Bifurcation and stability analysis of laminar flow in curved ducts

Machane, Werner

doi:10.1002/fld.2147

Cited by 4 publications

(4 citation statements)

References 42 publications

(72 reference statements)

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“…In figure 2 we show schematically a portion of the bifurcation diagram found. It incorporates branches 1-8 which are solutions found in previous studies (Boshier & Mestel 2014;Machane 2010;Daskopoulos & Lenhoff 1989;McConalogue & Srivastava 1968) and reproduced here and branches 9-15 which are found for the first time in the current work. To ease visualisation, the diagram represents branches projected onto the real K 2 axis.…”

Section: Constructing Complex Solution Branchessupporting

confidence: 80%

“…As expected, this problem is found to exhibit many features similar to that of the circular pipe. Machane (2010) found multi-vortex solutions for elliptic cross-sections of aspect ratio λ = 1.45, though his bifurcation diagram differs from that found for the circular cross-section. Interestingly, in his study he concluded that the number of vortices on a particular branch may gradually change without running through a singular point, suggesting that 'number of vortices' is not a robust diagnostic for distinguishing between solutions in the elliptic case.…”

Section: Pipes Of Elliptic Cross-sectionmentioning

confidence: 55%

“…Historically, Dean flow has been studied predominantly in circular pipes, but this is not necessary. In the final part of this paper, we perform analysis and numerics for elliptical cross-sections, as in Machane (2010). We use elliptic coordinates and obtain solutions by both a Dean series and DNS.…”

Section: Introductionmentioning

confidence: 99%

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Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

Boshier

Mestel

2017

J. Fluid Mech.

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Steady, incompressible flow down a slowly-curving circular pipe is considered, analyti- cally and numerically. Both real and complex solutions are investigated. Using high-order Hermite–Pad ́e approximants, the Dean series solution is analytically continued outside its circle of convergence where it predicts a complex solution branch for real, positive Dean number, K . This is confirmed by numerical solution. It is shown that other previously unknown solution branches exist for all K > 0, which are related to an unforced com- plex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokes equations in most geometries. By means of path continuation, numerical solutions are followed around the complex K -plane. The standard Dean two-vortex solution is shown to lie on the same hypersurface as the eigensolution and the four-vortex solutions found in the literature. Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case. There is an imaginary singularity limiting convergence of the Dean series, an unforced solution at K = 0 and nonuniqueness for K > 0, culminating in a real bifurcation

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Section: Constructing Complex Solution Branchessupporting

confidence: 80%

Section: Pipes Of Elliptic Cross-sectionmentioning

confidence: 55%

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Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

Boshier

Mestel

2017

J. Fluid Mech.

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“…Recently, Watanabe and Yanase (2013) obtained various bifurcation structures for the threedimensional case and found a connection between the branches. Bifurcation diagram from small to large aspect ratio for different aspect ratios have been also calculated by Mechane (2010). Using Euler-Newton continuation method, Yang and Wang (2001) pointed out the position of the symmetric and asymmetric solution of branches.…”

Section: Introductionmentioning

confidence: 99%

Investigation on Energy Distribution in Steady and Unsteady Flow Instabilities through a Bend Square Pipe

et al. 2021

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Fluid flow analysis through a bend pipe is extensively conducted in practical and cell separation operations. It is observed that flow behaviors in the bend pipe are influenced by some parameters such as curvature, aspect ratio, etc. As a result, various phenomena, steady solution branches, unsteady solutions, energy transfer are changed. In this paper, the acts of flows are performed together for fixed curvature, δ = 0.2, and Prandtl number, Pr = 7.0 (water). Here, for a wide variety of Dean numbers (100 ≤ Dn ≤ 1000) and three fixed Grashof numbers, Gr = 100, 500, and 1000; time-independent solutions with linear stabilities are investigated first where only the first steady branch exhibits linear stability out of two steady solution branches obtained. Then, different flow transitions between the required range of Dean numbers (Dn) and several Grashof numbers (Gr) are investigated using time-dependent solutions. Power spectrum density (PSD) is further revealed in order to gain a deeper understanding of periodic and multi-periodic flows. Flow velocity contours including axial flow (AF) and secondary flow (SF) and their temperature profiles (TP) are also exposed. The SFs reveal that two- to four-vortex flows are produced due to the turning of steady branch and the flow instabilities. Furthermore, the energy transfer between the cooled and heated sidewalls of the pipe is calculated. Finally, a link between centrifugal and body force with the energy transfer has been shown in this research which reveals that the fluid has merged that certainly rises the overall energy transfer.

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Separations and secondary structures due to unsteady flow in a curved pipe

2017

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Unsteady flows in highly curved geometries are of interest in many engineering applications and also in physiological flows. In this study, we use flow visualization and computational fluid dynamics to study unsteady flows in a highly curved tube ($\unicode[STIX]{x1D6FD}=0.3$) with square cross-section; here, $\unicode[STIX]{x1D6FD}$ is the ratio of the half edge length to the radius of curvature of the tube. To explore the combined effects of curvature and pulsatility, we use a single flow pulse of duration $T$ and peak area averaged axial velocity $U_{p(max)}$, which are independently varied to investigate a range of Dean and Womersley numbers. This range includes cases corresponding to flows in the ascending aorta. We observe radially inward moving secondary flows which have the structure of wall jets on the straight walls; their subsequent collision on the inner wall leads to a re-entrant radially outward moving jet. The wall jet arises due to an imbalance between the centrifugal force and the radial pressure gradient. During the deceleration phase, the low-axial-momentum fluid accumulated in the jet reverses direction and leads to flow separation near the inner wall. We use boundary layer equations to derive scales, which have not been reported earlier, for the secondary flow velocities, the wall shear stress components and the distance ($\hat{P}$) traversed by the secondary flow structures in the transverse plane. We show that $\hat{P}$ predicts the movement of vortical structures until collision. In the limit $\unicode[STIX]{x1D6FD}\rightarrow 0$, the Reynolds number based on this secondary flow velocity scale asymptotes to the secondary streaming Reynolds number proposed by Lyne (J. Fluid Mech., vol. 45 (01), 1971, pp. 13–31) in loosely curved pipes. The magnitude of the secondary flow velocity is high and ${\sim}40\,\%$ of $U_{p(max)}$ for physiological flow conditions. We show that the flow separation on the inner wall has origins in the secondary flow, which was reported in a few earlier studies, and is not due to the axial pressure gradient in the tube as proposed earlier. The wall shear stress components, hypothesized to be important in arterial mechanobiology, may be estimated using our scaling relations for geometries with different curvatures and varying pulsatilities.

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Bifurcation and stability analysis of laminar flow in curved ducts

Cited by 4 publications

References 42 publications

Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

Investigation on Energy Distribution in Steady and Unsteady Flow Instabilities through a Bend Square Pipe

Separations and secondary structures due to unsteady flow in a curved pipe

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