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Secondary Instabilities in Incompressible Axisymmetric Boundary Layers: Effect of Transverse Curvature

[+] Author and Article Information
N. Vinod

 Department of Mechanical Engineering, Indian Institute of Technology, Gandhinagar, Chandkheda, Ahmedabad 382 424, Indiavinod@iitgn.ac.in

Rama Govindarajan

 Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research,Jakkur, Bangalore 560 064, Indiarama@jncasr.ac.in

J. Fluids Eng 134(2), 024503 (Mar 19, 2012) (4 pages) doi:10.1115/1.4005767 History: Received April 01, 2011; Revised December 22, 2011; Published March 16, 2012; Online March 19, 2012

The secondary instability of the incompressible boundary layer in the axial flow past a cylinder is studied. The laminar flow is shown to be always stable at high transverse curvatures to secondary disturbances. Because the primary mode is stable as well, (Tutty , 2002, “Boundary Layer Flow on a Long Thin Cylinder,”. Phys. Fluids, 14 (2), pp. 628–637), this implies that the boundary layer on a thin long cylinder may undergo transition to turbulence by means very different from that on a flat plate. The azimuthal wavenumber of the least stable secondary modes (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.

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Grahic Jump Location
Figure 4

Growth rate of secondary disturbance modes with Ap  = 0.02. (a) n = 2, m+  = 4, and m−  = −2 at R = 3000; and (b) n = 3, m+  = 6, and m−  = −3 at R = 5000.

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Figure 3

Growth rate of secondary disturbance along the cylinder axis for (a) R = 2000, and (b) R = 5000. The other parameters are the same as in Fig. 2.

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Figure 2

Growth rate of secondary disturbance waves along the cylinder axis for the nonaxisymmetric mode n = 1, m−  = −1, with Ap  = 0.02 at R = 1000. The most unstable primary mode (α = 0.125 and n = 1) is shown by the dashed line.

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Figure 1

Schematic diagram showing the coordinate system

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