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Research Papers: Fundamental Issues and Canonical Flows

# Interactions of Vortices of a Square Cylinder and a Rectangular Vortex Generator Under Couette–Poiseuille Flow

[+] Author and Article Information
Dilip K. Maiti

Department of Mathematics,
Birla Institute of Technology & Science,
Pilani 333031, India
e-mail: d_iitkgp@yahoo.com

Rajesh Bhatt

Department of Mathematics,
Birla Institute of Technology & Science,
Pilani 333031, India
e-mail: rbhatt22@gmail.com

lCorresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received July 5, 2014; final manuscript received January 19, 2015; published online March 4, 2015. Assoc. Editor: Sharath S. Girimaji.

J. Fluids Eng 137(5), 051203 (May 01, 2015) (22 pages) Paper No: FE-14-1357; doi: 10.1115/1.4029631 History: Received July 05, 2014; Revised January 19, 2015; Online March 04, 2015

## Abstract

This study focuses on interactions of vortices generated by a family of eddy-promoting upstream rectangular cylinders (of different heights a* and widths b*) with the shear layers of a downstream square cylinder (of height A*) placed near a plane in an in-line tandem arrangement under the incidence of Couette–Poiseuille flow based nonuniform linear/nonlinear velocity profile. The dimensionless operational parameters are cylinders spacing distance S, ratio of heights $r2=a*/A*$ (≤1), aspect ratio $r1=b*/a*$ (≤1), Reynolds number Re (based on the velocity at height A* for Couette flow), $ReU2$ (based on the velocity at height 10A* for Couette–Poiseuille flow), and nondimensional pressure gradient P at the inlet. The governing equations are solved numerically through a pressure-correction-based iterative algorithm (SIMPLE) with the quadratic upwind interpolation for convective kinematics (QUICK) scheme for convective terms. The major issue of appearing multiple peaks in the spectrum of the fluctuating lift coefficient of the downstream cylinder is addressed and justified exhibiting the flow patterns. While considering the rectangular shape (for the upstream cylinder) and nonlinear velocity (at the inlet), the possibility of generating the unsteadiness in the steady wake flow of the downstream cylinder at a Re (based on height a*) less than the critical Re for the downstream cylinder is documented here. The dependence of flow characteristics of the downstream cylinder on the angle of incident linear velocity at specific S and r1 is also demonstrated here. It is observed that the discontinuous jump in the aerodynamic characteristics (due to a sudden change from one distinct flow pattern to the other in the critical spacing distance regime) is directly proportional to the height of the vortex generator. Increasing P under the same characteristic velocity causes the steady flow of cylinder(s) to convert to a periodic flow and reduces the critical spacing distance for the vortex generator.

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## Figures

Fig. 1

(a) Schematics of the two flow configurations (config-1 and config-2) depending upon inlet velocity profile. (b) Couette–Poiseuille flow based nonuniform linear/nonlinear incident velocity profiles for different P.

Fig. 6

Flow characteristics at Re = 100 for the case II (a* = 0.5A*: r2 = 0.5) for particular r1 = 0.1: (a) instantaneous equivorticity lines at S = 20, (b) instantaneous lift coefficient (CL) of both the cylinders against time in specific interval of time accompanied by their spectra at S = 20, and (c) spectra of fluctuating lift coefficient of the downstream cylinder for different S. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively. DN: downstream cylinder and UP: upstream cylinder.

Fig. 7

Proposed zone in the Sr1-plane for which unsteadiness in the steady flow of a square downstream cylinder can be generated at Re=100(<Rec−i−1.0−0.5): (a) r2 = 0.5 and (b) r2 = 1.0 by selecting the values of S and r1 from the shaded zone

Fig. 8

Streamlines/equivorticity lines for the square vortex generator of height 0.5A*(r2 = 0.5) at Re = 200: (a)–(d) instantaneous streamlines for different spacing distances, (e)–(h) equivorticity lines during one shedding cycle at S = 2.0, where T is the length of a period of the vortex generator. Spectra presented in (a)–(d) are related to the fluctuation of lift coefficient of the vortex generator. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively. (a) S = 1.0, (b) S = 1.5, (c) S = 2.0, (d) S = 2.5, (e) t = t0, (f) t = t0 + (T/4), (g) t = t0 + (T/2), and (h) t = t0 + (3T/4).

Fig. 2

The grid size distribution along x-direction (solid line) and y-direction (dotted line) for the case of upstream cylinder r2 = 1.0, r1 = 0.5, and S = 3. UP: upstream cylinder and DN: downstream cylinder.

Fig. 3

Instantaneous streamlines (left side) and equivorticity lines (right side) at Re = 100 for the case of a* = 0.5A* (r2 = 0.5) and different S and r1: (a) and (b) r1 = 1.0, (c) and (d) r1 = 0.5, and (e) and (f) r1 = 0.1. (a) S = 2.0, (b) S = 4.0, (c) S = 2.0, (d) S = 2.5, (e) S = 2.0, and (f) S = 2.5. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively. DN: downstream cylinder and UP: upstream cylinder.

Fig. 4

Instantaneous streamlines at Re = 100 for the case of a*=0.1A*(r2=0.1) and different S and r1: (a) and (b) r1 = 1.0 and (c) and (d) r1 = 0.1. (a) S = 2.0, (b) S = 4.0, (c) S = 2.0, (d) S = 4.0.

Fig. 5

Instantaneous equivorticity lines at Re = 100 for the case I (a* = A*: r2 = 1.0) for particular r1 = 0.75. (a) S = 3.5 and (b) S = 6.0. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively. DN: downstream cylinder and UP: upstream cylinder.

Fig. 9

Instantaneous lift coefficients of both the cylinders (dotted line: vortex generator and solid line: downstream cylinder) for different values of r2, r1, and S. Row shows the effect of height for particular shape of the vortex generator (first row: square, second row: rectangular of r1 = 0.5, and third row: rectangular of r1 = 0.1) at S = 3, 5, and 7. Column shows the effect of width for particular height of the vortex generator at S = 3, 5, and 7.

Fig. 10

The spectra of fluctuating lift coefficients of both the cylinders presented in Fig. 9

Fig. 11

Equivorticity lines for the cases of vortex generator of r1 = 0.5 during one shedding cycle of period T at S = 7.0 for Re = 200; (a)–(f): vortex generator of r2=1.0(a*=A*), (g)–(l): vortex generator of r2=0.5(a*=0.5A*). Here, t0 and t′0 are the time at which lift coefficient of the vortex generator of r2 = 1.0 and r2 = 0.5, respectively, attains its minimum value, and T is the period of the vortex generator of r2 = 1.0. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively. (a) t = t0, (b) t = t0 + (T/6), (c) t = t0 + (T/3), (d) t = t0 + (T/2), (e) t = t0 + (2T/3), (f) t = t0 + (5T/6), (g) t = t0′, (h) t =  t0′+ (T/6), (i) t =  t0′+ (T/3), (j) t =  t0′+ (T/2), (k) t =  t0′+ (2T/3), and (l) t =  t0′+ (5T/6).

Fig. 22

The spectra of fluctuating lift coefficient of a single rectangular cylinder of r1(=1.0,0.5, and 0.1) for different P (=0, 1, 3, and 5) at ReU2=1000

Fig. 23

Instantaneous equivorticity lines for a single cylinder of different r1 (=1.0, 0.5, and 0.1) for different P at ReU2=1000. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively.

Fig. 25

Instantaneous equivorticity lines and associated spectrum for the vortex generators of r1 = 0.5 and 0.1 with r2 = 1.0 at ReU2=1000 for different P. UP: upstream cylinder and DN: downstream cylinder. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively.

Fig. 12

The nondimensional normalized shedding frequency (St/St,0) as a function of spacing S at Re = 200 for the downstream cylinder in the presence of a vortex generator of different ratios r1 and r2. St,0=0.347, the value of an isolated square cylinder

Fig. 13

The normalized RMS of lift coefficient (CLrms/CLrms,0) as a function of spacing S at Re = 200 for the downstream cylinder in the presence of a vortex generator of different ratios r1 and r2. The normalized RMS of the lift coefficient of the vortex generator is also presented within the box. CLrms,0=0.39, the value of an isolated square cylinder.

Fig. 14

The normalized RMS of drag coefficient (CDrms/CDrms,0) as a function of spacing S at Re = 200 for the downstream cylinder in the presence of a vortex generator of different ratios r1 and r2. CDrms,0 = 0.1266, the value of an isolated square cylinder.

Fig. 15

Normalized time-averaged lift coefficient as a function of spacing at Re = 200 for different r1 and r2: (a) downstream square cylinder (CL¯/CL¯,0) and (b) sum of both the cylinders (CL¯,t/CL¯,0). CL¯,0=0.57, the value of an isolated square cylinder

Fig. 16

Normalized time-averaged drag coefficient as a function of spacing at Re = 200 for different r1 and r2: (a) downstream square cylinder (CD¯/CD¯,0) and (b) sum of both the cylinders (CD¯,t/CD¯,0). CD¯,0=2.985, the value of an isolated square cylinder

Fig. 17

Effect of height of the vortex generator of r1 = 0.5 on (a) time-averaged drag coefficient due to pressure (C¯DP) and due to shear (C¯DSh) on the downstream cylinder at S = 0.5 and 5.0 and (b) time-averaged surface pressure distribution (CP¯) on front (solid lines) and rear (dashed lines) faces of the downstream cylinder for S = 0.5

Fig. 18

The profiles of mean horizontal velocity component (u¯) along the vertical direction (y) at x = −3 and x = 1 for Re = 50 and 75 at fixed S = 3 and r1 = 0.5. (a) Gap flow of UP at x = −3 for Re = 50, (b) gap flow of DN at x = 1 for Re = 50, (c) gap flow of UP at x = −3 for Re = 75, and (d) gap flow of DN at x = 1 for Re = 75. DN: downstream cylinder and UP: upstream cylinder.

Fig. 19

(a) and (b) Instantaneous equivorticity lines at t = 410, and (c) time-averaged surface pressure distribution (CP¯) along the bottom face of the downstream cylinder and along the plane wall: for the case of vortex generator of r1 = 0.5 for r2 = 1.0 and 0.5 at Re = 175 and S = 3.0. (d)–(e) The time histories of the lift coefficient (CL) of both the cylinders at Re = 125, 150, and 175 for S = 3.0, r2 = 1.0, and r1 = 0.5. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively.

Fig. 20

Time-averaged coefficients as a function of Reynolds number for the downstream square cylinder for r2 = 1.0, 0.5, and aspect ratio r1 = 0.5 at S = 3: (a) drag coefficient (CD¯, CD,t¯) and (b) lift coefficient (CL¯, CL,t¯)

Fig. 21

(a) and (b) Instantaneous equivorticity lines, profile of mean horizontal velocity (u¯: (c) and (d)), and mean vertical velocity (v¯: (e) and (f)) along the vertical direction y for r1 = 0.1 at Re = 10 (left side) and ReU2=100 with P = 0 (right side)

Fig. 24

Spectrum along with the instantaneous equivorticity lines and pressure contours at ReU2=1000 for the square vortex generator (r1 = 1.0) of r2 = 1.0 for different P. In pressure contours, solid lines: positive pressure and dotted lines: negative pressure. UP: upstream cylinder and DN: downstream cylinder. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively.

Fig. 26

Instantaneous equivorticity lines and associated spectrum for the vortex generators of different r1 (=1.0, 0.5, and 0.1) with r2 = 0.5 at ReU2=1000 for different P. UP: upstream cylinder and DN: downstream cylinder. The black and white color in the grayscale flooded vorticity contour represents negative and positive vorticity, respectively.

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