Influence of higher order effects on the vortex instability of thermal boundary layer flow in a wedge shaped domain

We reconsider the onset of streamwise vortices in the thermal boundary layer flow induced by an inclined upward-facing heated semi-infinite surface placed within a Newtonian fluid. Particular emphasis is laid upon how the induced flow in the isothermal region outside the boundary layer affects the boundary layer itself at higher order, and how this, in turn, affects the stability criterion for the onset of vortices. We find that the stability criterion for thermal boundary layers in air is less susceptible to changes in external geometry than for boundary layers in water. In general, we conclude that the variation of the stability criterion with wedge angle (between the heated and the outer boundary surface) is too great for the theory to predict reliably where disturbances first begin to grow

Onset of convection in a porous layer with continuous periodic horizontal stratification, Part II:Three-dimensional convection

The onset of convection in a porous layer which is heated from below is considered. In particular we seek to determine the effect of spatially periodic variations in the permeability field on the identity of the onset mode as a function of both the period P of this variation and its amplitude A. A Floquet theory is assumed in order to ensure that the analysis is as general as possible. It is found that convection is always three-dimensional and that the critical Rayleigh number always decreases as either the period or the amplitude of the permeability variation increases. Furthermore, the corresponding Floquet exponent \u3bd is either 0 or 1, and the range of values of P over which \u3bd=1 corresponds to the favoured mode has been obtained as a function of A

Free convection in fluid-saturated porous media

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Linear instability of the isoflux Darcy-B\ue9nard problem in an inclined porous layer

The linear stability for convection in an inclined porous layer is considered for the case where the plane bounding surfaces are subjected to constant heat flux boundary conditions. A combined analytical and numerical study is undertaken to uncover the detailed thermoconvective instability characteristics for this configuration. Neutral curves and decrement spectra are shown. It is found that there are three distinct regimes between which the critical wavenumber changes discontinuously. The first is the zero-wavenumber steady regime which is well known for horizontal layers. The disappearance of this regime is found using a small-wavenumber asymptotic analysis. The second consists of unsteady modes with a nonzero wavenumber, while the third consists of a steady mode. Linear stability corresponds to inclinations which are greater than 32.544793\ub0 from the horizontal

Numerical investigation of the linear stability of a free convection boundary layer flow using a thermal disturbance with a slowly increasing frequency

Numerical simulations are performed to investigate the linear stability of a two-dimensional incompressible free convection flow induced by a vertical semi-infinite heated flat plate. A small-amplitude local temperature disturbance with a slowly increasing frequency is introduced on the surface near to the leading edge in order to generate disturbance waves within the boundary layer. The aim is to compare the response of the thermal boundary layer with that obtained by selecting discrete disturbance frequencies. In the present study, air is considered to be the working fluid for which the value of the Prandtl number is taken to be Pr=0.7. The computational results show that the disturbance decays initially until it reaches a critical distance, which depends on the current frequency of the disturbance. Thereafter the disturbance grows, but the growth rate also depends on the effective frequency of the disturbance. Comparisons with previous work using constant disturbance frequencies are given, and it is shown that the sine-sweep technique is an effective method for analyzing the instability of convectively unstable boundary layers

On the onset of convection in a highly permeable vertical porous layer with open boundaries

The unstable nature of buoyant flow in a vertical porous slab with a pure conduction temperature distribution is investigated. The permeable and isothermal boundaries are subject to a temperature difference, which is responsible for the basic stationary and parallel vertical flow in the slab. The momentum transfer is modeled by adopting the Darcy-Forchheimer law, thus including the quadratic form-drag contribution. The instability to small-amplitude perturbations is tested by parameterizing the basic stationary flow through the Darcy-Rayleigh number and the form-drag number. The modal analysis is carried out numerically with a pressure-temperature formulation of the governing equations for the perturbations. The neutral stability curves and the critical values of the wave number and of the Darcy-Rayleigh number are obtained for different prescribed values of the form-drag number

Darcy-Forchheimer flow with viscous dissipation in a horizontal porous layer: onset of convective instabilities

Parallel Darcy\u2013Forchheimer flow in a horizontal porous layer with an isothermal top boundary and a bottom boundary, which is subject to a third kind boundary condition, is discussed by taking into account the effect of viscous dissipation. This effect causes a nonlinear temperature profile within the layer. The linear stability of this nonisothermal base flow is then investigated with respect to the onset of convective rolls. The third kind boundary condition on the bottom boundary plane may imply adiabatic/isothermal conditions on this plane when the Biot number is either zero (adiabatic) or infinite (isothermal). The solution of the linear equations for the perturbation waves is determined by using a fourth order Runge\u2013Kutta scheme in conjunction with a shooting technique. The neutral stability curve and the critical value of the governing parameter R=Ge*Pe^2 are obtained, where Ge is the Gebhart number and Pe is the P\ue9clet number. Different values of the orientation angle between the direction of the basic flow and the propagation axis of the disturbances are also considered

Thermal receptivity of free convective flow from a heated vertical surface: nonlinear waves

In a previous paper (Int. J. Thermal. Sci., vol. 47, pp. 1382-1392, 2008), the authors performed a detailed numerical investigation of the linear instability of the thermal boundary layer flow over a vertical surface by introducing unsteady thermal disturbances near the leading edge and by solving numerically the fully elliptic linearized stability equations. The main aim of the present paper is to extend those results into the nonlinear regime by seeding the boundary layer with similar disturbances of finite amplitude. The ensuing nonlinear waves are found to exhibit a variety of behaviours, depending on the precise amplitude and period of the forcing. When the amplitude is sufficiently small, the linearized theory of the previous work is reproduced, but for larger amplitudes, cell splitting or cell merging may occur as waves travel downstream. Cell splitting takes place when disturbance frequencies are somewhat smaller than the most strongly amplified nondimensional disturbance frequency of 0.4 for which the boundary layer response, is at its greatest in terms of the surface rate of heat transfer (see Fig. 8 in previous paper). Cell merging takes place at frequencies what are approximately double that of the most strongly amplified disturbance frequency. Attention is focussed on fluids with a unit Prandtl number