Instability of spiral Poiseuille flow with either inner or outer cylinder rotation

  • Brockmann, Philipp (Tu Darmstadt SLA)
  • Vasanta Ram, Venkatesa Iyengar (Ruhr Universit√§t Bochum)
  • Jakirlic, Suad (Tu Darmstadt SLA)
  • Hussong, Jeanette (Tu Darmstadt SLA)

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Spiral Poiseuille Flow (SPF) refers to the flow between two concentric cylinders, induced by cylinder rotation and an axial pressure gradient. This flow can result in Tollmien-Schlichting type and centrifugal instabilities (Meseguer & Marques 2002, Meseguer & Marques 2005). The stability of the SPF, with either a rotating inner cylinder (IRSPF) or a rotating outer cylinder (ORSPF), is of great interest from both theoretical and practical perspectives. The IRSPF is characterized by a centrifugally unstable azimuthal flow, while the ORSPF is characterized by a centrifugally stable azimuthal flow. Despite this difference, we have found that the stability behavior of the IRSPF and ORSPF exhibits a remarkable similarity at low and intermediate swirls. However, both cases differ significantly at higher rotation rates. To study this problem, we formulated it using a curvature parameter and a swirl parameter, which describes the ratio between the azimuthal and axial velocity. Using linear stability analysis, we conducted extensive computations covering 77 (IRSPF) and 108 (ORSPF) different values of the curvature parameter ranging from 0.0025 to 0.785. The swirl parameter range considered was from 10$^{-5}$ to $10^5$ for both the IRSPF and the ORSPF. The results were used to generate phase maps, which fully cover the stability behavior of both flow cases as a function of curvature parameter and swirl parameter. For the first time, we introduced a method to analyze and identify instability mechanisms. Specifically, we analyzed the budgets of the Reynolds shear stresses and used the concept of the critical layer for non-axisymmetric disturbances. Using these identification methods, we were able to identify three regions associated with different instability characteristics in the phase maps. We found that within the first region, a Tollmien-Schlichting mechanism was present, while in the second region, a centrifugal instability mechanism was present for both the IRSPF and the ORSPF. The centrifugal instability mechanism led to a sharp decrease in the critical Reynolds number with increasing swirl parameter. In the third region, the stability behavior of the IRSPF and ORSPF differed significantly. It was shown that the underlying reason for this difference is that centrifugal effects have an opposite effect on the production of Reynolds shear stresses in the IRSPF and ORSPF within the third region.