Radiation and laminar forced convection of non-Newtonian fluid in a circular tube

Interaction of radiation with thermally developing laminar forced convection of power-law, non-Newtonian, absorbing, emitting, isotropically scattering, gray fluid through an isothermal circular tube with a black boundary is investigated. The energy equation is solved by an implicit finite-difference scheme, while the radiation part of the problem is solved by the collocation method. Results are presented for the effects of conduction-to-radiation parameter, single scattering albedo, optical thickness and the inlet temperature on the local Nusselt number along the tube for the case of power-law index n=1/3, 1 and 3, where the case n=1 corresponds to the Newtonian fluid.     

Conjugated laminar forced convection in ducts with periodic variation of inlet temperature

Conjugated laminar forced convection with parabolic velocity is studied for flow inside both parallel-plate and circular ducts subjected to a periodically varying inlet temperature. The analysis of this classs of problems lead to a Sturm-Liouville type complex eigenvalue problem for which no known solution is available. In this work, a new methodology is presented for a direct solution of such complex eigenvalue problems in the complex domain using the shooting method along with the Runge-Kutta method. The methodology is applicable for solving both laminar and turbulent flow problems; here we consider only the laminar flow with a parabolic velocity profile. The results clearly show that the slug-flow assumption overestimates the phase lag and the Nusselt number. Some benchmark results are also presented for the eigenquantities in tabular form.     

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.     

Numerical study of magnetohydrodynamic laminar flow separation in a channel with smooth expansion

The flow of an electrically conducting incompressible viscous fluid in a plane channel with smooth expansion in the presence of a uniform transverse magnetic field has been analysed. A solution technique for the governing magnetohydrodynamic equations in primitive variable formulation has been developed. A co‐ordinate transformation has been employed to map the infinite irregular domain into a finite regular computational domain. The governing equations are discretized using finite‐difference approximations in staggered grid. Pressure Poisson equation and pressure correction formulae are derived and solved numerically. It is found that with increase in the magnetic field, the size of the flow separation zone diminishes and for sufficiently large magnetic field, the separation zone disappears completely. The peak u‐velocity decreases with increase in the magnetic field. It is also found that the asymmetric flow in a symmetric geometry, which occurs at moderate Reynolds numbers, becomes symmetric with sufficient increase in the transverse magnetic field. Thus, a transverse magnetic field of suitable strength has a stabilizing effect in controlling flow separation, as also in delaying the transition to turbulence. Copyright © 2008 John Wiley & Sons, Ltd.     

MHD natural convection in porous media-filled enclosures

In this work, the magnetohydrodynamics （MHD） natural convection heat transfer problem inside a porous medium filled with inclined rectangular enclosures is investigated numerically. The boundary conditions selected on the enclosure are two adiabatic and two isothermal walls. The governing equations, continuity, and Forchheimer extension of the Darcy law and energy are transformed into dimensionless forms by using a set of suitable variables, and then solved by using a finite difference scheme. The governing parameters are the magnetic influence number, the Darcy Rayleigh number, the inclination angle, and the aspect ratio of the enclosure. It is found that the magnetic influence number and the inclination angle have pronounced effects on the fluid flow and heat transfer in porous media-filled enclosures.     

Experimental techniques in natural convection

Experimental techniques in natural convection heat transfer employed in the author's laboratory are introduced. The techniques are mostly related to visualization of flow, temperature field, and heat flux distribution in fluids. Three topics are presented, the first being natural convection in a horizontal rectangular liquid layer driven by surface tension and buoyancy. The patterns of flow were visualized by suspending fine aluminum flakes in the liquid. At the same time, the distribution of the temperature gradient in the liquid was visualized by an optical method making use of the refraction of light. The second topic is the onset of oscillatory convection in the Czochralski growth melt. In this case a forced flow due to rotation of the crystal and the vessel is superimposed on the buoyancy convection, resulting in an oscillatory flow under certain circumstances. The flow pattern and the temperature distribution in the liquid were visualized simultaneously by suspending in the liquid a microencapsulated temperature-sensitive liquid crystal. Periodical oscillation of the flow and the temperature was clearly recognized. The third topic is the rollover of double liquid layers that were stratified stably due to a density difference. A small-scale experiment was carried out to clarify the basic mechanism of rollover. The tracer method was used to visualize boundary layer flow along the vertical side wall and the shadowgraph technique to visualize the density distribution in the liquid layers. The article emphasizes the importance of visual observation in the investigation of natural convection phenomena.     

An efficient Legendre-Galerkin spectral method for the natural convection in two-dimensional cavities

An efficient numerical method is developed for solving the natural convection in two-dimensional cavities. The numerical scheme is proposed by using second-order projection scheme in time direction and Legendre-spectral in spatial variable of the incompressible flow. Finally, a series of numerical examples are presented to demonstrate the efficiency of our algorithm. The numerical strategies developed in this article could be readily applied to study other incompressible fluid problems.     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Numerical prediction for laminar forced convection heat transfer in parallel-plate channels with streamwise-periodic rod disturbances

A numerical study has been performed for the periodically fully-developed flow in two-dimensional channels with streamwise-periodic round disturbances on its two walls. To accurately describe the round disturbance boundary condition, a body fitted grid was used. The flow and heat transfer have been studied in the range of Reynolds number, Re=50–700, and Prandtl number Pr=0.71. The influences of disturbance parameters and Reynolds number on heat transfer and friction have been investigated in detail. Some of the solutions have been examined using both steady and unsteady finite difference schemes; and the same results have been obtained. The results show that different flow patterns can occur with different deployments of the disturbances. With appropriate configuration of the disturbances, the Nusselt number can reach a value four times greater than in a smooth channel at the same condition, with the penalty of a much greater pressure drop. On the other hand, if the disturbances are not deployed properly, augmentation of heat transfer cannot be acquired. © 1998 John Wiley & Sons, Ltd.     

Experimental investigation of chimney-enhanced natural convection in hexagonal honeycombs

The natural convective heat transfer performance of an aluminum hexagonal honeycomb acting as a novel heat sink for LED cooling is experimentally investigated. The concept of adding an adiabatic square chimney extension for heat transfer enhancement is proposed, and the effects of chimney shape, height, and diameter are quantified. The average N u_av of a heated honeycomb with straight chimney is significantly higher than that without chimney, and the enhancement increases with increasing chimney height. At a given chimney height, honeycombs with divergent chimneys perform better than those with convergent ones. For a fixed divergent angle, the N u_av number increases monotonically with increasing chimney height. In contrast, with the convergent angle fixed, there exists an optimal chimney height to achieve maximum heat transfer.     

Sharp entry and transition effects for laminar combined convection of water in vertical tubes

In previous work the authors showed a sharp entry to have a definite effect on laminar forced convection to water in a vertical circular tube. The study has been extended to situations of strong aiding natural convection, and new experimental data and numerical predictions are reported. The expected entry effect is confirmed, but it is found to be less marked in a strong combined convection field than in the previous forced convection study. The experimental data also include evidence of transition from laminar flow, and a possible criterion of transition is investigated, based on the axial location of minimum local Nusselt number. This is experimental and predictive data for this criterion are compared with the experimental correlation of Lawrence and Chato based on a criterion of temperature fluctuations     

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.     

Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions

A finite volume multigrid procedure for the prediction of laminar natural convection flows is presented, enabling efficient and accurate calculations on very fine grids. The method is fully conservative and uses second-order central differencing for convection and diffusion fluxes. The calculations start on a coarse (typically 10 × 10 control volumes) grid and proceed to finer grids until the desired accuracy or maximum affordable storage is reached. The computing times increase thereby linearly with the number of control volumes. Solutions are presented for the flow in a closed cavity with side walls at different temperatures and insulated top and bottom walls. Rayleigh numbers of 10⁴, 10⁵ and 10⁶ are considered. Grids as fine as 640 × 640 control volumes are used and the results are believed to be accurate to within 0–01%. Second-order monotonic convergence to grid-independent values is observed for all predicted quantities.     

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.     

The unstable modes of natural convection boundary layer

The instability of natural convection boundary layer around a vertical heated flat plate is analyzed theoretically in this paper. The results illustrate that the “loop” in the neutral curve is not a real loop but a twist of the curve in the frequency-wave number-Grashof number space, and there is only one unstable mode at small Prandtl numbers. Specially, when the Prandtl number is large enough two unstable modes will be found in the “loop” region. Along the amplyfying surface intersection the two unstable modes have the same Grashof number, wave number and frequency but different amplifying rates. Their instability characteristics are analyzed and the criterion for determining the existence of the multi-unstable modes is also discussed.     

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.     

Numerical prediction of laminar flow and heat transfer characteristics in a tube fitted with regularly spaced twisted-tape elements

By numerically solving the Navier-Stokes and energy equations in their three-dimensional parabolic form, the friction and heat transfer characteristics are predicted for laminar flow in a circular tube fitted with regularly spaced twisted-tape elements that are connected by circular rods. The predictions have agreed closely with the experimental data for water. It is shown that a strong L/D effect is associated with the configuration under consideration. By extending the predictions outside the range of experimental data, it is found that considerably enhanced thermohydraulic performance is achievable by increasing the number of turns on the tape elements, by reducing the connecting rod diameter, and at high fluid Prandtl numbers.     

Diffuse approximation method for solving natural convection in porous media

The diffuse approximation is presented and applied to natural convection problems in porous media. A comparison with the control volume-based finite-element method shows that, overall, the diffuse approximation appears to be fairly attractive.Nomenclature H height of the cavities - I functional - K permeability - p(M _i ,M) line vector of monomials - p ^T p-transpose - M current point - Nu Nusselt number - Ri inner radius - Ro outer radius - Ra Rayleigh number - x, y cartesian coordinates - u, v velocity components - T temperature - M vector of estimated derivatives - _t thermal diffusivity - coefficient of thermal expansion - practical aperture of the weighting function - scalar field - (M, M _i ) weighting function - streamfunction - kinematic viscosity