The stabilities of natural convection boundary layers along a tilt uniform-heat-flux plate were studied theoretically and numerically.For some Prandtl numbers

three-dimensional instability may have two critical modes

whose critical frequencies are close to the internal wave frequency and the value of two-dimensional T-S(Tolmien-Schlichting) wave

respectively.Specially

at proper tilt angles the three-dimensional wave mode is more unstable than the two-dimensional T-S wave.This result that seems contradict with the Squire Theorem is caused by the additional buoyancy terms in the governing equations.Direct numerical simulations confirmed the critical properties of the three-dimensional waves.Since hydrodynamic instability always exists at the first stage of turbulence transition process

current results may provide theoretical basis for optimal design of fins with high heat transfer efficiency.