Rethinking Pipe Flow Stability: Insights from a Meshless Global Analysis

Abstract

Despite extensive experimental evidence of turbulence in Hagen Poiseuille flow, linear stability analysis has not yet confirmed its instability. One challenge is the singularity introduced by the term 1/r in the center of the pipe, which complicates traditional stability approaches. In this study, we explore a global stability analysis using a meshless framework. Although this approach did not recover the expected unstable modes, it revealed a new set of modes with distinct characteristics from those observed in local stability analysis. We analyze these modes and their impact on transient energy growth, demonstrating the effectiveness of the global approach in capturing localized instabilities without requiring multiple simulations.

Figures (10)

• Figure 1: Sample point distribution generated using GMSHgmsh.
• Figure 2: Most unstable mode (with largest growth rate) from global stability analysis for different Reynolds number ($Re$) and streamwise wavenumber ($\alpha$).
• Figure 3: A 2 dimensional Fast Fourier Transform of the most unstable mode corresponding to the modes presented in \ref{['fig:modes_5x5']} from global stability analysis for different Reynolds number ($Re$) and streamwise wavenumber ($\alpha$). Here x-axis scales from 0 to 15 wavenumbers and y-axis scales from 0 to 1 radial distance from center of pipe in each figure.
• Figure 4: Transient growth curves at $Re = 3000$ and $\alpha = 1.0$ from the present meshless method compared with schmid1994optimal
• Figure 5: The optimal (a) perturbation and (b) response corresponding to the maximum growth at a $Re = 3000$ and $\alpha = 1.0$.