On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow

Abstract

In this work we study the efficiency of a spectral Petrov–Galerkin method for the linear and nonlinear stability analysis of the pipe or Hagen–Poiseuille flow. We formulate the problem in solenoidal primitive variables for the velocity field and the pressure term is eliminated from the scheme suitably projecting the equations on another solenoidal subspace. The method is unusual in being based on Chebyshev polynomials of selected parity for the radial variable, avoiding clustering of the quadrature points near the origin, satisfying appropriate regularity conditions at the pole and allowing the use of a fast cosine transform if required. Besides, this procedure provides good conditions for the time marching schemes. For the time evolution, we use semi-implicit time integration schemes. Special attention is given to the explicit treatment and efficient evaluation of the nonlinear terms via pseudospectral partial summations. The method provides spectral accuracy and the linear and nonlinear results obtained are in very good agreement with previous works. The scheme presented can be applied to other flows in unbounded cylindrical geometries.