Subcritical 3D modes employing Briggs’ method – exemplified using plane Couette flow

In various parallel shear flows such as plane Poiseuille or plane Couette flow, oblique structures are observed in subcritical transition. For plane Couette flow, the classical linear stability theory for temporal modes predicts unconditional stability, which is in stark contrast to experiments and DNS.

Presently, Squire’s transformation, originally based on the temporal stability framework and demonstrating that the critical Reynolds number for 2D modes is always lower than for 3D modes, is now extended to the spatio-temporal stability framework. Through this extension, spatio-temporally evolving 3D oblique modes are formulated. A key finding is that, unlike the classical Squire theorem, 3D modes can have critical Reynolds numbers lower than those of 2D modes. However, single spatial or spatio-temporal modes cannot be used in a meaningful way, as their perturbation energy diverges even at the initial state. To overcome this difficulty, the spatio-temporally evolving modes have to be used in the context of Briggs’ method. For this, the inverse Laplace and Fourier transforms have to be evaluated at saddle and pinch points in the complex planes of frequency and wave number.

The central difficulty of Briggs' method is to locate these points. For this, we present a new algorithm for the continuous deformation of the integration contours in the complex frequency and wave number plane. The spatio-temporal linear stability of plane Couette flow is analyzed using this algorithm.