Leakage effects in laminar duct flow: a numerical study

The influence of transverse leakage into a pressure-driven laminar flow in an infinitely long square duct is investigated. By a simple decomposition of the resulting three-dimensional pressure field, the leakage-induced secondary flow problem decouples from the primary flow problem. The numerical study reveals that two qualitatively different secondary flow patterns may occur, depending on the leakage flow rate. For a given streamwise pressure gradient it is observed that the axial mass flow rate may reduce by about 30 percent under certain leakage conditions, accompanied by a corresponding 50 percent increase in the Darcy-Weisbach friction factor.Nomenclature D duct height and width - D _h hydraulic diameter - D _i,j cell divergence - F dimensionless pressure force - h dimensionless slit height,H/D - H slit height - i,j indices - K streamwise kinematic pressure gradient - n summation index, time level - p dimensionless pressure - p pressure increment - P pressure - cross-sectional pressure variation - q dimensionless volumetric axial flow rate - Q leakage flow rate in m²/s - Re leakage Reynolds number,U ₀ D/ - Re _q primary flow Reynolds number - t time increment - x, y dimensionless cross-sectional coordinates - x, y cell widths - X, Y cross-sectional coordinates - z dimensionless axial coordinate - Z axial coordinate - u, v, w dimensionless velocity components - U, V, W velocity components - U ₀ leakage velocity,Q/H - V cross-sectional average velocity - W ₀ dummy velocity scale Greek letters density - kinematic viscosity - overrelaxation factor     

Leakage effects in laminar duct flow: a numerical study

The influence of transverse leakage into a pressure-driven laminar flow in an infinitely long square duct is investigated. By a simple decomposition of the resulting three-dimensional pressure field, the leakage-induced secondary flow problem decouples from the primary flow problem. The numerical study reveals that two qualitatively different secondary flow patterns may occur, depending on the leakage flow rate. For a given streamwise pressure gradient it is observed that the axial mass flow rate may reduce by about 30 percent under certain leakage conditions, accompanied by a corresponding 50 percent increase in the Darcy-Weisbach friction factor.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Experimental and numerical investigation of transition to turbulent flow and heat transfer inside a horizontal smooth rectangular duct under uniform bottom surface temperature

In this study, steady-state turbulent forced flow and heat transfer in a horizontal smooth rectangular duct both experimentally and numerically investigated. The study was carried out in the transition to turbulence region where Reynolds numbers range from 2,323 to 9,899. Flow is hydrodynamically and thermally developing (simultaneously developing flow) under uniform bottom surface temperature condition. A commercial CFD program Ansys Fluent 12.1 with different turbulent models was used to carry out the numerical study. Based on the present experimental data and three-dimensional numerical solutions, new engineering correlations were presented for the heat transfer and friction coefficients in the form of $ {\text{Nu}} = {\text{C}}_{2} {\text{Re}}^{{{\text{n}}_{ 1} }} $ and $ {\text{f}} = {\text{C}}_{3} {\text{Re}}^{{{\text{n}}_{3} }} $ , respectively. The results have shown that as the Reynolds number increases heat transfer coefficient increases but Darcy friction factor decreases. It is seen that there is a good agreement between the present experimental and numerical results. Examination of heat and mass transfer in rectangular cross-sectioned duct for different duct aspect ratio (α) was also carried out in this study. Average Nusselt number and average Darcy friction factor were expressed with graphics and correlations for different duct aspect ratios.     

Experimental and numerical study of turbulent flow and heat transfer inside hexagonal duct

Flow and heat transfer characteristics in transition and turbulent regions are studied experimentally and numerically in a horizontal smooth regular hexagonal duct under constant wall temperature boundary condition covering a range of Reynolds number from 2.3 × 10³ to 52 × 10³. Two types of k-omega (standard and shear stress transport (SST)) and three types of k-ε (standard, renormalization (RNG), and realizable) turbulence model are employed for transition and turbulent regions, respectively. Both average and fully developed Darcy friction factor and Nusselt number are presented as a function of Reynolds number. It is seen that k-omega SST and k-ε realizable turbulence models gave the best agreement with the experimental data in transition and turbulent regions, respectively. All the experimental results are correlated within an accuracy of ±13 % and ±7 % for Nusselt number and Darcy friction factor, respectively. Results obtained in this study are compared with circular duct results using hydraulic diameter.     

A numerical study on heat transfer of a ferrofluid flow in a square cavity under simultaneous gravitational and magnetic convection

Theoretical and Computational Fluid Dynamics - Thermomagnetic convection is based on the use of external magnetic fields to better control heat transfer fluxes in ferrofluids, finding important...     

Experimental studies on mixed convection heat transfer in laminar flow through a plain square duct

This paper reports the findings of experimental studies on combined free and forced convection through a plain square duct in laminar region. The test fluid flows through an inner square duct, hot water at high flow rate circulated through a annular channel formed between square duct and circular tube, in counter current fashion to attain a nearly uniform wall temperature conditions. The importance of mixed convection is judged by the value of the Richardson number (Ri). It was observed that at low Reynolds number, heat transfer was mainly governed by mixed convection. However at higher values of Reynolds number, heat transfer was significantly dominated by forced convection. It was found that Reynolds number higher than 1050 for water and 480 for ethylene glycol resulted in laminar forced convention heat transfer. The empirical correlation developed for Nusselt number in terms of Grashoff number and Graez number, was found to fit with experimental Nusselt number within ±11 and ±12?% for water and ethylene glycol respectively.     

Fully developed laminar flow and heat transfer in an arbitrarily shaped triangular duct

Hydrodynamic and thermal characteristics of the fully developed laminar flow and heat transfer in an arbitrarily shaped triangular duct are evaluated using a finite difference technique. The hydrodynamic information encompasses the friction factor, the length from the tube entrance necessary for complete hydrodynamic development, and the incremental pressure drop due to flow development in the entrance section. The Nusselt numbers in the case of an allover isothermal ductNu _T , as well as for a duct heated by an axial uniform heat flux while its transverse local periphery is at a constant temperatureNu _H , are presented. Comparison of the isosceles results with those from the work of Shah [1], Sparrow and Haji-Sheikh [2], and Schmidt and Newell [3] revealed a maximum difference of about –0.2% in theNu _Hi data, less than ±0.5% in theNu _T ,about +0.3% in the friction factor, a –0.47% in the incremental pressure drop, and around –1% in the developing entrance length. The deviations from the results of other authors become smaller as the triangular geometry approaches the equilateral.

---

Vollständig ausgebildete laminare Strömung und Wärmeübergang in einem willkürlich geformten dreieckigen Kanal

Zusammenfassung Mittels einer finiten Differenzenmethode wurde das hydrodynamische und thermische Verhalten einer vollständig ausgebildeten laminaren Strömung und der Wärmeübergang in einem willkürlich geformten dreieckigen Kanal untersucht. Hydrodynamische Erkenntnisse wurden über den Reibungskoeffizienten, die Einlauflänge bis zur vollständig ausgebildeten Strömung und den differentiellen Druckverlust im Einlaufgebiet gewonnen. Weiterhin wird sowohl die Nusselt-Zahl eines überall isothermen Kanals vorgestellt (Nu _T ), als auch die im Falle konstanter HeizflächenbelastungNu _Hi Ein Vergleich der eigenen Ergebnisse mit denen von Shah [1], Sparrow und Haji-Sheikh [2] sowie Schmidt und Newell [3] zeigt eine maximale Abweichung bei den Werten vonNu _Hi von ungefähr –0,2%, weniger als ±0,5% fürNu _T , ungefähr +0,3% beim Reibungskoeffizient, –0,47% beim differentiellen Druckverlust und etwa –1% bei der hydraulischen Einströmlänge. Die Abweichungen der Ergebnisse von anderen Autoren werden kleiner, wenn sich die Dreiecksgeometrie der Rechtecksgeometrie annähert.

---

Nomenclature D _h Equivalent Diameter=4A/p - f Friction Factor - K Incremental Pressure Drop - L _e Entrance Length - Nu _Hi Nusselt Number — Constant Axial Heat Flux, Isothermal Local Periphery - Nu _T Nusselt Number — Isothermal Duct - Re Reynolds Number     

Forced convection heat transfer on a heated bottom surface of cavity with different wall-height

Experiments to obtain the heat transfer characteristics of cavity, in which the downstream wall-heightD ₂ was changed from zero toD ₁ of upstream wall-height, have been performed. The vortex flow inside cavity was varied complicatedly depending on aspect-ratio of cavity and main flow velocity, and the flow pattern for cavity ofD ₂/D ₁=0.8 was altered entirely at theRe _H of about 1.5×10⁴. Three heat transfer regions ofNu _m versusRe _H were recognized for the cavity of large aspect-ratio. A close relation between those heat transfer behavior and approaching boundary layer flow was found. Heat transfer correlation was partially obtained for every cavities.     

Effect of cavity aspect ratio on flow and heat transfer characteristics in pipes: a numerical study

Using the standard k–ε turbulence model, a two-dimensional turbulent pipe flow was simulated with and without square cavities. Effect of cavity aspect ratio on flow and heat transfer characteristics was investigated. Uncertainty was approximated through experimental validation and grid independence. The simulation revealed circulation inside the cavities. Cavity boundaries were shown to contribute significantly toward turbulence production. Cavity presence was shown to enhance overall heat transfer through the wall, while increasing pressure drop significantly across the pipe. It was predicted that cavities with higher aspect ratio enhance heat transfer more while increasing pressure drop.     

A note on heat transfer in laminar flow through a gap

Summary The problem of heat transfer in laminar flow through a gap between two semi-infinite parallel plates at constant temperature was recently studied by Agrawal¹⁾. He solved this problem with the use of infinite Fourier sine series and derived an expression for the local temperature profile and the local Nusselt number as a function of the distance along the gap. A detailed solution for Péclèt number Pe=1 is given. Far enough from the entrance of the gap the local temperature profile of the fluid is almost independent of it's initial temperature. In this paper this limit temperature profile is expressed with the confluent hypergeometric function and the corresponding Nusselt number as a function of Pe is calculated.     

Three-dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts

The laminar convective flow and heat transfer in a duct with a trapezoidal cross-sectional area are studied numerically. The governing equations are solved numerically by a finite volume formulation in complex three-dimensional geometries using co-located variables and Cartesian velocity components. Details of the numerical method are presented. The accuracy of the method was also established by comparing the calculated results with the analytical and numerical results available in the open literature. The Nusselt numbers are obtained for the boundary condition of a uniform wall temperature whereas the friction factors are calculated for no-slip conditions at the walls. The asymptotic values of the Nusselt numbers, friction factors. incremental pressure drops, axial velocity and momentum rate and kinetic energy correction factors approach the available fully developed values. Various geometrical dimensions of the cross-section are considered.     

A numerical study of heat transfer during flow of melt in a cylindrical mould

A numerical analysis of transient heat transfer during the flow of a melt in a cylindrical mould is presented. The analysis includes thermal resistance at the melt-mould interface, and axial conduction inside both melt and mould. Energy equations are formulated in a domain that expands continuously due to the advance of the melt inside the empty mould, and solved by the finite difference method using a time-stepping procedure. Calculations are compared to existing analytic results. It is found that axial conduction in the melt can significantly influence the rate of heat loss from the flowing melt, and that analytic approximations, which neglect axial conduction, may give erroneous predictions for the rate of heat loss.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Velocity distributions in Taylor vortex flow with imposed laminar axial flow and isothermal surface heat transfer

Fully-developed flow in a concentric annulus formed by a stationary outer cylinder, which may be heated isothermally, and a rotatable inner cylinder has been studied experimentally by means of hot-wire anemometry techniques. Velocity profiles for the axial, tangential, and radial directions of flow have been obtained for adiabatic conditions and for wide and narrow annular gaps.It has been shown that the onset of Taylor vortex flow has a pronounced effect on the velocity profiles for all three directions. However, while the profiles for the axial and tangential directions are explicable, those for the radial direction are not so, at present. Also, it was found that heat transfer through the outer annular surface had a greater effect on the radial velocity profile than on the axial or tangential, but in the narrow gap case only.     

A numerical study of bifurcation in laminar flow in curved ducts

The bifurcation phenomenon whereby multiple-vortex secondary flow occurs in place of the normal two-vortex flow in laminar flow in curved ducts has previously been studied numerically by several researchers. However, the various results have been conflicting on many points. The present paper describes a set of numerical experiments conducted to study the effect of numerical accuracy on the solution. The results show that the transition from two- to four-vortex structure depends strongly on the differencing scheme and to a lesser extent on the grid size. The study also shows that as the Reynolds number of the flow increases, a two-vortex structure is re-established via a path which involves strongly asymmetric secondary flow patterns. These results are in agreement, at least qualitatively, with recent experimental theoretical and numerical results.     

Numerical study of the laminar thermocapillary flow in a square cavity

The hydrodynamic and heat-transfer processes in the problem of a laminar thermocapillary flow of a viscous incompressible fluid in a square cavity with isothermal vertical and isentropic horizontal surfaces are investigated numerically under the assumption that the gravity is absent, the free surface is flat, and the surface tension depends linearly on the temperature. Calculations were performed by a compact-difference method on irregular grids with a fifth-order accuracy for four Prandtl numbers (Pr=1, 16, 200, and 3000) as the Marangoni (Ma) number varies from 10² to 10⁴. The maximum local heat transfer versus theMa number is obtained. It is shown that, for thePr values considered, the maxima of the distribution of the horizontal velocity component on the surface is displaced to the cold boundary according to a law inversely proportional to theMa number. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 81–89, July–August, 1999.