Skip to main content Accessibility help

Theory of vortex nutation and amplitude oscillation in an inviscid shear instability

  • Akira Miura (a1) and Tetsuya Sato (a1)


In the first part of this paper the nonlinear development of the most unstable mode is numerically studied for a bounded shear layer with a hyperbolic-tangent profile. It is found that the vortex nutation, discovered by Zabusky & Deem (1971) for a jet profile, is a manifestation of strongly coupled oscillations in the vortex amplitude and the phase. In the second part, with the aid of the numerical result we devote ourselves to deriving coupled nonlinear equations that describe the amplitude oscillation, the vortex nutation and the momentum transport. The approximate oscillatory solution for the vortex amplitude and phase in the nonlinear stage is compared with the numerical solution and agreement is found.



Hide All
Al'Tshul’, L. M. & Karpman, V. I.1966 Theory of nonlinear oscillations in a collisionless plasma. Sov. Phys., J. Exp. Theor. Phys. 22, 1422.
Byers, J. A. 1966 Confined and nonconfined interchange instabilities obtained from nonlinear computer models. Phys. Fluids 9, 10381040.
Christiansen, J. P. & Zabusky, N. J. 1973 Instability, coalescence and fission of finite-area vortex structures. J. Fluid Mech. 61, 219243.
Levy, R. H. & Hockney, R. W. 1968 Computer experiments on low-density crossed-field electron beams. Phys. Fluids 11, 766771.
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel planes for finite disturbances. Proc. Roy. Soc. A 208, 517526.
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19, 543556.
Michalke, A. 1965 Vortex formation in a free boundary layer according to stability theory. J. Fluid Mech. 22, 371383.
Moore, R. E. & Saffman, P. G. 1971 Aircraft Wake Turbulence and its Detection. Plenum.
O'Neil, T.1965 Collisionless damping of nonlinear plasma oscillations. Phys. Fluids 8, 22552262.
Sato, H. & Kuriki, K. 1961 The mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow. J. Fluid Mech. 11, 321352.
Sato, T. 1975 Creation and annihilation of positive and negative pulses in an unstable macroplasma. Phys. Rev. Lett. 35, 223226.
Schade, H. 1964 Contribution to the nonlinear stability theory of inviscid shear layers. Phys. Fluids 7, 623628.
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 121.
Tanaka, H. 1975 Quasi-linear and non-linear interactions of finite amplitude perturbations in a stably stratified fluid with hyperbolic tangent shear. J. Met. Soc. Japan 53, 132.
Vahala, G. & Montgomery, D. 1971 Kinetic theory of two-dimensional magnetized plasma. J. Plasma Phys. 6, 425439.
Zabusky, N. J. & Deem, G. S. 1971 Dynamical evolution of two-dimensional unstable shear flows. J. Fluid Mech. 47, 353379.
MathJax is a JavaScript display engine for mathematics. For more information see


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed