On the mechanism of sinuous and varicose modes in three-dimensional viscous secondary instability of nonlinear Görtler rolls

Abstract

The present study begins with the solution of the linearized partial differential equations for viscous, secondary instabilities of nonlinearly developed primary Görtler rolls. A global energy balance verifies the amplification rates obtained from secondary instability analysis and confirms that the sinuous mode would dominate. This therefore explains the strong resemblance between the rms-streamwise u-velocity structure obtained by hot-wire measurements [Swearingen and Blackwelder, J. Fluid Mech. 182, 255 (1987)] in the cross-sectional yz plane and that of the sinuous mode. The u contours of the secondary disturbance have peaks at locations corresponding to the shoulder regions of the mushroom-like primary streamwise U-velocity structure and even more intense regions surrounding the mushroom stem near the wall. This structure is explained by detailed analysis of the energy-balancing mechanisms for u2/2 in yz plane: the Reynolds stress-conversion mechanism predominantly affects the primary spanwise rate of shear strain ∂U/∂z, whereas of less importance is the effect on the normal rate of shear strain ∂U/∂y; these are delicately balanced by the isotropizing mechanism of pressure rate of strain correlation and viscous dissipation. The appropriate pressure rates of strain correlation then provide the sources for the vertical v2/2 and spanwise w2/2 contributions to the kinetic energy. The sinuous mode can be identified with streamwise-propagating structures having spanwise waviness in flow visualization studies. The spanwise-antisymmetric w, which has intense regions in the upper and lower levels of the boundary layer, tends to wag the primary mushroom structure both at the upper domed region and in the stem region near the wall, not necessarily in phase, in the spanwise direction in the y,z plane, as shown in the flow-visualization work of Peerhossaini and Wesfreid [Intl. J. Heat Fluid Flow 9, 12 (1988)]. On the other hand, the infrequency of the varicose mode can be attributed to its relatively lower amplification rate for its most amplified mode. When observed, it is identifiable with the spanwise-symmetric streamwise propagating, repetitive ‘‘horseshoe’’ structures appearing in flow visualization. Energy balance analysis reveals that this mode is attributable mainly to the Reynolds stress-conversion mechanism associated with rate of strain ∂U/∂y in the high shear layer in the peak region, although the mechanism associated with ∂U/∂z is not dramatically smaller. Viscous dissipation, though not large enough to overcome the production mechanisms, is nevertheless prominent in the global balance as well as in the detailed structural balances in the yz plane. The secondary instability vorticity structure indicates that the sinuous mode is dominated by its vertical component ηs and the varicose mode by its spanwise component ζv.