The spatial stability of natural convection flow on inclined plates

The spatial stability of a natural convection flow on upward-facing, heated, inclined plates is revisited. The eigenvalue problem is solved numerically employing two methods: the collocation method with Chebyshev polynomials and the fourth-order Runge-Kutta method. Two modes, traveling waves and stationary longitudinal vortices, are considered. Previous theoretical models indicated that nonparallel effects of the mean flow are significant for the vortex instability mode, but most of them ignored the fact that the eigen-functions are dependent on the streamwise coordinate as well, in the present work, the method of multiple scales is applied to take the nonparallel flow effects into consideration. The results demonstrate the stabilizing character of the nonparallel flow effects. The vortex instability mode is also considered within the scope of partial differential equations. The results demonstrate dependence of the neutral point on the initial conditions but, farther downstream, the results collapse onto one curve. The marching method is compared with the quasi-parallel normal mode analysis and with theoretical results including correction to nonparallel flow effects. The marching method provides better agreement of theoretical and experimental growth rates.