Current Developments in Mathematical Sciences

Current Developments in Mathematical Sciences

Volume: 2

Liutex-based and Other Mathematical, Computational and Experimental Methods for Turbulence Structure

The knowledge of quantitative turbulence mechanics relies heavily upon the definition of the concept of a vortex in mathematical terms. This reference work introduces the reader to Liutex, which is ...
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Stability Analysis on Shear Flow and Vortices in Late Boundary Layer Transition

Pp. 59-114 (56)

Jie Tang


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.