We numerically study the Stokes eigen-modes in two dimensions on isosceles triangles with apex angle $\theta =\pi /3,$ π/2, and 2π/3 by using two spectral solvers, i.e., a Lagrangian collocation method with a weak formulation for the primitive variables and a Legendre–Galerkin method for the stream-function. We compute the first 6,400 Stokes eigen-modes. With 72 collocation points in each spatial dimension, the eigen-values λn for n ≤ 400 can be obtained with spectral accuracy and at least ten significant digits. We show the symmetry of the Stokes eigen-modes dictated by the geometry of the bounded flow domain. From the spectrally accurate data of the Stokes eigen-modes, the following features are observed. First, the n-dependence of the spectrum λn obeys the Weyl asymptotic formula in two dimensional space: ${\lambda }_n=C_{1}n+C_2\sqrt{n}+o\left(\sqrt{n}\right)$. Second, for an isosceles triangle with legs of unit length, the θ-dependence of the spectrum λn can be accurately approximated by λn (θ)/λn (π/2) ≈ 1/(sin θ), as a consequence the volume-dependence of the coefficient C 1 in the Weyl asymptotic formula. And third, a linear stream function-vorticity correlation is observed in the interior of the flow domain.

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weyl asymptotic formula space math altimgsi40gif overflowscroll mrow msub mimi minmi msub momo msub micmi mn1mn msub minmi momo msub micmi mn2mn msub msqrt minmi msqrt momo miomi mrow momo msqrt minmi msqrt momo mrow mrow math apex angle lagrangian collocation method isosceles triangles italiccitalic inf italicitalicdependence of weak formulation spectral accuracy linear stream functionvorticity correlation stokes eigenmodes primitive variables geometry bounded flow locpostninf italicitalicitalicitalicinf locpostninf italicitalicitalic2 accurate

Pierre Lallemand,Lizhen Chen, Li-Shi Luo, Gérard Labrosse,.Spectrally accurate Stokes eigen-modes on isosceles triangles. 132 (0),1-9.

Pierre Lallemand, et al."Spectrally accurate Stokes eigen-modes on isosceles triangles" 132

Pierre Lallemand,Lizhen Chen, Li-Shi Luo, Gérard Labrosse,"Spectrally accurate Stokes eigen-modes on isosceles triangles"