Kinetic linear stability theory for incompressible parallel shear flows with the full Boltzmann collision operator

Vassilis Theofilis

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arXiv · 2610.38714

Kinetic linear stability theory for incompressible parallel shear flows with the full Boltzmann collision operator

Vassilis Theofilis

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Abstract

Modal and non-modal linear stability analysis of incompressible shear flows is performed for the first time by linearizing the Boltzmann equation and its full collision operator. In this kinetic linear stability theory (kLST) the hard-sphere operator and its BGK, ES-BGK and Shakhov relaxation models share the same discretization, so that their results can be compared directly. Written in the peculiar velocity about a Chapman-Enskog base state, the collision operators are independent of the cross-stream coordinate and are formed once. Summation-by-parts Legendre collocation, weak diffuse-reflection wall conditions and a moment-consistent velocity derivative yield an energy-stable discrete operator, verified to spectral accuracy by a manufactured solution. Numerical solution of the four-dimensional problem, comparable in size to a TriGlobal problem (Theofilis 2011), is made feasible by the fast spectral method of Gamba, Haack, Hauck and Hu (2017) for the full operator. Modal kLST uses matrix-free shift-invert Arnoldi iteration, preconditioned by an exact relaxation-model inverse. Non-modal kLST is based on an energy norm containing Chu's disturbance energy, a kinetic Reynolds-Orr equation identifying the mean shear as the only energy source. Transient growth and pseudospectra follow from the same preconditioned solves. For plane Couette and Poiseuille flows, the mixing layer and the Blasius boundary layer at Reynolds numbers 10^2-10^5, all four operators recover the Orr-Sommerfeld eigenvalues (Orszag 1971; Mack 1976) in the incompressible limit to a few parts in 10^7 for the models and 10^6 for the full operator. The optimal growth of Butler and Farrell (1992) and the resolvent norms of Trefethen et al. (1993) are reproduced to 0.1%, while the pseudospectra converge where ill-conditioned eigenvalues do not.

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Vassilis Theofilis. 2026-10-28. Kinetic linear stability theory for incompressible parallel shear flows with the full Boltzmann collision operator. https://arxiv.org/abs/2610.38714

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