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Mixed convection in a square enclosure arising from thermal buoyancy force and a rotating flat plate is investigated in the paper. The moving boundary problem is solved with the implicit virtual boundary method on a fixed non-staggered Cartesian grid system. Computations are performed for Prandtl number 0.71 at various Rayleigh numbers Ra and rotational Reynolds numbers Re. For a rotor of length d = 0.6 rotating at Re = 430 , the numerical results show that thermal oscillation occurs when the Rayleigh number exceeds 0.55 × 10 6 . There is a critical Rayleigh number ( 0.13 × 10 6 ). Below that the rotor enhances the heat transfer. By contrast, the rotor would suppress the heat transfer beyond the critical Rayleigh number. In the case of Ra = 10 6 , the Nusselt number oscillates 23 cycles when the rotor makes 29 revolutions. For a longer rotor d = 0.9 rotating at Re = 430 , thermal oscillation exists for all Rayleigh numbers, and the critical Rayleigh number increases to 0.38 × 10 6 . In the case of Ra = 10 6 , the Nusselt number oscillates twice when the rotor makes one revolution.

Influence of cooling rate on ac LED (light-emitting diode) with multiple quantum wells is investigated in the present study. A semi-empirical model based on existing experimental measurements is proposed to describe the relationship between the current density and the forwards voltage drop across the p–n junction for LEDs with multiple quantum wells. The numerical model then is employed to compute the electrical and temperature fields on ac LED with multiple quantum wells under various cooling rate. The resulting temperature of the LED oscillates under ac current. The temperature increases due to the heat generated in the active layer of the LED when the electrical potential exceeds the threshold voltage. Otherwise, there is no electrical current and thus the temperature decreases due to the effect of the cooling device. Both light-emitting power and maximum temperature increase as expected when the applied ac electrical potential increases. Fortunately, the temperature of the LED can be efficiently controlled by increasing the cooling rate of the cooling device. Although increasing the cooling rate would decrease the light-emitting power, the influence is not significant.

The transition stage (self-accelerating necking process) generally takes place before a bubble collapses. However, the mechanism is still not well understood. In this article, seven existing experiments dealing with bubble formation on the orifice of a submerged vertical nozzle are examined by solving the Young–Laplace equation. Multiple solution modes are found. Bubble in solution mode 2 has a neck and thus taller than mode 1 at the same volume. The present numerical result along with an experiment from Longuet-Higgins et al. [J. Fluid Mech. 230, 365–390 (1991)] evidences an isometric transition from solution modes 1 to 2. This might account for the self-accelerating necking process before a bubble collapses. Surprisingly, all of the seven existing experiments agree excellently with the bubble shapes from the Young–Laplace equation without the dynamic effect even when the bubble growth rate in the experiment is 2.25 times as large as the critical value. The gas flow rate (the dynamic effect) seems to play a role only after the transition stage. © 2011 Canadian Society for Chemical Engineering

A liquid–air fountain flow due to the downward motion of a rectangular sleeve over a stationary piston is studied in the paper. Two-dimensional incompressible laminar flows are assumed to prevail in both air and liquid regions. A single set of governing equations over the entire physical domain including the liquid, the air, and the liquid–air interface (free surface) is solved with the extended weighting function scheme and the NAPPLE (nonstaggered APPLE) algorithm on a fixed nonstaggered Cartesian grid system. To ensure the required dynamic contact angle, the liquid meniscus near the sleeve wall is corrected by solving the force balance equation with the geometry method. This is equivalent to introducing a slip condition at the contact line, and thus successfully removes the stress singularity. Steady state solution of the velocity and the pressure as well as the shape of the free surface is obtained. The numerical result evidences the existence of a toroidal-like motion on the free surface postulated by Dussan [E.B. Dussan V., Immiscible liquid displacement in a capillary tube: the moving contact line, AIChE J. 23 (1977) 131–133], although it is quite weak and thin. The resulting free surface profile agrees with the existing experimental observation excellently. Influence of the piston on the flow is discussed.

A new numerical formulation is presented in this paper for incompressible viscous free surface flow without smearing the free surface. Laws of conservation for the entire physical domain, including both liquid and air, are formulated with a single set of governing equations. To properly handle the discontinuities of the physical properties across the free surface, an extended weighting function scheme is developed for the numerical solution on a fixed and nonstaggered Cartesian grid system. Unlike existing numerical methods, the force balance equation is imposed on the free surface through the use of the NAPPLE algorithm without smearing the free surface. During the solution procedure, a harmonic function referred as an “extrapolated velocity” from the liquid is computed based on the velocity solution at the grid points adjacent to the free surface on the liquid side. With a migration velocity interpolated from such a harmonic function, the free surface profile for the next time step is estimated. This gives rise to a smooth free surface profile and thus circumvents high frequency noises on the curvature of the resulting free surface. Furthermore, advancement of the free surface is not restricted to one grid mesh in a single time step. Performance of the present method is examined through three well-documented dam-breaking examples. The results reveal the existence of an induced vortex in a layer of air adjacent to the free surface. Good agreement between the computed water front and the experimental data is observed. Although only two-dimensional cases are demonstrated in this paper, concept of the present numerical method is equally applicable to three-dimensional problems with moving free surfaces. It also applies to convective heat and mass transfer problems such as filling process in gravity and die casting processing.

In the present investigation, an incompressible fluid flow across a bank of circular cylinders is studied and modeled as a non-Darcy flow through a porous medium. The continuity equation and the momentum equation in pore scale are solved on a Cartesian grid system. To circumvent the numerical difficulties arising from the flow domain of irregular shape, the weighting function scheme along with the APPLE algorithm and the SIS solver is employed. The Darcy-Forchheimer drag is then determined from the resulting volumetric flow rate under a prescribed pressure drop. The result indicates that the permeability approaches zero at the particular porosity of 0.2146 when the fluid flow across the cylinders becomes impossible. In addition, the pressure drag (Forchheimer drag) is found to contribute a significant resistance at large porosities and/or large granule Reynolds numbers. A correlation of Darcy-Forchheimer drag is proposed for 0.2146 ⩽ e ⩽ 1 and 0 ⩽ Red ⩽ 50.

Forced convection on a rotating cylinder cooled with an air jet is investigated in the present study. To properly handle the jet flow before and after hitting the cylinder, a computational domain having an irregular shape is employed. Results of streamlines, isotherms and Nusselt numbers are presented for jet Reynolds numbers of Re j = 100 , 500 and 1000 under various rotation Reynolds numbers in the range of 0 ≤ Re ω /Re j ≤ 1. It is interesting to find two solution modes in the present problem. For small rotation speeds, the jet flow is separated into two branches upon hitting the cylinder. Each branch has a separation point. With the aid of the rotating cylinder, one of the separation points moves downstream, while the other receives little influence. Hence, the overall heat transfer is enhanced. For high rotation speeds, however, this heat transfer behavior is reversed due to a layer of dead air round the cylinder. Nevertheless, a more uniform heat transfer is achieved. For both cases of Re j = 500 and 1000, dual solutions according to each of the two solution modes exist in a range of rotation Reynolds numbers. Like the transition from laminar to turbulent flow, the heat transfer characteristics jump from one solution mode to the other depending on the flow instability. However, the transition between the two solution modes is not clear for a jet Reynolds number Re j as small as 100.

Wave instability of mixed convection flow along an isothermal vertical flat plate is analyzed by the linear theory for the entire mixed convection regime (0 ⩽ χ ⩽ 1, χ = [1 + ( Gr x Re x 2 ) 1 4 ] −1 ) for fluids with Prandtl number of 0.7 and 7. In the analysis, the domain beyond the mainflow boundary layer edge (η ⩾ η∞) is solved analytically to provide boundary conditions at η = η∞ instead of those at η = ∞. This treatment is important in the use of the modified Thomas transformation when λ(λ = Re x 1 2 + Gr x 1 4 ) is small. Dual solutions of critical λ ∗ values are seen to exist in the range of 0 ⩽ χ ⩽ 0.5. The results show that the two limiting neutral stability curves, one for the Blasius flow (χ = 1 or Grx = 0, with λ ∗ = 290.6 ) and the other for the pure free convection flow (χ = 0 or Rex = 0, with λ ∗ = 33.33 ), correspond to two different modes.

Mixed convection along vertical cylinders and needles with uniform surface heat flux is investigated for the entire mixed convection regime. A single modified buoyancy parameter {chi} and a single curvature parameter {Lambda} are employed in the analysis such that a smooth transition from pure forced convection ({chi} = 1) to pure free convection ({chi} = 0) can be accomplished. For large values of the curvature parameter and/or Prandtl number, the governing transformed equations become stiff. Thus, a numerically stable finite-difference method is employed in the numerical solution in conjunction with the cubic spline interpolation scheme to overcome the difficulties that arise from the stiffness of the equations. Local Nusselt numbers are presented for 0.1 {le} Pr {le} 100 that cover 0 {le} {chi} {le} 1 ({infinity} {ge} {Omega}{sub x} {ge} 0) and 0 {le} {Lambda} {le} 50. For needles ({Lambda} {ge} 5), the local Nusselt numbers (Nu{sub x}/Re{sub x}{sup 1/2} + Gr{sub x}{sup 1/5}*) are found to be nearly independent of the buoyancy parameter {chi}. Correlation equations for the local Nusselt numbers are also presented.

A non-parallel flow analysis by an order-of-magnitude approach is performed to investigate the linear wave instability of mixed convection flow along an isothermal inclined flat plate. This analysis removes the previous restriction on the weak dependence of the streamwise variation of disturbance quantities. The resulting non-homogeneous, coupled equations for the momentum and temperature disturbances are solved by a superposition technique along with a modified Thomas transformation method. Critical Reynolds numbers are presented for inclination angles in the range of 0° ⩽ γ ⩽ 90° (with γ being measured from the horizontal) covering the bouyancy parameter range of −0.15 ⩽ Grx/Rex2 ⩽ 1 for Prandtl numbers of 0.7 and 7. It is found that the net effect of buoyancy force on the critical Reynolds number is essentially zero at γ = 1.05° when the plate is almost horizontal. For γ > 1.05°, an increase in the value of Grx/Rex2 stabilizes the flow. This behavior is reversed for γ

On emploie une methode de differences finies pour resoudre les equations d'Orr Sommerfeld modifiees, avec un couplage entre les fonctions d'amplitude pour la perturbation de vitesse et de temperature dans lequel le non-parallelisme de l'ecoulement principal est pris en compte