Stability Of Natural Convection Research Articles

Stability of natural convection in an inclined square duct with perfectly conducting side walls

We have performed three-dimensional linear stability analyses for natural convection in an inclined square duct. The duct is heated from the bottom, while the lateral walls are assumed to be perfectly thermal conducting. Three-dimensional transverse rolls whose axes are normal to the axis of the duct occur from the motionless state when the Rayleigh number exceeds a critical value and the duct is placed horizontally ( θ = 0°). However, it is found that when the duct is placed inclined ( θ = 0.01°), a two-dimensional longitudinal roll which is unchanged in the axis of the duct occurs and is stable if the Rayleigh number is small.

D213 同心二重円筒間内の多孔質媒体に生じる自然対流の線形安定性(OS-18 多孔質の新しい展開I)

D213 同心二重円筒間内の多孔質媒体に生じる自然対流の線形安定性(OS-18 多孔質の新しい展開I)

Stability of Natural Convection in a Cubic Cavity by Spectral Element Method

Three dimensional linear stability analysis for onset of natural convection flow in a cubic cavity is performed by using the spectral element method. The cubic cavity is placed horizontally and is heated from the bottom surface and cooled from the upper one. The lateral boundaries are assumed to be rigid and in a thermally medium state between conducting and insulating which can be established by introducing Biot number. The fluid is in a motionless state if the Rayleigh number is small. On the other hand, natural convection occurs if the Rayleigh number exceed a critical value. The critical Rayleigh numbers for the onset of the natural convection are evaluated numerically for various Biot numbers by constituting eigenvalue problem using linear stability theory. In addition, the velocity and temperature field at the critical state are obtained and discussed.

Electrohydrodynamic instability in a horizontal viscoelastic fluid layer in the presence of internal heat generation

Linear stability theory is applied to examine the effect of a vertical ac electric field on the stability of natural convection, which occurs in a dielectric viscoelastic fluid (Walters' liquid B' ) between two parallel horizontal plates with internal heat generation. The power-series method is used to obtain the eigenvalue equation, which is then solved numerically. Critical heat Rayleigh numbers and the critical wave number for the onset of instability are presented graphically as functions of the electric Rayleigh number for various values of the Prandtl number. PACS No.: 47.15Fe

Stability of natural convection between spherical shells: energy theory

The energy stability problem with respect to axisymmetric disturbances of the natural convection in the narrow gap between two spherical shells under the earth gravity is discussed. The results are compared with the results of the linear stability analysis for the same problem. The problem is solved for different fluids with Pr=0–100 and different radius ratios η=0.9,0.925,0.95. With the aid of the variational principle Euler–Lagrange equations are received, which have the form of an eigenvalue problem, that is solved by means of Galerkin–Chebyshev spectral method. The convergence problem and the dependence of the critical stability parameter on Prandtl number are discussed. The calculations show that there is a big difference between critical numbers for energy and linear stability theories for the small Prandtl numbers. For large Prandtl numbers this difference is very small.

Thermal convection in a porous toroidal thermosyphon

Stability of natural convection in porous media differs from that in Newtonian fluid due to the changes of the thermal inertia and the friction. This paper aims to investigate the natural convection and its stability in a toroidal thermosyphon filled with porous media. The flow and temperature fields were numerically simulated and compared with that of Newtonian fluid. The results predicted two obvious local reversal flows near the connections of heating and cooling halves. A one-dimensional model was proposed which described qualitatively the occurring and stability of the thermal convection. The model suggested that the flow stability depends largely on the Prandtl number. The global flow can probably be chaotic only if the Prandtl number is finite. Contradictions of the conclusions drawn by different researchers were also discussed.

The stability of natural convection in an inclined fluid layer in the presence of a temperature gradient and an a.c. electric field

Linear stability theory is applied to examine the effect of a normal a.c. electric field on the stability of the natural convection that occurs in a dielectric fluid layer between two inclined plates that are maintained at different temperatures. The power-series method is used to obtain the eigenvalue equation which is then solved numerically to obtain the stable and unstable solutions. Factors affecting the stability are discussed.

The stability of natural convection in an inclined fluid layer in the presence of a temperature gradient and an a.c. electric field

The stability of natural convection in an inclined fluid layer in the presence of a temperature gradient and an a.c. electric field

The Stability of Natural Convection in an Inclined Fluid Layer in the Presence of AC Electric Field

Linear stability theory is applied to the problem of the natural convection that occurs in an inclined fluid layer with uniformly distributed internal heat sources and subjected to the action of a normal ac electric field. It is assumed that the two plates between which the fluid layer is confined are maintained at constant and equal temperature. Both the dielectric constant and density of the fluid are assumed to be functions of temperature. The power series method is used to obtain the eigenvalue equation which is then solved numerically to obtain the stable and unstable solutions.

Effect of Imposed Wall Temperature Oscillations on the Stability of Natural Convection in a Square Enclosure

The present numerical study is directed toward buoyancy-driven laminar flows in a two-dimensional square enclosure with differential heating at the vertical walls. The top and bottom walls are insulated. A time-dependent temperature varying sinusoidal perturbation is imposed on the hot vertical wall. The cold vertical wall is maintained at a constant temperature. The fluid is air with a Prandtl number of 0.72. Computations were carried out at one imposed frequency, which is of the same order as the first natural frequency of the system. It was found that the perturbations destabilized the flow in that higher amplitudes lead to lower critical Rayleigh numbers for the flow transitions. Computations spanned four regimes: periodic, quasi-periodic with two frequencies, quasi-periodic with three-frequencies, and chaotic.

Linear stability of natural convection in a tall vertical slot with a moving sidewall

The effect of a shear force on the stability of natural convection in a tall vertical slot has been studied utilizing linear theory. Our results show that the stability of natural convection is significantly affected by the motion of the sidewall and that sidewall motion creates three types of instability. At small Prandtl numbers, the shear instability is dominant. At higher Prandtl numbers, the flow becomes unstable, creating a buoyant instability induced by the boundary layer near the fixed (unmoving) sidewall. When the sidewall moves slightly downward, the buoyant instability induced by the boundary layer near that moving wall occurs for Prandtl numbers near 10. The critical Prandtl number which marks the transition between the shear and buoyant modes is strongly dependent on the direction and speed of the sidewall movement.

Linear stability of natural convection in spherical annuli

Natural convection in a Boussinesq fluid filling the narrow gap between two isothermal, concentric spheres at different temperatures depends strongly on radius ratio, Prandtl number, and Grashof number. When the inner sphere has a higher temperature than the outer sphere, and for fixed values of radius ratio and Prandtl number, experiments show the flow to be steady and axisymmetric for sufficiently small Grashof number and quasi-periodic and axisymmetric for Grashof numbers greater than a critical value. It is our hypothesis that the observed transition is a flow bifurcation. This hypothesis is examined by solving an appropriate eigenvalue problem. The critical Grashof number, critical eigenvalues, and corresponding eigenvectors are obtained as functions of the radius ratio, Prandtl number, and longitudinal wavenumber. Critical Grashof numbers range from 1.18 × 104 to 2.63 × 103 as Prandtl number Pr increases from zero to 0.7, for radius ratios of 0.900 and 0.950. A transitional Prandtl number Prt exists such that for Pr < Prt the bifurcation is time-periodic and axisymmetric. For Pr > Prt the bifurcation is steady and non-axisymmetric with wavenumber twoA first approximation to the bifurcated flow is obtained using the critical eigenvectors. For Pr < Prt the bifurcation sets in as a cluster of relatively strong cells with alternating directions of rotation. The cells remain fixed in location, but pulsate with time. The cluster moves toward the top of the annulus as Pr increases toward Prt. An important feature of the non-axisymmetric bifurcation for Pr > Prt is a set of four cells located at each pole of the annulus in which the radial velocity alternates direction in moving from any one cell to an adjacent one. For fixed radius ratio, the average Nusselt number at criticality varies only slightly with Prandtl number.

Prandtl number effects on the stability of natural convection between spherical shells

An analysis of the effect of the Prandtl number on the linear stability of axisymmetric (m = 0) disturbances on steady natural convection contained between two concentric spherical shells when the gap is narrow are presented. The disturbance equations are solved using a truncated spectral series. Convergence of the series is examined. Prandtl numbers range from 0 to 100 while the relative gap-width is either 0.100, 0.075, or 0.050. Results confirm the hypothesis that experimentally observed changes in the basic motion for certain flow parameters are due to its instability and indicate that for any Prandtl number larger than a transition value, the unstable flows evolve to a steady pattern while for smaller Prandtl numbers the bifurcated flows are time periodic.

The effect of slab width on the stability of natural convection in confined saturated porous media

Stability boundaries separating different states of bimodal convection in a box of saturated porous material with impermeable bounding faces are determined. Two opposing vertical end walls are always insulated, while the thermal conditions on the other set of opposing vertical side walls range from insulated to perfectly conducting, as measured by the Biot number based on box height. The temperature difference between the hot lower plate and the cooler upper plate provides the mechanism for instability. The eigenvalue problem for the critical Rayleigh number is solved numerically over a range of box sizes and side-wall heat transfer conditions. For small values of the distance H2 between conducting side walls, agreement with previous asymptotic analysis is obtained. New results for boxes with planform dimensions both comparable to the box height show the existence of isolated regions rich in bimodal structure. These islands are separated by broad regions having modal properties identical to those found for conducting side walls in the limit H2→0. The numerical results for O(1) box dimensions continue to exhibit the stabilizing effect of side-wall heat transfer as previously observed for thin slabs.

The stability of natural convection in narrow-gap spherical annuli to axisymmetric disturbances

The linear stability with respect to axisymmetric disturbances of natural convection in narrow-gap, spherical annuli is investigated. The basic motion is an eight-order perturbation solution in the small parameter ε = 1- η, where η is the ratio of inner radius of the annulus to outer radius. The disturbance equations are reduced to a system of ordinary differential equations by means of a method of partial spectral expansions. These equations constitute an eigenvalue problem which is solved for the critical Rayleigh number as a function of η and Prandtl number, Pr. Cases considered are Pr = 0.1, 1, 10, and 100 for 0.900 ⩽ η ⩽ 0.995. A comparison with the experimental results found in the literature indicates that non-axisymmetric time periodic bifurcation will most likely take precedence over the case considered herein for Pr = 1,10. However, it appears that steady axisymmetric bifurcation is possible for Pr = 0.1.

The influence of side wall heat transfer on convection in a confined saturated porous medium

The effect of side wall heat transfer on the stability of natural convection in a vertically oriented finite slab of saturated porous material is considered. All six bounding faces are impermeable. A temperature contrast between the top and bottom horizontal surfaces provides the mechanism for destabilization. The narrow vertical end walls are perfectly insulated. The thermal conditions on the broad vertical side walls range from perfectly insulating to fully conducting as determined by the value of B, the Biot number based on slab height. An asymptotic analysis of the general solution is made in the narrow gap limit ε≪1, where ε is the cross-slab width-to-height ratio. In this case the relevant heat transfer variable is B̄=εB, the Biot number based on the narrow slab width. In the limit ε → 0 when B̄=O(1), including the case B̄→∞ that corresponds to a linear side wall temperature profile, tall, vertical, three-dimensional, finger-like cells are found at the critical Rayleigh number Rc=π2/ε2. In the limit B̄ → 0 corresponding to perfect insulation, one obtains two-dimensional, O(1) aspect ratio rolls with axes normal to the side walls at the critical value Rc=4π2. These two-dimensional rolls predominate only for B̄=O(ε2), and transition to tall narrow three-dimensional cells occurs when O(ε2)≪ B ≪ O(1).

Effects of lateral walls on the stability of natural convection in an inclined fluid layer.

The stability of natural convection in an inclined rectangular box fluid is investigated theoretically and experimentally. The Galerkin method is adopted to analyse the problem. It is found that, for small angles of inclination from the horizontal, the effects of the side walls give rise to the secondary flow in the form of stationary transverse rolls with horizontal axes parallel to the shorter side, instead of longitudinal rolls with axes oriented up the slope, predicted for the infinite fluid layer. The singular form of the stationary transverse rolls is experimentally confirmed for angles of inclination from the horizontal φ<10°15° for the aspect ratios (length of the shorter side to depth) Hy=2 and 4.

Thermal Stability of a Vertical Fluid Layer With Volumetric Energy Source

The effects of nonuniform volumetric energy sources on the thermal stability of natural convection in a vertical fluid layer are investigated using the linear theory of stability. The neutral states of stability are obtained for four values of Prandtl number, ranging from 0.71 to 100, and for different values of β in the range 0.1 ≤ β ≤ 10. The critical parameters and the energetics of the critical disturbances are determined for each case. For all values of Prandtl number and β studied here, the critical instabilities set in as traveling waves, moving in the direction of gravity. In the low range of Prandtl number, 0.71 to 5, the critical curves approach the same asymptotic value for β ≥ 10. The critical disturbances are found to be buoyancy driven for all Prandtl numbers and β considered here, except for the low range of Prandtl number, for which they become shear driven disturbances as β increases.

The Stability of Natural Convection in a Vertical Fluid Layer Having Side Walls of Different Temperatures and Internal Heat Generation

Linear stability theory is applied to the problem of the stability of natural convection in a vertical fluid layer. It is assumed that the two side walls are maintained at constant and different temperatures and the fluid layer is heated by uniformly distributed internal heat sources. The power series method is used to obtain the eigenvalue equation which is then solved numerically. The stability conditions are obtained for wide range of the Prandtl number. The numerical results show that when internal heat sources are present the instability takes the form of travelling waves in the critical state and the critical wave speeds are always negative.

The Stability of Natural Convection in a Vertical Layer of Dielectric Fluid in the Presence of a Horizontal ac Electric Field

Linear stability theory is applied to examine the effect of a horizontal ac electric field on the stability of natural convection which occurs in a dielectric fluid between two parallel vertical plates maintained at different temperatures. The power series method is used to obtain the eigenvalue equation which is then solved numerically. It is shown that when the electrical Rayleigh number L (defined in the text) is less than about 2130 the electric field has no effect on the stability of natural convection, and that when L exceeds this value the electric field and the natural convection flow are coupled strongly and they exhibit a complicated but interesting effect on the stability of the system. A rough criterion is given for applicability of the power series method to the present problem.