effect of Base Flow Variation on Flow Stability, J. Fluid Mech., 2003, vol. 476, pp. 293–302.

  7. 7

    Chomaz, J.M., Global Instabilities in Spatially Developing Flows: Non-Normality and Nonlinearity, Ann. Rev. Fluid Mech., 2005, vol. 37, pp. 357–392.

  8. 8

    Giannetti, F. and Luchini, P., Structural Sensitivity of the First Instability of the Cylinder Wake, J. Fluid Mech., 2007, vol. 581, pp. 167–197.

  9. 9

    Marquet, O., Sipp, D., and Jacquin, L., Sensitivity Analysis and Passive Control of Cylinder Flow, J. Fluid Mech., 2008, vol. 615, pp. 221–252.

  10. 10

    Zebib, A., Stability of Viscous Flow past a Circular Cylinder, J. Eng. Math., 1987, vol. 21, pp. 155–165.

  11. 11

    Jackson, C.P., A Finite-Element Study of the Onset of Vortex Shedding in Flow past Variously Shaped Bodies, J. Fluid Mech., 1987, vol. 182, pp. 23–45.

  12. 12

    Strykowski, P.J. and Sreenivasan, K.R., Sensitivity Analysis and Passive Control of Cylinder Flow, J. Fluid Mech., 1990, vol. 218, pp. 71–107.

  13. 13

    Kim, H.B. and Chang, K.S., Numerical Study on Vortex Shedding from a Circular Cylinder Influenced by a Nearby Control Wire, Int. J. Comp. Fluid Dyn., 1995, vol. 4, pp. 151–164.

  14. 14

    Mittal, S. and Raghuvanshi, A., Control of Vortex Shedding behind Circular Cylinder for Flows at Low Reynolds Numbers, Int. J. Num. Meth. Fluids, 2001, vol. 35, pp. 421–447.

  15. 15

    Morzynski, M., Afanasiev, K., and Thiele, F., Solution of the Eigenvalue Problems Resulting from Global Non-Parallel Flow Stability Analysis, Comp. Meth. Appl. Mech. Engin., 1999, vol. 169, pp. 161–176.

  16. 16

    Pralits, J.O., Brandt, L., and Giannetti, F., Instability and Sensitivity of the Flow around a Rotating Circular Cylinder, J. Fluid Mech., 2010, vol. 650, pp. 513–536.

  17. 17

    Meliga, P., Chomaz, J.M., and Sipp, D., Unsteadiness in the Wake of Disks and Spheres: Instability, Receptivity and Control Using Direct and Adjoint Global Stability Analyses, J. Fluids Struct., 2009, vol. 25, pp. 601–616.

  18. 18

    Theofilis, V., Barkley, D., and Sherwin, S., Spectral/Hp Element Technology for Global Flow Instability and Control, The Aero. J., 2002, vol. 106, pp. 619–625.

  19. 19

    Kitsios, V., Rodriguez, D., Theofilis, V., Ooi, A., and Soria, J., Biglobal Stability Analysis in Curvilinear Coordinates of Massively Separated Lifting Bodies, J. Comp. Phys., 2009, vol. 228, pp. 7181–7196.

  20. 20

    Zhang, W. and Samtaney, R., Biglobal Linear Stability Analysis on Low-Re Flow past an Airfoil at High Angle of Attack, Phys. Fluids, 2016, vol. 28, p. 044105.

  21. 21

    He, W., Yu, P., and Li, L.K., Ground Effects on the Stability of Separated Flow around a Naca 4415 Airfoil at Low Reynolds Numbers, Aero. Sci. Tech., 2019, vol. 72, pp. 63–76.

  22. 22

    He, W., Guan, Y., Theofilis, V., and Li, L.K., Stability of Low-Reynolds-Number Separated Flow around an Airfoil near a Wavy Ground, AIAA, 2019, vol. 57, pp. 29–34.

  23. 23

    Iorio, M.C., Gonzalez, L.M., and Ferrer, E., Direct and Adjoint Global Stability Analysis of Turbulent Transonic Flows over a Naca0012 Profile, Int. J. Num. Meth. Fluids, 2014, vol. 76, pp. 147–168.

  24. 24

    Sartor, F., Mettot, C., and Sipp, D., Stability, Receptivity, and Sensitivity Analyses of Buffeting Transonic Flow over a Profile, AIAA, 2015, vol. 53, pp. 1980–1993.

  25. 25

    Pando, M.F., Schmid, P.J., and Sipp, D., On the Receptivity of Aerofoil Tonal Noise: An Adjoint Analysis, J. Fluid Mech., 2017, vol. 812, pp. 771–791.

  26. 26

    Dauengauer, E.I., Mullyadzhanov, R.I., Timoshevskiy, M.V., Zapryagaev, I.I., and Pervunin, K.S., Flow around a Low-Aspect-Ratio Wall-Bounded 2D Hydrofoil: A LES/PIV Study, J. Phys.: Conf. Ser., 2018, vol. 1128, p. 012085.

  27. 27

    Fischer, P.F., Lottes, J.W., and Kerkemeier, S.G., Nek5000 Web Page, 2008;

  28. 28

    Patera, A.T., A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion, J. Comput. Phys., 1984, vol. 54, pp. 468–488.

  29. 29

    Peplinski, A., Schlatter, P., Fischer, P.F., and Henningson, D.S., Stability Tools for the Spectral-Element Code Nek5000: Application to Jet-In-Crossflow, New York: Springer, 2014.

  30. 30

    Noack, B.R. and Eckelmann, H., A Global Stability Analysis of the Steady and Periodic Cylinder Wake, J. Fluid Mech., 1994, vol. 270, pp. 297–330.

  31. 31

    Sreenivasan, K.R., Strykowski, P.J., and Olinger, D.J., Hopf Bifurcation, Landau Equation, and Vortex Shedding behind Circular Cylinders, in Conf. Proc., Am. Soc. of Mechanical Engineers, Fluids Engineering Division, vol. 52, 1987, pp. 1–13.

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The work was supported by RFBR, project nos. 18-38-20167 and 19-48-543036 (VR and RM for numerical simulations), and by RF Ministry of Education and Science, grant no. 075-15-2019-1923 (DS for global stability analysis). The development of numerical tools is performed under the state contract with IT SB RAS. The computational resources are provided by the Siberian Supercomputer Center SB RAS, Supercomputer Center of the Novosibirsk State University, and Joint Supercomputer Center RAS.

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Correspondence to V. O. Ryzhenkov.

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Ryzhenkov, V.O., Sozinov, D.A. & Mullyadzhanov, R.I. Effect of Geometry on Direct and Adjoint Linear Global Modes of Low Reynolds Number Laminar Flow over Body. J. Engin. Thermophys. 29, 576–581 (2020).

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