Mixed convection about a non-isothermal vertical surface in a porous medium

The problem of mixed convection about non-isothermal vertical surfaces in a saturated porous medium is analysed using boundary layer approximations. The analysis is made assuming that the surface temperature varies as an arbitrary function of the distance from the origin. A perturbation technique has been applied to obtain the solutions. Using the differentials of the wall temperature, which are functions of distance along the surface, as perturbation elements, universal functions are derived for various values of the governing parameter Gr/Re. Both aiding and opposing flows are considered. The universal functions obtained can be used to estimate the heat transfer and fluid velocity inside the boundary layer for any type of wall temperature variation. As a demonstration of the method, heat transfer results have been presented for the case of the wall temperature varying as a power function of the distance from the origin. The results have been studied for various combinations of the parameters Gr/Re and the power index m, taking both aiding and opposing flows into consideration. On comparing these results with those obtained by a similarity analysis, the agreement is found to be good.     

Mixed convection flow investigation in a rectangular horizontal tube stenosis via liquid crystal thermography and planar PIV

The temperature and velocity field in a horizontal convergent-divergent rectangular channel heated from below is studied experimentally for a Reynolds range 8-120, Grashof numbers from 0.44 × 10⁵ to 2.56 × 10⁵ and Richardson numbers from 3 to 4000, using water as working fluid. The duct aspect ratio (width/height) varies from 1 at its inlet to 2.28 at the stenosis neck, and both the upper and bottom walls are tilted with an angle of ±15.7° with respect to the horizontal. The temperature of the bottom wall is kept constant above that of the issuing fluid. The temperature field is recorded by liquid crystals in the vertical mid plane whereas the velocity field is measured in this plane as well as in four cross sections of the divergent passage by planar PIV, revealing the characteristics of the secondary velocity field. For all the examined cases the flow in the convergent passage is free of thermal plumes, and the thermal boundary layer is thin. In contrast, the divergent passage is characterized by a thermal plume which is shifted upstream with increasing Gr or reducing Re. Both transversal and longitudinal rolls emerge in this diffuser the strength of which depend on Re and Gr influencing the streamwise distribution of Nusselt which for low Re presents a minimum.     

The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder

This paper deals with a theoretical (numerical) analysis of the effects that blowing/injection and suction have on the steady mixed convection or combined forced and free convection boundary layer flows over a vertical slender cylinder with a mainstream velocity and a wall surface temperature proportional to the axial distance along the surface of the cylinder. Both cases of buoyancy forces aid and oppose the development of the boundary layer are considered. Similarity equations are derived and their solutions are dependent upon the mixed convection parameter, the non-dimensional transpiration parameter and the curvature parameter, as well as of the Prandtl number. Dual solutions for the previously studied mixed convection boundary layer flows over an impermeable surface of the cylinder are shown to exist also in the present problem for aiding and opposing flow situations.     

Non-darcy double-diffusive mixed convection from heated vertical and horizontal plates in saturated porous media

The non-darcy mixed convection flows from heated vertical and horizontal plates in saturated porous media have been considered using boundary layer approximations. The flows are considered to be driven by multiple buoyancy forces. The similarity solutions for both vertical and horizontal plates have been obtained. The governing equations have been solved numerically using a shooting method. The heat transfer, mass transfer and skin friction are reduced due to inertial forces. Also, they increase with the buoyancy parameter for aiding flow and decrease for the opposing flow. For aiding flow, the heat and mass transfer coefficients are found to approach asymptotically the forced or free convection values as the buoyancy parameter approaches zero or infinity.     

Heat transfer by combined free and forced convection in vertical tubes

The influence of free convection on forced convection heat transfer becomes important in laminar flows. Numerical methods have been applied for a study of mixed convection in vertical tubes for the following conditions: temperature-dependent fluid density, constant wall temperature and parabolic profile of axial velocity at the tube entrance. Both cases: heating and cooling have been considered.     

Viscous dissipation effects on fully developed combined free and forced non-Newtonian convection in a vertical tube

An investigation which includes the simultaneous effects of viscous dissipation and combined free and forced laminar non-Newtonian convection is presented. The problem under consideration is that of fully developed upflow in a vertical, circular tube which is heated with a constant wall heat flux. All properties are assumed to be constants in the analysis except for a temperature dependent density in the body force term which generates the free convection effects. The coupled continuity, momentum, and energy equations are solved using a finite difference technique. Numerical solutions are presented as a function of the parameters of the problem-flow behavior index n, Grashof number over Reynolds number ratio Gr/Re, and the Eckert number-Prandtl number product E Pr. The results show that heating due to viscous dissipation distorts the velocity profile, increases the friction factor, and decreases the Nusselt number.     

On the Reynolds-number independence of mixed convection in a vertical channel subjected to asymmetric wall temperatures with and without flow reversal

The problem of mixed convection in a vertical channel with asymmetric wall temperatures including situations of flow reversal is studied numerically. The SIMPLER algorithm with a staggered grid system is employed to solve the corresponding numerical equations formulated by the finite-volume method. A second-order upwind scheme is used to model the convective term, and a suitable grid distribution is introduced. The ranges of the parameters studied are 0 r_t 1, 1 Re 1000, and 0 Gr/Re 500.

The numerical results, with the streamwise coordinate scaled by the Reynolds number (Re), show that solutions for the velocity and temperature fields are independent of the Reynolds number when Re 50, even in the presence of flow reversal. These solutions, however, are dependent on r_t and Gr/Re. Subsequently, correlations are proposed for the bulk temperature distribution and the local Nusselt numbers along the hot wall and the cold wall.     

Mixed Convection Effect on Melting from a Vertical Plate in a Porous Medium

In the present work, the effect of mixed convection about vertical surfaces on the phenomenon of melting process in a fluid-saturated porous medium is analyzed on the basis of boundary layer approximations. Similarity solutions are obtained for aiding external flow. The final similarity equations are integrated numerically by use of the fourth-order Runge–Kutta method. Results are reported for the flow and thermal fields in the melt region. The melting phenomenon decreases the local Nusselt number at the solid–liquid interface.     

Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface

The mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent fluid is analyzed using similarity solution technique. Wall temperature and stretching velocity are assumed to have specific exponential function forms. The influence of buoyancy along with viscous dissipation on the convective transport in the boundary layer region is analyzed in both aiding and opposing flow situations. The flow is governed by the mixed convection parameter Gr/Re². The velocity and temperature inside the boundary layer are observed to be influenced by the parameters like Prandtl number Pr, Gebhart number Gb. Significant changes are observed in non-dimensional skin friction and heat transfer coefficients due to viscous dissipation in the medium. The flow and temperature distributions inside the boundary layer are analyzed and the results for non-dimensional skin friction and heat transfer coefficients are discussed through computer generated plots.     

Wärmeübergang bei turbulenter Ringspaltströmung und konstanter thermischer Randbedingung zweiter Art

An analysis has been carried out to determine the heat transfer characteristics for fully developed turbulent flow in concentric annular ducts with constant thermal boundary conditions of second kind. The energy equation has been solved analytically by separation of variables. Results for heat transfer in the thermal entrance region, in the thermal fully developed flow and thermal entrance-length results are presented over a wide range of Reynolds number (10⁴?Re?10⁶) and Prandtl-number (0?Pr?100) and compared to available empirical data.     

Adverse and favorable mixed convection heat transfer in a two-side heated square channel

Experimental heat transfer measurements and analysis for mixed convection in a vertical square channel are presented. Water flow directions are selected such that buoyancy assists or opposes the bulk flow pressure gradient. Unlike most previous experiments with symmetrically heated circular tubes, the present configuration uses an asymmetric heating condition (two sides heated and two sides insulated) and shows significant increase in the Nusselt number for both assisted and opposed flow conditions. Observed heat transfer coefficient distributions are different from the symmetrically heated channels; and this difference in heat transfer coefficient is attributed to the formation of buoyancy driven large-scale flow structures. In general, opposed flow shows higher heat transfer coefficients, and the Nusselt number ratio is observed to increase as Gr/Re or Gr/Re² ratios increase for both assisted and opposed flow conditions. A correlation based on the buoyancy parameter predicts the heat transfer pattern well in both assisted and opposed mixed convection. The range of Reynolds numbers discussed (Re=400–10,000) is of importance for direct numerical simulations and the details provided here can serve as the benchmark data required for complicated buoyancy affected turbulence simulations.     

Numerical analysis of a symmetrically heated vertical channel with obstruction

 In this paper, a numerical investigation of laminar natural convection flows in a vertical channel with obstructions is carried out. The main purpose was to analyze the effects of the locations of symmetric obstructions. The computations were performed in a two-dimensional domain and a symmetric uniform wall temperature has been taken as thermal boundary condition. The governing equations were solved using a control volume method and the SIMPLER algorithm for the velocity–pressure coupling was employed. The profiles of the local Nusselt number were given for three different locations of the obstructions. The variation of the average Nusselt number and inlet flow rate versus the modified Rayleigh number were investigated. The results demonstrated that the average Nusselt number decreases as the distance of the obstructions from the inlet increases. Received on 17 January 2000     

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid past nonlinear stretching surface

Sufficient conditions are found for the existence of similar solutions of the mixed convection flow of a Powell-Eyring fluid over a nonlinear stretching permeable sur- face in the presence of magnetic field. To achieve this, one parameter linear group trans- formation is applied. The governing momentum and energy equations are transformed to nonlinear ordinary differential equations by use of a similarity transformation. These equations are solved by the homotopy analysis method （HAM） to obtain the approximate solutions. The effects of magnetic field, suction, and buoyancy on the Powell-Eyring fluid flow with heat transfer inside the boundary layer are analyzed. The effects of the non- Newtonian fluid （Powell-Eyring model） parameters ε and δon the skin friction and local heat transfer coefficients for the cases of aiding and opposite flows are investigated and discussed. It is observed that the momentum boundary layer thickness increases and the thermal boundary layer thickness decreases with the increase in ε whereas the momentum boundary layer thickness decreases and thermal boundary layer thickness increases with the increase in δ for both the aiding and opposing mixed convection flows.     

Non-intrusive measurements of convective heat transfer in smooth- and rough-wall microchannels: laminar flow

The convective heat transfer behavior of laminar flow through a smooth- and two rough-wall microchannels is investigated by performing non-intrusive and spatially resolved measurements of fluid temperature via two-color fluorescent thermometry under constant heat flux conditions at three of the four microchannel walls. Pressure-drop measurements reveal that the apparent friction factors for all surfaces agree well with established macroscale predictions for laminar flow through rectangular ducts with the onset of transition at Re > Re_cr = 1,800 for smooth-wall flow and deviation from laminar behavior at progressively lower Re with increasing surface roughness. The local Nu for smooth-wall flow agrees well with macroscale predictions in both the thermally developing and developed regimes. With increasing roughness, while an enhancement in local Nu is noted for flow in the thermally developing regime, no measurable influence is noted upon attainment of a thermally developed state. These observations are supported by the examination of temperature profiles across the microchannel at various axial positions and Re, which suggest that the thermal boundary layer may be regenerated locally by roughness in the thermal entrance region of the flow resulting in an increased axial distance (compared to smooth-wall behavior) at which thermally developed flow is attained in the presence of roughness. Finally, estimates of the bulk Nu indicate enhancement in convective heat transfer over the smooth-wall case for laminar flow at higher Re while the smooth-wall bulk Nu data are found to agree well with macroscale predictions.     

Entrance flow in eccentric annular ducts

A control-volume-based solution of the complete set of Navier-Stokes equations for the laminar, three-dimensional developing flow in straight, eccentric, cylindrical annular ducts is described. Numerical results for velocity and pressure development, pressure defect and entrance lengths are presented for a wide range of duct parameters, i.e. relative eccentricity ? and radius ratio γ. The present results match very well with earlier numerical solutions for the limiting cases of developing flow in concentric ducts and fully developed flow in eccentric ducts. Comparison with earlier approximate results for developing flow in eccentric ducts indicates that the approximate model predicts the velocity and pressure development with an error of about 10%. However, the development length predicted by the approximate model is grossly in error. The pressure defect and development length in eccentric ducts are very high compared with their counterparts in concentric ducts. The pressure defect, development length and maximum velocity increase with the radius ratio for eccentric ducts, while the reverse is true for concentric ducts. Also, the apparent friction factor decreases as the eccentricity increases.     

Free Fluid Convection in a Closed Contour in a Heat-Conducting Half-Space

The steady-state convection of a fluid in a thin porous vertical ring located in a heat-conducting half-plane is considered. For this problem approximate equations are derived. For a circular ring an analytic solution is obtained. For an elliptic ring a numerical-analytic solution is found. The Nusselt number and the fluid flow rate as functions of the Rayleigh number, the aspect ratio, and the contour depth are investigated.Many studies have been devoted to fluid convection in a porous ring [1–3]. In [1] two-dimensional convection with an isothermal internal boundary was considered when a temperature stratification is given on the outer boundary. A feature of this problem is the fact that the ring is located inside an impermeable heat-conducting medium in which a thermal gradient directed vertically downward is specified at a large distance from the ring. In [2, 3] two-dimensional convection in an annular ring occupied by a porous medium was investigated. From the results obtained in these studies it follows that in the formulation considered the hydraulic approximation can be used with satisfactory accuracy. In the present study this question is discussed more concretely and the necessary estimates are found. The results obtained could be useful for investigating hydrothermal convection in the Earth's crust, which has important geophysical applications [4–6].     

Calculation of the flow within the channel of an axisymmetric entrance diffuser

One of the possible flow schemes within the channel of a supersonic axisymmetric entrance diffuser is considered. The channel is formed by the surfaces of the central body of the diffuser and its outer surface and has an annular section. Axisymmetric flow in the channel of such a diffuser is calculated in the presence of an oblique shock in front of the separation region formed at the corner of the central body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 164–168, January–February, 1981.We thank É. A. Ashratov for valuable comments and assistance in preparing the program and discussing the results of the computer calculations.     

Boundary effect on free convection flow in a porous medium at given heat transfer from a vertical surface

The system of momentum and energy conservation equations governing free convection flow near a vertical surface in a semi-infinite porous medium subject to the boundary conditions of the third kind is considered in the boundary layer approximation. Asymptotic expansions in the powers of the Darcy parameter Da are constructed. The dependence of flow parameters on the parameters Da and xGr is studied. Analytical solutions applicable throughout the entire flow region are constructed.     

The influence of a magnetic field on free convection in a finite vertical channel

An approximate analytical solution is presented for developing free convection flows of electrically conducting fluids between finite vertical channels which are subjected to a uniformly applied transverse magnetic field. Specifically, the basic approximation lies in the linearization of the governing boundary layer type of equations. It is demonstrated that the application of a transverse magnetic field reduces the induced flow rate in the channel and the heat transfer to the fluid.