Numerical analysis of natural convection around a vertical channel in a rectangular enclosure

Numerical computations were performed for the heat transfer and fluid flow characteristics of an internal vertical channel composed by a pair of parallel plates situated in a rectangular enclosure, with the inner plates and the bounding wall of the enclosure maintained at uniform but different temperatures. Natural convection occurred in the air which occupied the enclosure space. The plates were symmetrically arranged. The dimensionless channel widthS was varied parametrically. The Rayleigh numbers ranged from 10² to 10⁷. Static bifurcation was found in this configuration. The bifurcation is related to the flow pattern transition from single-vortex structure to double-vortex structure or vice versa. Comparison with the empirical correlations obtained for a vertical plate and a channel in an infinite space showed that the heat transfer process of the plates and the channel was deteriorated by the existence of the enclosure.     

Transient conjugated mixed-convective heat transfer in a vertical plate channel with one wall heated discretely

 This study presents a numerical solution of the unsteady conjugated mixed-convection heat transfer in a vertical plate channel with one wall suddenly subjected to either isoflux or isothermal discrete heat sources. The effects of the dimensionless heat source length H ₁, the dimensionless spacing between heat sources H ₂, the dimensionless channel length L, the dimensionless heated-plate thickness B _l, the wall-to-fluid conductivity ratio K and the ratio of Grashof number to Reynolds number Gr/Re on the interface heat flux, Nusselt number and bulk fluid temperature are discussed in detail. Results show that the discrete heating can cause the heat transfer direction conversely from the fluid to the heated plate during the transient period, which is more significant for the cases with larger L and H ₂. For the system with isoflux discrete heat sources, the time required to reach the steady-state is shorter for larger H ₂. While the trend is reverse for system with isothermal discrete heat sources. Additionally, a higher ratio of the input energy is axially conducted through the plate wall from heated sections to unheated regions for a larger H ₂ and B _l or smaller L. Received on 9 November 1998     

Effect of variable porosity on laminar convection in a uniformly heated vertical porous channel

The present investigation is devoted to the study of fully developed mixed convective flow through a vertical porous channel. The lateral variations of porosity and thermal diffusivity in the bed near the wall, are approximated by exponential functions. The correlation between permeability and porosity is brought through Kozney-Carman approximation. The volume averaged one dimensional low speed momentum equation proposed by Vafai is employed for the analysis of the problem. Results are obtained for steady heating of ascending cold fluid and steady cooling of ascending hot fluid. For the above physical situations it is observed that the heat transfer rate, and ratio of friction factor increases with increase in porous parameter, whereas the ratio of mass flow rate decreases with increase in porous parameter. The velocity profiles exhibit hydrodynamic channelling and peak velocity shifts towards the wall for higher values of the porous parameter. For steady heating of ascending could fluid increase in Rayleigh number enhances the heat transfer rate, and mass flow rate, while it reduces the ratio of friction factor. An opposite trend is observed for the case of steady cooling of ascending hot fluid.     

Numerical investigation of flame propagation and extinction in a vertical channel

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 18–26, July–August, 1992.     

Numerical investigation of a spatially developing turbulent natural convection boundary layer along a vertical heated plate

Large-eddy simulation (LES) on a spatially developing natural convection boundary layer along a vertical heated plate was conducted. The heat transfer rate, friction velocity, mean velocity and temperature, and second-order turbulent properties both in the wall-normal and the stream-wise direction showed reasonable agreement with the findings of past experiments. The spectrum of velocity and temperature fluctuation showed a -2/3-power decay slope and -2-power decay slope respectively. Quadrant analysis revealed the inclination on Q1 and Q3 in the Reynolds stress and turbulent heat flux, changing their contribution along the distance from the plate surface. Following the convention, we defined the threshold region where the stream-wise mean velocity takes local maximum, the inner layer which is closer to the plate than the threshold region, the outer layer which is farther to the plate than the threshold region. The space correlation of stream-wise velocity tilted the head toward the wall in the propagating direction in the outer layer; on the other hand, the correlated motion had little inclination in the threshold region. The time history of the second invariant of gradient tensor Q revealed that the vortex strength oscillates both in the inner and the outer layers in between the laminar and the transition region. In the turbulent region, the vortex was often dominant in the outer layer. Instantaneous three-dimensional visualization of Q revealed the existence of high-speed fluid parcels associated with arch-shape vortices. These results were considered as an intrinsic structure in the outer layer, which is symmetrical to the structure of canonical smooth/rough wall bounded layer flow in forced convection.     

Experimental observations of the thermal flow around a square obstruction on a vertical wall in a differentially heated cavity

The transient thermal boundary layer flow around a square obstruction placed at the middle of the hot wall in a differentially heated cavity is visualized using a shadowgraph technique. The results show that the thermal boundary layer flow, which is blocked by the obstruction, firstly forms an intrusion head under the obstruction (the lower intrusion head). Subsequently, the lower intrusion head bypasses the obstruction and reattaches to the down-stream boundary. During the reattachment process, a more complicated flow is induced, and eventually both the lower intrusion head and the thermal boundary layer destabilize. After the lower intrusion head is convected away, the thermal boundary layer flow re-stabilizes. At the quasi-steady state, the thermal boundary layer forms a double-layer structure, which is split into two sections by the obstruction. It is demonstrated that both the transient processes and the quasi-steady state flow structures of the thermal boundary layer are significantly altered by the obstruction in comparison with the case without the obstruction.     

Linearized analysis of magnetohydrodynamic entrance flow in a vertical channel with combined thermal convection

A linearized analysis is presented for the magnetohydrodynamic entrance flow with combined forced and free convection in a vertical, constant wall temperature parallel-plate channel. Numerical results are obtained for slug velocity profile at the entrance and for various Hartmann and Grashof Numbers. The results agree well with the finite difference numerical solutions obtained elsewhere. They demonstrate that the velocity development and pressure gradient in the channel entrance region are greatly influenced by the Hartmann Number and the Grashof Number. Increasing Hartmann Number decreases velocity entrance length while increasing Grashof Number increases it. Thermal development is also found to be dependent on the above mentioned parameters, but to a relatively minor extent.Nomenclature A _m constant defined by equation (23) - B ₀ applied magnetic field - C _n constant defined by equation (13) - E ₀ constant electric field - e nondimensional electric field parameter, E ₀/U _mB₀ - Gr Grashof Number, gL ³(T _w–T ₀)/ ² - L half-width of the channel - M Hartmann Number, B ₀ L(/)^1/2 - Nu Nusselt Number, (/y) _y=1/( _w– _m) - P pressure - Pr Prandtl Number, / - p nondimensional pressure parameter, (P–P ₀+ ₀ gX)/P ₀ U _m ² - Re Reynolds Number, U _m L/ - T temperature - T ₀ inlet temperature - T _w wall temperature - U velocity, X direction - U _m average velocity, (1/L) ₀ ^L UdY - u nondimensional form of U, U/U _m - u ₀(y) nondimensional inlet velocity - V velocity, Y direction - v nondimensional form of V, VL/ - X coordinate, axial direction - x nondimensional form of X, vX/L ² U _m - Y coordinate perpendicular to the channel - y nondimensional form of Y, Y/L - thermal diffusivity - _m eigenvalue defined by equation (25) - thermal expansion coefficient - _m eigenvalue defined by equation (24) - stretching factor, weighting function - nondimensional form of T, (T–T ₀)/(T _w–T ₀) - _m mean nondimensional temperature, ₀ ¹ udy - kinematic viscosity - magnetic permeability - mass density - electrical conductivity     

Numerical simulation of time-dependent buoyant flows in an enclosed vertical channel

A time-accurate Finite Volume method is used to investigate the two-dimensional buoyant flow in a closed cabinet containing two vertical heating plates. These are parallel, and form a channel at the centre of the cabinet enclosure. The cases of isothermal plates, and of uniform heat generation within them, are both considered for two values, 1×10⁵ and 1×10⁷, of the leading non-dimensional parameter, the Grashof number. Air (Pr = 0.71) is considered as the working fluid. Transient and long term thermal and flow behaviours are investigated. Steady-state solutions are asymptotically found at the lower Gr-value. However, time-dependent long-term solutions are predicted at Gr = 1×10⁷. Received on 20 May 1998     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Dual-tracer fluorescence thermometry measurements in a heated channel

The exponential growth of component density in microelectronics has renewed interest in compact and high heat flux thermal management technologies that can handle local heat fluxes exceeding 1 kW/cm². Accurate and spatially resolved thermometry techniques that can measure liquid-phase temperatures without disturbing the coolant flow are important in developing new heat exchangers employing forced-liquid and evaporative cooling. This paper describes water temperature measurements using dual-tracer fluorescence thermometry (DFT) with fluorescein and sulforhodamine B in laminar Poiseuille flow through polydimethyl siloxane-glass channels heated on one side. The major advantage of using the ratio of the signals from these two fluorophores is their temperature sensitivity of 4.0–12% per °C—a significant improvement over previous DFT studies at these spatial resolutions. For an in-plane spatial resolution of 30 μm, the average experimental uncertainties in the temperature data are estimated to be 0.3°C.     

Magnetoconvection in a vertical channel with heat source or sink

The problem of steady, laminar mixed convective flow and heat transfer of an electrically conducting fluid through a vertical channel with heat source or sink is analyzed. The effects of viscous and Ohmic dissipations are included in the energy equation. Both walls are kept either at the same or different temperatures such as isothermal-isothermal, isoflux-isothermal and isothermal-isoflux conditions. Analytical solutions are found using regular perturbation technique and numerical solutions are found using finite difference method. A selected set of graphical results illustrating the effects of various parameters involved in the problem on the flow as well as flow reversal situation and Nusselt numbers are presented and discussed. It is also found that both the analytical and numerical solutions agree very well for small values of the perturbation parameter.     

On the development of Taylor vortices in a vertical annulus with a heated rotating inner cylinder

A numerical study has been conducted to determine the heat transfer characteristics and flow patterns which develop around a rotating, heated vertical cylinder enclosed within a stationary concentric cylinder. A tall annulus (aspect ratio of 10) with fixed, adiabatic horizontal end-plates and a radius ratio of 0·5 has been considered. Furthermore, the effect that the introduction of buoyancy forces by heating the inner cylinder has on the development of the Taylor vortex flow is examined. It is observed that the formation of the Taylor vortices is delayed until the rotational parameter σ = Gr/Re² has a value below unity for any given Reynolds number Re which is above the critical value Re_crit for the formation of Taylor vortices in an isothermal flow. Also, the Taylor cells first appear at the top of the annulus. As σ is gradually decreased below unity, bifurcations to other states are observed. The final structure of the secondary flow is noticeably distorted in the mixed-convection mode, with the size of the Taylor cells varying greatly along the height of the annulus. This distortion diminishes as σ is further decreased, until the isothermal flow pattern is nearly recovered below σ = 0·01.     

Numerical modeling of a turbulent flow behind a heated grid

A numerical model of the dynamics of turbulence and temperature fluctuations behind a heated grid located in a wind tunnel is constructed on the basis of closed Kármán-Howarth and Corrsin equations. Results calculated by this model are in reasonable agreement with available experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 118–126, May–June, 2009.     

Numerical analysis of fluid flow and mass transfer in a channel with a porous bottom wall

The flow of a solution between parallel plates is considered. The bottom plate is porous, while the top one is an impermeable solid. A computer program based on the control volume approach was developed to analyse the flow and concentration fields. The effects of the slip at the porous wall on the velocity and particle concentration distributions were investigated. It was observed that as the slip increases, the concentration on the porous wall decreases and the maximum velocity moves towards the porous wall. The concentration on the porous wall increases in the flow direction. This increase in the particle concentration along the porous wall may cause a reduction of the porosity and hence a variation in the suction rate along the porous wall. In order to take this effect into account, a linearly varying transverse velocity along the porous wall was considered. The results were compared with the data available in the literature.     

Structure of turbulent two-dimensional channel flow with strongly heated wall

A laser Doppler velocimeter and a resistance thermometer were used to study velocity and temperature statistics in a strongly heated turbulent two-dimensional channel flow, with the wall temperature up to 700 °C and a Reynolds number of 14,000. Normalized mean velocity and mean temperature profiles were not significantly affected by the wall heating. Turbulent intensities of temperature fluctuation were also insensitive to the heat flux. However, turbulent intensities of velocity fluctuation were suppressed in the region away from the wall, whereas those near the wall were not changed noticeably by the wall heating. This phenomenon was explained by the balance of three parameters: turbulent production, viscous dissipation and intermittency.     

Magnetohydrodynamic mixed convection in a vertical channel

The problem of combined free and forced convective magnetohydrodynamic flow in a vertical channel is analysed by taking into account the effect of viscous and ohmic dissipations. The channel walls are maintained at equal or at different constant temperatures. The velocity field and the temperature field are obtained analytically by perturbation series method and numerically by finite difference technique. The results are presented for various values of the Brinkman number and the ratio of Grashof number to the Reynolds number for both equal and different wall temperatures. Nusselt number at the walls is determined. It is found that the viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. It is also found that the analytical and numerical solutions agree very well for small values of ε.