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.

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.

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.

The Dean equations extended to a helical pipe flow

1989 ◽  
Vol 203 ◽  
pp. 289-305 ◽  
Author(s):  
M. Germano
Keyword(s):  
Secondary Flow ◽  
Cross Section ◽  
Pipe Flow ◽  
Circular Cross Section ◽  
Order Effect ◽  
Displacement Effect ◽  
Elliptical Cross ◽  
First Order ◽  
Helical Pipe ◽  
Axial Pressure Gradient

In this paper the Dean (1928) equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter λ/[Rscr ] where λ is the ratio of the torsion τ to the curvature κ of the pipe axis and [Rscr ] the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit λ/[Rscr ] → ∞ (twisted pipes, provided only with torsion), the first-order ‘displacement’ effect of the walls on the secondary flow, analysed in detail by Choi (1988), is recovered.Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by Kao (1987) are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by Murata et al. (1981) is discussed.

Numerical Study of Turbulent Helical Pipe Flow With Comparison to the Experimental Results

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Anup Kumer Datta ◽  
Yasutaka Hayamizu ◽  
Toshinori Kouchi ◽  
Yasunori Nagata ◽  
Kyoji Yamamoto ◽  
...  
Keyword(s):  
Experimental Data ◽  
Turbulent Flow ◽  
Secondary Flow ◽  
Pipe Flow ◽  
Numerical Study ◽  
Circular Cross Section ◽  
Turbulence Models ◽  
Experimental Results ◽  
Helical Pipe ◽  
Wide Range

Turbulent flow through helical pipes with circular cross section is numerically investigated comparing with the experimental results obtained by our team. Numerical calculations are carried out for two helical circular pipes having different pitches and the same nondimensional curvature δ (=0.1) over a wide range of the Reynolds number from 3000 to 21,000 for torsion parameter β (=torsion /2δ  = 0.02 and 0.45). We numerically obtained the secondary flow, the axial flow and the intensity of the turbulent kinetic energy by use of three turbulence models incorporated in OpenFOAM. We found that the change to fully developed turbulence is identified by comparing experimental data with the results of numerical simulations using turbulence models. We also found that renormalization group (RNG) k−ε turbulence model can predict excellently the fully developed turbulent flow with comparison to the experimental data. It is found that the momentum transfer due to turbulence dominates the secondary flow pattern of the turbulent helical pipe flow. It is interesting that torsion effect is more remarkable for turbulent flows than laminar flows.

scholarly journals An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Süleyman Cengizci
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Numerical Procedure ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Convection Term ◽  
Linear Delay ◽  
Delay Differential

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

scholarly journals Flow through a helical pipe with rectangular cross-section

1970 ◽  
Vol 4 (2) ◽  
pp. 99-110
Author(s):  
Md Mahmud Alam ◽  
Delowara Begum ◽  
K Yamamoto
Keyword(s):  
Aspect Ratio ◽  
Iterative Method ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Main Tool ◽  
Numerical Approach ◽  
Dean Number ◽  
Helical Pipe ◽  
Flow Through ◽  
Marine Engineering

The effects of torsion, aspect ratio and curvature on the flow in a helical pipe of rectangular cross- section are studied by introducing a non-orthogonal helical coordinate system. Spectral method is applied as main tool for numerical approach where Chebyshev polynomial is used. The numerical calculations are obtained by the iterative method. The calculations are carried out for 0≤ δ ≤0.02, 1≤ λ ≤ 2.85, 1≤ γ ≤2.4, at Dn = 50 & 100 respectively, where d is the non-dimensional curvature, l the torsion parameter, g the aspect ratio and  Dn the pressure driven parameter (Dean number).DOI: http://dx.doi.org/10.3329/jname.v4i2.991 Journal of Naval Architecture and Marine Engineering Vol.4(2) 2007 p.99-110

SPECTRAL NUMERICAL SIMULATION OF MAGNETO-PHYSIOLOGICAL LAMINAR DEAN FLOW

2014 ◽  
Vol 14 (04) ◽  
pp. 1450047 ◽  
Author(s):  
O. ANWAR BEG ◽  
MD. MAINUL HOQUE ◽  
M. WAHIDUZZAMAN ◽  
MD. MAHMUD ALAM ◽  
M. FERDOWS
Keyword(s):  
Transport Model ◽  
Computational Simulation ◽  
Large Diameter ◽  
Experimental Studies ◽  
Circular Cross Section ◽  
Flow Characteristics ◽  
Axial Flow ◽  
Momentum Conservation ◽  
Dean Number ◽  
Dean Flow

A computational simulation of magnetohydrodynamic laminar blood flow under pressure gradient through a curved bio-vessel, with circular cross-section is presented. Electrical conductivity and other properties of the biofluid (blood) are assumed to be invariant. A Newtonian viscous flow (Navier–Stokes magnetohydrodynamic) model is employed which is appropriate for large diameter blood vessels, as confirmed in a number of experimental studies. Rheological effects are therefore neglected as these are generally only significant in smaller diameter vessels. Employing a toroidal coordinate system, the steady-state, three-dimensional mass and momentum conservation equations are developed. With appropriate transformations, the transport model is non-dimensionalized and further simplified to a pair of axial and secondary flow momenta equations with the aid of a stream function. The resulting non-linear boundary value problem is solved with an efficient, spectral collocation algorithm, subject to physically appropriate boundary conditions. The influence of magnetic body force parameter, Dean number and vessel curvature on the flow characteristics is examined in detail. For high magnetic parameter and Dean number and low curvature, the axial flow is observed to be displaced toward the center of the vessel with corresponding low fluid particle vorticity strengths. Visualization is achieved with the MAPLE software. The simulations are relevant to cardiovascular biomagnetic flow control.

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