On New Linear Sub-Critical Oblique Modes - an Extension of Squires Theorem for Spatial Instabilities

  • Oberlack, Martin (TU Darmstadt)
  • Yalcin, Alparslan (TU Darmstadt)
  • Laux, Jonathan (TU Darmstadt)

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In his fundamental theorem, Squire (1933) showed that for time-evolving perturbations, 2D instabilities occur at the smallest Reynolds number, which is usually called the critical Reynolds number. We recently showed that this is not necessarily so for spatially evolving 3D modes. For this we introduced both a complex streamwise and spanwise wavenumbers (see Turkac, 2019, Yalcin, 2022), which gives rise to oblique 3D modes. Such modes have a neutral stability line (NSL) in the x-z-plane, which is oblique to the main flow direction. Extending Squires idea by invoking symmetry methods in parameter space we show that oblique 3D instabilities at a Reynolds number below the critical 2D Reynolds number may exist. Using a direct numerical simulation and based on the extended approach, it is indeed possible to detect corresponding modes e.g. for the plane Couette flow. Other than for temporally evolving modes, however, for spatially evolving modes the additional condition of group velocity (GV) must be taken into account, which states that the GV has to propagate in the direction of the spatially increasing mode. In other words, for 3D instabilities the vector of the GV must thus point into the unstable region, i.e. cross the above mentioned NSL in the x-z-plane. We have worked out the above extension of Squire's theory based on Lie symmetries and will present it at the meeting as well as further details on the direction of the GV for the case of Couette flow and/or the Asymptotic Suction Boundary layer.