Stability Analysis on Shear Flow and Vortices in Late Boundary Layer Transition
Pp. 59-114 (56)
Turbulence is still an unsolved scientific problem, which has been regarded
as “the most important unsolved problem of classical physics”. Liu proposed a new
mechanism about turbulence generation and sustenance after decades of research on
turbulence and transition. One of them is the transitional flow instability. Liu believes
that inside the flow field, shear (dominant in laminar) is unstable while rotation (dominant
in turbulence) is relatively stable. This inherent property of flow creates the trend that
non-rotational vorticity must transfer to rotational vorticity and causes the flow transition.
To verify this new idea, this chapter analyzed the linear stability on two-dimensional
shear flow and quasi-rotational flow. Chebyshev collocation spectral method is applied
to solve Orr–Sommerfeld equation. Several typical parallel shear flows are tested as the
basic-state flows in the equation. The instability of shear flow is demonstrated by the
existence of positive eigenvalues associated with disturbance modes (eigenfunctions), i.e.
the growth of these linear modes. Quasi-rotation flow is considered under cylindrical
coordinates. An eigenvalue perturbation equation is derived to study the stability problem
with symmetric flows. Shifted Chebyshev polynomial with Gauss collocation points is
used to solve the equation. To investigate the stability of vortices in flow transition, a
ring-like vortex and a leg-like vortex over time from our Direct Numerical Simulation
(DNS) data are tracked. The result shows that, with the development over time, both ringlike
vortex and leg-like vortex become more stable as Omega becomes close to 1.
Shear flow, Stability analysis, Transition, Turbulence, Vortices.
Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019, USA.