Secondary flow structures in developing viscoelastic fluid flow through curved ducts with square cross section

Meccanica - The present paper numerically investigates viscoelastic fluid flow in the developing flow regime through both straight and 90-degree curve ducts. The aim is to investigate the effects...     

Numerical study of laminar fluid flow in a curved elliptic duct with internal fins

The fully developed laminar incompressible flow inside a curved duct of elliptical cross-section with four thin, internal longitudinal fins is studied using the improved CVP method. We present numerical results for the friction factor and an investigation of the effect of the fin height and the Dean number on the flow. It is found that the friction factor increases for large fins and for high Dean numbers and that in some cases, it has a strong dependence on the cross-sectional aspect ratio. The thermal results show that the heat transfer rate is enhanced by the internal fins and that it depends on the aspect ratio.     

Fluid flow in curved ducts

The advent of standard algorithms for the numerical solution of partial differential equations has given researchers a new tool for fluid flow calculations. In this paper, single-phase flow in curved ducts is numerically simulated by imposing a spatially varying centrifugal force on a fluid flowing in a straight tube. The resulting set of partial differential equations is solved using the HARWELL-FLOW3D computer program. Comparison with other numerical and experimental results shows that this simplified formulation gives accurate results. The model neglects certain geometric terms of the order d/D, the duct-to-coil diameter ratio. The effect of these terms is investigated by considering the flow in a 90° bend for large d/D. It is shown that while there may be significant error in the prediction of the local variables for large d/D, the circumference-averaged quantities are well predicted.     

Measurements of turbulent developing flow in a moderately curved square duct

Laser-Doppler measurements in the turbulent flow in a right-angled bend of square cross-section, radius/duct-width ratio 7.0, are presented and show the development of secondary circulation in cross-stream planes. Distribution of the streamwise and radial components of the mean velocity and turbulence intensity, and the corresponding Reynolds shear stress, are presented as contour plots and are intended for use in the further development of numerical flow prediction methods.     

Entry flow in weakly curved ducts

Summary The properties of steady incompressible flow in the entry region of curved ducts are studied theoretically for large Reynolds numbers. Under the assumption that the inviscid core flow may be approximated sufficiently well by a potential vortex the boundary layer equations are solved by means of integral methods. In order to improve the accuracy momentum as well as energy relationships are used.

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Einlaufströmung in schwach gekrümmten Rohren

Übersicht Die Eigenschaften einer stationären, inkompressiblen Strömung im Einlaufbereich gekrümmter Rohre werden für große Reynoldszahlen theoretisch untersucht. Unter der Voraussetzung, daß die reibungsfreie Kernströmung hinreichend genau durch einen Potentialwirbel angenähert werden kann, werden die Grenzschichtgleichungen mit Integralmethoden gelöst. Um die Genauigkeit zu erhöhen, werden sowohl Impuls- als auch Energiebeziehungen verwendet.

     

Experimental study of developing turbulent flow and heat transfer in ribbed convergent/divergent square ducts

The local heat transfer and pressure drop characteristics of developing turbulent flows of air in three stationary ribbed square ducts have been investigated experimentally. These are: ribbed square duct with constant cross-section (straight duct), ribbed divergent square duct and ribbed convergent square duct. The convergent/divergent duct has an inclination angle of 1°. The measurement was conducted within the range of Reynolds numbers from 10 000 to 77 000. The heat transfer performance of the divergent/convergent ducts is compared with the ribbed straight duct under three constraints: identical mass flow rate, identical pumping power and identical pressure drop. Because of the streamwise flow acceleration or deceleration, the local heat transfer characteristics of the divergent and convergent ducts are quite different from those of the straight duct. In the straight duct, the fluid flow and heat transfer become fully developed after 2–3 ribs, while in the divergent and convergent ducts there is no such trend. The comparison shows that among the three ducts, the divergent duct has the highest heat transfer performance, the convergent duct has the lowest, while the straight duct locates somewhere in between.     

Experimental and numerical study of developing turbulent flow and heat transfer in convergent/divergent square ducts

 The work reported in this paper is a systematic experimental and numerical study of friction and heat transfer characteristics of divergent/convergent square ducts with an inclination angle of 1^∘ in the two direction at cross section. The ratio of duct length to average hydraulic diameter is 10. For the comparison purpose, measurement and simulation are also conducted for a square duct with constant cross section area, which equals to the average cross section area of the convergent/divergent duct. In the numerical simulation the flow is modeled as being three-dimensional and fully elliptic by using the body-fitted finite volume method and the kɛ turbulence model. The uniform heat flux boundary condition is specified to simulate the electrical heating used in the experiments. The heat transfer performance of the divergent/convergent ducts is compared with the duct with uniform cross section under three constraints (identical mass flow rate, pumping power and pressure drop). The agreement of the experimental and numerical results is quite good except at the duct inlet. Results show that for the three ducts studied there is a weak secondary flow at the cross section, and the circumference distribution of the local heat transfer coefficient is not uniform, with an appreciable reduction in the four corner regions. In addition, the acceleration/deceleration caused by the cross section variation has a profound effect on the turbulent heat transfer: compared with the duct of constant cross section area, the divergent duct generally shows enhanced heat transfer behavior, while the convergent duct has an appreciable reduction in heat transfer performance. Received on 18 September 2000 / Published online: 29 November 2001     

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.     

Unsteady laminar flow developing in a curved duct

The flow developing in a tightly curved U-bend of square cross section has been investigated experimentally and via numerical simulation. Both long-time averages and time histories of the longitudinal (streamwise) component of velocity were measured using a laser-Doppler velocimeter. The Reynolds number investigated was Re = 1400. The data were obtained at different bend angles, θ, and were confined to the symmetry plane of the bend. At Re = 1400, the flow entering the bend is steady, but by θ = 90° it develops an oscillatory component of motion along the outer-radius wall. Autocorrelations and energy spectra derived from the time histories yield a base frequency of approximately 0.1 Hz for these oscillations. Flow-visualization studies showed that the proximity of the outer-radius wall served to damp the amplitude of the spanwise oscillations.

Numerical simulations of the flow were performed using both steady and unsteady version of the finite-difference elliptic calculation procedure of Humphrey et al. (1977). Although the unsteadiness observed experimentally does not arise spontaneously in the calculations, numerical experiments involving the imposition of a periodic time-dependent perturbation at the inlet plane suggest that the U-bend acts upon the incoming flow so as to damp the amplitude of the imposed oscillation while altering its frequency.

The oscillations observed experimentally, and numerically as a result of the periodic perturbation, have been linked to the formation of Goertler-type vortices of the outer-radius wall in the developing flow. The vortices, which develop as a result of the centrifugal instability of the flow on the outer-radius wall, undergo a further transition to an unsteady regime at higher flow rates.     

Laminar thermally developing flow inside right-angularly triangular ducts

An analytical approach based on the generalized integral transform technique is presented, for the solution of laminar forced convection within the thermal entry region of ducts with arbitrarily shaped cross-sections. The analysis is illustrated through consideration of a right triangular duct subjected to constant wall temperature boundary condition. Critical comparisons are made with results available in the literature, from direct numerical approaches. Numerical results for dimensionless average temperature and Nusselt numbers are presented for different apex angles.Nomenclature a,b sides of right triangular duct - A _c cross-sectional area of duct - c _p specific heat of fluid - D _h =4A _c /p hydraulic diameter, with P the wet perimeter - h(z) heat transfer coefficient at duct wall - k thermal conductivity - Pe=c _p D _h /k Peclet number - T(x, y, z) temperature distribution - T _o inlet temperature - T _w prescribed wall temperature - u(x, y); U(X, Y) dimensional and dimensionless velocity profile - average flow velocity - x; X dimensional and dimensionless normal coordinate (Fig. 1) - x ₁(y); X ₁(Y) dimensional and dimensionless position at irregular boundary (Fig. 1) - y; Y dimensional and dimensionless normal coordinate (Fig. 1) - z; Z dimensional and dimensionless axial coordinate Greek letters side of right triangular duct in X direction (dimensionless) - side of right triangular duct in Y direction (dimensionless) - density of fluid - (X, Y, Z) dimensionless temperature distribution - * apex angle of triangular duct (Fig. 1) - ** apex angle of triangular duct (Fig. 1)     

Pressure drop and heat transfer characteristics of turbulent flow in annular tubes with internal wave-like longitudinal fins

Measured were pressure drop and heat transfer characteristics with uniform axial heat input using air as the working fluid in both the entrance and fully developed regions of annular tubes with wave-like longitudinal fins. Five series of experiments were performed for turbulent flow and heat transfer in the annular tubes with number of waves equal to 4, 8, 12, 16 and 20, respectively. The test tube has a double-pipe structure with the inner blocked tubes as an insertion. The wave-like fins are in the annulus and span its full width. The friction factor and Nusselt number in the fully developed region were obtained. The friction factor and Nusselt number can be well corrected by a power-law correction in the Reynolds number range tested. In order to evaluate the thermal performance of the longitudinal finned tubes over a plain circular tube, comparisons were made under three conditions: (1) identical pumping power; (2) identical pressure drop and (3) identical mass flow. It was found that under the three constraints all the wave-like finned tubes can enhance heat transfer with the tube with wave number 20 being superior. Finally, discussion on the enhancement mechanism is conducted and a general correlation for the fully developed heat transfer is provided, which can cover all the fifty data of the five tubes with a mean deviation of 9.3%.     

Flow of second-order fluid in a curved duct with square cross-section

This paper investigates the inertial and creeping flow of a second-order fluid in a curved duct with a square cross-section. Numerical modeling is employed to analyze fluid flow, and the governing equations are discretized using the finite difference method on a staggered mesh. The marker-and-cell method is employed to allocate the parameters on the staggered mesh, and static pressure is calculated using the artificial compressibility approach. The effect of centrifugal force due to the curvature of the duct and the opposing effects of the first and second normal stress difference on the flow field are investigated. In addition, the order-of-magnitude technique is used to derive the force balance relations for the core region of flow. Based on these relations, the performance mechanism of centrifugal force and normal stress differences on the generation of secondary flows is considered. We also present an analytical relation for the axial velocity profile and flow resistance ratio of creeping flow. For this kind of flow, previous studies have investigated the effect of the first normal stress difference on the transition from one pair to two pairs of vortices while we show that the negative second normal stress difference has the opposite effect on this transition and can stabilize the flow.     

An exact analytical solution for creeping Dean flow of Bingham plastics through curved rectangular ducts

In this paper, an exact analytical solution for creeping flow of Bingham plastic fluid passing through curved rectangular ducts is presented for the first time. The closed form of axial velocity distribution, flow resistance ratio, and wall shear stress are derived using bounded Fourier transformation. An extensive investigation on mutual effects of Hedstrom number, curvature ratio, and aspect ratio is conducted. The results indicate that a drag reduction is caused in the flow field by increasing the Hedstrom number. It is shown that unlike the Newtonian creeping Dean flow, the critical aspect ratio (an aspect ratio in which the flow resistance ratio is independent from curvature ratio) does not exist at large enough Hedstrom numbers. Analytical solution also indicated that as Hedstrom number is increased, the value of Poiseuille number is enhanced, and unlike the Newtonian flows, the value of Poiseuille number is not zero at edges of cross section.     

Thermal and friction characteristics of turbulent flow through rectangular and square ducts with transverse ribs and wire-coil inserts

The heat transfer and the pressure drop characteristics of turbulent flow of air through rectangular and square ducts with internal transverse rib turbulators on two opposite surfaces of the ducts and with wire-coil inserts have been studied experimentally. Circular duct has also been used. The transverse ribs in combination with wire-coil inserts have been found to perform better than either ribs or wire-coil inserts acting alone. The flow friction and thermal characteristics are governed by duct aspect ratio, coil helix angle and wire diameter of the coil, rib height and rib spacing, Reynolds number and Prandtl number. Correlations developed for friction factor and Nusselt number have predicted the experimental data satisfactorily. It has been found that on the basis of constant pumping power, up to 35% heat duty increase occurs for the combined ribs and wire-coil inserts case compared to the individual ribs and wire-coil inserts cases in the measured experimental parameters space. On the constant heat duty basis, the pumping power has been reduced up to 20% for the combined enhancement geometry than the individual enhancement geometries.     

Thermohydraulics of turbulent flow through rectangular and square ducts with axial corrugation roughness and twisted-tapes with and without oblique teeth

The heat transfer and the pressure drop characteristics of turbulent flow of air (10,000 < Re < 100,000) through rectangular and square ducts with combined internal axial corrugations on all the surfaces of the ducts and with twisted-tape inserts with and without oblique teeth have been studied experimentally. The axial corrugations in combination with twisted-tapes of all types with oblique teeth have been found to perform better than those without oblique teeth in combination with axial corrugations. The heat transfer and the pressure drop measurements have been taken in separate test sections. Heat transfer tests were carried out in electrically heated stainless steel ducts incorporating uniform wall heat flux boundary conditions. Pressure drop tests were carried out in acrylic ducts. The flow friction and thermal characteristics are governed by duct aspect ratio, corrugation angle, corrugation pitch, twist ratio, space ratio, length, tooth horizontal length and tooth angle of the twisted-tape, Reynolds number and Prandtl number. Correlations developed for friction factor and Nusselt number have predicted the experimental data satisfactorily. The performance of the geometry under investigation has been evaluated. It has been found that on the basis of constant pumping power, up to 55% heat duty increase occurs for the combined axial corrugation and regularly spaced twisted-tape elements inserts with oblique teeth case compared to without oblique teeth twisted-tape inserts cases in the measured experimental parameters space. On the constant heat duty basis, the pumping power has been reduced up to 47% for the combined enhancement geometry than the individual enhancement geometries. However, full-length and short-length twisted-tapes with oblique teeth in combination with axial corrugations show only marginal improvements over the twisted-tapes without oblique teeth.     

Thermal and friction characteristics of laminar flow through rectangular and square ducts with transverse ribs and wire coil inserts

The heat transfer and the pressure drop characteristics of laminar flow of viscous oil (195 < Pr < 525) through rectangular and square ducts with internal transverse rib turbulators on two opposite surfaces of the ducts and with wire coil inserts have been studied experimentally. Circular duct has also been used. The transverse ribs in combination with wire coil inserts have been found to perform better than either ribs or wire coil inserts acting alone. The heat transfer and the pressure drop measurements have been taken in separate test sections. Heat transfer tests were carried out in electrically heated stainless steel ducts incorporating uniform wall heat flux boundary conditions. Pressure drop tests were carried out in acrylic ducts. The flow friction and thermal characteristics are governed by duct aspect ratio, coil helix angle and wire diameter of the coil, rib height and rib spacing, Reynolds number and Prandtl number. Correlations developed for friction factor and Nusselt number have predicted the experimental data satisfactorily. The performance of the geometry under investigation has been evaluated. It has been found that on the basis of constant pumping power, up to fifty per cent heat duty increase occurs for the combined ribs and wire coil inserts case compared to the individual ribs and wire coil inserts cases in the measured experimental parameters space. On the constant heat duty basis, the pumping power has been reduced up to forty per cent for the combined enhancement geometry than the individual enhancement geometries.     

Prediction of turbulent gas–solids flow in curved ducts using the Eulerian–Lagrangian method

The flow of particulate two‐phase flow mixtures occur in several components of solid fuel combustion systems, such as the pressurised fluidised bed combustors (PFBC) and suspension‐fired coal boilers. A detailed understanding of the mixture characteristics in the conveying component can aid in refining and optimising its design. In this study, the flow of an isothermal, dilute two‐phase particulate mixture has been examined in a high curvature duct, which can be representative of that transporting the gas–solid mixture from the hot clean‐up section to the gas turbine combustor in a PFBC plant. The numerical study has been approached by utilising the Eulerian–Lagrangian methodology for describing the characteristics of the fluid and particulate phases. By assuming that the mixture is dilute and the particles are spherical, the governing particle momentum equations have been solved with appropriately prescribed boundary conditions. Turbulence effects on the particle dispersion were represented by a statistical model that accounts for both the turbulent eddy lifetime and the particle transit time scales. For the turbulent flow condition examined it was observed that mixtures with small particle diameters had low interphase slip velocities and low impaction probability with the pipe walls. Increasing the particle diameters (>50 μm) resulted in higher interphase slip velocities and, as expected, their impaction probability with the pipe walls was significantly increased. The particle dispersion is significant for the smaller sizes, whereas the larger particles are relatively insensitive to the gas turbulence. The main particle impaction region, and locations most prone to erosion damage, is estimated to be within an outer duct length of two to six times the duct diameter, when the duct radius of curvature to the duct diameter ratio is equal to unity. Copyright © 1999 John Wiley & Sons, Ltd.