^{1,2,a)}and Alban Pothérat

^{1,b)}

### Abstract

An analysis of the successive regimes of the two-dimensional (2D) flow through a sharp 180° bend is performed by means of parametric numerical simulations where the Reynolds number Re and the opening ratio β (defined as the ratio of bend opening to the inlet width) vary in the respective ranges [0–2500] and [0.1–10]. In the outlet, the sequence of flow regimes is found to bear similarities with the flow behind a two-dimensional cylinder, despite being asymmetric by nature: when Re was increased, we found a laminar flow, then a flow with a first recirculation attached to the inside boundary, then one with a second recirculation attached to the top boundary. The onset of unsteadiness occurs through instability of the main stream and vortex shedding from the inside boundary. For β ⩽ 0.2, the flow is characterised by the dynamics of the jet generated at the very small turning part whereas for β ⩾ 0.3, it behaves rather like the flow behind an obstacle placed in a channel. This difference is most noticeable in the unsteady regimes where the vortex shedding mechanisms differ. While the former generates a more turbulent flow rich in small scale turbulence, the latter produces large structures of the size of the channel. In the turning part, further series of recirculation develop in each corner, akin to those identified by Moffatt [“Viscous and resistive eddies near a sharp corner,” J. Fluid Mech.18, 1 (Year: 1964)10.1017/S0022112064000015]. For β > 1 corresponding eddies merge to form a series of alternately rotating recirculating cells, which occupy the whole width of the turning part. We find that for β > 1, the effective opening ratio β*, which correspond to the area occupied by the mainstream while passing from the inlet to the outlet, tends towards a value of ≃0.7. The combination of regimes in the outlet and the turning part yields a wealth of flow regimes, which open interesting possibilities to tailor the design of 180° bends to suit particular applications involving mixing, heat, and mass transfer. Selected 3D simulations show that with a few noticeable exceptions, 2D dynamics determine the main features of the flow (drag and recirculation length), even in a wide bend, while 3D structure tends to slow down the shedding mechanism. 2D simulations are thus not only relevant to configurations where the flow is expected to be 2D (thin bend, MHD flows), but also to 3D flows where they can predict some of the global flow features at a low computational cost.

This work was supported by a Coventry University China Collaborative Fellowship and the Chinese Scholar Council.

I. INTRODUCTION

II. PROBLEM GEOMETRY AND GOVERNING EQUATIONS

III. NUMERICAL SETUP

IV. FLOW IN THE OUTLET PART OF THE BEND

A. Flow regimes

1. General features

2. Influence of the geometry on regimes I–IV

3. Geometry-dependent unsteady regimes for β ⩾ 0.3

4. Phase diagram

B. Dynamics of the outlet recirculation

1. General features

2. Influence of the geometry on the recirculation length

C. Flow coefficients

1. Drag and lift coefficients

2. Strouhal number

V. FLOW OF TURNING PART OF THE BEND

A. Flow regimes

B. Effective turning dimension of 180° sharp bend

VI. FLOW IN A 3D 180° SHARP BEND

VII. CONCLUSION

### Key Topics

- Rotating flows
- 58.0
- Flow instabilities
- 32.0
- Rheology and fluid dynamics
- 20.0
- Reynolds stress modeling
- 18.0
- Turbulent flows
- 17.0

## Figures

Key points of 180° bend duct have been studied: turning part shape (a), turning part width (b), divider thickness (c), and secondary flow recirculations (d).

Key points of 180° bend duct have been studied: turning part shape (a), turning part width (b), divider thickness (c), and secondary flow recirculations (d).

The geometry of 180° bend duct for the whole domain. Boundaries are labelled as follows: 1, top boundary (T); 2, inside boundary (I); 3, bottom boundary (B); 4, end wall boundary (E).

The geometry of 180° bend duct for the whole domain. Boundaries are labelled as follows: 1, top boundary (T); 2, inside boundary (I); 3, bottom boundary (B); 4, end wall boundary (E).

Details of the meshes around turning part area.

Details of the meshes around turning part area.

Sequence of flow regimes in the outlet part for β = 1. (a)–(c) Streamlines of steady flow. (d)–(e) Snapshot of vorticity contours taken from the unsteady flow. The outlet part is only represented up to 15 duct widths for clarity, when the simulated domain extends in fact to 30 duct widths. ω max stands for the dimensionless maximum vorticity which is normalised by U 0/a.

Sequence of flow regimes in the outlet part for β = 1. (a)–(c) Streamlines of steady flow. (d)–(e) Snapshot of vorticity contours taken from the unsteady flow. The outlet part is only represented up to 15 duct widths for clarity, when the simulated domain extends in fact to 30 duct widths. ω max stands for the dimensionless maximum vorticity which is normalised by U 0/a.

Flow regimes in outlet part of bend for β = 0.5. See legend of Fig. 4 . The outlet part is only represented up to 15 duct widths for clarity, when the simulated domain extends in fact to 30 duct widths.

Flow regimes in outlet part of bend for β = 0.5. See legend of Fig. 4 . The outlet part is only represented up to 15 duct widths for clarity, when the simulated domain extends in fact to 30 duct widths.

Flow regimes in outlet part of bend for β = 0.1. The outlet part is only represented up to 15 duct widths for clarity, when the simulated domain extends in fact to 30 duct widths.

Flow regimes in outlet part of bend for β = 0.1. The outlet part is only represented up to 15 duct widths for clarity, when the simulated domain extends in fact to 30 duct widths.

Streamlines of the steady flow at β = 0.2, showing the stages of the development of the inside recirculation within R1 at different values of Re.

Streamlines of the steady flow at β = 0.2, showing the stages of the development of the inside recirculation within R1 at different values of Re.

Relationship between the critical Reynolds number Rein for the appearance of the inside recirculation within R1, and β.

Relationship between the critical Reynolds number Rein for the appearance of the inside recirculation within R1, and β.

Phase diagram of 180° sharp bend: flow regimes are labelled by Roman characters followed by numbers. These, respectively, stand for the outlet flow regimes and the number of merged recirculations in the turning part section. A schematic representation of each combined flow regime is given for each distinct region of the phase diagram. Error bars span the interval between numerical simulations showing the “last stable” and those showing “first unstable” one.

Phase diagram of 180° sharp bend: flow regimes are labelled by Roman characters followed by numbers. These, respectively, stand for the outlet flow regimes and the number of merged recirculations in the turning part section. A schematic representation of each combined flow regime is given for each distinct region of the phase diagram. Error bars span the interval between numerical simulations showing the “last stable” and those showing “first unstable” one.

Sketch of separation and re-attachment points defining the locations of all recirculations.

Sketch of separation and re-attachment points defining the locations of all recirculations.

Time average of separation and re-attachment points in the outlet branch of the bend vs. Re. (a) β = 2, (b) β = 1, (c) β = 0.2, and (d) β = 0.1.

Time average of separation and re-attachment points in the outlet branch of the bend vs. Re. (a) β = 2, (b) β = 1, (c) β = 0.2, and (d) β = 0.1.

Variations of recirculation lengths ⟨L 1⟩ (top) and ⟨L 2⟩ (bottom) against their respective critical parameters for several fixed values of β.

Variations of recirculation lengths ⟨L 1⟩ (top) and ⟨L 2⟩ (bottom) against their respective critical parameters for several fixed values of β.

Sketch of L 1 and L 2: just before (top) and just after (bottom) the collapse of R1 (top), in the sense of increasing Reynolds.

Sketch of L 1 and L 2: just before (top) and just after (bottom) the collapse of R1 (top), in the sense of increasing Reynolds.

Variations of the length of the steady and unsteady recirculations with β for fixed values of Re.

Variations of the length of the steady and unsteady recirculations with β for fixed values of Re.

Force direction on inside boundary: C d (up) and C l (bottom).

Force direction on inside boundary: C d (up) and C l (bottom).

The relationship between C d with Re of inside boundary.

The relationship between C d with Re of inside boundary.

The relationship between C l with Re of inside boundary.

The relationship between C l with Re of inside boundary.

The relationship between S t with Re of β = 1 and β = 0.5. Black and white symbols, respectively, refer to shedding in regimes IV and V.

The relationship between S t with Re of β = 1 and β = 0.5. Black and white symbols, respectively, refer to shedding in regimes IV and V.

Snapshot of vorticity contours taken from unsteady flow for β = 1: (a) At Re = 1500, near the Maximum of S t (Re), the vortex street is still well defined; (b) at Re = 2000, past the Maximum of S t (Re), the interaction with the walls destroy the vortex street.

Snapshot of vorticity contours taken from unsteady flow for β = 1: (a) At Re = 1500, near the Maximum of S t (Re), the vortex street is still well defined; (b) at Re = 2000, past the Maximum of S t (Re), the interaction with the walls destroy the vortex street.

The relationship between S t with Re of β = 0.2.

The relationship between S t with Re of β = 0.2.

Flow regimes of the turning part section of 180° sharp bend.

Flow regimes of the turning part section of 180° sharp bend.

Time average of separation and re-attachment points in the turning part area of the bend vs. Re. (a) β = 0.5, (b) β = 1, (c) β = 3, and (d) β = 5.

Time average of separation and re-attachment points in the turning part area of the bend vs. Re. (a) β = 0.5, (b) β = 1, (c) β = 3, and (d) β = 5.

β* vs. Re for different β.

β* vs. Re for different β.

Sketch of 3D geometry.

Sketch of 3D geometry.

Details of the meshes for 3D geometry: (a) domain near the turning part and (b) mesh view from the end wall.

Details of the meshes for 3D geometry: (a) domain near the turning part and (b) mesh view from the end wall.

Velocity streamlines (a) for Re = 400 and iso-surfaces of z-vorticity [ω z = 0.025ω zmax ] and (b) for Re = 2000 at β = 1.

Velocity streamlines (a) for Re = 400 and iso-surfaces of z-vorticity [ω z = 0.025ω zmax ] and (b) for Re = 2000 at β = 1.

Velocity streamlines (a) for Re = 500 and iso-surfaces of z-vorticity [ω z = 0.027ω zmax ] and (b) for Re = 2000 at β = 5.

Velocity streamlines (a) for Re = 500 and iso-surfaces of z-vorticity [ω z = 0.027ω zmax ] and (b) for Re = 2000 at β = 5.

Iso-surfaces of z-vorticity for Re = 2000 at β = 1: (a) TC1 [ω z = 0.025ω zmax ] and (b) TC2 [ω z = 0.023ω zmax ].

Iso-surfaces of z-vorticity for Re = 2000 at β = 1: (a) TC1 [ω z = 0.025ω zmax ] and (b) TC2 [ω z = 0.023ω zmax ].

Velocity streamlines (a) for Re = 65 and iso-surfaces of z-vorticity [ω z = 0.027ω zmax ] and (b) for Re = 520 at β = 0.2.

Velocity streamlines (a) for Re = 65 and iso-surfaces of z-vorticity [ω z = 0.027ω zmax ] and (b) for Re = 520 at β = 0.2.

Snapshot of vorticity contours taken from the 2D simulation (a) and 3D simulation (b) for Re = 520 at β = 0.2. Even though the flow is strongly 3D, the general structure of the flow is very similar in both simulations, with in particular, a large 2D recirculation R1, acting as “wheel,” which conveys smaller structures generated by the instability of the jet that wraps around it. Values of C d , C l , L 1, and St found in the 2D and 3D simulations are accordingly very close to each other.

Snapshot of vorticity contours taken from the 2D simulation (a) and 3D simulation (b) for Re = 520 at β = 0.2. Even though the flow is strongly 3D, the general structure of the flow is very similar in both simulations, with in particular, a large 2D recirculation R1, acting as “wheel,” which conveys smaller structures generated by the instability of the jet that wraps around it. Values of C d , C l , L 1, and St found in the 2D and 3D simulations are accordingly very close to each other.

## Tables

Characteristics of different meshes and C d , St, and U errors at Re = 100 and 800 for β = 1.

Characteristics of different meshes and C d , St, and U errors at Re = 100 and 800 for β = 1.

Comparison of C d , C l , and S t for different resolutions and domain widths at Re = 2000 for β = 1. C d and C l are expressed per width of 2a. The superscript err stands for relative discrepancies to the reference case (C0).

Comparison of C d , C l , and S t for different resolutions and domain widths at Re = 2000 for β = 1. C d and C l are expressed per width of 2a. The superscript err stands for relative discrepancies to the reference case (C0).

Comparison of L 1, L 2, C d , C l , and discrepancy between 2D and 3D simulations. Quantities from 3D simulations are expressed per unit length.

Comparison of L 1, L 2, C d , C l , and discrepancy between 2D and 3D simulations. Quantities from 3D simulations are expressed per unit length.

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