The Onset and Nonlinear Development of Vortex Instabilities in a Horizontal Forced Convection Boundary Layer With Uniform Surface Suction

Abstract

We consider the flow and heat transfer caused by a strong external flow passing over a hot surface with uniform surface suction. When the Péclet number based on the external velocity is sufficiently large, the resulting thermal boundary layer develops in a nonsimilar manner until it attains an asymptotic state which is independent of the streamwise coordinate, x, when it is dominated by the surface suction. For sufficiently large, but moderate, values of the Darcy-Rayleigh number this boundary layer becomes unstable to streamwise vortex disturbances. We employ a parabolic solver to determine how such disturbances, when placed very close to the leading edge, evolve with distance downstream. Neutral stability is then defined to be when a suitable energy functional ceases to decay/grow as x increases. Thus a neutral curve may be mapped out based upon the behaviour of this functional. Given that the uniform asymptotic state is well known to admit subcritical instabilities, our linearised analysis is extended into the nonlinear domain and the effect of different magnitudes of disturbance is ascertained. It is found that a surprisingly rich variety of vortex pattern emerges which is sometimes sensitively dependent on the values of the governing parameters. These patterns include wavy vortices and abrupt changes in perceived wavelength.

Impact of Thermal Non-equilibrium on Weak Nonlinear Rotating Porous Convection

“…When h 0 32 0, it marks a transition between supercritical and subcritical convection. For this purpose, following Rees (2009), the linear and nonlinear oscillatory neutral curves are illustrated in Fig. 5a-f reliable with the condition ω 2 0 ≥ 0.…”

“…The stability of bifurcating equilibrium solution is studied by deriving the Ginzburg-Landau equations for both steady and oscillatory motions. Following Rees (2009), the nonlinear oscillatory solution curves are compared with the linear oscillatory ones to know the existence of subcritical/supercritical bifurcation and thereby quartic points (i.e. where the nonlinear oscillatory neutral curve branches off the linear curve) are located for considered governing parameters.…”

The effect of local thermal non-equilibrium on forced convection boundary layer flow from a heated surface in porous media

“…Papers by Rees and Bassom [17] and Rees [18][19][20] are concerned with various aspects of the instability of free convection boundary layers, while Rees [21] is concerned with linear instability and the nonlinear vortex development in a mixed convection flow where forced convection effects are strong compared with buoyancy. The situation described in [21] could be applied to the flow considered here if weak buoyancy forces are allowed to exist. As the flow develops downstream, the local Rayleigh number, which is proportional to the boundary layer thickness, also increases and eventually becomes sufficiently large that thermoconvective instability will occur.…”

The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium

“…fluid (Sutton [8]), then the bifurcation to convective flow is subcritical (Pieters and Schuttelaars [9], Rees [10]), meaning that strongly nonlinear flow exists at Rayleigh numbers below the linear threshold.…”

“…If the porous medium is layered, then it is possible to have bimodal convection, where the neutral curve has two minima, and also to have convection with a square planform immediately post onset (McKibbin and O'Sullivan [6], Rees and Riley [7]). If the layer has a constant vertical throughflow of fluid (Sutton [8]), then the bifurcation to convective flow is subcritical (Pieters and Schuttelaars [9], Rees [10]), meaning that strongly nonlinear flow exists at Rayleigh numbers below the linear threshold.…”