Classification of Pulsating Flow Patterns in Curved Pipes

1996 ◽  
Vol 118 (3) ◽  
pp. 311-317 ◽  
Author(s):  
Shigeru Tada ◽  
Shuzo Oshima ◽  
Ryuichiro Yamane
Keyword(s):  
Secondary Flow ◽  
Amplitude Ratio ◽  
Circular Cross Section ◽  
Volumetric Flow Rate ◽  
Flow Patterns ◽  
Womersley Number ◽  
Dean Number ◽  
Curved Pipes ◽  
Convection Dominated

The fully developed periodic laminar flow of incompressible Newtonian fluids through a pipe of circular cross section, which is coiled in a circle, was simulated numerically. The flow patterns are characterized by three parameters: the Womersley number Wo, the Dean number De, and the amplitude ratio β. The effect of these parameters on the flow was studied in the range 2.19 ≤ Wo ≤ 50.00, 15.07 ≤ De ≤ 265.49 and 0.50 ≤ β ≤ 2.00, with the curvature ratio δ fixed to be 0.05. The way the secondary flow evolved with increasing Womersley number and Dean number is explained. The secondary flow patterns are classified into three main groups: the viscosity-dominated type, the inertia-dominated type, and the convection-dominated type. It was found that when the amplitude ratio of the volumetric flow rate is equal to 1.0, four to six vortices of the secondary flow appear at high Dean numbers, and the Lyne-type flow patterns disappear at β ≥ 0.50.

Flow Visualization Experiments on Secondary Flow Patterns in an Isothermally Heated Curved Pipe

1987 ◽  
Vol 109 (1) ◽  
pp. 55-61 ◽  
Author(s):  
K. C. Cheng ◽  
F. P. Yuen
Keyword(s):  
Flow Visualization ◽  
Secondary Flow ◽  
Theoretical Models ◽  
Flow Patterns ◽  
Radius Of Curvature ◽  
Dean Number ◽  
Number Range ◽  
Curved Pipe ◽  
Curved Pipes ◽  
Numerical Predictions

Secondary flow patterns at the exit of a 180 deg bend (tube inside diameter d = 1.99 cm, radius of curvature Rc = 10.85 cm) are presented to illustrate the combined effects of centrifugal and buoyancy forces in hydrodynamically and thermally developing entrance region of an isothermally heated curved pipe with both parabolic and turbulent entrance velocity profiles. Three cases of upward, horizontal, and downward-curved pipe flows are studied for constant wall temperatures Tw=55–91°C, Dean number range K=22–1209 and ReRa=1.00×106–8.86×107. The flow visualization was realized by the smoke injection method. The secondary flow patterns shown are useful for future comparison with numerical predictions and confirming theoretical models. The results can be used to assess qualitatively the limit of the applicability of the existing correlation equations for laminar forced convection in isothermally heated curved pipes without buoyancy effects.

Flow Visualization Studies on Secondary Flow Patterns in Straight Tubes Downstream of a 180 deg Bend and in Isothermally Heated Horizontal Tubes

1987 ◽  
Vol 109 (1) ◽  
pp. 49-54 ◽  
Author(s):  
K. C. Cheng ◽  
F. P. Yuen
Keyword(s):  
Secondary Flow ◽  
Horizontal Tube ◽  
Flow Patterns ◽  
Radius Of Curvature ◽  
Straight Tube ◽  
Dean Number ◽  
Flowing Fluid ◽  
Number Range ◽  
Inside Diameter ◽  
Curved Pipes

Photographs are presented for secondary flow patterns in a straight tube (x/d = 0 ∼ 70) downstream of a 180 deg bend (tube inside diameter d = 2.54 cm, radius of curvature Rc = 12.7 cm) and in an isothermally heated horizontal tube (tube inside diameter d = 2.54 cm, heated length l = 46.2 cm) with free convection effects. Each test section is preceded by a long entrance length with air as the flowing fluid. For curved pipes, the Dean number range is K = 99 to 384. At the exit of the 180 deg bend, the onset of centrifugal instability in the form of an additional pair of Dean vortices near the central outer wall occurs at a Dean number of about K = 100. The developing secondary flow patterns in the thermal entrance region of an isothermally heated horizontal tube are shown for the dimensionless axial distance z = 0.8 × 10−2 to 1.83 (Re = 3134 ∼ 14) for a range of constant wall temperatures Tw = 55 ∼ 65° C with entrance air temperature at about 25° C. The secondary flow patterns shown are useful for future comparisons with predictions from numerical solutions.

Mixing by Time-Dependent Orbits in Spatiotemporal Chaotic Advection

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Mohammad Karami ◽  
Ebrahim Shirani ◽  
Mojtaba Jarrahi ◽  
Hassan Peerhossaini
Keyword(s):  
Secondary Flow ◽  
Lyapunov Exponents ◽  
Amplitude Ratio ◽  
Flow Patterns ◽  
Chaotic Advection ◽  
Mixing Enhancement ◽  
Flow Pulsation ◽  
Mean Flow ◽  
Homoclinic Connections ◽  
Vorticity Vector

The simultaneous effects of flow pulsation and geometrical perturbation on laminar mixing in curved ducts have been numerically studied by three different metrics: analysis of the secondary flow patterns, Lyapunov exponents and vorticity vector analysis. The mixer that creates the flow pulsation and geometrical perturbations in these simulations is a twisted duct consisting of three bends; the angle between the curvature planes of successive bends is 90 deg. Both steady and pulsating flows are considered. In the steady case, analysis of secondary flow patterns showed that homoclinic connections appear and become prominent at large Reynolds numbers. In the pulsatile flow, homoclinic and heteroclinic connections appear by increasing β, the ratio of the peak oscillatory velocity component of the mean flow velocity. Moreover, sharp variations in the secondary flow structure are observed over an oscillation cycle for high values of β. These variations are reduced and the homoclinic connections disappear at high Womersley numbers. We show that small and moderate values of the Womersley number (6 ≤ α ≤ 10) and high values of velocity amplitude ratio (β ≥ 2) provide a better mixing than that in the steady flow. These results correlate closely with those obtained using two other metrics, analysis of the Lyapunov exponents and vorticity vector. It is shown that the increase in the Lyapunov exponents, and thus mixing enhancement, is due to the formation of homoclinic and heteroclinic connections.

Torsion effect on fully developed flow in a helical pipe

1987 ◽  
Vol 184 ◽  
pp. 335-356 ◽  
Author(s):  
Hsiao C. Kao
Keyword(s):  
Secondary Flow ◽  
Circular Cross Section ◽  
Numerical Procedure ◽  
Expansion Method ◽  
Dean Number ◽  
Navier Stokes Equation ◽  
Order Unity ◽  
Narrow Region ◽  
Torsion Effect ◽  
Helical Pipe

Two approaches have been used to study the torsion effect on the fully developed laminar flow in a helical pipe of constant circular cross-section. The first approach is the series expansion method that perturbs the Poiseuille flow and is valid for low Dean numbers with both the dimensionless curvature and dimensionless torsion being much less than unity. The second is a numerical procedure that solves the complete Navier-Stokes equation and is applicable to intermediate values of the Dean number. The results obtained indicate that, as far as the secondary flow patterns are concerned, the presence of torsion can produce a large effect if the ratio of the curvature to the torsion is of order unity. In these cases the secondary flow, though still consisting of a pair of vortices, can be very much distorted. Under extreme conditions one vortex is so prevalent as to squeeze the second one into a narrow region. However, ordinarily the torsion effect is small and the secondary flow has the usual pattern of a pair of counter-rotating vortices of nearly equal strength. Concerning the flow resistance in the pipe the effect of torsion is always small in all the circumstances that have so far been considered.

Pulsating flow in a curved tube

1973 ◽  
Vol 59 (4) ◽  
pp. 693-705 ◽  
Author(s):  
R. G. Zalosh ◽  
W. G. Nelson
Keyword(s):  
Flow Field ◽  
Pressure Gradient ◽  
Secondary Flow ◽  
Cross Section ◽  
Circular Cross Section ◽  
Analytic Solutions ◽  
Radius Of Curvature ◽  
Dean Number ◽  
Tube Cross Section ◽  
Curved Tube

An analysis is presented of laminar fully developed flow in a curved tube of circular cross-section under the influence of a pressure gradient oscillating sinusoidally in time. The governing equations are linearized by an expansion valid for small values of the parameter (a/R) [Ka/ων]2, where a is the radius of the tube cross-section, R is the radius of curvature, ν is the kinematic viscosity of the fluid and K and ω are the amplitude and frequency, respectively, of the pressure gradient. A solution involving numerical evaluation of finite Hankel transforms is obtained for arbitrary values of the parameter α = a(ω/ν)½. In addition, closed-form analytic solutions are derived for both small and large values of α. The secondary flow is found to consist of a steady component and a component oscillatory at a frequency 2ω, while the axial velocity perturbation oscillates at ω and 3ω. The small-α flow field is similar to the low Dean number steady flow configuration, whereas the large-α flow field is altered and includes secondary flow directed towards the centre of curvature.

Axially invariant laminar flow in helical pipes with a finite pitch

1993 ◽  
Vol 251 ◽  
pp. 315-353 ◽  
Author(s):  
Shijie Liu ◽  
Jacob H. Masliyah
Keyword(s):  
Laminar Flow ◽  
Coordinate System ◽  
Secondary Flow ◽  
Flow Pattern ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Helical Flow ◽  
The Body ◽  
Dean Number ◽  
Orthogonal Coordinate System

Steady axially invariant (fully developed) incompressible laminar flow of a Newtonian fluid in helical pipes of constant circular cross-section with arbitrary pitch and arbitrary radius of coil is studied. A loose-coiling analysis leads to two dominant parameters, namely Dean number, Dn = Reλ½, and Germano number, Gn = Reη, where Re is the Reynolds number, λ is the normalized curvature ratio and η is the normalized torsion. The Germano number is embedded in the body-centred azimuthal velocity which appears as a group in the governing equations. When studying Gn effects on the helical flow in terms of the secondary flow pattern or the secondary flow structure viewed in the generic (non-orthogonal) coordinate system of large Dn, a third dimensionless group emerges, γ = η/(λDn)½. For Dn < 20, the group γ* = Gn Dn-2 = η/(λRe) takes the place of γ.Numerical simulations with the full Navier-Stokes equations confirmed the theoretical findings. It is revealed that the effect of torsion on the helical flow can be neglected when γ ≤ 0.01 for moderate Dn. The critical value for which the secondary flow pattern changes from two vortices to one vortex is γ* > 0.039 for Dn < 20 and γ > 0.2 for Dn ≥ 20. For flows with fixed high Dean number and A, increasing the torsion has the effect of changing the relative position of the secondary flow vortices and the eventual formation of a flow having a Poiseuille-type axial velocity with a superimposed swirling flow. In the orthogonal coordinate system, however, the secondary flow generally has two vortices with sources and sinks. In the small-γ limit or when Dn is very small, the secondary flow is of the usual two-vortex type when viewed in the orthogonal coordinate system. In the large-γ limit, the appearance of the secondary flow in the orthogonal coordinate system is also two-vortex like but its orientation is inclined towards the upper wall. The flow friction factor is correlated to account for Dn, A and γ effects for Dn ≤ 5000 and γ < 0.1.

On the steady laminar flow of an incompressible viscous fluid in a curved pipe of elliptical cross-section

1985 ◽  
Vol 158 ◽  
pp. 329-340 ◽  
Author(s):  
H. C. Topakoglu ◽  
M. A. Ebadian
Keyword(s):  
Secondary Flow ◽  
Cross Section ◽  
Circular Cross Section ◽  
Elliptical Cross Section ◽  
Radius Of Curvature ◽  
Elliptical Cross ◽  
Curved Pipe ◽  
Curved Pipes ◽  
Ellipticity Ratio ◽  
Explicit Expressions

A literature survey (Berger, Talbot & Yao 1983) indicates that laminar viscous flow in curved pipes has been extensively investigated. Most of the existing analytical results deal with the case of circular cross-section. The important studies dealing with elliptical cross-sections are mainly due to Thomas & Walters (1965) and Srivastava (1980). The analysis of Thomas & Walters is based on Dean's (1927, 1928) approach in which the simplified forms of the momentum and continuity equations have been used. The analysis of Srivastava is essentially a seminumerical approach, in which no explicit expressions have been presented.In this paper, using elliptic coordinates and following the unsimplified formulation of Topakoglu (1967), the flow in a curved pipe of elliptical cross-section is analysed. Two different geometries have been considered: (i) with the major axis of the ellipse placed in the direction of the radius of curvature; and (ii) with the minor axis of the ellipse placed in the direction of the radius of curvature. For both cases explicit expressions for the first term of the expansion of the secondary-flow stream function as a function of the ellipticity ratio of the elliptic section have been obtained. After selecting a typical numerical value for the ellipticity ratio, the secondary-flow streamlines are plotted. The results are compared with that of Thomas & Walters. The remaining terms of the expansion of the flow field are not included, but they will be analysed in a future paper.

Secondary Velocity Fields in the Conducting Airways of the Human Lung

2007 ◽  
Vol 129 (5) ◽  
pp. 722-732 ◽  
Author(s):  
Frank E. Fresconi ◽  
Ajay K. Prasad
Keyword(s):  
Secondary Flow ◽  
Oscillatory Flow ◽  
Flow Direction ◽  
Flow Fields ◽  
Velocity Fields ◽  
Secondary Flows ◽  
Womersley Number ◽  
Dean Number ◽  
Steady Streaming ◽  
Secondary Velocity

An understanding of flow and dispersion in the human respiratory airways is necessary to assess the toxicological impact of inhaled particulate matter as well as to optimize the design of inhalable pharmaceutical aerosols and their delivery systems. Secondary flows affect dispersion in the lung by mixing solute in the lumen cross section. The goal of this study is to measure and interpret these secondary velocity fields using in vitro lung models. Particle image velocimetry experiments were conducted in a three-generational, anatomically accurate model of the conducting region of the lung. Inspiration and expiration flows were examined under steady and oscillatory flow conditions. Results illustrate secondary flow fields as a function of flow direction, Reynolds number, axial location with respect to the bifurcation junction, generation, branch, phase in the oscillatory cycle, and Womersley number. Critical Dean number for the formation of secondary vortices in the airways, as well as the strength and development length of secondary flow, is characterized. The normalized secondary velocity magnitude was similar on inspiration and expiration and did not vary appreciably with generation or branch. Oscillatory flow fields were not significantly different from corresponding steady flow fields up to a Womersley number of 1 and no instabilities related to shear were detected on flow reversal. These observations were qualitatively interpreted with respect to the simple streaming, augmented dispersion, and steady streaming convective dispersion mechanisms.

Accuracy of Cotton-Mouton polarimetry in sheared toroidal plasma of circular cross-section

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Yury Kravtsov ◽  
Janusz Chrzanowski
Keyword(s):  
Simple Model ◽  
Phase Shift ◽  
Perturbation Method ◽  
Cross Section ◽  
Amplitude Ratio ◽  
Circular Cross Section ◽  
Shear Angle ◽  
Tokamak Plasma ◽  
Toroidal Plasma ◽  
Small Shear

AbstractThe Cotton-Mouton effect in sheared plasma with helical magnetic lines is studied on the basis of the equation for complex amplitude ratio (CAR). A simple model for helical magnetic lines in sheared plasma of toroidal configuration is suggested. The equation for CAR in the sheared plasma is solved by perturbation method, using the small shear angle deviations as is characteristic for tokamak plasma. It is shown that the inaccuracy in polarization measurements caused by deviations of the sheared angle amounts to some percentage of the shearless Cotton-Mouton phase shift. One suggested method is to subtract the “sheared” term, which may improve the accuracy of the Cotton-Mouton measurements in the sheared plasma.

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