Natural convection flow from an isothermal horizontal circular cylinder with temperature dependent viscosity

Free convection flow from an isothermal horizontal circular cylinder immersed in a fluid with viscosity proportional to an inverse linear function of temperature is studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity equations which are solved numerically by a very efficient implicit finite difference method together with Keller box scheme. Numerical results are presented by velocity and viscosity profiles of the fluid as well as heat transfer characteristics, namely the local heat transfer rate and the local skin-friction coefficients for a wide range of viscosity parameter (= 0.0, 0.5, 1.0, 2.0, 3.0,4.0) and the Prandtl number Pr (= 1.0, 7.0, 10.0, 15.0, 20.0, 30.0).     

Transient free convection from an isothermal horizontal circular cylinder

A numerical solution of the unsteady free convection on an isothermal horizontal circular cylinder whose temperature is suddenly increased is presented at large Grashof number. Solutions for simultaneous development of the momentum and thermal boundary-layers are obtained by using the local non-similarity solutions method, together with the difference-differential procedure. The transient approach of velocity and thermal fields to the ultimate steady state and the variations in time of wall shear and heat transfer coefficients were found. The results at the lower stagnation point of the cylinder for Prandtl number Pr=1.0 are seen to improve the previous study of Hermann [1].     

Numerical simulation of flow around a square cylinder in uniform-shear flow

This paper presents results obtained from a numerical simulation of a two-dimensional (2-D) incompressible linear shear flow over a square cylinder. Numerical simulations are performed, using the lattice Boltzmann method, in the ranges of 50⩽Re⩽200 and 0⩽K⩽0.5, where Re and K are the Reynolds number and the shear rate, respectively. The effect of the shear rate on the frequency of vortex shedding from the cylinder, and the lift and drag forces exerted on the cylinder are quantified together with the flow patterns around the cylinder. The present results show that vortex structure behind the cylinder is strongly dependant on both the shear rate and Reynolds number. When Re=50, a small K can disturb the steady state and cause an alternative vortex shedding with uneven intensity. In contrast, a large value of K will suppress the vortex shedding from the cylinder. When Re>50, the differences in the strength and size of vortices shed from the upper and lower sides of the cylinder become more pronounced as K increases. Vortex shedding disappears when K is larger than a critical value, which depends on Re. The flow patterns around the cylinder for different Re tend towards self-similarity with increasing K. The lift and drag forces exerted on the cylinder, in general, decrease with increasing K. Unlike a shear flow past a circular cylinder, the vortex shedding frequency past a square cylinder decreases with increasing the shear rate. A significant reduction of the drag force occurs in the range 0.15<K<0.3.     

Numerical investigation of forced convection heat transfer in unsteady flow past a row of square cylinders

This paper presents the unsteady laminar forced convection heat transfer from a row of five isothermal square cylinders placed in a side-by-side arrangement at a Reynolds number of 150. The numerical simulations are performed using a finite volume code based on the PISO algorithm in a collocated grid system. Special attention is paid to investigate the effect of the spacing between the cylinders on the overall transport processes for the separation ratios (spacing to size ratio) between 0.2 and 10. No significant interaction between the wakes is observed for spacing greater than four times the diameter at this Reynolds number. However, at smaller spacing, the wakes interact in a complicated manner resulting different thermo-hydrodynamic regimes. The vortex structures and isotherm patterns obtained are systematically presented and discussed for different separation ratios. In addition, the mean and instantaneous drag and lift coefficients, mean and local Nusselt number and Strouhal number are determined and discussed for various separation ratios. A new correlation is derived for mean Nusselt number as a function of separation ratio for such flows.     

Non-Newtonian effect on natural convection flow over cylinder of elliptic cross section

The non-Newtonian effect in the boundary layer flow over a horizontal elliptical cylinder is investigated numerically. A modified power-law viscosity model is used to correlate the non-Newtonian characteristics of the fluid flow. For natural convectionflows, the surface of the cylinder is maintained by the uniform surface temperature(UST)or the uniform heat flux(UHF) condition. The governing equations corresponding to theflow are first transformed into a dimensionless non-similar form using suit...     

Optical investigation of the heat transfer from a rotating cylinder in a cross flow

The heat transfer from a rotating cylinder in an air-cross flow was investigated by purely optical measuring techniques. Flow velocities were measured by a two-dimensional LDV both in the vicinity of the cylinder and in the boundary layer. A new optical device based on light-deflection in a temperature field was developed to examine local temperature gradients in the boundary layer of the rotating cylinder. Finally, a Michelson-interferometer was installed to produce real-time pictures of isothermal lines around the heated cylinder. The impact of rotation on flow patterns, boundary layer behaviour and heat transfer could be clearly identified. It appears that the velocity-ratio acts like an independent parameter, in that flow patterns correspond to this dimensionless number. Furthermore, it seems that rotation dominates over cross flow, both fluid-dynamically and thermally above = 2.This work was carried out at the University of the Federal Armed Forces in Munich/Germany.     

Unsteady mixed convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers

A two-dimensional numerical study is carried out to understand the influence of cross buoyancy on the vortex shedding processes behind two equal isothermal square cylinders placed in a tandem arrangement at low Reynolds numbers. The spacing between the cylinders is fixed with five widths of the cylinder dimension. The flow is considered in an unbounded medium, however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a finite volume method based on the PISO algorithm in a collocated grid system. The range of Reynolds number is chosen to be 50–150. The flow is unsteady laminar and two-dimensional in this Reynolds number range. The mixed convection effect is studied for Richardson number range of 0–2 and the Prandtl number is chosen constant as 0.71. The effect of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as overall drag and lift coefficients, local and surface average Nusselt numbers and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers.     

Numerical investigation of confined wakes behind a square cylinder in a channel

The structure of confined wakes behind a square cylinder in a channel is investigated via the numerical solution of the unsteady Navier–Stokes equations. Vortex shedding behind the cylinder induces periodicity in the flow field. Details of the phenomenon are simulated through numerical flow visualization. The unsteady periodic wake can be characterized by the Strouhal number, which varies with the Reynolds number and the blockage ratio of the channel. The periodicity of the flow is, however, damped in the downstream region of a long duct. This damping may be attributed to the influence of side walls on the flow structure.     

Efficient computation of natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder

In this work, the natural convection in a concentric annulus between a cold outer square cylinder and a heated inner circular cylinder is simulated using the differential quadrature (DQ) method. The vorticity‐stream function formulation is used as the governing equation, and the coordinate transformation technique is introduced in the DQ computation. It is shown in this paper that the outer square boundary can be approximated by a super elliptic function. As a result, the coordinate transformation from the physical domain to the computational domain is set up by an analytical expression, and all the geometrical parameters can be computed exactly. Numerical results for Rayleigh numbers range from 104 to 106 and aspect ratios between 1.67 and 5.0 are presented, which are in a good agreement with available data in the literature. It is found that both the aspect ratio and the Rayleigh number are critical to the patterns of flow and thermal fields. The present study suggests that a critical aspect ratio may exist at high Rayleigh number to distinguish the flow and thermal patterns. Copyright © 2002 John Wiley & Sons, Ltd.     

Forced‐convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers

This paper presents the results of a numerical study on the flow characteristics and heat transfer over two equal square cylinders in a tandem arrangement. Spacing between the cylinders is five widths of the cylinder and the Reynolds number ranges from 1 to 200, Pr=0.71. Both steady and unsteady incompressible laminar flow in the 2D regime are performed with a finite volume code based on the SIMPLEC algorithm and non‐staggered grid. A study of the effects of spatial resolution and blockage on the results is provided. In this study, the instantaneous and mean streamlines, vorticity and isotherm patterns for different Reynolds numbers are presented and discussed. In addition, the global quantities such as pressure and viscous drag coefficients, RMS lift and drag coefficients, recirculation length, Strouhal number and Nusselt number are determined and discussed for various Reynolds numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical investigation of flow over a two-dimensional square cylinder with a synthetic jet generated by a bi-frequency signal

Applied Mathematics and Mechanics - The flow around a square cylinder with a synthetic jet positioned at the rear surface is numerically investigated with the unsteady Reynolds-averaged...     

Numerical investigation of separated flow in front of a spiked cylinder

The problem of axisymmetric flow of a perfect heat conducting gas over a cylinder from the front of which a spike projects is solved at subsonic and supersonic velocities of the oncoming flow in the model of a viscous fluid. The formation and development of the flow separation due to the spike ahead of the front surface of the cylinder is investigated. The change in the gas-dynamic and geometrical parameters of the separated flow is studied as a function of the Mach and Reynolds numbers of the oncoming flow. The obtained results are compared with the available experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 162–166, October–December, 1981.     

Finite element solutions for laminar flow and combined convection around a square prism

The finite element solutions of the full Navier-Stokes and energy equations for steady laminar flow and combined convection around square prisms with attack angles of 0° and 45° are obtained for a gas having Pr=0.7. The variations of surface shear stress, local pressure and Nusselt number are obtained over the entire prism surface including the zone beyond the point of the separation. The predicted values of drag coefficients, the location of separation, average Nusselt number and the plots of velocity flow fields and isotherms are also presented. The trend of the present numerical results seems reasonable.

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Finite-Elemente-Verfahren für laminare Strömung und kombinierte Naturkonvektion um ein quadratisches Prisma

Zusammenfassung Es wird über Lösungen der Navier-Stokesund der Energiegleichungen mit Hilfe der Finite-Elemente-Methode für stationäre laminare Strömung, kombiniert mit Naturkonvektion, um ein quadratisches Prisma berichtet, wobei als Anströmwinkel 0° und 45° gewählt wurden und Gasströmung mitPr=0,7 angenommen wurde. Die Rechnung ergibt den Verlauf der Wandschubspannungen, des örtlichen Druckes und der Nusselt-Zahl über die gesamte Oberfläche des Prismas, einschließlich des Bereiches hinter dem Ablösepunkt. Weiterhin werden in dem Aufsatz Angaben gemacht über die Widerstandskoeffizienten, die Lage des Ablösepunktes, der mittleren Nusselt-Zahl sowie der Geschwindigkeits- und Temperaturfelder. Die numerischen Ergebnisse erscheinen im Trend vernünftig zu sein.

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Nomenclature a side length of square - C _f friction drag coefficient - C _p pressure drag coefficient - C _D total drag coefficient - F _f total friction drag force - F _P total pressure drag force - Gr Groshoff number,g (T _w -T )a ³/v ² - g gravitational acceleration - h local heat transfer coefficient - K thermal conductivity - L dimensionless location of surface from the front stagna tion point,L ^*/a - L ^* dimension location of prism surface - Lc location of separation - N _j shape function - Nu, local and average Nusselt numbers - M _l shape function - P dimensionless pressure,p ^*/u ² - P ^* pressure - p _x ^* x-componentP ^* - Pe Peclet number,Re Pr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,a u /v - s direction along the sides of prism - u dimensionlessX-direction component of velocity,u ^*/u - u ^* X-direction component of velocity - u free stream velocity - dimensionlessY-direction component of velocity,v*/u - ^* Y-direction component of velocity - X X-direction axis - x dimensionlessX-direction coordinate,x ^*/a - x ^* X- direction coordinate - Y Y-direction axis - y ^* dimensionless 7-direction coordinate,y ^*/a - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - attack angle - dynamic viscosity - kinematic viscosity,/ - density of fluid - _w dimensionless surface shear stress, _w ^* /u ² - _w ^* surface shear stress - _wx ^* x-component of _w ^/* - dimensionless temperature,      

A numerical and experimental study of natural convective heat transfer from an inclined isothermal square cylinder with an exposed top surface

Natural convective heat transfer from an isothermal inclined cylinder with a square cross-section which have an exposed top surface and is, in general, inclined at an angle to the vertical has been numerically and experimentally studied. The cylinder is mounted on a flat adiabatic base plate, the cylinder being normal to the base plate. The numerical solution has been obtained by solving the dimensionless governing equations subject to the boundary conditions using the commercial cfd solver, FLUENT. The flow has been assumed to be symmetrical about the vertical center-plane through the cylinder. Results have only been obtained for Prandtl number of 0.7. Values of inclination angle between 0° and 180° and a wide range of Rayleigh number and the dimensionless cylinder width, W = w/h, have been considered. The effects of Dimensionless widths, Rayleigh numbers, and inclination angles on the mean Nusselt number for the entire cylinder and for the mean Nusselt numbers for the various surfaces that make up the cylinder have been examined. Empirical equations for the heat transfer rates from the entire cylinder have been derived.     

Numerical investigation of transient natural convection heat transfer of freezing water in a square cavity

In this study, a transient heat transfer process of freezing water inside a two-dimensional square cavity has been investigated numerically. Water was used as a phase-change medium, and the numerical model has been created with control volume approach by using C++ programming language. To be able to accelerate the numerical calculations, CUT (Consistent-Update-Technique) algorithm has been implemented in the numerical code. Span-wise variations of the vertical component of the velocity have been represented in comparison with the experimental measurements from the literature at various vertical positions to examine the accuracy of the numerical scheme. The influence of natural convection has been considered by comparing the conduction and convection dominated solidification under same boundary conditions. Comparative results have been obtained regarding time-wise variations of the cold wall temperature and the dimensionless effectiveness. Moreover, the streamlines and isotherms have been represented to understand the differences between the conduction and convection driven phase change processes.Results indicate that natural convection becomes remarkable and has different forms at the initial periods of the phase change process. Increasing the effect of natural convection in the cavity increases the cooling rate of water. Near the density inversion temperature of water (4°C), temperature variations fluctuate and counter currents observed in the domain.     

Numerical prediction of combined laminar convection in isothermal vertical tubes via the method of lines

Simultaneous hydrodynamic and thermal development of a laminar flow due to superimposed natural and forced convection in an isothermal vertical pipe is investigated numerically. The distinct feature of the computational procedure is that it uses the method of lines, wherein the Navier-Stokes equations accounting for buoyancy and constant properties are reduced to a system of first order ordinary differential equations. The latter is readily solved by a standard fourth-order Runge-Kutta technique. The paper includes a detailed discussion of the proposed methodology, and in addition to this, it provides the distributions of velocities, pressure and temperature. All calculations are based on a coarse grid with ten lines uniformly distributed in the cross-stream direction of the pipe when the velocity profile at the entrance is assumed parabolic. Computed results accounting for both upflow and downflow situations are in good agreement with other more elaborate numerical investigations and also with available experimental data employing air and water as working fluids.Die simultane hydrodynamische und thermische Entwicklung einer laminaren Strömung infolge der Uberlagerung von freier und erzwungener Konvektion in einem isothermen senkrechten Rohr wird numerisch untersucht. Die besondere Eigenschaft der rechnerischen Prozedur ist, daß diese die Method of lines benutzt, worin die Navier-Stokes-Gleichungen für Antrieb bei konstanten Stoffwerten zu einem System einfacher Differentialgleichungen erster Ordnung reduziert werden. Das letztere ist ohne weiteres mit der Runge-Kutta-Technik 4. Ordnung zu lösen. Die Untersuchung beinhaltet eine detaillierte Diskussion der vorgeschlagenen Methodik und in Anfügung an diese liefert es die Verteilung von Geschwindigkeiten, Druck und Temperatur. Alle Kalkulationen basieren auf einem groben Gitter mit 10 Linien, die einheitlich in der Querstromrichtung des Rohres verteilt sind, wenn das Geschwindigkeitsprofil am Eingang des Rohres parabolisch angenommen wird. Die errechneten Ergebnisse, die sowohl für Aufwärts- als auch Abwärtsströmungen gelten, stimmen gut mit anderen aufwendigen Untersuchungen und ebenso mit verfügbaren experimentellen Daten, die Luft und Wasser als Arbeitsfluide gebrauchen, überein.     

Numerical and experimental investigation of thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct with rotating inner cylinder

Thermal convection for an incompressible Herschel-Bulkley fluid along an annular duct, whose inner cylinder is rotating and outer is at rest, is analyzed numerically and experimentally. The outer cylinder is heated at constant heat flux density and the inner one is assumed adiabatic. The first part of this study deals with the effect of the rheological behavior of the fluid and that of the rotation of the inner cylinder on the flow field and heat transfer coefficient. All the physical properties are assumed constant and the flow is assumed fully developed. The critical Rossby number Ro_c = (R₁Ω/U_d)_c, for which the dimension of the plug flow is reduced to zero is determined with respect to the flow behavior index, the radius ratio and the Herschel-Bulkley number for axial flow. The rotation of the inner cylinder induces a decrease of the axial velocity gradient at the outer cylinder thereby reducing the heat transfer between the heated wall and the fluid. The second part of this study introduces the variation of the consistency K with temperature and analyzes the evolution of the flow pattern and heat transfer coefficient along the heating zone. Two cases are distinguished depending on the Rossby number: (i) Ro < Ro_c, the plug flow dimension increases along the heating zone; (ii) Ro < Ro_c, the decrease of K with temperature leads to the reappearance of the plug flow. For high angular velocities, it is possible to have a plug zone attached to the outer cylinder. Finally, a correlation is proposed for the Nusselt number. It shows clearly that the effect of thermodependency of K on the heat transfer becomes more important with increasing rotational velocity of the inner cylinder.     

Numerical simulation of steady-state and transient natural convection in an isothermal open cubic cavity

In this work the numeric results, of the steady-state and transient heat transfer by natural convection in a horizontal isothermal open cubic cavity are presented. The most important assumptions in the mathematical formulation are two, the flow is laminar and the Boussinesq approximation is valid. The conservation equations in primitive variables are solved using the finite volume method and the SIMPLEC algorithm. The advective terms are approximated by the SMART scheme and the diffusive terms are approximated using the central differencing scheme. The results are obtained for a Rayleigh number range from 10⁴ to 10⁷.The numerical model predicted flow instabilities and Nusselt number oscillations for high Rayleigh numbers.