Instability of streaks in pipe flow of shear-thinning fluids.
Résumé
This study is motivated by recent experimental results dealing with the transition to turbulence in a pipe flow of shear-thinning fluids, where a streaky flow with an azimuthal wave number n=1 is observed in the transitional regime. Here, a linear stability analysis of pipe flow of shear-thinning fluids modulated azimuthally by finite amplitude streaks is performed. The shear-thinning behavior of the fluid is described by the Carreau model. The streaky base flows considered are obtained from two-dimensional direct numerical simulation using finite amplitude longitudinal rolls as the initial condition and by extracting the velocity field at time t(max), where the amplitude of the streaks reaches its maximum, denoted by A(max). It is found that the amplitude A(max) increases with increasing Reynolds number as well as with increasing amplitude E(0) of the initial longitudinal rolls. For sufficiently large streaks amplitude, streamwise velocity profiles develop inflection points, leading to instabilities. Depending on the threshold amplitude A(c), two different modes may trigger the instability of the streaks. If A(c) exceeds approximately 41.5% of the centerline velocity, the instability mode is located near the axis of the pipe, i.e., it is a "center mode." For weaker amplitude A(c), the instability mode is located near the pipe wall, in the region of highest wall normal shear, i.e., it is a "wall mode." The threshold amplitude A(c) decreases with increasing shear-thinning effects. The energy equation analysis indicates that (i) wall modes are driven mainly by the work of the Reynolds stress against the wall normal shear and (ii) for center modes, the contribution of the normal wall shear remains dominant; however, it is noted that the contribution of the Reynolds stress against the azimuthal shear increases with increasing shear-thinning effects.