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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.

Stresses distribution between the interfaces of the materials is critical for structures like thin film layers, organic/inorganic light emitting diode layers and various kinds of modern optical-electronics components with stacking materials properties. The classical Aleck's model ( 1949 ) can be considered as a very simple and a practical method to analyze these kinds of structures. In order to deal with it, the integration scheme is used to solve the displacement equations with zero displacements on the clamped edge including the clamped corner. The condition of zero traction is imposed at all of the grid points on the free surface except for the clamped corner of layers. The results show a stress singularity at the clamped corner. This agrees with an existing finite element solution performed on a very fine grid. By contrast, the known Kantorovich solution for Aleck problem from Blech and Kantor (1984) did not predict such stress singularity. As remarked by Blech and Kantor, the Kantorovich solution posed difficulties in convergence in the immediate vicinity of the clamped corner, thus is questionable to apply the results for the layer structures through this model.

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.

The NAPPLE algorithm for incompressible viscous flow on Cartesian grid system is extended to nonorthogonal curvilinear grid system in this paper. A pressure-linked equation is obtained by substituting the discretized momentum equations into the discretized continuity equation. Instead of employing a velocity interpolation such as pressure-weighted interpolation method (PWIM), a particular approximation is adopted to circumvent the checkerboard error such that the solution does not depend on the under-relaxation factor. This is a distinctive feature of the present method. Furthermore, the pressure is directly solved from the pressure-linked equation without recourse to a pressure-correction equation. In the use of the NAPPLE algorithm, solving the pressure-linked equation is as simple as solving a heat conduction equation. Through two well-documented examples, performance of the NAPPLE algorithm is validated for both buoyancy-driven and pressure-driven flows.

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 this paper, the integration scheme is employed to solve a coupling problem of transient heat conduction and displacement due to thermal deformation. Based on the resulting displacement, gap or contact pressure on the interface of two thermoelastic bodies in imperfect contact is estimated. Such information is then used to determine the interfacial heat transfer between the two thermoelastic bodies. This again affects the temperature distribution and thus the thermal deformation. In the course of heat transfer and thermal deformation, effect of thermal rectification also is taken into account. Numerical solutions of transient and steady-state quantities including gap formation, normal stress, and temperature are demonstrated for a pair of stainless steel and aluminum with conforming shapes. Such an analysis of conduction-deformation interaction based on a finite-difference-like scheme has not been studied in the past.

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.

Growth and detachment of carbon dioxide bubbles on a horizontal porous surface in water is investigated in this paper. Uniform carbon dioxide injection on the porous surface is considered. The Young–Laplace equation is solved with the geometry method to yield the bubble shape. The dynamic pressure on the bubble surface due to bubble expansion is neglected. Multi-solution modes are found. Based on the characteristics of the solution modes, it is postulated that the bubble grows with a monotonically increasing base area until the maximum value is reached according to the fundamental solution mode. After that, the bubble jumps toward the secondary solution mode at a constant volume, and then detaches from the surface. The increasing rate of the bubble volume due to mass diffusion/convection is evaluated by solving the momentum and concentration equations. Evolution of the bubble shape then is determined corresponding to the variation of the bubble volume. The numerical results indicate that hydrophobic surface produces large bubble and thus gives rise to better efficiency for dissolved gas removal.

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.

There are still many unanswered questions related to the problem of a capillary surface rising in a tube. One of the major questions is the evolution of the liquid meniscus shape. In this paper, a simple geometry method is proposed to solve the force balance equation on the liquid meniscus. Based on a proper model for the macroscopic dynamic contact angle, the evolution of the liquid meniscus, including the moving speed and the shape, is obtained. The wall condition of zero dynamic contact angle is allowed. The resulting slipping velocity at the contact line resolves the stress singularity successfully. Performance of the present method is examined through six well-documented capillary-rise examples. Good agreements between the predictions and the measurements are observable if a reliable model for the dynamic contact angle is available. Although only the capillary-rise problem is demonstrated in this paper, the concept of this method is equally applicable to free surface flow in the vicinity of a contact line where the capillary force dominates the flow.

A numerical simulation for a filling process in an open tank is performed in this paper. A single set of governing equations is employed for the entire physical domain covering both water and air regions. The great density jump and the surface tension existing at the free surface are properly handled with the extended weighting function scheme and the NAPPLE algorithm. There is no need to smear the free surface. Through the use of a properly defined boundary condition, the method of “extrapolated velocity” is seen to provide accurate migrating velocity for the free surface, especially when the water front hits a corner or a vertical wall. In the present numerical procedure, the unsteady term of the momentum equation is discretized with an implicit scheme. Large time-steps thus are allowed. The numerical results show that when the water impinges upon a corner, a strong pressure gradient forms in the vicinity of the stagnation point. This forces the water to move upward along the vertical wall. The water eventually falls down and generates a gravity wave. The resulting free surface evolution is seen to agree well with existing experimental data. Due to its accuracy and simplicity, the present numerical method is believed to have applicability for viscous free-surface flows in industrial and environmental problems such as die-casting, cutting with water jet, gravity wave on sea surface, and many others.

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