The Torsion Effect on Fully Developed Laminar Flow in Helical Square Ducts

1993 ◽  
Vol 115 (2) ◽  
pp. 292-301 ◽  
Author(s):  
Wen-Hwa Chen ◽  
Ray Jan
Keyword(s):  
Laminar Flow ◽  
Secondary Flow ◽  
Friction Factor ◽  
Stokes Equations ◽  
Axial Flow ◽  
Outer Wall ◽  
Square Duct ◽  
Torsion Effect ◽  
Inclined Angle ◽  
Curvature And Torsion

The continuity equation and Navier-Stokes equations derived from a non-orthogonal helical coordinate system are solved by the Galerkin finite-element method in an attempt to study the torsion effect on the fully developed laminar flow in the helical square duct. Since high-order terms of curvature and torsion are considered, the approach is also applicable to the problems with finite curvature and torsion. The interaction effects of curvature, torsion, and the inclined angle of the cross section on the secondary flow, axial velocity, and friction factor in the helical square duct are presented. The results show that the torsion has more pronounced effect on the secondary flow rather than the axial flow. In addition, unlike the flow in the toroidal square duct, Dean’s instability of the secondary flow, which occurs near the outer wall in the helical square duct, can be avoided due to the effects of torsion and/or inclined angle. In such cases, a decrease of the friction factor is observed. However, as the pressure gradient decreases to a small value, the friction factor for the toroidal square duct is also applicable to the helical square duct.

Fully Developed Laminar Flow in Curved Rectangular Channels

1976 ◽  
Vol 98 (1) ◽  
pp. 41-48 ◽  
Author(s):  
K. C. Cheng ◽  
Ran-Chau Lin ◽  
Jenn-Wuu Ou
Keyword(s):  
Laminar Flow ◽  
Secondary Flow ◽  
Friction Factor ◽  
Stokes Equations ◽  
Outer Region ◽  
Navier Stokes ◽  
Dean Number ◽  
Square Channel ◽  
Aspect Ratios ◽  
Rectangular Channels

The Navier-Stokes equations are solved by a numerical method for steady, fully developed, incompressible, laminar flow in curved rectangular channels considering the curvature ratio effect in the formulation. Solutions are obtained for aspect ratios 1, 2, 5 and 0.5 and Dean number ranges from 5 to 715, for example, for the case of square channel. It is found that an additional counter-rotating pair of vortices appears near the central outer region of the channel in addition to the familiar secondary flow at a certain higher Dean number depending on the aspect ratio. This phenomenon is consistent with Dean’s centrifugal instability problem and the secondary flow patterns with two pairs of counter-rotating vortices have not been reported in the past. The correlation equations for the friction factor are developed. The friction factor results are compared with the available theoretical and experimental results for the case of curved square channel and the agreement is found to be good.

G307 Experimental Study of the Flow in Helical Channel Rotors : Torsion Effect on the Secondary Flow and Friction Factor

2010 ◽  
Vol 2010 (0) ◽  
pp. 565-566
Author(s):  
Akira TANBA ◽  
Yasutaka HAYAMIZU ◽  
Shigeru OHTSUKA ◽  
Shinichi MORITA ◽  
Shinichiro YANASE ◽  
...  
Keyword(s):  
Experimental Study ◽  
Secondary Flow ◽  
Friction Factor ◽  
Helical Channel ◽  
Torsion Effect

Laminar flow in a two-dimensional plane channel with local pressure-dependent crossflow

2007 ◽  
Vol 593 ◽  
pp. 463-473 ◽  
Author(s):  
P. HALDENWANG
Keyword(s):  
Laminar Flow ◽  
Stokes Equations ◽  
Biological Fluids ◽  
Plane Channel ◽  
Axial Flow ◽  
Two Dimensional ◽  
Permeate Flux ◽  
Local Pressure ◽  
Porous Walls ◽  
Pressure Independent

Long ducts (or pipes) composed of transpiring (e.g. porous) walls are at the root of numerous industrial devices for species separation, as tangential filtration or membrane desalination. Similar configurations can also be involved in fluid supply systems, as irrigation or biological fluids in capillaries. A transverse leakage (or permeate flux), the strength of which is assumed to depend linearly on local pressure (as in Starling's law for capillary), takes place through permeable walls. All other dependences, as osmotic pressure or partial fouling due to polarization of species concentration, are neglected. To analyse this open problem we consider the simplest situation: the steady laminar flow in a two-dimensional channel composed of two symmetrical porous walls.First, dimensional analysis helps us to determine the relevant parameters. We then revisit the Berman problem that considers a uniform crossflow (i.e. pressure-independent leakage). We expand the solution in a series of Rt, the transverse Reynolds number. We note this series has a rapid convergence in the considered range of Rt (i.e. Rt ≤ O(1)). A particular method of variable separation then allows us to derive from the Navier–Stokes equations two new ordinary differential equations (ODE), which correspond to first and second orders in the development in Rt, whereas the zero order recovers the Regirer linear theory. Finally, both new ODEs are used to study the occurrence of two undesirable events in the filtration process: axial flow exhaustion (AFE) and crossflow reversal (CFR). This study is compared with a numerical approach.

Fully developed intermittent flow in a curved tube

1997 ◽  
Vol 347 ◽  
pp. 263-287 ◽  
Author(s):  
YUTAKA KOMAI ◽  
KAZUO TANISHITA
Keyword(s):  
Secondary Flow ◽  
Axial Velocity ◽  
Axial Flow ◽  
Outer Wall ◽  
Intermittent Flow ◽  
Cycle Period ◽  
Straight Tube ◽  
Flow Profile ◽  
Dean Number ◽  
Curved Tube

Fully developed intermittent flow in a strongly curved tube was numerically simulated using a numerical scheme based on the simpler method. Physiological pulsatile flow in the aorta was simulated as intermittent flow, with a waveform consisting of a pulse-like systolic flow period followed by a stationary diastolic period. Numerical simulations were carried out for the following conditions: Dean number κ=393, frequency parameter α=4–27, curvature ratio δ=1/2, 1/3 and 1/7, and intermittency parameter η=0–1/2, where η is the ratio of a systolic time to the cycle period. For α=18 and 27 the axial-flow profile in a systolic period becomes close to that of a sinusoidally oscillatory flow. At the end of the systole, a region of reversed axial velocity appears in the vicinity of the tube wall, which is caused by the blocking of the flow, similar to blocked flow in a straight tube. This area is enlarged near the inner wall of the bend by the curvature effect. Circumferential flow accelerated in a systole streams into the inner corner and collides at the symmetry line, which creates a jet-like secondary flow towards the outer wall. The region of reversed axial velocity is extended to the tube centre by the secondary flow. The development of the flow continues during the diastolic period for α higher than 8, and the flow does not completely dissipate, so that a residual secondary vortex persists until the next systole. Accordingly, the development of secondary flow in the following systolic phase is strongly affected by the residual vortex at the end of the previous diastolic phase, especially by stationary diastolic periods. Therefore, intermittent flow in a curved tube is strongly affected by the stationary diastolic period. For η=0 and 1/5, the induced secondary flow in a systole forms additional vortices near the inner wall, whereas for η=1/3 and 1/2 additional vortices do not appear. The characteristics of intermittent flow in a curved tube are also strongly affected by the length of the diastolic period, which represents a period of zero flow.

scholarly journals Numerical Flow Analysis in a Rotating Square Duct and a Rotating Curved-Duct

2000 ◽  
Vol 6 (1) ◽  
pp. 1-9
Author(s):  
Je Hyun Baekt ◽  
Chang Hwan Ko
Keyword(s):  
Laminar Flow ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Centrifugal Force ◽  
Coriolis Force ◽  
Axial Velocity ◽  
Numerical Study ◽  
Axial Direction ◽  
Square Duct ◽  
Rotation Rates

A numerical study is conducted on the fully-developed laminar flow of an incompressible viscous fluid in a square duct rotating about a perpendicular axis to the axial direction of the duct. At the straight duct, the rotation produces vortices due to the Coriolis force. Generally two vortex cells are formed and the axial velocity distribution is distorted by the effect of this Coriolis force. When a convective force is weak, two counter-rotating vortices are shown with a quasi-parabolic axial velocity profile for weak rotation rates. As the rotation rate increases, the axial velocity on the vertical centreline of the duct begins to flatten and the location of vorticity center is moved near to wall by the effect of the Coriolis force. When the convective inertia force is strong, a double-vortex secondary flow appears in the transverse planes of the duct for weak rotation rates but as the speed of rotation increases the secondary flow is shown to split into an asymmetric configuration of four counter-rotating vortices. If the rotation rates are increased further, the secondary flow restabilizes to a slightly asymmetric double-vortex configuration. Also, a numerical study is conducted on the laminar flow of an incompressible viscous fluid in a90°-bend square duct that rotates about axis parallel to the axial direction of the inlet. At a90°-bend square duct, the feature of flow by the effect of a Coriolis force and a centrifugal force, namely a secondary flow by the centrifugal force in the curved region and the Coriolis force in the downstream region, is shown since the centrifugal force in curved region and the Coriolis force in downstream region are dominant respectively.

Numerical Investigation of Laminar Flow in a Helical Pipe Filled With a Fluid Saturated Porous Medium: The Sensitivity of Secondary Flow to Parameter Variations

2004 ◽  
Author(s):  
Liping Cheng ◽  
Andrey V. Kuznetsov
Keyword(s):  
Porous Medium ◽  
Laminar Flow ◽  
Secondary Flow ◽  
Saturated Porous Medium ◽  
Axial Flow ◽  
Flow In Porous Media ◽  
Darcy Law ◽  
Helical Pipe ◽  
Fluid Saturated Porous Medium ◽  
Flow Inertia

Laminar flow in a helical pipe filled with a fluid saturated porous medium is investigated numerically. The analysis is based on a full momentum equation for the flow in porous media that accounts for the Brinkman and Forchheimer extensions of the Darcy law as well as for the flow inertia. Accounting for the flow inertia is shown to be important for predicting secondary flow in a helical pipe. The effects of the Darcy number, the Forchheimer coefficient as well as the curvature and torsion of the helical pipe on the axial flow velocity and secondary flow are investigated numerically.

Numerical Procedure for the Laminar Developed Flow in a Helical Square Duct

2005 ◽  
Vol 127 (1) ◽  
pp. 136-148 ◽  
Author(s):  
V. D. Sakalis ◽  
P. M. Hatzikonstantinou ◽  
P. K. Papadopoulos
Keyword(s):  
Laminar Flow ◽  
Friction Factor ◽  
Cross Section ◽  
Numerical Procedure ◽  
Governing Equations ◽  
Axial Pressure ◽  
Dean Number ◽  
Square Duct ◽  
Orthogonal Coordinate System ◽  
Axial Pressure Gradient

The incompressible fully developed laminar flow in a helically duct of square cross section is studied expressing the governing equations in terms of an orthogonal coordinate system. Numerical results are obtained with the described continuity, vorticity, and pressure (CVP) numerical method using a colocation grid for all variables. Since there are not approximations, the interaction effects of curvature, torsion and axial pressure gradient on the velocity components and the friction factor are presented. The results show that the torsion deforms substantially the symmetry of the two centrifugal vortices of the secondary flow, which for large values of torsion combined with small curvature tend to one vortex covering the whole cross section. The friction factor decreases for torsion in the range 0 to 0.1 and increases as the torsion increases further, a behavior which is more profound as the Dean number increases. Our results are stable for the calculated Dean numbers.

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.

Efficient Estimation of the Friction Factor for Forced Laminar Flow in Axially Symmetric Corrugated Pipes

2010 ◽  
Author(s):  
Patricio I. Rosen Esquivel ◽  
Jan H. M. ten Thije Boonkkamp ◽  
Jacques A. M. Dam ◽  
Robert M. M. Mattheij
Keyword(s):  
Laminar Flow ◽  
Friction Factor ◽  
Stokes Equations ◽  
Efficient Estimation ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Cfd Simulations ◽  
Navier Stokes Equations ◽  
Pressure Profiles ◽  
Adequate Accuracy

In this paper we present an efficient method for calculating the friction factor for forced laminar flow in arbitrary axially symmetric pipes. The approach is based on an analytic expression for the friction factor, obtained after integrating the Navier-Stokes equations over a segment of the pipe. The friction factor is expressed in terms of surface integrals over the pipe wall, these integrals are then estimated by means of approximate velocity and pressure profiles computed via the method of slow variations. Our method for computing the friction factor is validated by comparing the results, to those obtained using CFD techniques for a set of examples featuring pipes with sinusoidal walls. The amplitude and wavelength parameters are used for describing their influence on the flow, as well as for characterizing the cases in which the method is applicable. Since the approach requires only numerical integration in one dimension, the method proves to be much faster than general CFD simulations, while predicting the friction factor with adequate accuracy.

Large Eddy Simulation of Flow and Heat Transfer in an Internal Cooling Duct With High Blockage Ratio 45° Staggered Ribs

2005 ◽  
Author(s):  
Aroon K. Viswanathan ◽  
Danesh K. Tafti ◽  
Samer Abdel-Wahab
Keyword(s):  
Heat Transfer ◽  
Secondary Flow ◽  
Flow Direction ◽  
Outer Wall ◽  
Internal Cooling ◽  
Square Duct ◽  
Square Channel ◽  
Heat Transfer Augmentation ◽  
Eddy Simulation ◽  
Numerical Predictions

Numerical predictions of a hydrodynamic and thermally developed turbulent flow are presented for a unit period of a stationary duct with square ribs aligned at 45° to the main flow direction. The rib height to channel hydraulic diameter (e/Dh) is 0.375 and the rib pitch to rib height (P/e) is 10. The domain under consideration is a rectangular passage of aspect ratio 1:2.5 with 45° ribs on the top and bottom walls arranged in a staggered fashion. The computations are carried out for a bulk Re of 27,000. The rib geometry introduces a strong secondary flow along the rib. A large helical vortex develops behind the rib which breaks down before it reaches the outer wall. This results in higher heat transfer at the inner wall as compared to the outer wall, which is in contrast to the trend observed in a square channel with low blockage ribs. In a square duct with low blockage ribs the secondary flow has two counter-rotating cells which do not change direction through the channel. However in this case only one rotating cell is observed in this case, which changes direction as it passes over successive ribs. The average friction and the heat transfer augmentation ratios are consistent with the experimental results [1], predicting values within 15% of the measured quantities.

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