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Research Papers: Fundamental Issues and Canonical Flows

Numerical Analysis of Laminar-Drag-Reducing Grooves

[+] Author and Article Information
A. Mohammadi

Department of Mechanical and
Materials Engineering,
The University of Western Ontario,
London ON N6A 5B9, Canada
e-mail: amoham69@alumni.uwo.ca

J. M. Floryan

Department of Mechanical and
Materials Engineering,
The University of Western Ontario,
London ON N6A 5B9, Canada
e-mail: mfloryan@eng.uwo.ca

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 28, 2014; final manuscript received October 13, 2014; published online December 3, 2014. Assoc. Editor: Mark F. Tachie.

J. Fluids Eng 137(4), 041201 (Apr 01, 2015) (12 pages) Paper No: FE-14-1106; doi: 10.1115/1.4028842 History: Received February 28, 2014; Revised October 13, 2014; Online December 03, 2014

The performance of grooves capable of reducing shear drag in laminar channel flow driven by a pressure gradient has been analyzed numerically. Only grooves with shapes that are easy to manufacture have been considered. Four classes of grooves have been studied: triangular grooves, trapezoidal grooves, rectangular grooves, and circular-segment grooves. Two types of groove placements have been considered: grooves that are cut into the surface (they can be created using material removal techniques) and grooves that are deposited on the surface (they can be created using material deposition techniques). It has been shown that the best performance is achieved when the grooves are aligned with the flow direction and are symmetric. For each class of grooves, there exists an optimal groove spacing, which results in the largest drag reduction. The largest drag reduction results from the use of trapezoidal grooves and the smallest results from the use of triangular grooves for the range of parameters considered in this work. Placing the same grooves on both walls increases the drag reduction by up to four times when comparing with grooves on one wall only. The predictions remain valid for any Reynolds number as long as the flow remains laminar.

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References

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Figures

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Fig. 1

Channel with grooved walls

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Fig. 2

Geometry of triangular grooves: (a)—triangular grooves cut into the surface of the lower wall and (b)—triangular grooves deposited onto the surface of the lower wall

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Fig. 3

Effects of symmetry of triangular grooves: (a) variations of the groove shape from an isosceles triangle with a1/λ = 0.5, b1/λ = 0.5 to a right triangle with a1/λ = 1, b1/λ = 0 and (b) variations of the normalized modification friction factor f1/f0 as a function of the groove wave number β and the shape parameter a1/λ (see Fig. 2(a)) for grooves placed at the lower wall with b1 = λ − a1, d1 = 0, and S = 0.4. Black (gray) isolines identify zones with drag reduction (increase).

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Fig. 4

Variations of the normalized modification friction factor f1/f0 as a function of the groove wave number β and the amplitude S for symmetric triangular grooves with a1 = b1 = π/β and zero spacing (d1 = 0) placed at the lower wall. Black (gray) isolines identify zones with the drag reduction (increase).

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Fig. 5

Variations of the normalized modification friction factor f1/f0 as a function of the spacing d1 between symmetric triangular grooves placed at the lower wall. Results for grooves with S = 0.2, 0.4, 0.8, and 1.6 are displayed in (a)–(d), respectively. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the surface of the lower wall. Asterisks mark the optimal distance between the grooves.

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Fig. 6

Variations of channel geometry associated with change of fraction of a typical channel segment occupied by the cut. The total segment length is 2π/β and the distance between the cuts is d1.

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Fig. 7

Variations of the normalized modification friction factor f1/f0 as a function of the groove spacing expressed as a fraction of the groove wavelength d1/λ for symmetric triangular grooves placed at the lower wall. Results presented in (a)–(d) correspond to S = 0.2, 0.4, 0.8, and 1.6, respectively. Asterisks mark the optimal distance between the grooves. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the surface of the lower wall.

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Fig. 8

Geometry of channels with triangular grooves placed at both walls. ztip denotes the relative spanwise positioning of the upper and lower groove systems. (a) and (b) Grooves that are either cut into or deposited onto the surface of the walls, respectively.

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Fig. 9

Variations of the normalized modification friction factor f1/f0 as a function of the relative spanwise position ztip of the grooves on the upper and lower walls. (a)–(c)Results for S = 0.2, 0.4, and 0.8, respectively, with solid (dashed) lines referring to grooves cut into (deposited onto) the wall. The spacing between grooves is the same on both walls. (a) Results for grooves cut into the wall with (d1/λ, β) = (0.33, 0.1), (0.24, 0.5), (0.13, 0.9) and grooves deposited onto the wall with (d1/λ, β) = (0.32, 0.1), (0.25, 0.5), (0.15, 0.9). (b) Results for grooves cut into the wall with (d1/λ, β) = (0.34, 0.1), (0.24, 0.5), (0.12, 0.9) and grooves deposited onto the wall with (d1/λ, β) = (0.32, 0.1), (0.26, 0.5), (0.17, 0.9). (c) Results for grooves cut into the wall with (d1/λ, β) = (0.35, 0.1), (0.23, 0.5), (0.1, 0.9) and grooves deposited onto the wall with (d1/λ, β) = (0.31, 0.1), (0.27, 0.5), (0.2, 0.9).

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Fig. 10

Variations of the normalized modification friction factor f1x/f0x as a function of the groove wave number α˜ and the groove inclination angle ϕ for a channel with triangular grooves placed at the lower wall with d1 = 0, S = 0.2, and Re = 5 (a) and Re = 500 (b). Solid, dashed and dotted lines correspond to grooves with a1 = b1 = π/α˜, a1 = 2π/α˜ and b1 = 0, and a1 = 0 and b1 = 2π/α˜ (see the Appendix and Fig. 2 for definitions of parameters). Black (gray) isolines identify drag reduction (increase). The reader should note that α˜ defines the groove wave number in the direction orthogonal to the groove ridge; α˜ = β for the longitudinal grooves. The reader may also note that in (a), dashed and dotted lines overlap within the figure resolution.

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Fig. 11

Geometry of trapezoidal grooves: (a)—trapezoidal grooves cut into the surface of the lower wall and (b)—trapezoidal grooves deposited onto the surface of the lower wall

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Fig. 12

Variations of the normalized modification friction factor f1/f0 as a function of the groove spacing d2 for symmetric trapezoidal grooves placed at the lower wall. Results presented in (a) and (b) correspond to S = 0.2 and 0.8, respectively, and a2 = b2 = c2. Solid (dashed) lines correspond to grooves cut into (deposited onto) the surface of the lower wall. Asterisks mark the optimal distance between the grooves.

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Fig. 13

Variations of the normalized modification friction factor f1/f0 as a function of the groove base c2/λ and the groove spacing d2/λ for trapezoidal grooves with a2 = b2 = (λ − c2 − d2)/2 cut into the lower wall. (a)–(f) Grooves with (S, β) = (0.2, 0.1), (0.2, 0.5), (0.2, 0.9), (0.8, 0.1), (0.8, 0.5), and (0.8, 0.9), respectively. Asterisks identify combinations of (c2, d2) which produce the maximum drag reduction. Black (gray) isolines identify drag reduction (increase).

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Fig. 14

The same as in Fig. 13 but with the grooves placed on both walls exactly opposite to each other (ztip = 0)

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Fig. 15

Geometry of rectangular grooves: (a)—rectangular grooves cut into the surface of the lower wall and (b)—rectangular grooves deposited onto the surface of the lower wall

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Fig. 16

Variations of the normalized modification friction factor f1/f0 as a function of the groove spacing d3 for rectangular grooves placed at the lower wall. Results for grooves with the heights S = 0.2, 0.4, 0.8, and 1.6 are displayed in (a)–(d), respectively. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the surface of the lower wall. Asterisks mark the optimal distance between the grooves.

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Fig. 17

Variations of the normalized modification friction factor f1/f0 as a function of the groove spacing expressed as a fraction of the groove wavelength d3/λ for rectangular grooves placed at the lower wall. Results for the groove heights S = 0.2, 0.4, 0.8, and 1.6 are displayed in (a)–(d), respectively. Asterisks identify the optimal groove spacing. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the surface of the lower wall.

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Fig. 18

The same as in Fig. 17 but with grooves placed on both walls exactly opposite to each other (ztip = 0). (a)–(c) The groove heights S = 0.2, 0.4, and 0.8, respectively. Asterisks denote the optimal groove spacing. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the wall surface.

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Fig. 19

Geometry of grooves made of circular segments: (a)—grooves cut into the lower wall with c4≥2S and radius of curvature R defined by Eq. (11) and (b)—the same as in (a) but deposited onto the lower wall

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Fig. 20

Variations of the normalized modification friction factor f1/f0 as a function of the groove spacing d4 for the circular-segment grooves placed at the lower wall. Results for groove heights S = 0.2, 0.4, 0.8, and 1.6 are displayed in (a)–(d), respectively. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the wall surface. Asterisks identify the optimal distance between the grooves.

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Fig. 21

Variations of the normalized modification friction factor f1/f0 as a function of the groove spacing expressed as a fraction of the groove wavelength d4/λ for circular segment grooves placed at the lower wall. Results for the groove heights S = 0.2, 0.4, 0.8, and 1.6 are displayed in (a)–(d), respectively. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the wall surface. Asterisks identify the optimal groove spacing.

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Fig. 22

The same as in Fig. 21 but with the grooves placed on both walls exactly opposite to each other (ztip = 0). Results for groove heights S = 0.2, 0.4, and 0.8 are displayed in (a)–(c), respectively. Solid (dashed) lines correspond to grooves that are cut into (deposited onto) the wall surface. Asterisks identify the optimal groove spacing.

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Fig. 23

Channel with grooved walls. The (x,y,z)-coordinate system is flow-oriented and the (x˜,y,z˜)-system is groove-oriented. The inclination angle ϕ shows the relative orientation of both systems.

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