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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.

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 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.

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.

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.

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.

Flow structure around a rising clean microbubble in a vertical square tube is investigated on a fixed nonstaggered Cartesian grid system. A single set of three dimensional governing equations over the entire physical domain including the liquid, the air, and the bubble surface (the liquid-air interface) is formulated without smearing the bubble surface. There are three step functions in the problem, namely, the density jump, the viscosity jump, and the pressure jump across the bubble surface. A modified NAPPLE algorithm is proposed to handle the pressure jump on a fixed nonstaggered Cartesian grid system, while the density and viscosity jumps appearing in the momentum equations are solved with the extended weighting function scheme. This strategy prevents the occurrence of “parasitic current” on the bubble surface. Solutions of flow structure (velocity and pressure) are obtained for bubble diameters of less than 500 micrometers rising in quiescent pure water. The numerical result evidences that the distorted microbubble under study is of the oblate spheroid type. The distortion parameter is only 1.0006 when the bubble diameter is 250 micrometers. However, the surface tension force due to bubble distortion generates a large pressure jump across the bubble surface that dominates the flow around the bubble. It plays a role similar to the non-transpiration condition on a solid surface. The resulting terminal velocity of the bubble agrees excellently with the existing analytical solutions and numerical results.Copyright © 2007 by ASME

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.

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