Toroidal Pipe Research Articles

Modal stability analysis of toroidal pipe flow approaching zero curvature

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$ , the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$ . Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$ . This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.

Analyzing the Influence of Dean Number on an Accelerated Toroidal: Insights from Particle Imaging Velocimetry Gyroscope (PIVG)

Computational Fluid Dynamics (CFD) simulations were utilized in this study to comprehensively explore the fluid dynamics within an accelerated toroidal vessel, specifically those central to Particle Imaging Velocimetry Gyroscope (PIVG) technology. To ensure the robustness of our simulations, we systematically conducted grid convergence studies and quantified uncertainties, affirming the stability, accuracy, and reliability of our computational grid and results. Comprehensive validation against experimental data further confirmed our simulations’ fidelity, emphasizing the model’s fidelity. As the PIVG is set up to address the primary flow through the toroidal pipe, we focused on the interaction between the primary and secondary flows to provide insights into the relevant dynamics of the fluid. In our investigation covering Dean numbers (De) from 10 to 70, we analyzed diverse aspects, including primary flow, secondary flow patterns, pressure distribution, and the interrelation between primary and secondary flows within toroidal structures, offering a comprehensive view across this range. Our research indicated stability and fully developed fluid dynamics within the toroidal pipe under accelerated angular velocity, particularly for low De. Furthermore, we identified an optimal Dean number of 11, which corresponded to ideal dimensions for the toroidal geometry with a curvature radius of 25 mm and a cross-sectional diameter of 5 mm. This study enhances our understanding of toroidal fluid dynamics and highlights the pivotal role of CFD in optimizing toroidal vessel design for advanced navigation technologies.

Floquet stability analysis of pulsatile flow in toroidal pipes

The linear temporal stability of the fully developed pulsatile flow in a torus with high curvature is investigated using Floquet theory. The baseflow is computed via a Newton-Raphson iteration in frequency space to obtain basic states at supercritical Reynolds numbers in the steady case for two curvatures, δ=0.1 and 0.3, exhibiting structurally different linear instabilities for the steady flow. The addition of a pulsatile component is found to be overall stabilizing over a wide range of pulsation amplitudes, in particular for high values of curvature. The pulsatile flows are found to be at most transiently stable with large intracyclic growth rate variations even at small pulsation amplitudes. While these growth rates are likely insufficient to trigger an abrupt transition at the parameters in this work, the trends indicate that this is indeed likely for higher pulsation amplitudes, similar to pulsatile flow in straight pipes. At the edge of the considered parameter range, subharmonic eigenvalue orbits in the local spectrum of the time-periodic operator, recently found in pulsating channel flow, have been confirmed also for pulsatile flow in toroidal pipes, underlining the generality of this phenomenon. Published by the American Physical Society 2024

Trajectory dynamics of particles in accelerated toroidal pipe: A computational study using CFD-DPM simulations

In this study, we investigate the behavior of solid particles of varying sizes within an accelerated toroidal filled with fluid using computational fluid dynamics- discrete phase model (CFD-DPM) simulations. A comprehensive grid independence study and validation were conducted to ensure the accuracy of our simulations. We focus on particles with 10-, 50-, and 100-microns diameters. Our simulations indicate that the primary flow (relative tangential velocity) of both the fluid and the particles follow the input acceleration. We also examined the secondary flow within the toroidal. Smaller particles (10 microns) closely follow the fluid flow with minimal lag, while larger particles (50 and 100 microns) exhibit a noticeable lag due to the inertia effect. Furthermore, the displacement from the initial position is also dependent on the particle size, with smaller particles traveling further distances. However, despite the ongoing acceleration of the fluid and the toroidal, the largest particles (100 microns) eventually settle and stop due to their high inertia (mass), and their velocity begins to decrease during the acceleration of the toroidal. These findings highlight the significant role of particle size in determining particle behavior within a toroidal, paving the way for future research and Particle Imaging Velocimetry Gyroscope (PIVG) application.

Chiral symmetry breaking and entropy production in Dean vortices

In toroidal pipes, the secondary flow in cross section is a mirror symmetric pair of counter-rotating axially oriented Dean vortices. This mirror symmetry is broken in helical pipes. We investigate in detail the mirror symmetry breaking in these secondary flows in going from toroidal to helical geometries. We quantify the degree of mirror symmetry breaking in helical flows by calculating both an (i) order-parameter − 1 ≤ χ ≤ 1 that measures the net integrated chirality of vortices in section and (ii) the entropy production due to both viscous shear forces and convection for Dean vortices as the Dean number and pitch of the helix are varied. We prove that the entropy production due to convective processes is always greater than that due to viscous shear, for stationary incompressible flows in the absence of body forces. For the same pipe radius and pipe curvature, fluid density, viscosity, and entrance flows, the vortex entropy production in the stationary state is minimized for helical conduits (for a given Dean number) with respect to that of toroidal pipes (zero pitch). The dissipation in the fluid flow due to Dean vortices decreases in going from a toroidal to a helical geometry, while the chiral order parameter tends to χ = ± 1 for finite values of the pitch as the Dean number is decreased.

Sound attenuation characteristics of acoustically lined curved pipes

Boundary element solutions of the Helmholtz integral equation for the sound attenuation of acoustically lined curved pipes are presented. The curved pipes are in the form of cylindrical and toroidal coordinates. The main objective of the study is a comparative analysis of sound attenuation characteristics of lined toroidal pipes. Although toroidal pipes with circular cross-section are used extensively in industrial and agricultural applications, they have been taken little attention in literature. One reason may be the impossibility of theoretical solution since wave equation is not separable in toroidal coordinates. A numerical method should be utilized in the solution; in fact, not many researchers are familiar with toroidal coordinates on the contrary of other coordinate systems, such as cylindrical coordinates. The use of toroidal coordinates and generation of a pipe in these coordinates are illustrated herein. Sound waves in the pipes are produced by the vibrating inlet surface and resulting sound pressure distribution on the interior surfaces of side walls and anechoic exit are computed. Sound transmission loss of the pipes is represented by the ratio of inlet to exit pressures, as sound attenuation spectra. The results of cylindrical and toroidal pipes with the same cross-sectional area are compared. The sound attenuation performance of lined toroidal pipes is compared in terms of bend sharpness as a measure of curvature. The comparison is extended to cover the hard-walled and lined toroidal pipes to conclude their common and/or different acoustic behaviours.

Investigation of Newtonian and Power-Law Blood Flow Models in a 180° Curved Pipe at Low to Medium Shear Rate

The purpose of this brief paper is to obtain quantitative information on mean velocity profile in ideal vasculature (i.e. straight and toroidal pipes) at low to medium shear rates. To shed the light on the significance of considering blood shear-thinning properties, the power-law model is compared to the commonly used Newtonian viscosity hypothesis. Validated CFD models of blood flow were established and parameterized to solve steady incompressible blood flow under Reynolds number of 50~200. The calculations of the Reynolds number and boundary conditions adopted the shear-thinning index of the power-law models to provide physically correct benchmark for the comparison presented herein. Velocity profiles for Newtonian and non-Newtonian fluid flow are described and sketched. Shear rate values had a range of 20~200 s-1 which represents the physiological range found in cerebral vasculature. This study provides the means to estimate the effect of the non-Newtonian properties of the blood on the flow patterns. It is clearly shown that the difference between Newtonian and power-law blood flow models is not significantly affected by Reynolds number for the current range of shear rate. The differences identified in the pressuredrop per unit length and average wall-shear stress were found to be of significant values. The difference between the Newtonian and power-law model (case1) in the pressure drop per unit length for the straight pipe was 386 while for the curved pipe was 371. These differences increased to 538 at Re=200 for the straight pipe and reached 603 for the curved pipe. This research suggests that the non-Newtonian effects of cerebral blood flow should be considered in the respective CFD models.

A comparative study on instability of steady flows in helical pipes

A computational study of three-dimensional instability of steady flows in a helical pipe of arbitrary curvature and torsion is carried out for the first time. The problem is formulated in Germano coordinates in two equivalent but different forms of the momentum equations so that results obtained using both formulations cross verify each other. An additional formulation in the cylindrical coordinates is applied for a limiting case of the toroidal pipe. The calculations are performed by the finite volume and finite difference methods. Grid independence of the results is established for both steady flows, the eigenvalues associated with the linear stability problem, and the critical parameters. The calculated steady flows agree well with experimental measurements and previous numerical results. The computed critical Reynolds numbers corresponding to the onset of oscillatory instability agree well with the most recent experimental results, but disagree with the earlier ones. Novel results related to the parametric stability study are reported.

Critical Point for Bifurcation Cascades and Featureless Turbulence.

In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyze the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures (pipe-to-torus diameter ratio) and supercritical transition at high curvatures, as was previously documented. We unveil an additional step in the bifurcation cascade and provide evidence that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emerging through subcritical transition mechanisms.

The vanishing of strong turbulent fronts in bent pipes

Isolated patches of turbulence in transitional straight pipes are sustained by a strong instability at their upstream front, where the production of turbulent kinetic energy (TKE) is up to five times higher than in the core. Direct numerical simulations presented in this paper show no evidence of such strong fronts if the pipe is bent. We examine the temporal and spatial evolution of puffs and slugs in a toroidal pipe with pipe-to-torus diameter ratio$\unicode[STIX]{x1D6FF}=D/d=0.01$at several subcritical Reynolds numbers. Results show that the upstream overshoot of TKE production is at most one-and-a-half times the value in the core and that the average cross-flow fluctuations at the front are up to three times lower if compared to a straight pipe, while attaining similar values in the core. Localised turbulence can be sustained at smaller energies through a redistribution of turbulent fluctuations and vortical structures by the in-plane Dean motion of the mean flow. This asymmetry determines a strong localisation of TKE production near the outer bend, where linear and nonlinear mechanisms optimally amplify perturbations. We further observe a substantial reduction of the range of Reynolds numbers for long-lived intermittent turbulence, in agreement with experimental data from the literature. Moreover, no occurrence of nucleation of spots through splitting could be detected in the range of parameters considered. Based on the present results, we argue that this mechanism gradually becomes marginal as the curvature of the pipe increases and the transition scenario approaches a dynamical switch from subcritical to supercritical.

Comparative Analysis of the Discharge Coefficients of Critical Nozzles

We perform the comparative and statistical analyses of the discharge coefficients of critical nozzles with toroidal branch pipes and discuss the main approaches aimed at the evaluation of the discharge coefficients. We present the equations for the dependences used to describe the behavior of the discharge coefficients of critical nozzles as functions of the Reynolds number. By using an example of critical Venturi nozzles, we study the factors affecting the accuracy of measurements of the gas flow rate.

Flow topology of rare back flow events and critical points in turbulent channels and toroidal pipes

A study of the back flow events and critical points in the flow through a toroidal pipe at friction Reynolds number Reτ ≈ 650 is performed and compared with the results in a turbulent channel flow at Reτ ≈ 934. The statistics and topological properties of the back flow events are analysed and discussed. Conditionally-averaged flow fields in the vicinity of the back flow event are obtained, and the results for the torus show a similar streamwise wall-shear stress topology which varies considerably for the spanwise wall-shear stress when compared to the channel flow. The comparison between the toroidal pipe and channel flows also shows fewer back flow events and critical points in the torus. This cannot be solely attributed to differences in Reynolds number, but is a clear effect of the secondary flow present in the toroidal pipe. A possible mechanism is the effect of the secondary flow present in the torus, which convects momentum from the inner to the outer bend through the core of the pipe, and back from the outer to the inner bend through the pipe walls. In the region around the critical points, the skin-friction streamlines and vorticity lines exhibit similar flow characteristics with a node and saddle pair for both flows. These results indicate that back flow events and critical points are genuine features of wall-bounded turbulence, and are not artifacts of specific boundary or inflow conditions in simulations and/or measurement uncertainties in experiments.

Characterisation of the steady, laminar incompressible flow in toroidal pipes covering the entire curvature range

This work is concerned with a detailed investigation of the steady (laminar), incompressible flow inside bent pipes. In particular, a toroidal pipe is considered in an effort to isolate the effect of the curvature, δ, on the flow features, and to compare the present results to available correlations in the literature. More than 110 000 numerical solutions are computed, without any approximation, spanning the entire curvature range, 0 ≤ δ ≤ 1, and for bulk Reynolds numbers Re up to 7 000, where the flow is known to be unsteady. Results show that the Dean number De provides a meaningful non-dimensional group only below very strict limits on the curvature and the Dean number itself. For δ>10−6 and De > 10, in fact, not a single flow feature is found to scale well with the Dean number. These considerations are also valid for quantities, such as the Fanning friction factor, that were previously considered Dean-number dependent only. The flow is therefore studied as a function of two equally important, independent parameters: the curvature of the pipe and the Reynolds number. The analysis shows that by increasing the curvature the flow is fundamentally changed. Moderate to high curvatures are not only quantitatively, but also qualitatively different from low δ cases. A complete description of some of the most relevant flow quantities is provided. Most notably the friction factor f for laminar flow in curved pipes by Ito [J. Basic Eng. 81:123–134 (1959)] is reproduced, the influence of the curvature on f is quantified and the scaling is discussed. A complete database including all the computed solutions is available at www.flow.kth.se.

Swirl-switching phenomenon in turbulent flow through toroidal pipes

Direct numerical simulations (DNSs) are performed to investigate turbulent flows in toroidal pipes with two different curvatures. By means of proper orthogonal decomposition (POD), dominant coherent structures in the flow are identified. The most energetic structures in the strongly curved pipes (with curvature κ=0.1 and κ=0.3 and Reb=11,700 based on bulk velocity and diameter) are very similar in structure for both configurations studied. The dominant modes (shape and frequency) appear to match the coherent structures responsible for the swirl switching phenomenon found earlier in experimental and numerical studies of turbulent flow in spatially developing 90° bends, with which the current results are compared. The fact that turbulence in the toroidal pipe features low-frequency coherent structures very similar to swirl switching is relevant as it may challenge the current hypothesis regarding the origin of swirl switching, which is usually connected to the conditions in the upstream straight section. However, the toroidal pipe is homogeneous in the streamwise direction and does as such not feature a straight part.

Modal instability of the flow in a toroidal pipe

The modal instability encountered by the incompressible flow inside a toroidal pipe is studied, for the first time, by means of linear stability analysis and direct numerical simulation (DNS). In addition to the unquestionable aesthetic appeal, the torus represents the smallest departure from the canonical straight pipe flow, at least for low curvatures. The flow is governed by only two parameters: the Reynolds number $\mathit{Re}$ and the curvature of the torus ${\it\delta}$, i.e. the ratio between pipe radius and torus radius. The absence of additional features, such as torsion in the case of a helical pipe, allows us to isolate the effect that the curvature has on the onset of the instability. Results show that the flow is linearly unstable for all curvatures investigated between 0.002 and unity, and undergoes a Hopf bifurcation at $\mathit{Re}$ of about 4000. The bifurcation is followed by the onset of a periodic regime, characterised by travelling waves with wavelength $\mathit{O}(1)$ pipe diameters. The neutral curve associated with the instability is traced in parameter space by means of a novel continuation algorithm. Tracking the bifurcation provides a complete description of the modal onset of instability as a function of the two governing parameters, and allows a precise calculation of the critical values of $\mathit{Re}$ and ${\it\delta}$. Several different modes are found, with differing properties and eigenfunction shapes. Some eigenmodes are observed to belong to groups with a set of common characteristics, deemed ‘families’, while others appear as ‘isolated’. Comparison with nonlinear DNS shows excellent agreement, confirming every aspect of the linear analysis, its accuracy, and proving its significance for the nonlinear flow. Experimental data from the literature are also shown to be in considerable agreement with the present results.

The effect of seam imperfections on the unsteady flow within a fluid-filled torus

AbstractWe consider the behaviour of the flow within a fluid-filled torus when there is a sudden change in the rotation rate of the torus. Experimental work on this problem by Madden & Mullin (J. Fluid Mech., vol. 265, 1994, p. 217) demonstrated a flow with a rich and complex dynamics. In particular, planar (top-down) flow visualisation images show a well-defined laminar band at both the inner and outer bend of the toroidal pipe. Hewitt et al. (J. Fluid Mech., vol. 688, 2011, pp. 88–119) demonstrated the existence of finite-time singularities in the resulting viscous boundary layers, and linked the post-singularity structure to one of the laminar bands identified in experiments (Madden & Mullin J. Fluid Mech., vol. 265, 1994, p. 217; del Pino et al.Phys. Fluids, vol. 20 (12), 2008, 124104). The second band (or laminar front) identified by Madden & Mullin was conjectured by Hewitt et al. to be the result of a centrifugal instability, perhaps generated by small imperfections in the experimental apparatus. Here we explore this conjecture further, demonstrating that a small seam imperfection can generate substantial secondary motion but with considerably different dynamics than the centrifugally driven instability of Hewitt et al.

On the influence of gravitational and centrifugal buoyancy on laminar flow and heat transfer in curved pipes and coils

The effects of gravitational and centrifugal buoyancy on laminar flow and heat transfer in curved and helical pipes were investigated by numerical simulation. Six dimensionless numbers characterizing the problem were identified, and an analysis was conducted on the possible combinations of signs of the gravitational and centrifugal buoyancy effects. Two distinct Richardson numbers were introduced in order to quantify the importance of the two types of buoyancy, and it was shown that, in the case of heating from the wall, a maximum realizable value of the centrifugal Richardson number exists which is a linear function of the curvature δ (ratio of pipe radius a to curvature radius c). Detailed results were obtained for δ=0.9, torsion λ (ratio of reduced pitch H/(2π) to curvature radius c)=0 (toroidal pipe) or 0.4 (helical pipe), Re=100, Pr=1 and gravitational and centrifugal Richardson numbers Rig, Ric each varying from −0.1 to +0.1. A complex interaction between the two forms of buoyancy was found to exist. In the helical geometry, provided |Rig|≈|Ric| they exhibited effects of the same order. The lowest values of both the friction coefficient and the mean Nusselt number were obtained in the presence of positive gravitational and centrifugal buoyancy, while the highest values were obtained when both forms of buoyancy were negative; the reason for this behavior was identified in the different degree of coupling between the distributions of axial velocity and temperature. In the toroidal geometry, a simpler behavior was predicted due to the presence of top–bottom symmetry; both the friction coefficient and the mean Nusselt number were found to decrease with the intensity of centrifugal buoyancy and to be little affected by gravitational buoyancy in the range of Rig investigated.

Experimental investigation of transitional flow in a toroidal pipe

AbstractThe flow instability and further transition to turbulence in a toroidal pipe (torus) with curvature ratio (tube-to-coiling diameter) 0.049 is investigated experimentally. The flow inside the toroidal pipe is driven by a steel sphere fitted to the inner pipe diameter. The sphere is moved with constant azimuthal velocity from outside the torus by a moving magnet. The experiment is designed to investigate curved pipe flow by optical measurement techniques. Using stereoscopic particle image velocimetry, laser Doppler velocimetry and pressure drop measurements, the flow is measured for Reynolds numbers ranging from 1000 to 15 000. Time- and space-resolved velocity fields are obtained and analysed. The steady axisymmetric basic flow is strongly influenced by centrifugal effects. On an increase of the Reynolds number we find a sequence of bifurcations. For $\mathit{Re}= 4075\pm 2\hspace{0.167em} \% $ a supercritical bifurcation to an oscillatory flow is found in which waves travel in the streamwise direction with a phase velocity slightly faster than the mean flow. The oscillatory flow is superseded by a presumably quasi-periodic flow at a further increase of the Reynolds number before turbulence sets in. The results are found to be compatible, in general, with earlier experimental and numerical investigations on transition to turbulence in helical and curved pipes. However, important aspects of the bifurcation scenario differ considerably.

Transition to turbulence in toroidal pipes

AbstractIncompressible flow in toroidal pipes of circular cross-section was investigated by three-dimensional, time-dependent numerical simulations using a finite volume method. The computational domain included a whole torus and was discretized by up to ${\ensuremath{\sim} }11. 4\ensuremath{\times} 1{0}^{6} $ nodes. Two curvatures $\delta $ (radius of the cross-section/radius of the torus), namely 0.3 and 0.1, were examined; a streamwise forcing term was imposed, and its magnitude was made to vary so that the bulk Reynolds number ranged between ${\ensuremath{\sim} }3500$ and ${\ensuremath{\sim} }14\hspace{0.167em} 700$. The results were processed by different techniques in order to confirm the spatio-temporal structure of the flow. Consecutive transitions between different flow regimes were found, from stationary to periodic, quasi-periodic and chaotic flow. At low Reynolds number, stationary flow was predicted, exhibiting a symmetric couple of Dean vortex rings and a strong shift of the streamwise velocity maximum towards the outer wall. For $\delta = 0. 3$, between $\mathit{Re}= 4556$ and $\mathit{Re}= 4605$ a first transition occurred from stationary to periodic flow, associated with a supercritical Hopf bifurcation and giving rise to a travelling wave which took the form of a varicose streamwise modulation of the Dean vortex ring intensity. A further transition, associated with a secondary Hopf bifurcation, occurred between $\mathit{Re}= 5042$ and $\mathit{Re}= 5270$ and led to a quasi-periodic flow characterized by two independent fundamental frequencies associated with distinct travelling waves, the first affecting mainly the Dean vortex rings and similar to that observed in purely periodic flow, the second localized mainly in the secondary flow boundary layers and manifesting itself as an array of oblique vortices produced at the edge of the Dean vortex regions. Both the periodic and the quasi-periodic regimes were characterized by an instantaneous anti-symmetry of the oscillatory components with respect to the equatorial midplane of the torus. For $\delta = 0. 1$, between $\mathit{Re}= 5139$ and $\mathit{Re}= 5208$ a direct (‘hard’) transition from steady to quasi-periodic flow occurred. Hysteresis was also observed: starting from a quasi-periodic solution and letting the Reynolds number decrease, both quasi-periodic and periodic stable solutions were obtained at Reynolds numbers below the critical value. A further decrease in $\mathit{Re}$ led to steady-state solutions. This behaviour suggests the existence of a subcritical Hopf bifurcation followed by a secondary Hopf bifurcation. The resulting periodic and quasi-periodic flows were similar to those observed for the higher curvature, but the travelling modes were now instantaneously symmetric with respect to the equatorial midplane of the torus. Also, the further transition from quasi-periodic to chaotic flow occurred with different modalities for the two curvatures. For $\delta = 0. 3$, a centrifugal instability of the main flow in the outer region occurred abruptly between $\mathit{Re}= 7850$ and $\mathit{Re}= 8160$, while a further increase of $\mathit{Re}$ up to 13 180 did not cause any relevant change in the distribution and intensity of the fluctuations. For $\delta = 0. 1$ the transition to chaotic flow was gradual in the range $\mathit{Re}= 6280$ to 8160 and affected mainly the inner region; only a further increase of $\mathit{Re}$ to 14 700 caused fluctuations to appear also in the outer region.

Experimental study of the flow in helical circular pipes: Torsion effect on the flow velocity and turbulence

An objective of the present paper is to experimentally clarify the torsion effect on the flow in helical circular pipes. We have made six helical circular pipes having different pitches and common non-dimensional curvature δ of about 0.1. The torsion parameter β0, which is defined by β0 = τ/(2δ)1/2 with non-dimensional torsion τ, are taken to be 0.02, 0.45, 0.69, 1.01, 1.38 and 1.89 covering from small to very large pitch. The velocity distributions and the turbulence of the flow are measured using an X-type hot-wire anemometer in the range of the Reynolds number from 200 to 20000. The results obtained are summarized as follows: The mean secondary flow pattern in a cross section of the pipe changes from an ordinary twin-vortex type as is seen in a curved pipe without torsion (toroidal pipe) to a single vortex type after one of the twin-vortex gradually disappears as β0 increases. The circulation direction of the single vortex is the same as the direction of torsion of the pipe. The mean velocity distribution of the axial flow is similar to that of the toroidal pipe at small β0, but changes its shape as β0 increases, and attains the shape similar to that in a straight circular pipe when β0 = 1.89. It is also found that the critical Reynolds number, at which the flow shows a marginal behavior to turbulence, decreases as β0 increases for small β0, and then increases after taking a minimum at β0 ≈ 1.4 as β0 increases. The minimum of the critical Reynolds number experimentally obtained is about 400 at β0 ≈ 1.4.