Temporal Stability of Channel Flow at Low Peclet Number

A phenomenon called thermal striping, consisting of quasi-periodic temperature oscillations of Ο(100 °C), is of major concern in liquid-metal-cooled small modular nuclear reactors. While it is believed to be caused by a shear-flow instability, the physical mechanism is largely unknown, and its onset is difficult to predict. We consider plane Poiseuille flow with stable density stratification in the wall-normal direction as a model for a heated, wall-bounded shear flow. The linear temporal stability eigenvalue problem is then solved. The analysis shows that, in the limit of small Peclet number, ≲ 1, the flow is stable, in the sense of negative eigenvalues, when a modified Richardson number, R = Ri·Pe, is ≳ 0.332. We further develop a semi-analytic solution of the Low-Peclet-number equations (LPNE) of Lignières [Astron. Astrophys. 348 (1999)], which contain R as the natural parameter controlling buoyancy forces in the momentum equation. A perturbation series to first-order in R is obtained for the temporal stability eigenvalue problem of the LPNE. The perturbation in the eigenvalue is found to be strictly negative for all wavenumbers/Reynolds numbers indicating the effect of stratification is purely stabilizing.