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Research Papers: Fundamental Issues and Canonical Flows

# Flow Past Two Freely Rotatable Triangular Cylinders in Tandem Arrangement

[+] Author and Article Information
Shizhao Wang

Assistant Professor  The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China e-mail: wangsz@lnm.imech.ac.cn

Luoding Zhu

Associate Professor  Department of Mathematical Sciences, IN University-Purdue University, Indianapolis Indianapolis, IN 46202 e-mail: lzhu@math.iupui.edu

Xing Zhang1

Associate Professor e-mail: zhangx@lnm.imech.ac.cn

Guowei He

Professor and Director e-mail: hgw@lnm.imech.ac.cn The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China

1

Corresponding author

J. Fluids Eng 133(8), 081202 (Aug 19, 2011) (12 pages) doi:10.1115/1.4004637 History: Received September 05, 2010; Revised July 13, 2011; Accepted July 15, 2011; Published August 19, 2011; Online August 19, 2011

## Abstract

In this paper we investigate the interaction of two freely rotatable triangular cylinders that are placed in tandem in a laminar flow. To study how the spacing between the two cylinders may influence the dynamic behavior of the cylinders and vortical structure of the flow, we have performed a series of numerical simulations of the two-cylinder-flow system. In all the simulations, the dimensionless moment of inertia and Reynolds number are fixed to 1.0 and 200, respectively. Four cases with the spacing ratio (L/D) of 2.0, 3.0, 4.0, and 5.0 are studied. With the increase of spacing, three different states of motion of the system are found. At L/D = 2.0, oscillatory rotation (swinging in both directions) is observed. At L/D = 3.0 both cylinders exhibit quasi-periodic autorotations. At L/D = 4.0 and 5.0, a more complicated pattern (irregular autorotation) is observed. For each case, the time history of angular velocity, the phase portrait (angular acceleration versus angular velocity,) and the spectra of the moments of forces on both cylinders are plotted and analyzed. The vortical structures in the near and far wake are visualized. Physical interpretations for various phenomenon observed are presented whenever possible.

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## Figures

Figure 17

Time history of the angular velocities of the two tandem cylinders at (a) L/D = 4.0; (b) L/D = 5.0. Solid line represents the front cylinder and dashed line represents the rear cylinder. The autorotations of the triangular cylinders for L/D = 4.0 and L/D = 5.0 are similar. For both cases, the triangular cylinders rotate irregularly. While the front cylinder rotates counterclockwise constantly, the rear cylinder alternates its rotating directions from time to time: it rotates counterclockwise for some time, pauses, and switches its direction and rotates clockwise for some time.

Figure 18

Phase portraits (angular velocity versus angular velocity) of the two tandem cylinders. (a) Front cylinder, L/D = 4.0; (b) rear cylinder, L/D = 4.0; (c) front cylinder, L/D = 5.0; (d) rear cylinder, L/D = 5.0. For both cases, the autorotations of the front and rear cylinders are irregular. No periodicity can be identified on the phase portraits.

Figure 1

A schematic diagram of the setup of the problem. Two hinged equilateral triangular cylinders are placed in tandem in a uniform free stream of velocity U. D is the diameter of the circumcircle of the triangular cylinder. L is the distance between the circumcenters of the two cylinders.

Figure 2

The size of the computational domain. The computational domain is 37D–40D by 50D, with 10D from the inlet and 25D from the outlet. (The figure is not to scale.)

Figure 3

Computational mesh: (a) around one cylinder, including the background mesh (in blue color) and the moving mesh (in red color); (b) zoom in near the top corner. Quadrilateral elements are deployed near the surface of the cylinder to capture the flow features in the boundary layer; (c) locally refined mesh used in the wake region (the background mesh is in blue color, and moving mesh in red color).

Figure 4

Angular velocity versus time for a single cylinder. The angular velocity of the autorotation triangular cylinder is of multifrequency and quasi-periodic.

Figure 5

Phase portrait (angular acceleration versus angular velocity) for a single cylinder. A multifrequency and quasi-periodic solution is obtained.

Figure 6

Power spectrum of the moment of force on the single cylinder. Five frequencies are involved in the autorotation: 0.05, 0.20, 0.25, 0.45, and 0.50.

Figure 7

Power spectrum of the moment of force on a fixed cylinder with different relative positions to the upstream flow. Three different positions are shown: (a) “back-to-flow” configuration (▷), the dominant vortex shedding frequency is 0.2; (b) “face-to-flow” configuration (◁), the dominant vortex shedding frequency is 0.25; (c) “side-to-flow” configuration (Δ), the dominant vortex shedding frequency is 0.225.

Figure 8

Snapshots of vorticity contours during the autorotation of a single cylinder. The vortical structure resembles the regular Karman vortex street in the wake of a stationary bluff body. But, the vortex street slightly tilts up in the near wake. The distances between the vortices are not uniform because of the interaction of the autorotation and vortex shedding. (a)–(f) corresponds to t = 155, 160, 165, 170, 175, and 180 in Fig. 4, respectively. The slowdown of the rotation at t = 166 generates a gap in the contours of vorticity in (e). This gap separates the vortices into two groups.

Figure 9

Time history of the angular velocities of the two tandem cylinders in the case of L/D = 2.0. Solid line represents the front cylinder and dashed line represents the rear cylinder. Instead of autorotation, both cylinders exhibit oscillatory rotation about their axles. The front and rear cylinders oscillate at the same frequency but in antiphase. The amplitude of the rear one is about four times larger than that of the front one.

Figure 10

Phase portraits (angular acceleration versus angular velocity) of the two tandem cylinders in the case of L/D = 2.0; (a) front cylinder; (b) rear cylinder. Both the oscillating of the front and rear cylinders are periodic. The open circles indicate the equilibrium positions for the point-to-point configuration.

Figure 11

Power spectra of the moments of force of the two cylinders in the case of L/D = 2.0; (a) front cylinder; (b) rear cylinder. The dominant oscillating frequency is 0.14 for either cylinder.

Figure 12

The instantaneous vorticity contours for the two tandem cylinders in the case of L/D = 2.0. The vortex shedding behind the front cylinder is suppressed due to the closeness of the two. A vortex street only exists behind the rear cylinder.

Figure 13

Time history of the angular velocities of the two tandem cylinders in the case of L/D = 3.0. Solid line represents the front cylinder and dashed line represents the rear cylinder. Both the front and rear cylinders experience multiperiodic autorotation. The average angular velocity of the font cylinder is approximately the same as that of the single cylinder, and the rear cylinder rotates at a lower speed.

Figure 14

Phase portraits (angular acceleration versus angular velocity) of the two tandem cylinders in the case of L/D = 3.0; (a) front cylinder; (b) rear cylinder. Both the autorotations of the front and rear cylinders are multiperiodic.

Figure 15

Power spectra of the moments of force of the two cylinders in the case of L/D = 3.0; (a) front cylinder; (b) rear cylinder. Both the autorotations of the front and rear cylinder are of multifrequency. The frequencies of the highest peaks are 0.24 and 0.16 for the front and rear cylinder, respectively.

Figure 16

The instantaneous vorticity contours for the two tandem cylinders in the case of L/D = 3.0. The vortex shedding from the front cylinder is fully recovered and a vortex exists in the gap between the two cylinders. A vortex street exists behind the rear cylinder.

Figure 19

Power spectra of moments of force of the two cylinders. (a) Front cylinder, L/D = 4.0; (b) rear cylinder, L/D = 4.0; (c) front cylinder, L/D = 5.0; (d) rear cylinder, L/D = 5.0. For both cases, the frequency of the highest peak is 0.25 for the front cylinder and 0.18 for the rear one. The autorotations of the rear cylinder become very noisy, and more frequencies are excited comparing with that of the front ones.

Figure 20

The instantaneous vorticity contours for the two tandem cylinders in the case of L/D = 4.0. The vortex layer shedding from the front cylinder rolls up and forms a vortex street in the gap between the two cylinders. The wake structures behind the rear cylinder are irregular because of the interaction of the two cylinders.

Figure 21

The instantaneous vorticity contours for the two tandem cylinders in the case of L/D = 5.0. The front cylinder forms a vortex street in the gap between the two cylinders. The wake behind the rear cylinder is an interaction of the vortices shedding from the two cylinders. The flow structures are similar to the L/D = 4.0 case.

Figure 22

The estimator as a function of the number of time steps for the data from the time series of moments of force in the case of L/D = 5.0: (a) front cylinder; (b) rear cylinders. [The solid lines show the results for embedding dimension m = 4, 5, and 6 at five different initial distances and the dashed lines are the reference lines with the slope of ln(1.3) and ln(1.4), respectively.] (c) The estimator as a function of the number of time steps for the data from the logistic map. [The solid lines show the results for embedding dimension m = 2 and 3 at five different initial distances. The dashed line is the reference line with the slope of ln(2.0).]

Figure 23

The test of time step independence for an autorotating single cylinder. The large time step dt=0.01 causes a phase and amplitude difference. The difference between the dt=0.005 case and dt=0.0025 case is negligible. The time step dt=0.005 is used in the present simulations.

Figure 24

The test of grid size independence for an autorotating single cylinder. For the coarse mesh, the triangle is 54 meshed with line element, the first layer thickness of the boundary-layer mesh is 0.015D. The corresponding parameters for the present mesh are 105 and 0.01, for the fine mesh 210 and 0.05. The result on the present mesh is comparable to that on the finer mesh.

Figure 25

The test of computational domain independence for an autorotating single cylinder. There are no difference between the result on a 50D by 100D mesh and that on a 35D by 50D mesh. The computational domain used in the present work is 35D by 50D.

Figure 26

The independence test of the initial angular velocity. Three different initial angular velocities are specified: ω0 = 1.5, ω0 = 1.0, and ω0 = 0.5. After the transient process the initial effect disappears and the same quasi-periodic solutions are obtained.

Figure 27

The independence test of the initial angular velocity for two tandem cylinders with L/D=2.0. (a) The initial angular velocities for both cylinders are counterclockwise. (b) The initial angular velocity for the front cylinder is counterclockwise, and the rear one is clockwise. (c) The initial angular velocities for both cylinders are zero. The oscillations of the two cylinders are independent of the initial angular velocities.

Figure 28

The independence test of the initial angular velocity for two tandem cylinders with L/D=3.0. (a) The initial angular velocities for both cylinders are counterclockwise. (b) The initial angular velocity for the front cylinder is counterclockwise, and the rear one is clockwise. The autorotations of the two cylinders are independent of the initial angular velocities.

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