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.

Instability and transitions of flow in a curved square duct: the development of two pairs of Dean vortices

1996 ◽  
Vol 314 ◽  
pp. 227-246 ◽  
Author(s):  
Philip A. J. Mees ◽  
K. Nandakumar ◽  
J. H. Masliyah
Keyword(s):  
Secondary Flow ◽  
Flow Pattern ◽  
Stokes Equations ◽  
Velocity Profiles ◽  
Navier Stokes ◽  
Flow State ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Measured Velocity ◽  
Dean Vortices

Steady developing flow of an incompressible Newtonian fluid in a curved duct of square cross-section (the Dean problem) is investigated both experimentally and numerically. This study is a continuation of the work by Bara, Nandakumar & Masliyah (1992) and is focused on flow rates between Dn = 200 and Dn = 600 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension; the curvature ratio, R/a, is 15.1).Numerical simulations based on the steady three-dimensional Navier – Stokes equations predict the development of a 6-cell secondary flow pattern above a Dean number of 350. The 6-cell state consists of two large Ekman vortices and two pairs of small Dean vortices near the outer wall that result from the primary instability that is of centrifugal nature. The 6-cell flow state develops near θ = 80° and breaks down symmetrically into a 2-cell flow pattern.The apparatus used to verify the simulations had a duct dimension of 1.27 cm and a streamwise length of 270°. At a Dean number of 453, different velocity profiles of the 6-cell flow state at θ = 90° and spanwise profiles of the streamwise velocity at every 20° were measured using a laser-Doppler anemometer. All measured velocity profiles, as well as flow visualization of secondary flow patterns, are in very good agreement with the simulations, indicating that the parabolized Navier – Stokes equations give an accurate description of the flow.Based on the similarity with boundary layer flow over a concave wall (the Görtler problem), it is suggested that the transition to the 6-cell flow state is the result of a decreasing spanwise wavelength of the Dean vortices with increasing flow rate. A numerical stability analysis shows that the 6-cell flow state is unconditionally unstable. This is the first time that detailed experiments and simulations of the development of a 6-cell flow state are reported.

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.

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.

Some numerical experiments on developing laminar flow in circular-sectioned bends

1985 ◽  
Vol 154 ◽  
pp. 357-375 ◽  
Author(s):  
J. A. C. Humphrey ◽  
H. Iacovides ◽  
B. E. Launder
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Streamwise Velocity ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Numerical Predictions

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

The effects of secondary flow on the laminar dispersion of an injected substance in a curved tube

1970 ◽  
Vol 315 (1520) ◽  
pp. 99-113 ◽  
Keyword(s):  
Secondary Flow ◽  
Cross Section ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Flux Ratio ◽  
Original Method ◽  
Asymptotic Solutions ◽  
Navier Stokes ◽  
Inviscid Fluid ◽  
Mean Velocity

The numerical solution by McConalogue & Srivastava (1968) of Dean’s simplified Navier–Stokes equations for the laminar flow of an inviscid fluid through a tube of circular cross-section of radius a , coiled in a circular arc of radius L , and valid for k in the range (16.6, 77.1), where k = Re √( a / L ), Re the Reynolds number, is compared with experiment, correlated to the asymptotic solutions for k > 100, and extended to study the convective axial dis­persion of a substance injected into the tube. The variation of the calculated flux ratio agrees closely with White’s (1929) measurements of the inverse quantity over the same range, and the field patterns for the upper end of the range establish the validity of the two basic assumptions of the asymptotic solutions. The original method is extended to calculate the mean axial velocity of a typical particle of the fluid and to present the statis­tical distribution of mean velocity over the particles of a substance injected as a thin disk uniformly over the cross section of the tube. These distributions are used to display the varia­tion with k of the shape of indicator concentration-time curves. The expected effect of secondary flow, in producing a more uniform distribution of velocity over the fluid than in Poiseuille flow, is evident.

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.

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.

Calculation of the steady flow through a curved tube using a new finite-difference method

1980 ◽  
Vol 99 (3) ◽  
pp. 449-467 ◽  
Author(s):  
S. C. R. Dennis
Keyword(s):  
Finite Difference ◽  
Steady Flow ◽  
Stokes Equations ◽  
Steady Motion ◽  
Circular Cross Section ◽  
Navier Stokes ◽  
Dean Number ◽  
Numerical Work ◽  
Curved Tube ◽  
Flow Through

A numerical method is described which is suitable for solving the equations governing the steady motion of a viscous fluid through a slightly curved tube of circular cross-section but which is also applicable to the solution of any problem governed by the steady two-dimensional Navier–Stokes equations in the plane polar co-ordinate system. The governing equations are approximated by a scheme which yields finite-difference equations which are of second-order accuracy with respect to the grid sizes but which have associated matrices which are diagonally dominant. This makes them generally more amenable to solution by iterative techniques than the approximations obtained using standard central differences, while preserving the same order of accuracy.The main object of the investigation is to obtain numerical results for the problem of steady flow through a curved tube which corroborate previous numerical work on this problem in view of a recent paper (Van Dyke 1978) which tends to cast doubt on the accuracy of previous calculations at moderately high values of the Dean number; this is the appropriate Reynolds-number parameter in this problem. The present calculations tend to verify the accuracy of previous results for Dean numbers up to 5000, beyond which it is difficult to obtain accurate results. Calculated properties of the flow are compared with those obtained in previous numerical work, with the predictions of boundary-layer theory for large Dean numbers and with the predictions of Van Dyke (1978).

scholarly journals Viscous Analysis of Three-Dimensional Turbomachinery Flows on Body Conforming Grids Using an Implicit Solver

1991 ◽  
Author(s):  
K. F. Weber ◽  
R. A. Delaney
Keyword(s):  
Coordinate System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Turbulence Effects ◽  
Implicit Solver ◽  
Cartesian Coordinate ◽  
Solution Accuracy

A 3-D Navier-Stokes analysis for turbomachinery flows on C- or O-type grids is presented. The analysis is based on the Beam and Warming implicit algorithm for solution of the unsteady Navier-Stokes equations and is derived from an early version of the ARC3D flow code developed at NASA Ames Research Center. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the z-axis, and then mapped to a general body-fitted coordinate system. All viscous terms are calculated and the turbulence effects are modelled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked body-conforming grids. Modifications made to convert ARC3D from external flow to internal turbomachinery flows and to improve solution accuracy are given in detail. The body-conforming grid construction procedure is also presented. Calculations for several rotor flows have been made, and results of code experimental verification studies are presented. Comparisons of the solutions obtained on the C- and O-type grids are also presented, with particular attention to shock resolution.

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.

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