Forced Convection Flow Research Articles

Three-dimensional needle network model for dendritic growth with fluid flow

We present a first implementation of the Dendritic Needle Network (DNN) model for dendritic crystal growth in three dimensions including convective transport in the melt. The numerical solving of the Navier-Stokes equations is performed with finite differences and is validated by comparison with a classical benchmark in fluid mechanics for unsteady flow. We compute the growth behavior of a single equiaxed crystal under a forced convective flow. As expected, the resulting dendrite morphology differs strongly from the case of the purely diffusive regime and from similar two-dimensional simulations. The resulting computationally efficient simulations open the way to studying mechanisms of microstructure selection in presence of fluid flow, using realistic alloy and process parameters.

Turbulent Heat Transfer Characteristics of a W-Baffled Channel Flow - Heat Transfer Aspect

In this work, the thermal behavior of a turbulent forced-convection flow of air in a rectangular cross section channel with attached W-shaped obstacles is investigated. The continuity, momentum and energy equations employed to control the heat and velocity in the computational domain. The turbulence model of k-ε is employed to simulate the turbulence effects. The finite volume method with SIMPLE algorithm is employed as the solution method. The results are reported temperature, local and average Nusselt numbers, and mean velocity contours. The subject is relevant and important for industrial applications.

3D Numerical study of the effect of aspect ratio on mixed convection air flow in upward solar air heater

A 3D Numerical study of mixed convection air flow in upward solar air heater with large spanwise aspect ratio (A = 10 to 40) was performed using CFD commercial code Fluent 14.5 (ANSYS). The main objective of this study is to investigate the channel height's effect (aspect ratio) on flow pattern and heat transfer in upward solar air heater in the particular case of low Re and high aspect ratio. The bottom plate (absorber) was submitted to Constant Heat Flux (CHF) in the range of 200 to 1000 W/m2 and Reynolds number was varied from 50 to 1000. Our results are in concordance with most of authors conclusions about Poiseuille–Rayleigh–Benard flows. In mixed convection, increasing heat flux enhances heat transfer unlike forced convection flows. Simulation results of flow visualizations and Nusselt number calculations have shown that depending on Ri*, the velocity and temperature distributions in SAH vary greatly with the channel's height. The obtained results were different from previous studies. Indeed, our investigation of channel's height was achieved for the same heat flux but different Grashof numbers. For low channel's heights (high aspect ratio), increasing heat flux has not a significant effect but for higher channel's heights, an augmentation of heat flux enhances buoyancy effects in the flow and causes high turbulence. Also, increasing Reynolds number in low channel's heights (high A), can enhance substantially heat transfer. For higher channel's heights (low A), increasing Reynolds number decreases Ri* and thus buoyancy forces. Heat transfer is reduced and so Nusselt number. The obtained results may be very useful for engineers in designing and testing solar collectors.

A Unified Treatment for Local Thermal Nonequilibrium Free, Forced, and Mixed Convective Boundary Layer Flows Over an Arbitrarily Shaped Nonisothermal Body in a Fluid Saturated Porous Medium

Abstract A unified integral solution procedure has been proposed to analyze all possible Darcian local thermal nonequilibrium (LTNE) free, forced, and mixed convective boundary layer flows, commonly encountered in porous media engineering applications. The heated body may be arbitrarily shaped, and its temperature may vary over the surface. The integral energy equation for the solid phase yields an algebraic equation between the dimensionless fluid thermal boundary layer thickness and its ratio to the solid-phase counterpart, while the integral energy equation for the fluid phase reduces to a first order ordinary differential equation in terms of the dimensionless fluid thermal boundary layer thickness. This set of the equations for determining the local Nusselt number of our primary interest proved to be valid for all possible Darcian cases of LTNE free, forced, and mixed convective boundary layer flows over an arbitrarily shaped nonisothermal body in a fluid saturated porous medium. Asymptotic expressions for the cases of arbitrary shapes were also obtained analytically for both leading edge and far downstream regions. The results are found to agree well with available direct numerical integration results. Furthermore, the regime map has been constructed to show the boundary layer transition point from the LTNE to equilibrium. The proposed unified method is found quite useful when designing thermal engineering systems associated with fluid saturated porous media.

Cu–Al2O3–H2O hybrid nanofluid flow with melting heat transfer, irreversibility analysis and nonlinear thermal radiation

We have investigated the influence of hybrid nanoparticles on various physical quantities in a water-based hybrid nanofluid involved in a steady and fully developed forced convective flow generated over a stretched surface. Nonlinear thermal radiation and melting heat transfer analysis are featured in this work. To obtain the solution of the governing equations, a standard transformation and numerical procedure are implemented. Then, a comprehensive discussion of the effects of the flow regime on several governing parameters is presented. The results indicated that increasing magnetic strength $$M$$ and nanoparticle volume fraction $$\phi_{1}$$ lead to a thicker thermal boundary layer. A similar trend takes place with increasing nonlinear thermal radiation while the reverse is noticed for Eckert number. The entropy generation rate increases with the increase in Brinkman number and Bejan number reduces with increasing Eckert number. The obtained results of this model closely match with those available in the literature as a limiting situation. It is demonstrated that hybrid nanofluids exhibit lower entropy generation rates. The results of this study are of importance in the assessment of the effect of some essential design parameters on heat transfer and, consequently, in the optimization of industrial processes.

Laminar hydrodynamic and thermal entrance lengths for simultaneously hydrodynamically and thermally developing forced and mixed convective flows in horizontal tubes

Correlations to predict the hydrodynamic and thermal entrance lengths in a smooth horizontal tube heated at a constant heat flux are often restricted to forced convection conditions. The reason being that most scholars do not distinguish very well between forced and mixed convection conditions. Furthermore, most correlations have been developed for hydrodynamically fully developed and thermally developing flows. The purpose of this study was to experimentally investigate the hydrodynamic and thermal entrance lengths for laminar flows in smooth horizontal tubes heated at a constant heat flux. Simultaneously hydrodynamically and thermally developing flows were investigated in both forced and mixed convection conditions. Two smooth circular test sections with inner diameters of 4 mm and 11.5 mm, and maximum length-to-diameter ratios of 1373 and 872, respectively, were used. Heat transfer and pressure drop measurements were taken at Reynolds numbers between 460 and 3200 at different heat fluxes. A large database with a total of 513 mass flow rate measurements, 43501 temperature measurements, and 1665 pressure drop measurements were used. The test fluid was water and the Prandtl number ranged between 3 and 8. For the first time, correlations were developed to calculate the hydrodynamic and thermal entrance lengths for simultaneously hydrodynamically and thermally developing flow in smooth horizontal tubes. The correlations were valid for tubes heated at a constant heat flux in both forced and mixed convection conditions. For ease of use, the correlations were developed in terms of the local, inlet, and bulk fluid properties. Furthermore, the mixed convection correlations were developed in terms of both the Grashof and modified Grashof numbers.

Stimulation of cell growth and neurogenesis using protein-functionalized microfibrous scaffolds and fluid flow in bioreactors

Microfibrous viscose-rayon scaffolds functionalized with poly-L-lysine and laminin were used to culture pheochromocytoma (PC12) cells in recirculation bioreactors under bulk laminar flow conditions. To determine the effects of continuous forced convective flow on cell viability, growth, and neurogenesis, the cultures were exposed to fluid-induced shear stresses of magnitude 2.7 × 10–5 to 4.6 × 10–3 Pa for up to 12 days. The results were compared with those obtained using static scaffold and static monolayer (2D) control cultures. Cell growth rates were 1.5- to 1.8-fold greater in the bioreactor scaffold cultures relative to the controls without flow. After treatment of the cells with nerve growth factor, development and extension of neurite networks and expression of several neural differentiation markers including β3-tubulin, shootin1, and EphA2 were also substantially enhanced in the presence of fluid convection. In the control cultures without fluid flow, cell growth and differentiation were greater in the static scaffold cultures than in the static 2D monolayers. This work demonstrates the significant beneficial effects of features of the physical culture environment, such as hydrodynamic conditions and cell attachment surface geometry, in regulating culture performance. Flow devices capable of supporting long-term cell culture while delivering controlled levels of hydrodynamic shear are a valuable practical tool for promoting nerve tissue development and regeneration in vitro.

Radiation and Chemical Reaction Effects on MHD Convective Flow over a Porous Plate through a Porous Medium with Heat Generation

The present paper analyzes the radiation and chemical reaction effects on a steady laminar forced convection flow of a viscous incompressible electrical conducting fluid over a flat plate embedded in a porous medium in the presence of heat generation. The governing equations are transformed by using similarity transformation and resultant dimensionless equations are solved numerically using the Runge-Kutta method with Shooting technique. The effects of various governing parameters on velocity, temperature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are shown in figures and tables and analyzed in details.

Similarity Requirements for Mixed Convective Boundary Layer Flow over Vertical Curvilinear Porous Surfaces with Heat Generation/Absorption

Similarity requirements for three dimensional combined forced and free convective laminar boundary layer flows over the porous inclined vertical curvilinear surfaces with buoyancy effects and heat absorption/generation effects are investigated theoretically. The potential flow in the mainstream and Gabriel lame coefficients outside of the boundary layer are the function of ξ,η. Hence, the external velocity components (Ue, Ve) and Gabriel lame coefficients h1,h2,h3 are independent of ζ. Here, h3ξ,η=1 has been set such that ζ represents actual distance measured normal to the surface. Similarity requirements for an incompressible fluid are sought on the basis of detailed analyses in order to reduce the governing partial differential equations into a set of ordinary differential equations. Finally, different possible cases are exhibited in a tabular form with the inclusion of ΔT variations for onward flow study that are helpful to the future researchers for the flow over the orthogonal curvilinear surfaces.

Numerical study of thermo-hydraulic characteristics for forced convective flow through wavy channel at different Prandtl numbers

The present work numerically investigates the thermo-fluidic and entropy generation characteristics for laminar forced convective flow through wavy channel at different Prandtl number (Pr). Results are presented for the following range of parameters: Reynolds number $$5 \le \text{Re} \le 200$$ , Prandtl number $$0.72 \le \Pr \le 100$$ , dimensionless amplitude $$0.3 \le \alpha \le 0.7$$ and dimensionless wavelength $$0.5 \le \lambda \le 1.5$$ . It is observed that with increase in Pr, the thickness of the thermal boundary layer at trough region decreases slowly for smaller Re, whereas at higher Re, the rate of decrement is higher. The average Nusselt number increases with Pr for all amplitude, wavelength and Reynolds number. The relative heat transfer enhancement compared to equivalent plane channel is presented in terms of enhancement ratio (ER), and it shows a non-monotonic variation of ER with Pr at lower Re and a monotonic one at higher Re. The combined alteration of rate of heat transfer and pressure drop as compared to plane channel is enumerated by performance factor (PF), and the variation of PF with Pr shows non-monotonic behaviour at lower Re and monotonic one at higher Re. The variation of PF shows non-monotonic variation with Re for higher Pr and for smaller wavelength, whereas it monotonically decreases for all Pr at higher wavelength. Thermal entropy generation contribution is higher over the viscous one for all the cases considered. The local thermal entropy generation distribution varies with Re, Pr and geometrical configuration of the channel. For smaller amplitude (α = 0.3), the total entropy generation is minimum in the considered range of Re and Pr.

Numerical and experimental analysis of a planar jet with heated co-flow at medium and low Prandtl-number values

In the present work, we combine experiments and numerical simulations of a planar jet with heated co-flow with medium (air) and low-Prandtl (He-Xe gas mixture) fluids. Jets are recognized as representative test cases to be investigated in large components of pool-type liquid metal-cooled nuclear systems, like the Multi-purpose hYbrid Research Reactor for High-tech Applications (MYRRHA), currently under design at SCK•CEN. The present planar jet configuration mimics a closed wind tunnel that is designed and operated at VKI to generate an experimental database for velocity and temperature fields of a turbulent forced-convection flow regime. The performed experiments combine the Particle Imaging Velocimetry (PIV) (in characteristic planes) and thermocouple (single point) measurements. In parallel with experiments, comprehensive numerical simulations have been performed within the RANS modeling framework. Next to the standard eddy-viscosity based two-equation k-ε model, an extended variant based on the low-Reynolds elliptic relaxation concept (so-called ζ-f model) has been applied too. To investigate the low-Prandtl effects on the heat transfer, series of the turbulent heat transfer models have been applied, ranging from a conventional constant turbulent Prandtl number to a more elaborate kθ-εθ model. The combination of the low-Reynolds ζ-f and kθ-εθ models was explored for the first time in the content of nuclear engineering applications. The focus of the numerical studies is to address in details the effects of low-Prandtl fluid in the strongly forced convection flow (central planar cold jet) in presence of a strong shear (hot co-flow). We demonstrate the importance of the proper specification of the inlet boundary conditions in numerical simulations to mimic correctly experimentally observed asymmetrical distributions of the cross-wise profiles of stream-wise velocity, turbulent kinetic energy and temperature. Finally, the minor differences in results between the assumed constant turbulent Prandtl number and more advanced kθ-εθ model of the turbulent heat flux confirmed the overly dominant mechanisms of the strong convection and molecular diffusion in the present configuration.

Forced convective flow and heat transfer past an unconfined blunt headed cylinder at different angles of incidence

In the present paper, a numerical investigation has been carried out to study the forced convective flow and heat transfer characteristics past a blunt-headed cylinder in crossflow. Employing air as an operating fluid, calculations are carried out for a range of Reynolds number (Re) from 40 to 160. The angle of incidence is varied in the range of 0∘ ≤ α ≤ 180 ∘. The thermofluid features of flow and heat transport are analysed in detail for different angles of incidence. To analyse the aerodynamic characteristics, several parameters such as drag and lift coefficients, moment coefficient, Strouhal number, recirculation length, and local time-averaged vorticity flux have been calculated. Furthermore, a stability analysis has been undertaken by using the Stuart Landau equation to enumerate the critical Reynolds number at each angle of incidence. Heat transfer characteristics are studied by computing local and time-averaged values of Nusselt numbers. When compared to a rectangular cylinder, a blunt-headed cylinder exhibits an enhanced heat transfer rate. In the end, an entropy generation analysis has been carried out to study the effects of Re and angle of incidence on the efficiency of thermofluid transport characteristics.

Force convective flow over a porous and stretching (shrinking) sheet of variable thickness

The classical problems of natural and forced convection flows are reformulated and combined into a single model by introducing new variables for the field quantities. The proposed model is characterized by additional features of physical interest. A generalized problem of force convection flow and heat transferred is solved for a vertical stretching (shrinking) and porous (impermeable) sheet of variable thickness. The generalization gives rise to new problems, which are obtained in the next fourth section. It is assumed that the stretching/shrinking and porous velocities are variable such that both suction and injection can take place through the porous surface. The variable surface velocity may have linear, nonlinear, exponential and power law forms. Analysis of viscous flow and heat transfer is accomplished by considering force convection boundary layer flow over a stretching (shrinking) and porous surface of variable thickness. The nonlinear problem of partial differential equation is simplified by considering the boundary layer approximations and reduced it into nonlinear ODEs. The ODEs are obtained by introducing unusual and generalized similarity transformations for the stream function and similarity variables. Final ODEs are characterized by suction (injection), stretching (shrinking), convection parameters and Prandtl number. The ODEs are solved numerically, and effects of all existing parameters are studied on flow and heat transfer characteristics. Comparisons of the new problem and its solutions are established with the studies to demonstrate the applicability, validity and high accuracy of the present approach.

Optimal heating strategy for minimization of peak temperature and entropy generation for forced convective flow through a circular pipe

Fundamental insight on the minimization of entropy generation and maximum temperature is important for efficient performance and proper design of thermal systems. One of the key factors that dictate the entropy generation and temperature within a thermal system is the applied heat source and therefore appropriate heating strategy that minimizes the peak temperature and entropy generation needs to be determined. In this work, non-uniform distribution of heat flux applied to the wall of a circular tube undergoing laminar forced convective flow of a high Prandtl number fluid has been investigated numerically to optimize the heating strategy which results in minimum entropy generation and peak temperature as compared to the uniform heating. Various heat flux configurations are evaluated for the same amount of total heat rate and comparative assessment between the different cases has been made in terms of peak temperature and entropy generation. In terms of achieving both minimum peak temperature and entropy generation simultaneously, the periodic heat flux condition (sinusoidal with number of waves greater than or equal to 100) can be recommended as the optimum heating strategy for typical thermal system.

Hydrodynamic and thermal analysis of water, ethylene glycol and water-ethylene glycol as base fluids dispersed by aluminum oxide nano-sized solid particles

PurposeThe purpose of this paper is to carry out a hydrodynamic and thermal analysis of turbulent forced-convection flows of pure water, pure ethylene glycol and water-ethylene glycol mixture, as base fluids dispersed by Al2O3 nano-sized solid particles, through a constant temperature-surfaced rectangular cross-section channel with detached and attached obstacles, using a computational fluid dynamics (CFD) technique. Effects of various base fluids and different Al2O3 nano-sized solid particle solid volume fractions with Reynolds numbers ranging from 5,000 to 50,000 were analyzed. The contour plots of dynamic pressure, stream-function, velocity-magnitude, axial velocity, transverse velocity, turbulent intensity, turbulent kinetic energy, turbulent viscosity and temperature fields, the axial velocity profiles, the local and average Nusselt numbers, as well as the local and average coefficients of skin friction, were obtained and investigated numerically.Design/methodology/approachThe fluid flow and temperature fields were simulated using the Commercial CFD Software FLUENT. The same package included a preprocessor GAMBIT which was used to create the mesh needed for the solver. The RANS equations, along with the standard k-epsilon turbulence model and the energy equation were used to control the channel flow model. All the equations were discretized by the finite volume method using a two-dimensional formulation, using the semi-implicit method for pressure-linked equations pressure-velocity coupling algorithm. With regard to the flow characteristics, the interpolation QUICK scheme was applied, and a second-order upwind scheme was used for the pressure terms. The under-relaxation was changed between the values 0.3 and 1.0 to control the update of the computed variables at each iteration. Moreover, various grid systems were tested to analyze the effect of the grid size on the numerical solution. Then, the solutions are said to be converging when the normalized residuals are smaller than 10-12 and 10-9 for the energy equation and the other variables, respectively. The equations were iterated by the solver till it reached the needed residuals or when it stabilized at a fixed value.FindingsThe result analysis showed that the pure ethylene glycol with Al2O3 nanoparticles showed a significant heat transfer enhancement, in terms of local and average Nusselt numbers, compared with other pure or mixed fluid-based nanofluids, with low-pressure losses in terms of local and average skin friction coefficients.Originality/valueThe present research ended up at interesting results which constitute a valuable contribution to the improvement of the knowledge basis of professional work through research related to turbulent flow forced-convection within channels supplied with obstacles, and especially inside heat exchangers and solar flat plate collectors.

Mathematical modeling of air pollution in city tunnels and evaluating mitigation strategies

A mаthemаtical model has been developed that is able to describe the environment of the city tunnels being affected by the road traffic, natural and forced air convective flows. The mathematical model describes the peculiarities of the traffic flows on one-lane roads with a satisfactory accuracy. The model well matches experimental data on traffic flows. Multidimensional calculations of the influence of cars on the airflow in tunnels are performed. The numerical model for simulating exhaust gas emissions by automobiles and their accumulation in a tunnel and evolution with traffic induced air flow was developed. The results of numerical investigations make it possible providing recommendations for transport researchers and policy-makers. In particular, it was shown that in the presence of long tunnels on automobile roads it is necessary to choose the traffic arrangement avoiding the necessity for vehicles to come to a full stop and then accelerate in tunnels. This could happen in the presence of traffic lights or other type of traffic regulation near the exit of the tunnel. Thus it is necessary in arranging traffic regulation avoiding placing traffic lights in the proximity of tunnel exit for the waiting cars line to be shorter than the distance from the tunnel exit. In venting the tunnel, the direction of the wind should coincide with the direction of traffic flow. Attempts to arrange venting in the opposite direction for high blockage ratio of a tunnel by vehicles could result in bringing to a worse situation with air pollution.

Revisiting Forced Convection Flow Through a Porous Medium Saturated Channel Using Singular Perturbation Analysis

Abstract Accounting for the fact that thermal conductivity of fluid is much less than the thermal conductivity of solid in most of the porous medium-related applications, this study applies perturbation approach in analyzing forced convection through a parallel plate channel under local thermal nonequilibirum (LTNE) condition by denoting the thermal conductivity ratio of fluid to solid as the small parameter, suggesting leading order solutions to solve the two-equation energy model, by incorporating Darcy model and Brinkman model for large porous medium shape factor, respectively, in the presence of heat generation in both fluid and solid. This study provides important fluid temperatures, solid temperatures, and heat transfer coefficient approximations, which enables further analysis on the fluid and solid temperature gradient at the boundary and hence delineate the roles of thermal conductivities and interfacial heat transfer in LNTE mode. The results signify competition between the heat conduction from the wall through fluid conduction and interfacial heat transfer from solid to fluid in the thermal boundary layer. The effect of thermal boundary layer is intensified with the attendant increase in porous medium shape factor and heat generation in solid. The results for Brinkman model also establish conditions for temperature bifurcations to take place whereby in such cases, an increase in viscous dissipation in fluid attributes to the detachment of thermal boundary layer as the porous medium shape factor, S decreases. The phenomenon caused by insufficient convection rate to overcome viscous dissipation bears much resemblance to the separation point in the momentum boundary layer.

Magnetic field effects on forced convection flow of a hybrid nanofluid in a cylinder filled with porous media: a numerical study

The magnetic field can serve as a proper controlling parameter for heat transfer and fluid flow; it can be also employed to maximize the thermodynamic efficiency in various fields. Nanofluids and porous inserts are among the conventional approaches of heat transfer enhancements. Porous media, in addition to improving the heat transfer, can enhance the pressure drop. This research presents a numerical investigation on the magnetohydrodynamics forced convection effects of Al2O3–CuO–water nanofluid inside a partitioned cylinder within a porous medium. The calculations were carried out for a broad range of governing parameters. The nanofluid flow is modeled as a two-phase flow using two-phase mixture model, and the Darcy–Brinkman–Forchheimer equation is employed to model fluid flow in porous media. Simulation was also conducted under the laminar flow regime by finite volume method. Furthermore, the thermal boundary condition of constant uniform heat flux was imposed on the cylinder walls. The average Nusselt number as well as the performance evaluation criteria (PEC) were examined for diverse Darcy numbers (0.0001 < Da < 0.1) and Hartmann numbers (0 < Ha < 40). The results indicate the significant impact of Hartmann and Darcy number enhancement on the elevation of heat transfer coefficient. Additionally, incorporation of nanoparticles to the base fluid increased the PEC in all cases. Moreover, the PEC reached to its maximum value in configurations involving permeable porous media (i.e., a medium with Da = 0.1 and Ha = 40).

Analysis of Convective Transport of Temperature-Dependent Viscosity for Non-Newtonian Erying Powell Fluid: A Numerical Approach

Non-Newtonian is a type of fluid that does not comply with the viscosity under the Law of Newton and is being widely used in industrial applications. These include those related to chemical industries, cosmetics manufacturing, pharmaceutical field, food processing, as well as oil and gas activities. The inability of the conventional equations of Navier–Stokes to accurately depict rheological behavior for certain fluids led to an emergence study for non-Newtonian fluids’ models. In line with this, a mathematical model of forced convective flow on non-Newtonian Eyring Powell fluid under temperature-dependent viscosity (TDV) circumstance is formulated. The fluid model is embedded with the Newtonian heating (NH) boundary condition as a heating circumstance and is assumed to move over a stretching sheet acting vertically. Using appropriate similarity variables, the respective model was converted into ordinary differential equations (ODE), which was later solved utilizing the Keller box approach. The present model is validated by comparing the existing output in literature at certain special limiting cases, where the validation results display a firm agreement. The current outputs for the proposed model are shown in tabular and graphical form for variation of skin friction plus Nusselt number, velocity and temperature distribution, respectively.

Characterisation of the heat transfer in displaced enhancement devices by means of inverse problem approach applied to IR images

ABSTRACT Displaced enhancement devices are inserts positioned inside pipes that increase the heat transfer rate. This study presents an original application of an inverse analysis technique to experimental infrared temperature data to estimate the local convective heat flux for forced convection flow in a pipe in which this kind of device is present. Butterfly-shaped inserts have been investigated to determine their local and global thermal performance. The insert produces a distortion in the velocity profile that causes an irregular distribution of the wall heat flux along the circumferential coordinate. Several studies have investigated displaced enhancement devices, but they generally considered only the overall heat transfer performance. The present work aims to fill this gap by presenting and testing an experimental procedure for estimating the local convective heat flux in pipes equipped with this type of insert. The experimental results obtained in the present investigation are useful for the design of heat exchangers for processes in which product temperature must be carefully controlled (e.g. pharmaceutical and food industries). These types of devices induce significant spatial variation of the heat flux, and quantitative data of this aspect is of primary importance.