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Review Article

Turbulent Flows in Curved Pipes: Recent Advances in Experiments and Simulations

[+] Author and Article Information
Athanasia Kalpakli Vester

Department of Mechanics,
Linné FLOW Centre & CCGEx,
Royal Institute of Technology (KTH),
Stockholm SE-100 44, Sweden
e-mail: sissy@mech.kth.se

Ramis Örlü

Department of Mechanics,
Linné FLOW Centre & CCGEx,
Royal Institute of Technology (KTH),
Stockholm SE-100 44, Sweden
e-mail: ramis@mech.kth.se

P. Henrik Alfredsson

Department of Mechanics,
Linné FLOW Centre & CCGEx,
Royal Institute of Technology (KTH),
Stockholm SE-100 44, Sweden
e-mail: phal@mech.kth.se

1Corresponding author.

Manuscript received January 29, 2016; final manuscript received July 11, 2016; published online September 19, 2016. Assoc. Editor: Herman J. H. Clercx.

Appl. Mech. Rev 68(5), 050802 (Sep 19, 2016) (25 pages) Paper No: AMR-16-1012; doi: 10.1115/1.4034135 History: Received January 29, 2016; Revised July 11, 2016

Curved pipes are essential components of nearly all the industrial process equipments, ranging from power production, chemical and food industries, heat exchangers, nuclear reactors, or exhaust gas ducts of engines. During the last two decades, an interest on turbulent flows in such conduits has revived, probably due to their connection to technical applications such as cooling systems of nuclear reactors (e.g., safety issues due to flow-induced fatigue) and reciprocating engines (e.g., efficiency optimization through exhaust gas treatment in pulsatile turbulent flows). The present review paper, therefore, is an account on the state-of-the-art research concerning turbulent flow in curved pipes, naturally covering mostly experimental work, while also analytical and numerical works are reviewed. This paper starts with a historical review on pipe flows in general and specifically on flows through curved conduits. In particular, research dealing with the effect of curvature on transition to turbulence, work dealing with pressure losses in curved pipes, as well as turbulence statistics are summarized. The swirl-switching phenomenon, a specific structural phenomenon occurring in turbulent curved pipe flows, which has interesting fundamental as well as practical implications, is reviewed. Additional complications, with respect to flow through bends, namely, entering swirling flow and pulsating flow, are reviewed as well. This review closes with a summary on the main literature body as well as an outlook on future work that should be performed in order to tackle open questions remaining in the field.

Copyright © 2016 by ASME
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References

Figures

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Fig. 1

Different pipe configurations: (a) straight pipe, (b) 90 deg bend, (c) helix, and (d) torus, showing how some parameters are defined. L, pipe length; D=2R, pipe diameter; Rc, curvature radius; c, coil radius; and 2πb, coil pitch. The torsion of a coil is defined as τ=b/(c2+b2).

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Fig. 2

(a) Pipe bend with the cross-sectional plane A–A denoted as well as the outer and inner walls of the pipe bend. (b) Top view of the cross-sectional cut (A–A) at the exit of the pipe bend showing the defined Cartesian coordinate system.

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Fig. 3

(a) W. R. Dean (1896–1973). Reproduced with permission of Annual Review of Fluid Mechanics, Volume 10, 1–11, 1978 [34] Copyright by Annual Reviews, http://www.annualreviews.org (b) A schematic of the Dean vortices as sectional streamlines. Originally published in Dean, W. R., “Note on the motion of fluid in a curved pipe”, Philosophical magazine, 1927, vol. 4, pp. 208–223 [12]. Reprinted by permission of the publisher Taylor & Francis Ltd, http://tandfonline.com

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Fig. 4

The referee report by Taylor on Dean's 1928 paper on curved channel flow [34]. The report should answer six question (only the first five were answered). Reprinted with permission by the Royal Society.

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Fig. 5

A drawing illustrating the tea-leaf effect. (Reproduced with permission from Einstein [43]. Copyright 1926 by Springer).

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Fig. 6

Data from the experiments by Taylor [15] and White [16] in helical pipes. + Speed at which White's curve indicates “first appearance” of turbulence. These values were shortly after confirmed through the experiments by Adler [42]. ⌾ Lowest speed at which the flow appears completely turbulent according to Taylor's data.  ⊡ Highest speed at which the flow is “quite steady,” according to Taylor. The pipe diameter and curvature diameter are given as d and D, respectively, in the figure (hence their ratio yields γ). Reprinted with permission from Taylor [15]. Copyright 1929 by the Royal Society.

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Fig. 7

Visualization of relaminarization in coiled pipes for D = 19.1 mm, Rc = 90 mm, and Re=4×103. Reproduced with permission from Sreenivasan and Strykowski [55]. Copyright 1983 by Springer. Caption of figure in the original publication: “Laminarization in coiled pipes. Pipe diameter 2a=1.91 cm, radius of curvature rc=9.0 cm, Reynolds number, R = 4050”.

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Fig. 8

Comparison of available experimental and numerical data with the neutral curve by Canton et al. [19]. The envelope of families F4S and F5A is shown with the black line and intersect at δ=0.016 (which here denotes the curvature ratio, γ), and the two dotted lines highlight their continuation. The data by Cioncolini and Santini [65], which refer to the first discontinuity in their friction measurements, the data from Sreenivasan and Strykowski [55] which they refer to as “conservative lower critical limit,” and that by Noorani and Schlatter [67] referring to sublaminar drag are also shown. The interpolants by Kühnen et al. [62] are also shown: with a dashed (blue) line the fit for the supercritical transition and with a dotted (blue) line the fit for subcritical transition. Reproduced with permission from Canton et al. [19]. Copyright 2016 by Cambridge University Press.

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Fig. 9

Neutral curve in the γ –Re plane (γ is denoted in the figure as δ) formed by the envelopes of lines for 0.002≤γ≤1 by Canton et al. [19]. Each line corresponds to the neutral curve of one mode. Five families (black and blue) and three isolated modes (green) are shown. Continuous lines denote symmetric modes, whereas dashed lines indicate antisymmetric ones. The curves are not interpolated, i.e., they are segments connecting computed solutions with Δγ=O(10−3), and the uncertainty of the Reynolds number is ±10−4. Reproduced with permission from Canton et al. [19]. Copyright 2016 by Cambridge University Press.

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Fig. 10

Pressure distribution along pipeline containing a 90 deg pipe bend (γ=0.27), where lu and ld denote the pipe length upstream and downstream the bend, respectively. The Reynolds number is here denoted as R, whereas R is the curvature radius and r is the pipe radius. Reprinted with permission from Ito [76]. Copyright 1960 by ASME.

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Fig. 11

Fanning friction factor, f, as function of Reynolds number, where • is for straight pipe data, and ° is for coiled pipe data for γ=0.00964 (data obtained by Ref. [65]). Laminar and turbulent correlations by Ito [52] are shown by - - - (for straight pipe) and — (for curved pipe). DNS results by Noorani et al. [81] are shown for fully developed turbulent flow (Re=5.3×103, 6.9×103, and 12×103) in straight pipe (■) and curved pipe with γ=0.01 (□). DNS from Noorani and Schlatter are also shown: laminar flows obtained in curved pipes are indicated with ♦, whereas the symbol ◊ illustrates sublaminar drag of the flow in weakly turbulent curved pipe with Re=3.4×103 and γ=0.01. Reprinted with permission from Noorani and Schlatter [67]. Copyright 2015 by AIP Publishing.

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Fig. 12

Different pipe bend shapes: (a) 90 deg bend, (b) U-bend, (c) double bend (or S-bend), and (d) out-of-plane bend

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Fig. 13

Contours of longitudinal mean velocity, scaled with the bulk velocity on the horizontal axis along a 180 deg bend and its downstream straight pipe section until a downstream location of z/d=5. Minus sign denotes upstream location from the bend. The angles along the bend are also indicated. Reprinted with permission from Sudo et al. [84]. Copyright 2000 by Springer. Caption of figure in the original publication: “Contours of longitudinal mean velocity W¯/Wa on the horizontal plane”.

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Fig. 14

(a) Mean and (b) r.m.s. streamwise velocity profiles measured along the horizontal axis by means of a single hot-wire probe, for three Reynolds numbers (based on pipe diameter), Re=1.4×104, 2.4×104, and 3.4×104 at 0.67D downstream of a 90 deg curved pipe. Reprinted with permission from Kalpakli Vester et al. [141]. Copyright 2016 by Springer. Caption of figure in the original publication: “Reynolds number effect on the a mean streamwise velocity and b r.m.s. profile normalized by the bulk velocity”.

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Fig. 15

Mean flow velocity contours for locations on the upstream straight section (negative z/d, where d is the pipe diameter), different bend angles ϕ, and locations downstream the bend (positive z/d). Streamwise velocity contours (top), scaled by the bulk velocity and secondary flow velocity vectors (bottom), where the length of D/8 corresponds to Wb. The left and right sides of each figure show the inside and outside walls in the bend, respectively. Reprinted with permission from Sudo et al. [87]. Copyright 1998 by Springer. Caption of figure in the original publication: “Time mean flow velocities in cross section. (Top figure: Longitudinal velocity contours, where numerical values are for W¯/Wa. Bottom figure: secondary flow velocity vectors, where the length of d/8 corresponds to Wa. The left and right sides of each figure show the inside and outside walls in the bend, respectively).” (a) z′/d = −1, (b) φ = 0 deg, (c) φ = 30 deg, (d) φ = 60 deg, (e) φ = 90 deg, (f) z/d = 1, (g) z/d = 2, (h) z/d = 5, and (i) z/d = 10.

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Fig. 16

Comparison of friction factors in fully developed turbulent flow. Experimental data: ■, Schlichting [3]; •, Ito [52], γ=0.038; ▲, Ito [52], γ=0.06, and — predictions using k−ε model by Patankar et al. [101], where a denotes the pipe radius and R the curvature radius. Reproduced with permission from Patankar et al. [101]. Copyright 1975 by Cambridge University Press.

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Fig. 17

Secondary flow pattern in a curved pipe: (a) laminar flow with parabolic inlet, γ=0.077, De = 277.5, and Re=103 and (b) turbulent flow with fully developed inlet, γ=0.077 and Re=5×104. Reprinted with permission from Anwer and So [17]. Copyright 1993 by Springer. Caption of figure in the original publication: “Secondary flow pattern in a curved pipe: a laminar flow with parabolic inlet, α=0.077, De = 277.5, and Re = 1, 000; b turbulent flow with fully developed inlet, u = 0, 077, De = 13, 874 and Re = 50, 000”.

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Fig. 18

Comparison of the nondimensional mean velocity magnitude evaluated in the domain's symmetry plane at a position of 0.67D behind the pipe elbow for Re=2.4×104. Reprinted with permission from Röhrig et al. [100]. Copyright 2015 by Elsevier.

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Fig. 19

Snapshots of the in-plane velocity field showing counterclockwise Dean cell. (a) Vortex as obtained from PIV for Re=3.4×104. (b) Vortex captured by the LES of Ref. [100] for the same flow parameters as the PIV shown to the left. Small-scale fluctuations have been suppressed in both cases using a spatial low-pass filtering operation. Reprinted with permission from Röhrig et al. [100]. Copyright 2015 by Elsevier.

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Fig. 20

Azimuthal distribution of the local axial wall shear stress normalized by its averaged value.  ........ Straight pipe, - - - - Re=1.2×104 and γ=0.01, - ⋅ - Re=5.3×103 and γ=0.01, and –⋅– Re=1.2×104 and γ=0.1. Reprinted with permission from Noorani et al. [81]. Copyright 2013 by Elsevier.

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Fig. 21

Isocontours of stream-function at (a) Re=1.2×104, γ=0.01 and (b) Re=1.2×104, γ=0.1. The symbol • indicates the center of the counter-rotating mean Dean vortices. Reprinted with permission from Noorani et al. [81]. Copyright 2013 by Elsevier.

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Fig. 22

Visualization of the swirl switching as captured by different authors: (a) velocity vector field for two snapshots within the cycle of the swirl switching downstream a 90 deg bend for Re = 5 × 103 and γ=0.5. The blue-filled circle marks the saddle point, and the red line indicates roughly the plane of symmetry of the flow. Reprinted with permission from Brücker [130]. (b) Reconstructed instantaneous flow fields from the first six POD modes downstream a 90 deg bend for Re = 3.4 × 104 and γ=0.32. Magnitude of the in-plane components is shown as the background contour map and their direction as vectors. The highest magnitude of the in-plane components on average is 0.3Wb, while instantaneously it is 0.5Wb. Reprinted with permission from Kalpakli and Örlü [110]. Copyright 2013 by Elsevier. (c) Flow field from the most energetic POD modes and the mean, from DNS in a torus with Re = 1.2 × 104 and γ=0.3. Isocontours of the streamfunction Ψ projected on top of its pseudocolors where red represents positive and blue indicates negative values of the streamfunction. Reprinted with permission from Noorani and Schlatter [121].

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Fig. 23

Surface plot of the streamwise component of conditional fluctuating velocity vectors, ŵ, at z/D=−3.1 (minus sign denotes upstream location from the bend). The value is normalized by the bulk velocity, Wb. The surfaces of the constant value of ŵ=±0.003Wb are superposed. Reprinted with permission from Sakakibara and Machida [128]. Copyright 2012 by AIP Publishing.

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Fig. 24

First three POD modes (from top to bottom) for various stations (5, 12, and 18D) downstream the bend (from left to right) for Re=2.5×104. The inner and outer stagnation points are indicated with open and filled circles, respectively. Reproduced with permission from Hellström et al. [126]. Copyright 2013 by Cambridge University Press.

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Fig. 25

(a) First POD mode for 0.67D station from the bend with γ=0.36 and for Re=2.4×104. Reproduced with permission from Kalpakli and Örlü [110]. Copyright 2013 by Elsevier. (b) Same as before but the results are from DNS data in a torus for γ=0.3 and Re=1.2×104. Reprinted with permission from Noorani and Schlatter [121].

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Fig. 26

Flow structures at increasing swirl numbers, S, as captured by Stereo-PIV experiments. Background contour maps show the streamwise velocity component scaled by the bulk velocity, whereas the in-plane velocities are shown as the imposed vectors. Reproduced with permission from Kalpakli and Örlü [110]. Copyright 2013 by Elsevier.

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Fig. 27

The effect of the swirl number on the (a) mean and (b) r.m.s. streamwise velocity profiles normalized by the bulk velocity at Re=2.4×104 and at 0.67D distance from the bend. Reprinted with permission from Kalpakli Vester et al. [141]. Copyright 2016 by Springer. Caption of figure in the original publication: “The swirl number effect on the a outer-scaled streamwise r.m.s. and b turbulence intensity at ReD=24,000”.

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Fig. 28

Top: Contour plots of the velocity fields obtained through reconstruction using the four most energetic POD modes. z/D=0.2, Re=2.4×104, γ=0.4, and α= 41. Five pulsating flow fields are shown for the corresponding phase angles indicated (1–5) in the centerline signal of the streamwise velocity (middle subplot). Streamwise velocity scaled by the bulk speed is shown as the background contour map, whereas the in-plane components are shown as the vectors. Bottom: the first, second, and third POD modes (from left to right) shown as “cross-sectional” streamlines. Reprinted with permission from Kalpakli et al. [180]. Copyright 2013 by Elsevier.

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