Flow over and forced convection heat transfer in Newtonian fluids from a semi-circular cylinder

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in unconfined flowing Newtonian fluids have been investigated numerically. The governing equations, namely, continuity, Navier–Stokes and energy, have been solved in the steady flow regime over wide ranges of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr ? 100). Prior to the investigation of drag and heat transfer phenomena, the critical values of the Reynolds number for wake formation (0.55 < Re^c < 0.6) and for the onset of vortex shedding (39.5 < Re_c < 40) have been identified. The corresponding values of the lift coefficient, drag coefficient, and Strouhal number are also presented. After establishing the limit of the steady flow regime, the influence of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr = 0.72, 1, 10, 50 and 100) on the global flow and heat transfer characteristics have been elucidated. Detailed kinematics of the flow is investigated in terms of the streamline and vorticity profiles and the variation of pressure coefficient in the vicinity of the cylinder. The functional dependence of the individual and total drag coefficients on the Reynolds number is explored. The Nusselt number shows an additional dependence on the Prandtl number. In addition, the isotherm profiles, local Nusselt number (Nu_L) and average Nusselt number (Nu) are also presented to analyze the heat transfer characteristic of a semi-circular cylinder in Newtonian media.     

Heat transfer and pressure drop in furrowed channels with transverse and skewed sinusoidal wavy walls

This comparative study examines the detailed Nusselt number (Nu) distributions, pressure drop coefficients (f) and thermal performance factors (η) for two furrowed rectangular channels with transverse and skewed sinusoidal wavy walls. Detailed heat transfer measurements over these transverse and skewed sinusoidal wavy walls at the Reynolds numbers (Re) = 1000, 1500, 2000, 5000, 10,000, 15,000, 20,000, 25,000 and 30,000 are performed using the steady-state infrared thermo-graphic method. Impacts of Re on Nu and f for two tested furrowed channels with transverse and skewed waviness are individually examined. In addition to the macroscopic mixing between the near-wall recirculations and core flows due to the shear layer instabilities in each wavy channel, the secondary flows tripped by the skewed wall-waves further elevate heat transfer performances and distinguish their Nu distributions from those over the transverse wavy wall. The area-averaged Nusselt numbers ($\overline{\mathit{Nu}}$) for two tested furrowed channels with transverse and skewed waviness with 5000 < Re < 30000 fall, respectively, in the ranges of 3.45–3.71 and 3.98–4.2 times of the Dittus–Boelter levels. A set of $\overline{\mathit{Nu}}$ and f correlations for each tested furrowed channel is individually derived using Re as the controlling parameter. By way of comparing the thermal performance factors (η) with a selection of rib-roughened channels, the η factors for the present skewed wavy channel are compatible with those in the channel roughened by the compound V-ribs and deepened scales due to the relative low pressure drop penalties with the equivalent heat transfer augmentations to those offered by V-ribs.     

Flow over and forced convection heat transfer around a semi-circular cylinder at incidence

Wake dynamics and forced convective heat transfer characteristics past a semi-circular cylinder at incidence have been investigated numerically. Utilizing air as an operating fluid computations are carried out for wide ranges of the Reynolds number (80 ? Re ? 180) and angle of incidences (0 ? α ? 180°). Angle of incidence reveals three flow separation zones. Structure properties of shear layer and vortex motions on each flow separation zones are analyzed critically. Functional dependence of drag (C_D), lift (C_L), and moment (C_M) coefficients on the angle of incidence is explored and analyzed in detail. Increase in angle of incidence increases streamline curvature. A structural similarity is observed between the contours of vorticity and the corresponding isotherms. Strouhal number shows a decreasing trend up to certain values of α and thereafter it increases marginally. A new correlation of Strouhal number as a function of Re and α has been established for the present range of Reynolds numbers. At the singularity points a sudden jump in local Nusselt number distribution is observed. The trend of variation of average Nusselt number with α is similar to that of Strouhal number variation. The average Nusselt number is found to vary as ${\mathit{Re}}^{0.529}(1+\alpha {)}^{-0.0476}$.     

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in power-law fluids in the steady flow regime

The continuity, momentum and energy equations describing the flow and heat transfer of power-law fluids over a semi-circular cylinder have been solved numerically in the two-dimensional steady flow regime. The influence of the Reynolds number (Re), Prandtl number (Pr) and power-law index (n) on the local and global flow and heat characteristics have been studied over wide ranges of conditions as follows: 0.01 ? Re ? 30, 1 ? Pr ? 100 and 0.2 ? n ? 1.8. The variation of drag coefficient and Nusselt number with the Reynolds number, Prandtl number and power-law index is shown over the aforementioned ranges of conditions. In addition, streamline and isotherm profiles along with the recirculation length and distribution of pressure coefficient and Nusselt number over the surface of the semi-circular cylinder are also presented to gain further insights into the nature of the underlying kinematics. The wake size (recirculation length) shows almost linear dependence on the Reynolds number (Re ? 1) for all values of power-law index studied herein. The drag values show the classical inverse variation with the Reynolds number, especially for shear-thinning fluids at low Reynolds numbers. The point of maximum pressure coefficient is found slightly displaced from the front stagnation point for highly shear-thinning fluids, whereas for shear-thickening and Newtonian fluids, it coincides with the front stagnation point. For fixed values of the Prandtl number and Reynolds number, the rate of heat transfer decreases with the gradual increase in power-law index; this effect is particularly striking at high Prandtl numbers due to the thinning of the thermal boundary layer. Conversely, as expected, shear-thinning behavior facilitates heat transfer and shear-thickening impedes it. The effect of power-law index on both momentum and heat-transfer characteristics is seen to be appreciable at low Reynolds numbers and it gradually diminishes with the increasing Reynolds number.     

Two-dimensional unsteady forced convection heat transfer in power-law fluids from a cylinder

Forced convection heat transfer characteristics of a cylinder (maintained at a constant temperature) immersed in a streaming power-law fluids have been studied numerically in the two-dimensional (2-D), unsteady flow regime. The governing equations, namely, continuity, momentum and thermal energy, have been solved using a finite volume method based solver (FLUENT 6.3) over wide ranges of conditions (power law index, 0.4 ? n ? 1.8; Reynolds number, 40 ? Re ? 140; Prandtl number, 1 ? Pr ? 100). In particular, extensive numerical results elucidating the influence of Reynolds number, Prandtl number and power-law index on the isotherm patterns, local and average Nusselt numbers and their evolution with time are discussed in detail. Over the ranges of conditions considered herein, the nature of flow is fully periodic in time. The heat transfer characteristics are seen to be influenced in an intricate manner by the value of the Reynolds number (Re), Prandtl number (Pr) and the power-law index (n). Depending upon the value of the power-law index (n), though the flow transits from being steady to unsteady somewhere in the range ～33 < Re < 50, the fully periodic behavior is seen only beyond the critical value of the Reynolds number (Re). As expected, the average Nusselt number increases with an increase in the values of Reynolds and/or Prandtl numbers, irrespective of the value of the flow behavior index. A strong influence of the power-law index on both local and time-averaged Nusselt numbers was observed. Broadly, all else being equal, shear-thinning behavior (n < 1) promotes heat transfer whereas shear-thickening behavior (n > 1) impedes it. Furthermore, this effect is much more pronounced in shear-thinning fluids than that in shear-thickening fluids.     

Laminar natural convection in a square cavity: Low Prandtl numbers and large density differences

Steady natural convection at low Prandtl numbers caused by large density differences in a square cavity heated through the side walls is investigated numerically and theoretically. An appropriate dimensionless parameter characterizing the density differences of the working fluid is identified by the Gay-Lussac number. The Boussinesq assumption is achieved when the Gay-Lussac number tends to zero. The Nusselt number is derived for the ranges in Rayleigh number 10 ? Ra ? 10⁸, in Prandtl number 0.0071 ? Pr ? 7.1 and in Gay-Lussac number 0 ? Ga < 2. The effects of the Rayleigh, Prandtl and Gay-Lussac numbers on the Nusselt number are discussed on physical grounds by means of a scale analysis. Finally, based on physical arguments, a heat transfer correlation is proposed, valid for all Prandtl and Gay-Lussac number ranges addressed.     

Onset of entrainment in transitional round fountains

It is of fundamental interest to understand the behavior of transitional fountains with intermediate Froude and Reynolds numbers, together with the associated entrainment and turbulence. In this work, the transient behavior of axisymmetric fountains with 1 ? Fr ? 8 and 200 ? Re ? 800 is studied by direct numerical simulation. It is found that at Re ? 200, there is little entrainment present at the upflow–downflow interface and at the downflow–ambient interface, even for a value of Fr as high as 8; however, at Re > 200, entrainment is present at these interfaces and the extent increases with Re, which clearly demonstrates that entrainment is strongly dependent on Re whereas the contribution from the Fr effect is relatively much smaller. The DNS results also show that z_m, which is the maximum fountain penetration height, fluctuates, even when the flow reaches full development, due to the entrainment at the upflow–downflow and the downflow–ambient interfaces, and the averaged z_m scales with ${\mathit{Fr}}^{\frac{3}{2}}{\mathit{Re}}^{\frac{1}{4}}$ for 1 < Fr ? 8 and 100 ? Re ? 800.     

Confined flow and heat transfer across a triangular cylinder in a channel

In this paper, fluid flow and heat transfer across a long equilateral triangular cylinder placed in a horizontal channel is studied for Reynolds number range 1–80 (in the steps of 5) and Prandtl number of 0.71 for a fixed blockage ratio of 0.25. The governing Navier-Stokes and energy equations along with appropriate boundary conditions are solved by using a commercial CFD solver FLUENT (6.3). The computational grid is created in a commercial grid generator GAMBIT. The flow and temperature fields are presented by stream-line and isotherm profiles, respectively. The wake/recirculation length, mean drag coefficient and average Nusselt number, etc. are calculated for the above range of conditions studied here. The critical value of the Reynolds number (i.e., transition to transient) is found to lie between Re = 58 and Re = 59. The average Nusselt number and the wake length increase with increasing value of the Reynolds number; however, the mean drag coefficient decreases with increasing value of the Reynolds number. Finally, simple correlations for wake length, mean drag coefficient and average Nusselt number are obtained for the range of conditions studied here.     

Analysis of entropy generation on mixed convection in square enclosures for various horizontal or vertical moving wall(s)

Entropy generation during the mixed convection process have been studied in a square enclosure for various moving horizontal (cases 1a–1d) or vertical wall(s) (cases 2a–2c) where the bottom wall of the cavity is isothermally hot, side walls are cold, and the top wall is adiabatic. Simulations have been performed for Prandtl number Pr = 0.026 and 7.2, Reynolds number Re = 10 − 100, and Grashof number Gr = 10³ − 10⁵. Results show that, in the case of the horizontally moving wall(s) (cases 1a–1d), the overall heat transfer rate $\left(\overline{N{u}_b}\right)$ and total entropy generation (S_total) are identical for cases 1a–1d and the cup-mixing temperature (θ_cup) is high for case 1b at Pr = 0.026, Re = 100, and Gr = 10⁵. Similarly, in the case of the vertically moving wall(s) (cases 2a–2c), $\overline{N{u}_b}$ and S_total are identical for cases 2a–2c with the maximum θ_cup occurring for the case 2a. At Pr = 7.2, Gr = 10⁵, and Re = 10, case 1a and case 1c are preferable for horizontally moving wall(s) and either of case 2a–2c is preferable for vertically moving wall(s). At Pr = 7.2, Gr = 10⁵, and Re = 100, case 1d may be preferable for the horizontally moving wall(s) and case 2a may be preferable for the vertically moving wall(s).     

Coriolis and rotating buoyancy effect on detailed heat transfer distributions in a two-pass square channel roughened by 45° ribs at high rotation numbers

Heat transfer measurements from a rotating two-pass square channel with two opposite leading and trailing walls roughened by 45° parallel ribs arranged in the staggered manner are performed to examine the effects of Reynolds (Re), rotation (Ro) and buoyancy (Bu) numbers on local and area-averaged Nusselt numbers (Nu and $\overline{\mathit{Nu}}$). Full-field Nu distributions over the two rib-roughened leading and trailing walls are measured at the conditions of 4000 ? Re ? 16,000, 0 ? Ro ? 0.8 and 0.0015 ? Bu ? 0.93 (0.05 ? Δρ/ρ ? 0.1) using the infrared thermography which allows for the detailed examination of the Coriolis and rotating buoyancy effects on Nu distributions over the rotating ribbed surface. Selected heat transfer data in term of Nu ratio between rotating and stationary levels illustrates the influences of rotation on local and area-averaged heat transfer performances. Area-averaged $\overline{\mathit{Nu}}$ for the turn region and the inlet and outlet ribbed legs of the rotating two-pass channel are parametrically analyzed to devise a set of empirical heat transfer correlations that permits the evaluation of the interdependent and individual effects of Re, Ro and Bu on $\overline{\mathit{Nu}}$.     

Mixed convection flow and heat transfer across a square cylinder under the influence of aiding buoyancy at low Reynolds numbers

In this paper, mixed convection flow and heat transfer around a long cylinder of square cross-section under the influence of aiding buoyancy are investigated in the vertical unconfined configuration (Reynolds number, Re = 1–40 and Richardson number, Ri = 0–1). The semi-explicit finite volume method implemented on the collocated grid arrangement is used to solve the governing equations along with the appropriate boundary conditions. The onset of flow separation occurs between Re = 1–2, between Re = 2–3 and between Re = 3–4 for Ri = 0, 0.5 and 1, respectively. The flow is found to be steady for the range of conditions studied here. The friction, pressure and total drag coefficients are found to increase with Richardson number, i.e., as the influence of aiding buoyancy increases drag coefficients increase at the constant value of the Reynolds number. The temperature field around the obstacle is presented by isotherm contours at the Prandtl number of 0.7 (air). The local and average Nusselt numbers are calculated to give a detailed study of heat transfer over each surface of the square cylinder and an overall heat transfer rate and it is found that heat transfer increases with increase in Reynolds number and/or Richardson number. The simple expressions for the wake length and average cylinder Nusselt number are obtained for the range of conditions covered in this work.     

Effect of power-law fluid behavior on momentum and heat transfer characteristics of an inclined square cylinder in steady flow regime

The governing equations describing the momentum and heat transfer phenomena of power-law non-Newtonian fluids over a heated square cylinder at 45° of incidence in the two-dimensional (2-D) steady flow regime are solved numerically. Extensive results on the detailed structure of the flow and temperature fields as well as on the gross engineering parameters are presented over the following ranges of conditions: 0.2 ? n ? 1; 0.1 ? Re ? 40 and 0.7 ? Pr ? 100. At low Reynolds numbers, the flow remains attached to the surface of the cylinder. This seems to occur for all values of power-law index, at least up to about Re = 1. On the other hand, twin standing vortices were seen to form at Re = 10 for all values of power-law index considered herein. The influence of the Reynolds number and power-law index is delineated on the detailed structure of the flow field (streamlines), wake characteristics and surface pressure distribution as well as on the value of drag coefficients. Similarly, the effect of Prandtl number is studied on forced convective heat transfer for the two commonly encountered boundary conditions, namely, constant temperature or constant heat flux prescribed on the surface of the cylinder. Using the computed numerical results, simple heat transfer correlations are obtained in terms of the Nusselt number as a function of the pertinent governing parameters thereby enabling the prediction of the rate of heat transfer between the fluid and the immersed cylinder. In addition, variation of the local Nusselt number on the surface of the inclined of square cylinder and representative isotherm plots are also presented to elucidate the effect of Reynolds number, Prandtl number and power-law index on the heat transfer phenomenon.     

Forced convection heat transfer from an elliptical cylinder to power-law fluids

Forced convection heat transfer to incompressible power-law fluids from a heated elliptical cylinder in the steady, laminar cross-flow regime has been studied numerically. In particular, the effects of the power-law index (0.2 ? n ? 1.8), Reynolds number (0.01 ? Re ? 40), Prandtl number (1 ? Pr ? 100) and the aspect ratio of the elliptic cylinder (0.2 ? E ? 5) on the average Nusselt number (Nu) have been studied. The average Nusselt number for an elliptic cylinder shows a dependence on the Reynolds and Prandtl numbers and power-law index, which is qualitatively similar to that for a circular cylinder. Thus, heat transfer is facilitated by the shear-thinning tendency of the fluid, while it is generally impeded in shear-thickening fluids. The average Nusselt number values have also been interpreted in terms of the usual Colburn heat transfer factor (j). The functional dependence of the average Nusselt number on the dimensionless parameters (Re, n, Pr, E) has been presented by empirically fitting the numerical results for their easy use in process design calculations.     

Prandtl number dependence of laminar natural convection heat transfer in a horizontal cylindrical enclosure with an inner coaxial triangular cylinder

A numerical study of laminar convection heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure is performed to investigate the Prandtl number effect on flow and heat transfer characteristics. The Prandtl number over several orders of magnitude (10^?2 < Pr < 10³) as well as different aspect ratios (AR = 1.2 and 2.0) and different Rayleigh numbers (Ra = 10³, 10⁴, 10⁵, and 10⁶) are considered. The finite volume approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The computed flow patterns and temperature fields are shown by means of streamlines and isotherms, respectively, and the local and average heat transfer coefficients are also presented. It is found that the flow and heat transfer characteristics for a low Prandtl number fluid (Pr = 0.03) are unique and they are almost independent of Prandtl number when Pr ? 0.7. The entire spectrum of Prandtl number investigated can be divided into three sections based on the variations of average heat transfer coefficients. In each section, correlating equations of the average Nusselt number to the Rayleigh number are proposed with the maximum deviation less than 3%.