The incompressible boundary layer in the axial flow past a cylinder has been shown Tutty et. al. ([11]) to be stabler than a two-dimensional boundary layer, with the helical mode being the least stable. In this paper the secondary instability of this flow is studied. The laminar flow is shown here to be always stable at high transverse curvatures to secondary disturbances, which, together with a similar observation for the linear modes implies that the flow past a thin cylinder is likely to remain laminar. The azimuthal wavenumber of the pair of least stable secondary modes (m + and m − ) are related to that of the primary (n) by m + = 2n and m − = −n. The base flow is shown to be inviscidly stable at any curvature.Keywords Hydrodynamic stability · Boundary layer PACS 47.15.Cb · 47.15.Fe · 47.20.Lz
IntroductionAt low to moderate freestream disturbance levels, the first step in the process of transition to turbulence in a boundary layer is that at some streamwise location, the laminar flow becomes unstable to linear disturbances. While this instability and the events that follow have been investigated in great detail for two-dimensional boundary layers during the past century, much less work has been done on its axisymmetric counterpart, the incompressible boundary layer in the flow past a cylinder, notable exceptions being the early and approximate linear stability analysis of Rao[7] and the more recent and accurate one of Tutty et. al. [11]. In Rao's work, the equations were not solved directly and the stability estimates had severe limitations. Tutty et. al. [11] showed that non-axisymmetric modes are less stable than axisymmetric ones. The critical Reynolds number was found to be 1060 for the n = 1 mode and 12439 for n = 0. The instability is thus of a different character from that in two-dimensional boundary layers, since Squire's (1933) theorem, stating that the first instabilities are twodimensional, is not applicable in this case. The expected next stage of the process of transition to turbulence, namely the secondary modes of instability, have not been studied before, to our knowledge, although the turbulent flow over a long thin cylinder has been studied by Tutty [10], who computed the meanflow quantities N. Vinod Engineering Mechanics Unit
This paper presents linear biglobal stability analysis of axisymmetric boundary layer over a circular cone. An incompressible flow over a sharp circular cone is considered with zero angle of attack. The base flow velocity profile is fully non-parallel and non-similar. Linearized Navier-Stokes (LNS) equations are derived for disturbance flow quantities using the standard procedure. The LNS equations are discretized using Chebyshev spectral collocation method. The governing equations along with boundary conditions form a general eigenvalues problem. The numerical solution of general eigenvalues problem is obtained using ARPACK subroutine, which uses Arnoldis iterative algorithm. The global temporal modes are computed for the range of Reynolds number and semi-cone angles(α)for the axisymmetric mode(N=0). The flow is found temporally and spatially stable for 1° semi-cone angle and the range of Reynolds numbers considered. However, flow becomes temporally unstable and spatially stable with the increase in semi-cone angle(α). The wave-like behaviour of the disturbances is found at small semi-cone angles (α).
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