Fundamental Issues and Canonical Flows

Steady Laminar Axisymmetrical Nozzle Flow at Moderate Reynolds Numbers: Modeling and Experiment

[+] Author and Article Information
X. Grandchamp, Y. Fujiso, B. Wu

GIPSA-lab, UMR CNRS 5216,  Grenoble University, Grenoble, 38000 France

A. Van Hirtum1

GIPSA-lab, UMR CNRS 5216,  Grenoble University, Grenoble, 38000 Franceannemie.vanhirtum@gipsa-lab


Corresponding author.

J. Fluids Eng 134(1), 011203 (Feb 24, 2012) (13 pages) doi:10.1115/1.4005690 History: Received February 02, 2011; Revised November 15, 2011; Published February 23, 2012; Online February 24, 2012

Flow through an axisymmetrical parameterized contraction nozzle of limited size with area contraction ratio 21.8 and total length 6 cm is studied for moderate Reynolds numbers 300 < Re < 20,200. The transverse flow profiles at the nozzle exit are characterized by hot film anemometry for two different spatial step sizes. The flow at the exit is laminar and uniform in its core. Boundary layer characteristics at the nozzle exit are estimated from the transverse velocity profiles. Flow throughout the nozzle is modeled by implementing Thwaites laminar axisymmetrical boundary layer solutions in an iterative algorithm for which both universal functions, describing the shape factor and skin friction parameters respectively, are altered by adding a constant. The value of the constants is determined by fitting the modified universal functions to tabulated values reported in Blevins (Blevins, R., 1992, Applied Fluid Dynamics Handbook. Krieger, Malabar, FL.). The model is validated on the measured data. Adding nonzero constants to the universal functions improves the prediction of boundary layer characteristics so that the range of Reynolds numbers for which the discrepancy with experimental findings is less than 4% is extended from Re > 3000 to Re > 1000. Therefore, the studied contraction nozzle is of use for applications requiring a small nozzle with known low turbulence flow at the exit such as moderate Reynolds number free jet studies or bio fluid mechanics (respiration, speech production,…) and the flow at the exit of the nozzle can be accurately described by a simple boundary layer algorithm for Re > 1000.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Illustration of parameterized axisymmetrical nozzle geometry, D(x) = 2R(x), obtained from matching at x = xm an upstream cubic (1) (dashed line) and a downstream cubic (2) (thin full line) with parameters D1  = 100 mm, D2  = 25 mm, L = 60 mm, and xm  = 52mm. The longitudinal x-axis corresponds to the main streamwise direction and the y-axis to the transverse direction.

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Figure 2

Illustration of (a) H(λ) for cH  = 0 (c = 0) and cH  = 0.35 (c ≠ 0) and (b) S(λ) for cS  = 0 (c = 0) and cS  = −0.02 (c ≠ 0)

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Figure 3

(a) A sketch of the apparatus: [a] air supply, [b] pressure regulator, [c] valve, [d] mass flow meter, [e] divergent, [f] uniform pipe, [g] convergent nozzle with parameters D1 =100mm, D2 =21.4mm, L=60mm and xm =52mm, [h] hot film, [i] positioning system, [j] IFA 300. (b) Detail of the experimental nozzle and velocities of interest: centerline velocity Uc (x, y = 0), centerline velocity at the inlet U1  = Uc (x = 0, y = 0) and centerline velocity at exit U2  = Uc (x = L, y = 0). The transverse velocity at the nozzle exit Ue (y) is measured at a small distance from the nozzle exit (x − L)/D2  < 0.04. The complete wall to wall exit velocity profile is measured with a transverse spatial step size of Δy = 0.1 mm further labeled ‘measured exit profile’. A second partial profile is measured near the wall covering the boundary layer with a precise transverse spatial step size Δy = 0.01 mm further labeled “measured exit boundary layer” profile.

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Figure 4

(a) Comparison of measured exit velocity profiles Ue (y) obtained with spatial step Δy = 0.1 mm (symbols) and measured exit boundary layer profiles Δy = 0.01 mm (dots) for different Reynolds numbers Re in the range −0.5 ≤ R/D2  ≤ −0.3. (b) Measured normalized transverse exit velocity profiles Ue (y)/U2 for Δy = 0.1mm and comparison with parabolic, 1/7 power law, uniform profile with vanishing momentum thickness δ2  = 0 and top hat profile with momentum thickness δ2  = 0.004D2 .

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Figure 5

Measured turbulence intensities TU  × 100 [%] of exit velocity profiles for Δy = 0.1 mm. In the inner plot, measured centerline turbulence intensities TU (y = 0) are presented as function Reynolds number Re.

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Figure 6

Boundary layer characterization of the measured mean boundary layer profiles for Δy = 0.01 mm: (a) comparison with Blasius profile for a laminar boundary layer. (b) Comparison of U+ (y+ ) with linear law of the wall U+  = y+ for a laminar boundary layer.

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Figure 7

Illustration of modeled streamwise flow development through the contraction nozzle with parameters D1 =100 mm, D2 =21.4 mm, L=60 mm and xm =52 mm, corresponding to area contraction ratio CR = 21.8, for cH  = 0 and cS  = 0 as function of different Reynolds numbers in the range 300 ≤ Re ≤ 17,000 (symbols). The vertical dashed-dotted line indicates the matching point of the cubics x = xm . The scaled radius of the nozzle is indicated by a solid thick line.

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Figure 8

Modeled and measured centerline velocity at the nozzle exit normalized by the inlet centerline velocity U2 /U1 as function of Reynolds number Re for (a) fixed inlet diameter D1  = 100 mm, (b) inlet diameter D1  = 100 mm and D1  = 50 mm, (c) fixed area contraction ratio CR = 4.5 and (d) fixed area contraction ratio CR = 21.8. The ratio U2 /U1 for an ideal fluid yields the area contraction ratio CR (dashed line labeled ideal). Measured centerline velocities for D1  = 100 mm and D2  = 21.4 mm are indicated (measured). Modeled values are obtained for cH  = 0 and cS  = 0, denoted cH,S  = 0, except in Fig. 8 where also results for cH  = 0.35 and cS  = −0.02, labeled cH,S  ≠ 0, are shown.

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Figure 9

Comparison of modeled and experimental assessed normalized boundary layer characteristics δ1 /D2 (Fig. 9), δ2 /D2 (Fig. 9), H (Fig. 9) and λ (Fig. 9) at the exit of the nozzle with parameters D1 =100 mm, D2 =21.4 mm, L=60 mm, and xm =52 mm as function of Reynolds number. The influence of model coefficients cH,S and spatial step size Δy in the transverse exit profile is illustrated for δ1 and δ2 . Zero model constants cH  = 0 and cS  = 0 is denoted cH,S  = 0 whereas non zero model constants cH  = 0.35 and cS  = −0.02 is denoted cH,S  ≠ 0. Quantities estimated from transverse profiles using Δy = 0.1mm are labeled “measured exit profile” and transverse profiles using Δy = 0.01 mm are labeled “measured exit boundary layer.” In Fig. 9 also the theoretical value H = 2.59 for Blasius laminar profile is shown.




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