Transient natural convection between horizontal concentric cylinders: Natural convection between concentric cylinders

The initial phase of transient natural convection between two horizontal concentric cylinders is investigated. Short-time solutions to the momentum and energy equations are obtained using the method of matched asymptotic expansions. The solutions for the velocity and temperature are expressed in terms of three expansions reflecting the existence of three distinct regions in the initial flow field, and four different patterns of motion may be distinguished, corresponding to four different types of thermal boundary conditions.     

Dual steady solutions in natural convection between horizontal concentric cylinders

Dual steady solutions in natural convection in an annulus between two horizontal concentric cylinders are numerically investigated for a fluid of Prandtl number 0.7. It is found that, when the Rayleigh number based on the gap width exceeds a certain critical value, dual steady two-dimensional (2-D) flows can be realized: one being the crescent-shaped eddy flow commonly observed and the other the flow consisting of two counter-rotating eddies and their mirror images. The critical Rayleigh number decreases as the inverse relative gap width increases.     

Conjugate natural convection between horizontal concentric cylinders filled with a porous medium

A conjugate problem of natural convection between two horizontal concentric, isothermal cylinders filled with a fluid-saturated porous medium is studied. The flow field and energy equations are solved under the conditions of equality in temperature but with jump in heat flux at the fluid-solid interface. Numerical results by a finite-difference technique are presented for a large variation in the parameters entering the problem. A comparison of the solution with non-conjugate problem is also given.Das konjugierte Problem der freien Konvektion zwischen zwei horizontalen, konzentrischen, isothermen Zylindern, die mit einem fluid-gesättigten porösen Medium gefüllt sind, wurde hier untersucht.Das Strömungsfeld und die Energiegleichung sind unter den Bedingungen gelöst worden, daß die Temperatur gleich ist, aber ein Wärmeflußsprung am Übergang von der Flüssigkeits- zur Festphase stattfindet. Die numerischen Ergebnisse werden mit dem Finite-Differenzen-Verfahren für viele Parametervariationen dargestellt. Zudem wurde ein Lösungsvergleich mit einem nicht konjugierten Problem gegeben.     

Effect of induced magnetic field on natural convection in vertical concentric annuli

In the present paper,we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field.The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account.The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity,induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli.The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables.It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder,while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder.These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases.It is also found that as the Hartmann number increases,there is a flattening tendency for both the velocity and the induced magnetic field.The influence of the induced magnetic field is to increase the velocity profiles.     

Transient laminar free convection in horizontal cylinders

A numerical study of laminar natural convection inside uniformly heated, partially or fully filled horizontal cylinders is made. A coordinate transformation which simplifies the discretization of the equations of motion and energy is utilized. The resulting system of partial differential equations with their boundary conditions is solved using central differences for various Prandtl and Grashof numbers for two different grid sizes. The flow in completely filled cylinders for which experimental data are available is predicted. Close agreement between steady-state predictions and experiments is obtained for temperature and velocity profiles as well as for the streamline contours and isotherms. The technique is further demonstrated by solving the transient natural convection flow inside a partially filled horizontal cylinder with an adiabatic free surface and subjected to uniform wall heating.

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Laminare freie Konvektion in horizontalen Zylindern

Zusammenfassung Es wurde eine numerische Berechnung der laminaren, freien Konvektion in gleichmäßig beheizten, teilweise oder ganz gefüllten, horizontalen Zylindern durchgeführt. Dabei wird eine Koordinatentransformation benützt, welche die Diskretisierung der Bewegungs- und der Energiegleichung vereinfacht. Das so resultierende System von partiellen Differentialgleichungen wird, zusammen mit seinen Randbedingungen, unter Verwendung einer Differenzenmethode für verschiedene Prandtl und Grashof-Zahlen sowie für zwei verschiedene Gittergrößen gelöst. Für den vollständig gefüllten Zylinder, für den experimentelle Daten verfügbar sind, wird die Strömung vorhergesagt. Dabei wird für stationäre Zustände gute Übereinstimmung zwischen Rechnung und Experiment erzielt. Dies gilt sowohl für den Verlauf der Stromlinien als auch für den der Isothermen. Das Verfahren wird weiterhin am Beispiel der Berechnung instationärer, freier Konvektion in einem partiell gefüllten, horizontalen Zylinder demonstriert, wobei eine adiabate, freie Oberfläche und gleichmäßige Beheizung der Wand angenommen sind.

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Nomenclature g acceleration due to gravity, m/s² - Gr _R ^* modified Grashof number =gqR⁴/kv² - Gr _R Grashof number =gTR³/v² - H heat function vector, dimensionless - k thermal conductivity, W/mK - L(Y) cord length associated with coordinateY, dimensionless - Pr Prandtl number=v/ - q wall heat flux, W/m² - R radius, m - r(X, Y,Z) distance of a boundary point from the reference axis, dimensionless - S vector derived from the flow field solution, dimensionless - T temperature, K - T _w wall temperature, K - T reference temperature, K - t time, s - u, v velocity components inx, y directions, m/s - U, V dimensionless velocity components inX- and Y-direction normalized withU - U reference velocity=gqR²/k or gTR, m/s - V velocity vector, dimensionless - W vorticity vector, dimensionless - W vorticity, dimensionless - x, y, z cartesian coordinates, m - X, Y, Z cartesian coordinates normalized with a reference length, dimensionless Greek letters thermal diffusivity, m²/s - coefficient of thermal expansion, K^–1 - ,,, non-dimensional coordinates in the transformed domain - non-dimensional temperature =(T–T)_k/q^R or T–T/T_w–T - v kinematic viscosity, m²/s - non-dimensional time=v/R² Gr_Rt or v/R² G _R ^* t - angle measured from the bottom of the cylinder, rads - ^* angle measured from the axis on (– ) plane, rads - heat potential, dimensionless - angle of incidence of the heat flux vector, rads - non-dimensional stream function - vector potential, dimensionless - grid size, dimensionless - ² Laplacian operator - gradient vector     

Improved perturbation solutions for laminar natural convection on a vertical cylinder

The method of extended perturbation series is applied to solve for laminar natural convection from an isothermal, thin vertical cylinder. The series in terms of the transverse curvature parameterξ extended to five terms and is subsequently improved by applying the Shanks transformation twice. The validity of the solution is extended up toξ=10 and possibly even beyond. Up toξ=10, the results for wall shear as well as the local and average Nusselt numbers agree very closely with those of local nonsimilarity and finite difference solutions. The ease of computation coupled with high accuracy makes the present approach far more attractive than the currently popular local nonsimilarity and finite difference methods. Its success with the present problem should motivate applications to a host of nonsimilar boundary layer flows.     

Effect of vertical separation distance on laminar natural convective heat transfer over two vertically spaced equitemperature horizontal cylinders

A numerical study of laminar natural convective heat transfer in air from a pair of equitemperature horizontal cylinders placed one above the other in a vertical plane was carried out. Prime attention was focused on how heat transfer characteristics of the upper cylinder are affected by center-to-center separation distance between two cylinders (CCD). The study was limited to Rayleigh number ranging from 2×10⁴ to 2×10⁵. The vertical center-to-center separation distance between the cylinders was varied from two to ten cylinder diameters. A finite difference scheme based on the integration of the governing equations over finite cells was used. Temperature profiles around two cylinders, and heat transfer coefficients for each cylinder were obtained. The effect of center-to-center separation distance between the cylinders on heat transfer from the upper cylinder was considered. For assessing the accuracy of the current computational approach the results are compared with the experimental results reported by Sparrow and Niethammer [10].     

A numerical study of three-dimensional laminar natural convection in a vented enclosure

A three-dimensional numerical investigation of steady laminar natural convection in vented enclosures is carried out. A discrete flush-type heat source mounted on the substrate is used to simulate an electronic component. Four different vent locations are investigated. Combined natural convection in the air and conduction in the heat source, the substrate, and the enclosure walls are solved. Solutions are obtained for Rayleigh numbers ranging from 10⁴ to 10⁶, different substrate thermal conductivity ratios, and varied vent sizes. The calculation domain is extended beyond the cubic enclosure in x-, y-, and z-directions. Appropriate boundary conditions are prescribed on the extended computational domain. The resulting flow and temperature patterns are discussed. Also, the local and overall heat transfer from the heat source and the substrate, in terms of Nusselt numbers and the surface temperatures, are presented to illustrate the vent effects.     

Non-Darcian effects on conjugate natural convection between horizontal concentric cylinders filled with a porous medium

An analysis is presented of steady conjugate free convection between two horizontal concentric cylinders filled with a fluid-saturated porous medium; the innermost cylinder surface is maintained at a high temperature and the outermost cylinder surface at a lower one. The velocity-pressure-gradient relation is taken to be nonlinear, with departure from the linear Darcy situation measured by a parameter F₀. The investigation is based on the numerical solution, by a finite-difference method, of the full momentum and energy equations. The streamline and isotherm patterns as well as the local and mean Nusselt numbers are plotted for several physical parameters to show some of the flow and heat transfer characteristics. It is found that all parameters play an important role in the flow and heat transfer characteristics. The model can be applied to a variety of engineering problems.     

Flow separation in laminar natural convection

Theoretical and experimental studies of flow separation in laminal natural convection are presented. Since classical boundary-layer theory cannot determine separation on curved walls in natural convection, two extensions of the classical boundary-layer theory are discussed: boundary-layer theory of higher order and double-deck theory. Both theories are applied to experiments on a vertical flat plate with humps.     

Analysis of conjugate laminar natural convection between finitely top heating vertical channel flow

Conjugate laminar natural convection between finitely conducting vertical parallel plates is studied numerically. Since the heat transfer coefficient along the plate is undetermined and is strongly dependent on both the channel fluid flow and the thermal properties of the plate. The energy equation for the plate and the thermal boundary layer equation for the fluid between the parallel plates are coupled and solved by employing the finite-difference technique. Results of the distribution of local heat transfer coefficient, local heat flux and temperature along the plate are presented. The effects of Grashof number, fluid-to-wall conductivity ratio, and channel height-to-width ratios on heat transfer phenomena between the plates have been discussed. The tested range of Grashof number was up to 10⁶, the fluid-to-wall conductivity ratio of 0, 0.01 and 0.02, and channel height-to-width ratio varied between 0.5 and 5. These results indicate that conduction has a significant influence on the total heat transfer rate, particularly at high Grashof numbers and high fluid-to-wall conductivity ratios. When the length of the plates increases, the total heat transfer rate firstly increases and then maintains at a constant value.Die laminare freie Konvektion zwischen endlichen, wärmeleitenden, vertikalen und parallelen Wänden wird hier numerisch erforscht. Da der Wärmeübertragungskoeffizient längs der Wand nicht eindeutig bestimmt werden kann, ist er vom Kanalfluidstrom und den thermischen Eigenschaften der Wand stark abhängig. Die Energiegleichungen für die Wand und die Gleichung der Grenzfläche für das Fluid zwischen den parallelen Wänden sind zusammengefügt und für die Verwendung von Finite-Differenzenverfahren aufgelöst worden. Das Ergebnis der Verteilung des lokalen Wärmeübergangskoeffizienten, des lokalen Wärmestroms und der Temperatur längs der Wand wird dargestellt. Die Einwirkungen der Grashof-Zahl, das Leitfähigkeitsverhältnis von Fluid zu Wand und das Verhältis zwischen Höhe und Breite des Kanals bei dem Wärmeübergangsphänomen zwischen den Wänden sind besprochen worden.Der untersuchte Bereich der Grashof-Zahl erstreckt sich bis zu 10⁶, das Verhältnis der Wärmeleitfähigkeit von Fluid zu Wand lag bei 0, 0,01 und 0,02 und das Höhen-Breite-Verhältnis des Kanals variierte zwischen 0,5 und 5. Dieses Ergebnis weist auf den bedeutenden Einfluß der Wärmeleitfähigkeit auf den gesamten Wärmeübertragungswert bei besonders hohen Grashof-Zahlen und hohem Wärmeleitfähigkeitsverhältnis von Fluid zu Wand hin. Wenn die Länge der Wände zunimmt, dann steigt als erstes der gesamte Wärmeübertragungswert und behält danach einen konstanten Wert bei.     

Combined heat and mass transfer by laminar natural convection from a vertical plate

Simultaneous heat and mass transfer in buoyancy-induced laminar boundary-layer flow along a vertical plate is studied for any ratio of the solutal buoyancy force to the thermal buoyancy force by using a new similarity transformation. The effects of the buoyancy ratio and Lewis number on the rates of heat and mass transfer are presented explicitly for most practical gaseous solutions (Pr=0.7, 0.21≤Sc≤2.1) and aqueous solutions (Pr=7, 140≤Sc≤1400). Very accurate correlations of the mass transfer and heat transfer rates are developed for the cases of single and combined buoyancy forces.     

Developing laminar convection in a vertical double-passage channel

 The effect of the presence of a thin, perfectly conductive baffle on the development of laminar convection in a vertical channel has been investigated numerically. The channel has different constant wall temperatures which are higher than those at the entrance. Velocity and temperature profiles have been presented. The effect of the different parameters on heat transfer in the channel has been discussed. The occurrence of flow reversal has been observed in some cases but examination of this phenomenon has been considered to be beyond the scope of the present work. For long channels, the numerical solutions approach the fully developed flow analytical solution. Finally, the results showed that higher values of Nu_h can be obtained when the baffle is near the hot wall. Received on 23 May 2000     

竖直平板间自然对流的湍流频谱特性

根据直接数值模拟的计算结果，对竖直平板间湍流自然对流的脉动动能、速度及温度等物理量的时间序列进行频谱分析．结果表明，流动达到充分发展状态后，小尺度到大尺度的能谱很宽，计算的分辨率足够．从能谱分布可以观察到含能区、惯性子区和耗散区的存在，文中对各区的特性进行分析．由于该流动的强各向异性，惯性子区很窄．并讨论了法向位置对脉动动能的影响以及大尺度结构的特性．     

Numerical study of natural convection flow in a vertical slot

The stability of convective motion, generated by a lateral temperature difference across a vertical slot, is studied numerically over a range ofGr=5000 to 1.5 × 10⁵,Pr=0.01 to 10, andA=8,16 and 20. Various cellular flow structures and temperature patterns are illustrated. Several branches of solutions characterized by different numbers of the cells in the flow patterns as well as by both steady and unsteady multicellular patterns are found for low-Prandtl-number fluid in the vertical slot. Meanwhile, the behaviors of the temperature variation and heat transfer are also discussed. The project supported by the National Natural Science Foundation of China (59776011) and by the Returnee from Abroad Funding of Academia Sinica.     

Numerical feasibility study of utilizing nanofluids in laminar natural convection inside enclosures

Laminar natural convective heat transfer of nanofluids inside an enclosure is numerically investigated considering the thermal dispersion effect of the nanoparticles. Feasibility of applying nanofluids instead of pure liquids in natural convective, which is a discrepancy issue between the previous numerical and experimental works, is examined. Results confirm the previous experimental results of general deterioration in heat transfer rate. Discussions, justifications and correlations for average Nusselt number are presented.     

Effects of wetted walls on laminar natural convection between vertical parallel plates with asymmetric heating

A numerical analysis is carried out to investigate the effect of latent heat transfer, in association with the evaporation or condensation of the water vapor on the wetted channel walls which are under asymmetric heating, on the natural convection heat transfer. Major nondimensional groups identified are Gr _T , Gr _M , Pr and Sc. Results for Nusselt and Sherwood numbers are specifically presented for an air-water system under various heating conditions to illustrate latent heat transfer during the evaporation and condensation processes. The effects of the channel length and wetted wall temperatures on the momentum, heat and mass transfer are examined in great detail.     

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.