Stability characteristics of the spiral Poiseuille flow induced by inner or outer wall rotation

Abstract

In this paper we reveal, that the stability behavior of the spiral Poiseuille flow in an annular gap with rotating inner or outer cylinder exhibits a striking similarity at low and intermediate rotation rates, while both cases differ significantly at higher rotation rates. Based on the work of Vasanta Ram (2019) we formulate the flow problem by means of the inner swirl parameter (Si), the outer swirl parameter (So) and the curvature parameter (ϵ). Six different values of the curvature parameter {0.005; 0.025; 0.111; 0.25; 0.333; 0.785} and a swirl range of 10−5<Si;So<105 are investigated. We reveal that for the rotating outer cylinder case three characteristic jumps of the critical axial and azimuthal wavenumbers can be identified. Similar features can be identified for the inner rotation case. Based on this, we define three regimes delimited for both the rotating inner and rotating outer cylinder which enables us to perform a systematic comparison. The first regime transition occurs at low values of the swirl parameter (Si; So>10−4) and is accompanied by a rapid decrease of the critical Reynolds number. Our computations reveal that this regime transition does not occur for a small curvature parameter of ϵ= {0.005; 0.025} in the outer rotation case. The second wavenumber jump occurs at higher swirl parameters of around Si>2.5 and So>1.6. While the critical Reynoldsnumber decreases monotonously in the inner cylinder rotating case, we show that in the case of rotating outer cylinder a rapid increase of Rec is observed as So increases further. To gain an insight into the mechanisms involved that induce such wavenumber jumps, a detailed comparative study on the associated Reynolds shear stress distributions is performed. For larger ϵ it is shown that the distributions of the Reynolds shear stresses change significantly over the first jump exhibiting similar features for both flow cases, leading to the conclusion that in both cases the same centrifugal instability is triggered here. Instead, the distributions of the Reynolds shear stresses before and after the second jump are fundamentally different for inner and outer rotation. While the Reynolds shear stresses spread over the whole gap height for the inner rotation case, they are nonzero only close to a small regime near the inner cylinder for the outer rotation case at high rotation rates. Overall, analyzing the shear stresses, we reveal that a similar instability arises in the Spiral Poiseuille flow with inner and outer cylinder rotation at intermediate swirls, but at higher swirl outer rotation seems to give rise to an instability associated with a critical layer that is inherently different to the inner rotation case.