Normal mode analysis of the fully developed free convection flow in a vertical slot with open to capped ends

The fully developed free convection flow in a differentially heated vertical slot with open to capped ends investigated recently by Bühler (Heat Mass Transf 39:631–638, 2003) and Weidman (Heat Mass Transf Online First, February 2006) is revisited in this paper. A new method of solution of the corresponding fourth order boundary value problem, based on its reduction to “normal modes” by a complex matrix similarity transformation is presented. As a byproduct of the method, some invariant relationships involving the heat flux and the shear stress in the flow could be found.     

Conjugate natural convection in a square porous cavity

 Steady-state conjugate natural convection in a square cavity filled with a porous medium is studied numerically in this paper. The enclosure consists of two horizontal conductive walls of finite thickness and two vertical walls at different uniform temperatures. The focus is on the role of solid-fluid conductivity ratio, k, on the flow and heat transfer characteristics and the average Nusselt number, , over the vertical hot and cold walls of the cavity for a limited set of particular parameters. It was shown that the interface temperature, θ_w, along the top of the solid wall decreases with the increase in the wall conductivity k. Also, the values of decreases with the increase of the values of the parameter k. Comparison with known results from the open literature when the wall thickness of the horizontal solid walls is neglected (non-conjugate problem) is excellent. Received on 4 April 2000     

Magnetohydrodynamics convective flow in a vertical slot through a porous medium

An analysis is presented with magnetohydrodynamics natural convective flow of a viscous Newtonian fluid saturated porous medium in a vertical slot. The flow in the porous media has been modeled using the Brinkman model. The fully-developed two-dimensional flow from capped to open ends is considered for which a continuum of solutions is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. The asymptotic behaviour and the volume flux are analyzed and incorporated graphically for the three-parameter family of solution.     

Large eddy simulation of a confined square cavity with natural convection based on compressible flow equations

Large eddy simulation of natural convection in a confined square cavity is described. The use of a complex compressible code with an artificial acoustic stiffness correction method, allows the use of higher time steps for a faster time and statistical convergence. We consider a broadly studied experimental case, consisting of a natural convective flow in a confined square cavity, with vertical walls heated at different rates (active walls), set at Ra = 1.58 × 10⁹. Turbulent boundary layers developing on the active walls and a vertical stable stratification characterize the mean flow. It is shown here that the results of this study match the experimental results reported in literature; for instance, mean velocity results. Although results for rms velocity fluctuations are barely over-predicted, the peak region is properly represented, while the greatest disagreements are found in the turbulent heat flow rate (velocity–temperature correlations). Turbulent structures were identified using different visualization methods and statistical studies. The authors found that the boundary layers on the active walls almost reach the fully turbulent regime, tending toward the laminar regime along the horizontal walls.     

Stability properties of the vertical boundary layers in differentially heated cavities

In the present study, the two-dimensional (2-D) stability properties of the vertical boundary layers in a cavity that is differentially heated over two opposing vertical walls is considered. The study is performed by introducing artificial, controlled perturbations at the base of the vertical boundary layer along the hot cavity wall and by following the evolution of these disturbances. For small initial perturbations, the evolution is governed by linear effects. This method accurately predicts the frequency of the bifurcation, which occurs for (much) larger Rayleigh numbers. Convective instability sets in for Rayleigh numbers much smaller than those at which the absolute instability (i.e., the bifurcation) occurs, and these Rayleigh numbers are in reasonable agreement with those for the boundary layer along a plate. The absolute instability does not result from the first wave which becomes unstable. For small Prandtl numbers (≤ 2), the unstable waves which lead to the absolute instability are shear-driven, and a single frequency is introduced in the flow after the bifurcation. For larger Prandtl numbers, the unstable waves are buoyancy driven and no single-frequency unsteady flow is observed after the bifurcation.     

The effects of forced boundary conditions on flow within a cubic cavity using digital particle image thermometry and velocimetry (DPITV)

Buoyancy-driven convection within a cavity, whose sidewalls are heated and cooled, is a problem of great interest, because it has applications in heat transfer and mixing. Most studies to date have studied one of two cases: the steady-state case or the development of the transient flow as it approaches steady state. Our main concern was to study the response of the cavity to time-varying thermal boundary conditions. We therefore decided to observe the flow phenomena within a convection cavity under sinusoidal thermal forcing of the sidewalls. To map the flow properly, it is necessary to have simultaneous kinematic and thermal information. Therefore, the digital particle image thermometry and velocimetry (DPITV) is used to acquire data. Implementing this technique requires seeding the flow with encapsulated liquid crystal particles and illuminating a cross section of the flow with a sheet of white light. Extraction of the thermal and kinematic content is in two parts. For the first, the liquid crystals will reflect different colors of the visible spectrum, depending on the temperatures to which they are subjected. Therefore, calibrating their color reflection with temperature allows for the extraction of the thermal content. For the second part, the kinematic information is obtained through the use of a digital cross-correlation particle image velocimetry technique. With the use of DPITV, the flow within a convection cavity is mapped and studied under steady forcing and sinusoidally forced boundary conditions at the Brunt-Väisälä frequency. For the sinusoidally forced case, three cases are studied. In the first, the heating between the two walls is in phase. In the second, the heating between the two walls is 180° out of phase. In the third, the heating between the two walls is 90° out of phase. For steady forcing, the thermal plots show that the flow develops a linearly stratified profile within the center of the cell. At the sidewalls, however, owing to forcing, hot/cold thermal boundary layers develop at the left/right walls. These hot/cold thermal boundary layers then turn around the upper-left/lower-right corners and develop into intrusion layers that extend across the top and bottom walls. The vorticity and streamlines show that the bulk of the fluid motion is concentrated around the walls, whereas the fluid within the center of the cell remains stationary. For the sinusoidally forced cases, the thermal plots show the existence of many thermal “islands,” or pockets of fluid where the temperature is different with respect to its surroundings. The vorticity plots show that the center of the cell is mostly devoid of vorticity and that the vorticity is mainly confined to the sidewalls, with some vorticity at the top and bottom walls. For the 0° forcing, the streamlines show the development of two counterrotating rollers. For the 180° forcing, the streamlines show the development of only one roller. Finally, for the 90° forcing, the streamlines show the development of both a two-roller and a one-roller system, depending on the position within the forcing cycle.     

Effect of the Source Term on Steady Free Convection Boundary Layer Flows over an Vertical Plate in a Porous Medium. Part I

The Darcy free convection boundary layer flow over a vertical flat plate is considered in the presence of volumetric heat generation/absorption. In the present first part of the paper it is assumed that the heat generation/absorption takes place in a self-consistent way, the source term q ^′′′≡ S of the energy equation being an analytical function of the local temperature difference T − T _∞. In a forthcoming second part, the case of the externally controlled source terms S = S(x,y ) will be considered. It is shown that due to the presence of S, the physical equivalence of the up- and downflows gets in general broken, in the sense that the free convection flow over the upward projecting hot plate (“upflow”) and over its downward projecting cold counterpart (“downflow”) in general become physically distinct. The consequences of this circumstance are examined for different forms of S. Several analytical solutions are given. Some of them describe algebraically decaying boundary layers which can also be recovered as limiting cases of exponentially decayingones. This asymptotic phenomenon is discussed in some detail.     

Experimental study of the instability of convective boundary layers under the action of vibration

The wave instability of convective boundary layers in a horizontal cylindrical layer of ethanol under the action of vertical hamonic high-frequency vibration is studied. A strong destabilizing effect of the vibration on the stability of the convective boundary layers is detected. In the plane of the gravity and vibration Rayleigh numbers (Ra and R _V ), the excitation limit of the wave instability is determined. The periods of the temperature oscillations caused by the waves in the boundary layers near the inner and outer cavity boundaries are studied as functions of the Rayleigh numbers. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 32–40, May–June, 1998.     

On the stability of the natural convection flow in a square cavity heated from the side

The stability of the steady laminar natural-convection flow of air (Prandtl number 0.71) and water (Prandtl number 7.0) in a square cavity is calculated by numerically solving the unsteady, two-dimensional Navier-Stokes equations. The cavity has a hot and cold vertical wall and either conducting or adiabatic horizontal walls. The flow looses its stability at a lower Rayleigh number in the case of conducting horizontal walls than in the case of adiabatic horizontal walls. The flow of water is more stable than the flow of air. Directly above the critical Rayleigh number the unsteady flow shows a single-frequency oscillation. Air in the case of adiabatic horizontal walls is an exception and shows two frequencies. The instabilities in the cavity seem to be related to well-known elementary instability mechanisms. In the case of conducting and adiabatic horizontal walls the instability seems to be related to a Rayleigh/Bénard and a Tollmien-Schlichting instability respectively. The second instability for air in the case of adiabatic horizontal walls seems to be related to an instability after a hydraulic jump.     

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics. (Accepted July 2, 1998)     

Buoyant MHD flows in a vertical channel: the levitation regime

Buoyant magnetohydrodynamic (MHD) flows with Joulean and viscous heating effects are considered in a vertical parallel plate channel. The applied magnetic field is uniform and perpendicular to the plates which are subject to adiabatic and isothermal boundary conditions, respectively. The main issue of the paper is the levitation regime, i.e., the fully developed flow regime for large values of the Hartmann number M, when the hydrodynamic pressure gradient evaluated at the temperature of the adiabatic wall is vanishing. The problem is solved analytically by Taylor series method and the solution is validated numerically. It is found that the fluid velocity points everywhere and for all values of M downward. For small M’s, the velocity field extends nearly symmetrically (with respect to the mid-plane) over the whole section of the channel between the adiabatic and the isothermal walls. For large values of M, by contrast, the fluid levitates over a broad transversal range of the channel, while the motion becomes concentrated in a narrow boundary layer in the neighborhood of the isothermal wall. Accordingly, the fluid temperature is nearly uniform in the levitation range and decreases rapidly within the boundary layer in front of the isothermal wall. It also turns out that not only the volumetric heat generation by the Joule effect, but also that by viscous friction increases rapidly with increasing values of M, the latter effect being even larger than the former one for all M.     

Temperature distribution in a Newtonian fluid injected between two semi-infinite plates

The analysis of the temperature distribution in time and place of a hot heat-conducting Newtonian fluid injected between two cooled parallel plates is presented. The 2-dimensional flow has a free flow front moving with constant velocity. The kernel of the fluid remains almost at the inlet temperature, but at the walls boundary layers occur with steeply descending temperature. The inner solutions inside these boundary layers are determined. To this end, the total region is divided into three distinct regions: the region G_I far behind the flow front, the flow front region G_II, and the intermediate region G_III between G_I and G_II. The asymptotics owing to each region are presented. The fundamental small parameter here is the thickness-to-length ratio of the 2-dimensional flow region. In most of the cases, similarity solutions are found. In the flow front region, for the formulation of the inner solution a Wiener–Hopf technique is used. Via matching procedures, the separate boundary layers are linked to each other to form one global boundary layer for the whole front region. All calculations in this paper are performed by analytical means, and all results are in analytical form. Comparison of our results with numerical solutions shows perfect agreement.     

An experimental study of the effect of wall temperature nonuniformity on natural convection in an enclosure heated from the side

An experimental study of natural convection in a parallelepipedal enclosure induced by a single vertical wall is described. The upper half of this wall was warm and the lower half cold. The other enclosure walls were insulated. The temperature and flow measurements were performed in the high Rayleigh number regime (10¹⁰<Ra<5×10¹⁰) by using water as the working fluid. The Rayleigh number was based on the enclosure height and the temperature difference between the warm and the cold part of the driving wall. The flow field featured two flat cells, one filled with warm fluid along the top horizontal wall, and the other filled with cold fluid along the bottom horizontal wall. Each of these cells was surrounded by an additional cell as tall as half the enclosure height. The above flow structure prohibited extensive thermal contact between warm and cold fluid, thus limiting the role of convection on the heat transfer process in the cavity. The findings of this study differ significantly from the findings of previous studies based on the ‘classical’ enclosure model possessing two isothermal vertical walls, the one warm and the other cold, and support the view that the use of ‘more realistic‘ temperature boundary conditions in enclosure natural convection needs careful examination.     

Numerical heat transfer in a cavity with a solar control coating deposited to a vertical semitransparent wall

A transient two‐dimensional computational model of combined natural convection, conduction, and radiation in a cavity with an aspect ratio of one, containing air as a laminar and non‐participating fluid, is presented. The cavity has two opaque adiabatic horizontal walls, one opaque isothermal vertical wall, and an opposite semitransparent wall, which consists of a 6‐mm glass sheet with a solar control coating of SnS–Cu_xS facing the cavity. The semitransparent wall also exchanges heat by convection and radiation from its external surface to the surroundings and allows solar radiation pass through into the interior of the cavity. The momentum and energy equations in the transient state were solved by finite differences using the alternating direction implicit (ADI) technique. The transient conduction equation and the radiative energy flux boundary conditions are coupled to these equations. The results in this paper are limited to the following conditions: 10⁴≤Gr≤10⁶, an isothermal vertical cold wall of 21°C, outside air temperatures in the range 30°C≤T₀≤40°C and incident solar radiation of AM2 (750 W m⁻²) normal to the semitransparent wall. The model allows calculation of the redistribution of the absorbed component of solar radiation to the inside and outside of the cavity. The influences of the time step and mesh size were considered. Using arguments of energy balance in the cavity, it was found that the percentage difference was less than 4 per cent, showing a possible total numerical error less than this number. For Gr=10⁶ a wave appeared in the upper side of the cavity, suggesting the influence of the boundary walls over the air flow inside the cavity. A Nusselt number correlation as a function of the Rayleigh number is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

A numerical study of 3D natural convection in a cube: effects of the horizontal thermal boundary conditions

A high-resolution, three-dimensional finite-difference numerical study of natural convection flows of a viscous fluid in a differentially heated cubical box is reported. The vertical sidewalls of the enclosure are maintained at constant temperatures of different values. The other vertical walls (the end walls) are thermally insulated. For the horizontal walls, two kinds of thermal boundary conditions are specified: adiabatic and perfectly conducting. Computations have been performed for an air-filled cavity for Rayleigh numbers of 10⁵ and 10⁶. The specific effects of the horizontal thermal boundary conditions on the flow structure are examined in detail. In the case of conducting walls, heat transfer through the horizontal walls enhances the convective flow activities. The numerically predicted velocity and temperature profiles in the symmetry planes are consistent with previous experimental measurements and computations.     

Boundary layer structure of oscillatory open-channel shallow flows over smooth and rough beds

The boundary layer structure of oscillatory shallow open channel flows has been studied in a wide flume. Fluorescence solution was released at a porous rough bed through a diffuser covered by gravel of 0.5 cm grain size. A planar laser-induced fluorescence (PLIF) system was used to visualise the dye plumes in both vertical and horizontal planes for a qualitative understanding of the roles of large-scale flow structures in mass transport. A variety of tests were conducted for a range of oscillatory periods (30–240 s), water depths (3–16 cm) and velocity amplitudes (0.027–0.325 m/s), which cover a wide range of oscillatory flows with Reynolds numbers Re _a varied from 0.3 × 10⁴ (laminar) to 2.1 × 10⁶ (fully turbulent). For quantitative investigation, a novel technique, namely combined laser-induced fluorescence (LIF) and 2D laser Doppler velocimetry (LDV) (LIF/LDV), was developed and used to measure the velocity and solute concentration simultaneously in a vertical plane over 50 cycles. From the dye plumes revealed by the PLIF in transitional flows, there are different patterns of flow structure and solute transport with three representative stages of acceleration, deceleration and flow reversal. In the acceleration stage, turbulence was suppressed with dye layers adhering to the surface with little vertical mass transport. In the deceleration stage, flame-like turbulent structures occurred when turbulence generation was prominent. This was investigated quantitatively by recording the percentage occurrence of the adhered smooth layers per cycle. For those smooth bed cases with Re _a < 1.8 × 10⁵, the adhered smooth dye layers type of boundary layer occupied 100% of the oscillation period. Over a sufficiently high Re _a , a rough bed can generate fully turbulent oscillatory flows without the appearance of adhering dye layers. Between these two extremes, a transitional flow regime occurs in a wide range of flow conditions: Re _a > 2.7 × 10⁴ over the rough bed and Re _a > 8.3 × 10⁶ over a smooth bed.     

A numerical study based on a weakly compressible formulation for thermosolutal convection in vertical cavities

Laminar thermosolutal convection in cavities with uniform, constant temperature and mass fraction profiles at the vertical side is studied numerically. The study is conducted in the case where an inert carrier gas (species “1”) present in the cavity is not soluble in species “2”, and do not diffuse into the walls. A mass flux of species “2” into the cavity occurs at the hot vertical wall and a mass flux out of the cavity occurs at the opposite cold wall. The weakly compressible model proposed in this work was used to investigate the flow fields, and heat and mass transfer in cavities filled with binary mixtures of ideal gases. The dimensionless form of the seven governing equations for constant thermophysical properties, except density, show that the problem formulation involves ten dimensionless parameters. The results were validated against numerical results published in the literature for purely thermal convection, and thermodynamic predictions for transient thermosolutal flows. A parametric study has been performed to investigate the effects of the initial conditions, molecular weight ratio, Lewis number, and aspect ratio of the cavity for aiding or opposing buoyancy forces. For the range of parameters considered, the results show that variations in the density field have larger effects on mass transfer than on heat transfer. For opposing buoyancy forces, the numerical simulations predict complex flow structures and possible chaotic behavior for rectangular vertical cavities according to the value of the molecular weight ratio.     

侧加热腔体内重力波演化过程的数值模拟

利用二维数值模拟的方法研究了侧加热腔体内的自然对流．基于数值模拟结果,描述了水平热入侵流(intrusion)的整个演化过程,并对该过程的物理机制进行了讨论．结果表明：当热入侵流抵达腔体冷壁后,由于冷壁无法卷入所有的热入侵流,热入侵流在冷上角堆积并产生一个反向流动,在冷壁边界层附近形成一个顺时针涡,该涡在浮力效应驱动下可返回热壁,并在腔体的冷热壁之间形成了腔体尺度的流体振荡,即内重力波．     

Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures

Numerical computations and experiments were carried out for a buoyant flow of liquid metal (mercury in the experiments) in a long vertical enclosure of square cross-section, in the presence of a uniform horizontal magnetic field. A strong emphasis is put on the case of a magnetic field perpendicular to the applied temperature gradient for two reasons: (1) the MHD damping is smaller than with any other orientation, and (2) the quasi-two-dimensionality of the flow in this case yields a quite efficient velocity measurement technique. The enclosure is heated by a thermally controlled flow of water from one of the vertical walls and cooled by a similar technique from the facing wall. Those two walls are good thermal conductors (thick copper plates in the experiments), whereas the four other walls are thermally insulating. All walls are electrically insulated from the fluid. In this paper, as well as in the companion paper by Tagawa et al. (Eur. J. Mech. B Fluids 21 (4) (2002) 383–398), we model analytically the Hartmann layers present along the walls perpendicular to the magnetic field. This modeling, which yields boundary conditions for the core flow without any meshing of the thin layers, is quite accurate when Hartmann layers are stable. The numerical results are in fairly good agreement with the experimental data. They namely reveal how the heat flux and the fluid flow organization depend on the magnetic field.     

Steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 4°C: case of variable wall temperature

The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as x ^m and the velocity outside boundary layer varies as x ^{2 m} , where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U _∞ oriented in the upward or downward direction, while the ambient temperature is T _∞ = T _m (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.