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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.

The effect of anisotropy on transport phenomena in anisotropic porous media is studied in this paper. For convenience, a bank of circular cylinders which can be treated as an anisotropic porous medium is employed such that both Darcy-Forchheimer drag and effective thermal conductivity can be accurately determined. Two problems including a forced flow and a natural convection are illustrated to investigate the effect of anisotropy on fluid flow and heat transfer through a bank of circular cylinders. The solutions reveal that inclination of the cylinder bundle could give rise to an influence of more than 100% on the heat transfer rate for both forced convection and natural convection. Hence, the anisotropy of an anisotropic porous medium should not be ignored.

Based on the microscale temperature, the effective thermal conductivity in the principal direction of an anisotropic porous medium is evaluated. A correlation of the effective thermal conductivity then is employed to predict the volume-averaged temperature of a two-dimensional nonhomogeneous anisotropic porous medium. Through the example, the present model is found to possess an excellent performance even when the principal direction of the anisotropic medium is not parallel to the physical coordinates. The results also reveal that using the classic mixing rule could lead to a significant error when the thermal conductivity ratio of the solid and fluid is large. The present numerical procedure is believed to be able to determine the effective thermal conductivity for most porous media as long as their microstructure is regular and known.

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.

Transient conjugate heat transfer relating the heat conduction inside a solid body of arbitrary shape and the natural convection around the solid is studied in the present investigation. A single set of governing equations covering both solid and fluid is solved by using the weighting function scheme along with the NAPPLE algorithm and the SIS solver. All of the computations are performed on a Cartesian grid system. Numerical results of velocities and isotherms are presented for Pr = 7 and Gr = 10 7 and 10 8 . An interesting observation on formation, growth, merging and decay of several plumes is discussed. The solution shows that there is a dead fluid inside the cavity under the solid body due to an upward temperature gradient. The maximum temperature thus occurs there in a late stage of the heat transfer process. An increase in the Grashof number is found to enhance the heat transfer. Through this study, the present numerical method is shown to have a good performance for conjugate heat transfer on solid body of arbitrary shape without the need of a body-fitted grid system.

A latent heat method is modified in the present study to properly handle the latent heat evolved in the solidification process of a binary alloy system. The eutectic state ( C E , T E ) is imposed on the eutectic front such that no species equation is needed in the pure solid region. Both momentum and species equations in the irregular domain consisting of the mushy zone and the liquid region are successfully solved by using the weighting function scheme along with the NAPPLE algorithm. To trace the interface of the mushy zone and the pure liquid region, an interpolation technique is proposed. Two examples were conducted to examine the performance of the present method. Satisfactory agreement with the existing experimental results is observed on the total volume of the mushy zone.

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.

The discretized pressure-linked equation can be proven to possess a singular coefficient matrix. This implies that the pressure solution corresponding to a given velocity would not exist unless this velocity precisely satisfies the continuity equation. In the present investigation, an artificial source term is added to the pressure-linked equation. This additional term will generate an extra pressure called the ‘artificial pressure’ for each updated velocity to compensate for their nonzero dilation. The use of the artificial pressure is equivalent to creating an extra mass for the need of the updated velocity in satisfying the continuity equation. This treatment guarantees the existence and uniqueness of the pressure solution such that the pressure can be directly solved without recourse to the conventional pressure correction equation. Based on the concept of artificial pressure, the APPLE (Artificial Pressure for Pressure-Linked Equation) and the NAPPLE (Nonstaggered APPLE) algorithms are developed for incompressible flows. Through two well-known examples, the APPLE algorithm is found to produce essentially the same results with only about 40% CPU time as compared with the SIMPLER algorithm. Due to its simplicity, the NAPPLE algorithm has a potential use for problems with arbitrarily shaped domain as long as the grid size is not large.

An enthalpy formulation is proposed in the present investigation for a phase change material (PCM) having a distinct freezing temperature. The latent heat is separated from the sensible heat such that there exists a dependent variable (the sensible heat) that is a continuous function over the entire physical domain. Inside each control volume, the latent heat is rigorously evaluated from the fraction of the liquid phase to achieve an even latent heat evolution. In the dimensionless transformation, the characteristic time is denned in terms of the Stefan number. The coefficients of the unsteady terms thus are always less than unity. This will achieve a good numerical stability for any Stefan number. In addition, this particular transformation makes the present enthalpy formulation applicable to single phase problems if an infinite Stefan number is assigned. To account for a thermal diffusivity jump at the liquid-solid interface, a modified weighting function scheme is developed. Through a few examples, the present enthalpy formulation is seen to produce an accurate and smooth liquid-solid interface for PCM having a distinct freezing point.

A numerical formulation based on the weighting function scheme is developed in the present investigation for convection-conduction phase change problems. The solid phase is regarded as a liquid having an infinite viscosity. Such a treatment allows the stream-vorticity formulation to apply over the entire physical domain including the liquid-solid interface. The weighting function scheme then is used to handle the sharp viscosity jump at the liquid-solid interface. A problem dealing with pure tin is employed to test the performance of the present method. The predicted interface profiles are found to agree with available experimental data qualitatively. The computation effort required by the present numerical technique, however, is less than 2% ofthat needed by a two-region method for this same problem. Unlike the existing single-region methods, the present numerical technique produces smooth streamlines and isotherms even in the vicinity of the solidification front.

The present investigation studies the cooling of a continuous moving sheet of finite thickness. The effect of the buoyancy force is also taken into account. The temperature distribution along the solid-fluid interface is determined by solving a conjugate heat transfer problem that consists of heat conduction inside the sheet and induced mixed convection adjacent to the sheet surface. For a better numerical stability, the weighting function scheme along with an axial coordinate transformation is employed to solve the transformed boundary layer equations. Three parameters are found to exist in the present investigation. They are the Prandtl number of the fluid Pr, the buoyancy parameter Ω and the heat capacity ratio C. Numerical results including the Biot number, the surface temperature and the overall heat transfer rate of the sheet are presented for 0.7 ⩽ Pr ⩽ 100, 0 ⩽ Ω ⩽ 10 and 0.1 ⩽ C ⩽ 1. The buoyancy force is seen to have a significant effect on the results. The heat capacity ratio, however, is the most important parameter. Based on the present results, it is concluded that using a liquid as the cooling medium could obtain a better cooling performance than using a gas. This is because the liquid has a larger heat capacity than a gas. The Prandtl number has only a minor effect.

It is very difficult to fabricate tunable optical systems having an aperture below 1000 micrometers with the conventional means on macroscopic scale. Krogmann et al. (J. Opt. A 8, S330-S336, 2006) presented a MEMS-based tunable liquid micro-lens system with an aperture of 300 micrometers. The system exhibited a tuning range of back focal length between 2.3mm and infinity by using the electrowetting effect to change the contact angle of the meniscus shape on silicon with a voltage of 0-45 V. However, spherical aberration was found in their lens system. In the present study, a numerical simulation is performed for this same physical configuration by solving the Young-Laplace equation on the interface of the lens liquid and the surrounding liquid. The resulting meniscus shape produces a back focal length that agrees with the experimental observation excellently. To eliminate the spherical aberration, an electric field is applied on the lens. The electric field alters the Young-Laplace equation and thus changes the meniscus shape and the lens quality. The numerical result shows that the spherical aberration of the lens can be essentially eliminated when a proper electric field is applied.

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.

This paper presents a finite element solution to the problem of low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed to be laminar and fully developed throughout the pipe and the fluid temperature is kept uniform atX = —∞. The pipe wall is adiabatic at X ≤ 0 and cooled convectively at X ≥ 0. The solutions include temperature distributions and Nusselt numbers for the parameters, Bi = 0.04, 0.4, 4, 20 and Pe = 1, 3, 5, 10, 20, 30, which are in excellent agreement with the existing analytic solution except in the region near the singular point Δ A temperature discrepancy in the analytic solution at this point is physically impossible. The finite element method overcomes this mathematical difficulty and shows a greater value in the Nusselt number due to a higher wall temperature at X ≥ 0. On presente, dans le travail, une solution utilisant une methode d'elements finis pour le probleme de l'ecoulement d'un fluide a faibles nombres de Peclet dans la region d'entree thermique d'un tube rond. On suppose que la vitesse est laminaire et etablie d'un bout a l'autre du tube et que la temperature du fluide est maintenue uniforme a X = — ∞. La paroi du tube est adiabatique a X ≤ 0 et elle est refroidie d'une maniere convective a X ≥ 0. Les solutions comprennent des distributions de temperatures et des nombres de Nusselt pour les parametres Bi = 0.04, 0.4, 4, 20 et Pe = 1, 3, 5, 10, 20, 30, elles concordent tres bien avec la solution analytique existante, sauf dans la region a proximite du point singulier X = 0, r = 1; il est physiquement impossible d'obtenir une difference de temperature dans la solution analytique en ce point. La methode des elements finis remedie a cette difficulte mathematique et fournit un nombre Nusselt de aleur plus grande, par suite d'une temperature plus elevee de la paroi a X ≥0.

The combined effects of axial conduction and solidification on heat transfer and pressure drop in pipe flows are investigated by the use of a modified Galerkin finite element method. To allow for the upstream heat conduction, the domain of study is extended from X = -∞ to X = -∞ as has been done in previous analyses. As a preliminary study on the effect of axial conduction, the present investigation assumes a superheat ratio that is sufficiently large (To > Tf or Tw ≈ Tf) such that solidification begins at a location near X = 0. For numerical convenience, the infinite domain -∞ ≤ X ≤ ∞ is transformed onto a finite domain -1 ≤ z ≤ 1. The energy equation for the liquid-phase is then solved by a modified Galerkin finite element method. For better numerical stability, a procedure is proposed for controlling the numerical error that might propagate from the singular point (X, R) = (O, Ro). The profile of the solid-liquid interface, the heat transfer rate and the pressure drop are presented for various values of Peclet number, Pe = 1, 3, 5, 10 and 30, and for the modified superheat ratio, c = 0.1, 0.5, 5.0, and ∞. A l'aide d'une methode d'elements finis de type Galerkin modifie, nous avons etudie les effets combines de la conduction axiale et de la solidification sur le transfert de chaleur et la perte de charge dans des ecoulements dans des tubes. Pour permettre la conduction de la chaleur en amont, le champ d'etude a ete etendu de X = ∞ a X = -∞ comme nous l'avons fait dans les analyses anterieures. Comme etude preliminaire sur l'effet de la conduction axiale, les presentes recherches supposent un rapport de surchauffe qui soit suffisamment grand (To ≤ Tf ou Tw ≈ Tf) pour permettre a la solidification de demarrer a un point pres de X = 0. Pour faciliter le calcul numerique, le domaine infini -∞ ≤ X ≤ ∞ est transforme en un domaine fini -1 ≤ z ≤ 1. L'equation d'energie pour la phase liquide est alors resolue par une methode d'elements finis de type Galerkin modifie. Pour obtenir une meilleure stabilite numerique, nous proposons une methode pour contrǒler l'erreur numerique qui peut se propager du point singulier (X, R) = (0, Ro). Le profil de l'interface solides-liquide, le taux de transfert de chaleur et la perte de charge sont presentes pour diverses valeurs du nombre de Peclet, Pe = 1, 3, 5, 10 et 30 et pour le rapport modifie de surchauffe, c = 0, 1, 0,5, 5,0 et ∞.

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