Ducts Of Arbitrary Cross Section Research Articles

Incompressible flow in a duct with an arbitrary cross-section

In this paper, the problem of finding the velocity profile of fully developed laminar flows through ducts of arbitrary cross-sections is investigated. The method which is used is based on a perturbative approach, although some nonperturbative exact results are also obtained. The main quantity which is addressed is the specific conductivity. This is a dimensionless quantity, which is independent of the size of the duct and the properties of the fluid, depending on only the shape of the duct. Some examples are studied is detail, for which the numerical values of the specific conductivity are determined, in the form of perturbative or exact results.

Generalized Lévêque Solution for Ducts of Arbitrary Cross Section

Abstract The asymptotic limit for perimeter averaged convection is generalized for short ducts of arbitrary cross section. A correction factor to Lévêque's original analysis is derived in terms of the state of wall shear stress under conditions of fully developed flows for walls of constant temperature (T) and constant heat flux (H1 and H2). This analysis is performed for four duct geometries: elliptic, rhombic, rectangular, and regular polygons. The magnitude of this correction is greatest for the H2 wall condition and for ducts having walls with acute corners. The results of this analysis can be incorporated into a generalized correlation for the full Graetz problem in ducts of arbitrary cross section.

Correlations for Convection in Hydrodynamically Developing Laminar Duct Flow

Abstract A generalized correlation for combined entry convection in ducts of arbitrary cross section has been developed. The correlation is constructed for the average Nusselt number using knowledge of fully developed transport constants. The general correlation reproduces the first principle solutions for the well-established round and parallel plate duct geometries to within ±5% for both constant temperature and constant heat flux wall conditions when Pr ≥ 0.7. A survey of the literature demonstrates that the new generalized correlation performs as well or better than existing correlations, which are expressed for specific geometries and wall conditions. The new correlation is generally in good agreement with the first principle solutions of less common duct geometries so long as the duct has a convective surface equal to the wetted perimeter. The new correlation is not recommended for ducts having small aspect ratios that pinch the flow when convection is prescribed by the H2 constant heat flux wall condition.

Correlations for the Graetz problem in convection – Part 2: For ducts of arbitrary cross-section

A general correlation is proposed for the laminar Graetz problem associated with convection in ducts of arbitrary cross-section. The correlation assumes knowledge of fully developed transport constants, specifically the product of fully developed friction factor and Reynolds number, and the fully developed value of Nusselt number, whose values are particular to the duct geometry and wall condition. The new correlation is suitable for ducts having a convective surface equal to the wetted perimeter, and for ducts having a wall condition of either constant temperature or constant heat flux. For the majority of solutions to the Graetz problem found in the literature for various duct geometries, the new correlation is generally found to be within ±5% of first principle calculations.

Generalized formulation for estimating pressure drop in fully-developed laminar flow in singly and doubly connected channels of non-circular cross-sections

Fully developed internal laminar flow through both, singly- and doubly-connected ducts of arbitrary cross-sections, is investigated using a two-dimensional semi-analytical technique in which the condition at the inner/outer arbitrarily defined peripheral boundary of the ducts is matched by a collocation technique. This boundary collocation technique has been applied and validated for (a) singly-connected ducts of standard cross-sectional shapes such as square, circular, etc., (b) a variety of complicated singly-connected duct cross-sectional geometries and, (c) two principal variations of the doubly-connected ducts (i) annular duct having an arbitrary outside perimeter while the internal core perimeter is circular, and (ii) annular duct having a circular outside perimeter while the internal core perimeter is arbitrarily shaped. The proposed model is only a function of geometrical parameters of the duct cross-section. Fluid velocity contours, shear stress distribution along the duct walls and the friction factor (Poiseuille number) has been estimated with this technique and validated against existing solutions for several standard cross-sections and found to be in excellent agreement. Several arbitrary shapes have been considered for analysis. As most micro-fabrication techniques normally do not produce mathematically well-defined or ideal geometries, an experimental test case is also presented wherein, the application of the present methodology in accurately predicting the fully-developed velocity profile, and therefore the net pressure drop, in a microchannel etched on a silicon substrate, is discussed. This simple and universally applicable semi-analytical technique substantially improves the overall predictions for real-time geometries. The critical nuances of successful application of this technique are outlined.

Nonlinear waves and shocks in a rigid acoustical guide

A model is developed for the propagation of finite amplitude acoustical waves and weak shocks in a straight duct of arbitrary cross section. It generalizes the linear modal solution, assuming mode amplitudes slowly vary along the guide axis under the influence of nonlinearities. Using orthogonality properties, the model finally reduces to a set of ordinary differential equations for each mode at each of the harmonics of the input frequency. The theory is then applied to a two-dimensional waveguide. Dispersion relations indicate that there can be two types of nonlinear interactions either called "resonant" or "non-resonant." Resonant interactions occur dominantly for modes propagating at a rather large angle with respect to the axis and involve mostly modes propagating with the same phase velocity. In this case, guided propagation is similar to nonlinear plane wave propagation, with the progressive steepening up to shock formation of the two waves that constitute the mode and reflect onto the guide walls. Non-resonant interactions can be observed as the input modes propagate at a small angle, in which case, nonlinear interactions involve many adjacent modes having close phase velocities. Grazing propagation can also lead to more complex phenomena such as wavefront curvature and irregular reflection.

A computational mode-matching approach for sound propagation in three-dimensional ducts with flow

A finite element (FE) mode-matching approach is presented for duct acoustics with flow and circumferentially varying liners. The primary application of the method is the prediction of sound attenuation in turbofan inlets and bypass ducts including the effects of liner splices. A fully numerical procedure has been developed to determine the acoustic modes in ducts of arbitrary cross-section and mean flow profile. A numerical mode-matching method is proposed using a modified matching technique in order to deal more accurately with liner discontinuity with flow. An analytic radiation model can also be integrated with the mode-matching procedure in order to obtain far-field directivity for tones and broadband multimode noise. The accuracy of the mode-matching method is demonstrated by comparison with exact solutions and high-resolution FE solutions. The influence of non-axisymmetric liners on noise radiation is also illustrated.

A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts

Starting from a multiple-scales approach to model sound transmission through a slowly varying duct of arbitrary cross section, an explicit analytical solution is derived for a mode undergoing cut-on cut-off transition. The solution is a composite one, removing the singularity that appears in the WKB approximation by encompassing the inner boundary-layer solution with the behaviour far upstream and downstream. The solution should not only prove to be a useful benchmark for computational aeroacoustic models, but also enable a designer to continue to use multiple-scales theory to examine sound transmission even when cut-on cut-off transitional modes are present.

The attenuation of the higher-order cross-section modes in a duct with a thin porous layer

A numerical method for sound propagation of higher-order cross-sectional modes in a duct of arbitrary cross-section and boundary conditions with nonzero, complex acoustic admittance has been considered. This method assumes that the cross-section of the duct is uniform and that the duct is of a considerable length so that the longitudinal modes can be neglected. The problem is reduced to a two-dimensional (2D) finite element (FE) solution, from which a set of cross-sectional eigen-values and eigen-functions are determined. This result is used to obtain the modal frequencies, velocities and the attenuation coefficients. The 2D FE solution is then extended to three-dimensional via the normal mode decomposition technique. The numerical solution is validated against experimental data for sound propagation in a pipe with inner walls partially covered by coarse sand or granulated rubber. The values of the eigen-frequencies calculated from the proposed numerical model are validated against those predicted by the standard analytical solution for both a circular and rectangular pipe with rigid walls. It is shown that the considered numerical method is useful for predicting the sound pressure distribution, attenuation, and eigen-frequencies in a duct with acoustically nonrigid boundary conditions. The purpose of this work is to pave the way for the development of an efficient inverse problem solution for the remote characterization of the acoustic boundary conditions in natural and artificial waveguides.

Semianalytical solutions of laminar fully developed pulsating flows through ducts of arbitrary cross sections

A semianalytical analysis of fully developed pulsating flows in pipes of noncircular cross section is presented. The flow is assumed to be pressure gradient driven. Details of the analytical treatment of the flow are presented and it is shown that the analysis can be employed for any arbitrary cross-sectional shape. Special considerations are given to laminar pipe flows with circular cross sections and results of the present analysis are compared with those of conventional analytical treatments of the flow. Comparison of the velocity distribution, mass flow-rate pulsation, ratio of amplitudes of the mass flow rate and the pressure gradient, and the corresponding phase lag obtained by the present treatment show excellent agreement with data in the literature. Similar comparisons are also performed for oscillating flow through ducts of square and rectangular cross sections. Numerical data are introduced to confirm the resultant analysis. Finally, a few test cases are presented for sinusoidally pulsating flows through ducts of rectangular and elliptical cross sections with varying aspect ratios.

Sound propagation in slowly varying lined flow ducts of arbitrary cross-section

Sound transmission through ducts of constant cross-section with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion, where the modes are eigenfunctions of the corresponding Laplace eigenvalue problem along a duct cross-section. A natural extension for ducts with cross-section and wall impedance that are varying slowly (compared to a typical acoustic wavelength and a typical duct radius) in the axial direction is a multiple-scales solution. This has been done for the simpler problem of circular ducts with homentropic irrotational flow. In the present paper, this solution is generalized to the problem of ducts of arbitrary cross-section. It is shown that the multiple-scales problem allows an exact solution, given the cross-sectional Laplace eigensolutions. The formulation includes both hollow and annular geometries. In addition, the turning point analysis is given for a single hard-wall cut-on, cut-off transition. This appears to yield the same reflection and transmission coefficients as in the circular duct problem.

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions.

EMISSION OF SOUND FROM TURBULENCE CONVECTED BY A PARALLEL MEAN FLOW IN THE PRESENCE OF A CONFINING DUCT

An approximate method for calculating the noise generated by a turbulent flow within a semi-infinite duct of arbitrary cross-section is developed. It is based on a previously derived high-frequency solution to Lilley's equation, which describes the sound propagation in a transversely sheared mean flow. The source term is simplified by assuming the turbulence to be axisymmetric about the mean flow direction. Numerical results are presented for the special case of a ring source in a circular duct with an axisymmetric mean flow. They show that the internally generated noise is suppressed at sufficiently large upstream angles in a hard-walled duct, and that acoustic liners can significantly reduce the sound radiated in both the upstream and downstream regions, depending upon the source location and Mach number of the flow.

General equations for fully developed fluid flow and heat transfer characteristics in complex geometries

This paper presents general equations for laminar fully developed friction factor, incremental pressure drop and Nusselt number of Newtonian fluid flowing in ducts of arbitrary cross sections. These general equations are obtained using some definitions, e.g. equivalent diameter, number of imaginary circles inside specific geometry and the diameter of surrounded circle. The comparison of the results obtained from these equations with the theoretically calculated values available in the literature for various ducts indicates the accuracy of this investigation.

Application of chaos theory to ray tracing in ducts

To predict the radar cross section of large ducts and cavities it is common to employ ray-tracing methods of one kind or another. Electromagnetic engineers traditionally employ a first-order ray theory based on the Deschamps formulation (1972). This is not the only formulation and it is shown that the semi-classical tangent-plane methods employed in quantum physics offer some advantages. The special case of straight ducts of arbitrary cross section in considered where the existing theory of billiard dynamics shows that such ray tracing is in general a chaotic process. In a duct or cavity the average rate of increase of ray divergence or change in ray intensity is thus proportional to the Lyapunov exponent. A relationship is provided between the accuracy of the geometry and/or mesh size, the angle of incidence, the duct length and the Lyapunov exponent for a straight duct. This establishes a computational limit on the ability to make numerically deterministic predictions using shooting-and-bouncing or related ray-tracing methods. For a sufficiently long duct of general cross section, away from normal incidence, it is not possible to achieve convergence using these methods.

Flow of non-Newtonian fluids through tubes with abrupt expansions and contractions (square wave tubes)

The steady and dynamic shear properties of three different non-Newtonian fluids (an Acrysol TT615 solution, a 50% creosote-in-water emulsion and a Separan AP-30 solution) are determined. Their flow behaviour through tubes with abrupt expansions and contractions (`square wave tubes') is investigated. The steady shear viscosity of these fluids can be described with a modified Quemada model. The geometric parameter method proposed previously by Kozicki et al. for duct of arbitrary cross section can be extended to the present systems with the use of a volume-average diameter. Flow in square wave tubes can be readily obtained from the straight capillary data by means of a shift factor ( a+ b). The shift factor is related to a pressure-average diameter. Conversely, using the pressure-average diameter eliminates the necessity of the shift factor. Corrections of inertial and entrance effects from Newtonian results can be approximately applied to non-Newtonian results. These corrections are relatively insignificant as compared with the effects of fluid elasticity and elongational-viscosity for viscoelastic fluids.

Laminar Fluid Transients in Conduits of Arbitrary Cross Section

Conduit cross sections of nonelementary shape are the rule in biological fluid systems. Because of mathematical and experimental difficulties, however, relatively little is known about the dynamics of flow in these vessels, particularly under time-varying conditions, although knowledge of the fluid motion therein in terms of properties such as driving head, velocity, and wall shear stress is clearly essential for proper diagnosis of a flow problem. This paper presents an analytical method for solving problems of unsteady laminar flow in long ducts of arbitrary cross section. The implied assumption of parallel flow renders the mathematical situation linear, but the proposed method is far from conventional because it allows consideration of almost any cross-sectional configuration conceivable. This novel approach is made possible by joining a steady-state superposition method for creating new conduits from known basic shapes with a regular perturbation scheme found suitable for dealing with fluid transients...

An Equation for Laminar Flow Heat Transfer for Constant Heat Flux Boundary Condition in Ducts of Arbitrary Cross-Sectional Area

We can calculate the heat transfer in ducts of arbitrary cross section with the definition of the equivalent diameter only in turbulent flow. In laminar flow, it is not sufficient to define an equivalent diameter, because the boundary layer of each wall is influenced by another wall. Therefore, one needs additional quantities to describe the heat transfer and pressure drop. This is shown for pressure drop calculations by Yilmaz and heat transfer for constant wall temperature by Yilmaz and Cihan. In these works, by using other quantities, it was possible to obtain general equations for pressure drop and heat transfer for constant wall temperature. In the present work, an equation for heat transfer for the boundary condition of constant heat flux at the wall for laminar developed flow in ducts of arbitrary cross-sectional area is given. 14 refs., 1 fig., 4 tabs.

Generalized Reynolds number for the flow of power law fluids in cylindrical ducts of arbitrary cross-section

The pressure drop prediction requires the generalization of the Reynolds number for laminar flow of pseudoplastic fluids in cylindrical ducts. This generalized Reynolds number is now well established for very simple cross-sections, such as circular tubes or infinite parallel plane plates. To our knowledge, there is no generalization available for other cross-sections due to the complexity of the velocity profiles. The aim of this paper is to propose a generalized Reynolds number for cylindrical ducts of arbitrary cross-section. This new Reynolds number involves only one geometrical parameter, which can be easily determined theoretically or experimentally. This Reynolds number was successfully compared with definitions found in the literature. Moreover, this generalized Reynolds number seems to be in good agreement with experimental studies carried out in plate heat exchangers.

General equation for heat transfer for laminar flow in ducts of arbitrary cross-sections

This paper presents the general equations for heat transfer calculations for constant wall temperature in laminar developed flow in ducts of arbitrary cross sections. The results obtained from these equations compare well with the theoretically-calculated values available in the literature for circular, rectangular, triangular, elliptical and parallel plate ducts. The maximum and minimum deviations by these comparisons are −8.7 and +8.0% respectively.