Structures of nonlinear disturbances arising from shear flow instabilities

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As an example of typical shear flows we consider internally heated convective flows in an inclined channel with two parallel plates of infinite extent. The strength of the shear is parameterized by the Grashof number which measures the intensity of the homogeneously distributed heat source. The stability of the basic shear flow, that can have two inflectional points in its velocity profile, Is examined and subsequent nonlinear states are followed numerically. In the case of vertical orientation of the channel, the secondary flow in a form of two-dimensional traveling transverse vortices emerges from the basic state via a Hopf bifurcation regardless of the value of the Prandtl number, Pr, of the fluid, as the Grashof number is increased. We find that as the Grashof number is further increased the secondary flow loses its stability also in a Hopf bifurcation for the fluid with Pr=0 so that the tertiary flow is represented by three-dimensionality in space and quasi-periodicity in time. In the case of water (Pr=7.0) the secondary instability is described by a quasi-periodic mode as in the case of the fluid with Pr=0, whereas the instability is characterized by a phase-locked mode in the case of air (Pr=0.71).The extension of the investigation to cases with inclined orientation of the channel shows that perturbations in the form of longitudinal vortices, in contrast to the transverse vortices in the vertical orientation, are most dangerous for a wide range of the value of inclination angles. The most surprising finding is that the inclination of the channel angle by one degree from the vertical orientation is enough to change the spatial structure of disturbances which are responsible for the instability of the basic state.