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Research Papers: Fundamental Issues and Canonical Flows

A Numerical Study of the Impact of Wavy Walls on Steady Fluid Flow Through a Curved Tube

[+] Author and Article Information
Sean D. Peterson

e-mail: peterson@mme.uwaterloo.ca
Mechanical and Mechatronics Engineering,
University of Waterloo,
200 University Avenue West,
Waterloo, ON, N2L 3G1, Canada

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received April 13, 2012; final manuscript received October 27, 2012; published online May 23, 2013. Assoc. Editor: Sharath S. Girimaji.

J. Fluids Eng 135(7), 071207 (May 23, 2013) (13 pages) Paper No: FE-12-1194; doi: 10.1115/1.4023662 History: Received April 13, 2012; Revised October 27, 2012

In this paper, we discuss the impact of a wavy-walled pipe cross-section on steady flow in a curved tube at moderate Dean numbers and small tube radius-to-radius-of-curvature ratios. Parameters investigated include the protrusion height, the number of protrusions around the tube circumference, and the pipe curvature. This work extends a previous analytical investigation that employed a double perturbation expansion to elucidate the flow field as a function of these parameters. Due to the rapid growth in the solution complexity as the number of terms in each expansion increases, the analytical work is relegated to small wall perturbations and low Dean numbers. These barriers are removed in the present study by numerically solving the Navier–Stokes equations at Dean numbers up to 2500. The impact on the axial and secondary flow structures are emphasized, along with the resulting wall shear stress distributions.

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References

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Figures

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Fig. 1

Model geometry and coordinate system definitions

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Fig. 2

Hexahedral mesh in the (a) pipe cross-section and (b) on a portion of the symmetry plane which has 600 divisions in the stream-wise direction for ε = 0.1, n = 5, and δ = 1.25 × 10−2

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Fig. 3

Axial velocity contours (a) and secondary flow structure (b) for K = 100 for ε = 0.1, n = 5, and δ = 1.25 × 10−2. In (a), the solid lines and dashed lines represent contour lines for the wavy-walled pipe and circular pipe, respectively. For the secondary flow structures (b), the flow is carried from the inner wall toward the outer wall along the horizontal centerline.

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Fig. 4

Effect of the inclusion of the wavy wall on the (a) axial (Cfa) and (b) circumferential (Cfϕ) skin friction for K = 100 for circular pipe and ε = 0.1, n = 5, and δ = 1.25 × 10−2

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Fig. 5

Contour plot of the axial vorticity for the circular pipe (a) and the wavy-walled pipe (b) with ε = 0.1 and n = 5 for K = 100 and δ = 1.25 × 10−2. The spacing between contour levels is given in brackets underneath each figure (min: interval: max).

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Fig. 6

Axial velocity contours (left column) and secondary flow structures (right column) for (a) K = 500, (b) 1000, and (c) 2500 for ε = 0.05, n = 5, and δ = 1.25 × 10−2. For the axial velocity contours (left column), the solid lines indicate the wavy-walled pipe with ε = 0.05 and the dashed lines indicate the circular pipe. For the secondary flow structures (right column), the flow is carried from the inner wall toward the outer wall along the horizontal centerline.

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Fig. 7

Profile of the v component of velocity along the y-axis for ε = 0.05, n = 5, and δ = 1.25 × 10−2 at various Dean numbers. Each profile is normalized by its maximum velocity value (K = 500—circular pipe: 5.67 × 10−4 m/s, ε = 0.05: 5.70 × 10−4 m/s; K = 2500—circular pipe: 9.20 × 10−4 m/s, ε = 0.05: 9.24 × 10−4 m/s), and the position along the y-axis is normalized by the pipe radius a.

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Fig. 8

(a) Axial and (b) circumferential WSS for the circular pipe and (c) axial and (d) circumferential WSS for ε = 0.05, n = 5, and δ = 1.25 × 10−2. All WSS profiles are normalized by their maximum values. The normalized protrusion heights are also plotted for positional reference in the wavy wall cases. Note that φ=0 corresponds to the outer wall.

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Fig. 9

Contour plot of the axial vorticity for (a) K = 100 and (b) K = 2500 for ε = 0.05, n = 5, and δ = 1.25 × 10−2. The spacing between contour levels is given in brackets underneath each figure (min: interval: max).

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Fig. 10

(a) Contour plot of zero vorticity region, and (b) thickness of the boundary layer region aω0 normalized by the tube radius rtube for ε = 0.05, n = 5, and δ = 1.25 × 10−2

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Fig. 11

Nondimensional circulation of the secondary flow as a function of the Dean number for various protrusion heights at n = 5 and δ = 1.25 × 10−2

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Fig. 12

Cross-stream velocity profiles for (a) K = 500 and (b) K = 2500 for various protrusion heights with n = 5 and δ = 1.25 × 10−2

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Fig. 13

Comparison of the numerically predicted axial (first row) and cross-stream velocity profiles (second row) with the perturbation solution of Peterson [18] for K = 1 (first column) and K = 100 (second column) with n = 5 and δ = 1.45 × 10−4

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Fig. 14

Effect of ε on the (a) axial and (b) circumferential skin friction for K = 500, n = 5, and δ = 1.25 × 10−2

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Fig. 15

Zero vorticity contour for various protrusion heights at K = 2500 and δ = 1.25 × 10−2

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Fig. 16

The wall shape (a) for pipe cross-section with n = 5 versus n = 8. Effect of n on the (b) axial and (c) circumferential skin friction for K = 500, ε = 0.05, and δ = 1.25 × 10−2 (n = 0 represents a circular pipe).

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