^{1,2,3}, A. Pollard

^{1,a)}, J. Mi

^{2,a)}, F. Secretain

^{1}and H. Sadeghi

^{1}

### Abstract

The flow in the near-to-intermediate field of a jet emanating from a long square pipe is investigated using hot-wire anemometry. The data include distributions of the mean and high order turbulence moments over 8000 < Re < 50 000 along the jet centreline. It is demonstrated that the far-field rates of the mean velocity decay and spread, as well as the asymptotic value of the streamwise turbulent intensity, all decrease as Re increases for Re ≤ 30 000; however, they become approximately Re-independent for Re > 30 000. It follows that the critical Reynolds number should occur at Recr = 30 000. Attention is given to the exponents associated with the compensated axial velocity spectra that show that the inertial subrange calculated according to isotropic, homogeneous turbulence emerges at x/D e = 30 for all Re; however, if the scaling exponent is altered from m = −5/3 to between −1.56 < m < −1.31 the “inertial” range emerges at lower values of Re and the exponent is Re dependent. It is also found that the exponent agrees very well with Mydlarski and Warhaft correlation m = (5/3)(1−3.15R λ −2/3), where R λ is the Taylor Reynolds number, obtained for turbulence decay behind a grid.

M.X. gratefully acknowledges the support of Graduate School of Peking University for his visit to Queen's University. J.M. acknowledges the support of National Science Foundation of China (NSFC) (Grant Nos. 10921202 and 11072005). A.P. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.

I. INTRODUCTION

II. EXPERIMENTAL DETAILS

III. PRESENTATION AND DISCUSSION OF RESULTS

A. Inlet conditions

B. The mean velocity field

C. The fluctuating velocity field

IV. CONCLUSION AND RECOMMENDATION FOR FURTHER WORK

### Key Topics

- Reynolds stress modeling
- 34.0
- Turbulent flows
- 32.0
- Duct flows
- 9.0
- Nozzle jets
- 7.0
- Velocity measurement
- 7.0

##### G01F

## Figures

Schematic of the experimental arrangement (left) and three-dimensional square jet nozzle exit, jet notation and coordinate system (right).

Schematic of the experimental arrangement (left) and three-dimensional square jet nozzle exit, jet notation and coordinate system (right).

Inflow conditions at x/D e = 0.05 for different Reynolds numbers. (a) Normalized mean velocity; (b) turbulence intensity. Note that the inset panel indicates location (Y) used for integral calculations (see Eqs. (4) and (5) ).

Inflow conditions at x/D e = 0.05 for different Reynolds numbers. (a) Normalized mean velocity; (b) turbulence intensity. Note that the inset panel indicates location (Y) used for integral calculations (see Eqs. (4) and (5) ).

Dependences of (a) displacement thicknesses (δ) and (b) momentum thickness (θ) on Reynolds number and integral upper limit (Y), Figure 2 and Eqs. (4) and (5) .

Dependences of (a) displacement thicknesses (δ) and (b) momentum thickness (θ) on Reynolds number and integral upper limit (Y), Figure 2 and Eqs. (4) and (5) .

Streamwise variations of U j /U c for Re = 8000 ∼ 50 000.

Streamwise variations of U j /U c for Re = 8000 ∼ 50 000.

Dependence on Re of K U (open symbol) and x U /D e (closed symbol).

Dependence on Re of K U (open symbol) and x U /D e (closed symbol).

Lateral distributions of U/U c for Re = 8000–50 000.

Lateral distributions of U/U c for Re = 8000–50 000.

Streamwise variations of R 1/2 /D e for Re = 8000 ∼ 50 000.

Streamwise variations of R 1/2 /D e for Re = 8000 ∼ 50 000.

Dependence of the spreading rate (K Y ) and virtual origin (x R /D e ) on Re d .

Dependence of the spreading rate (K Y ) and virtual origin (x R /D e ) on Re d .

Normalized streamwise RMS velocity u ′/U c along the centreline.

Normalized streamwise RMS velocity u ′/U c along the centreline.

Re dependence of the asymptotic value of turbulence intensity.

Re dependence of the asymptotic value of turbulence intensity.

Lateral distributions of u ′/U c for (a) Re = 8000, (b) = 12 000, (c) Re = 20 000; (d) Re = 30 000, (e) Re = 50 000.

Lateral distributions of u ′/U c for (a) Re = 8000, (b) = 12 000, (c) Re = 20 000; (d) Re = 30 000, (e) Re = 50 000.

Streamwise evolution of skewness for Re = 8000–50 000.

Streamwise evolution of skewness for Re = 8000–50 000.

Streamwise evolution of flatness for Re = 8000–50 000.

Streamwise evolution of flatness for Re = 8000–50 000.

Dependence of asymptotic value of S *and F *.

Dependence of asymptotic value of S *and F *.

Centreline spectra Φ u of velocity fluctuation u between x/D e = 1 and x/D e = 10 for (a) Re = 8000, (b) Re = 12 000, (c) Re = 20 000, (d) Re = 30 000, (e) Re = 50 000.

Centreline spectra Φ u of velocity fluctuation u between x/D e = 1 and x/D e = 10 for (a) Re = 8000, (b) Re = 12 000, (c) Re = 20 000, (d) Re = 30 000, (e) Re = 50 000.

Dependence of the Strouhal number St = f p D e /U j on Re. Note that the values of the St are those of f * that correspond to peaks of Φ u between x/D e = 2 and x/D e = 5 as indicated in Fig. 14 by the dashed lines.

Dependence of the Strouhal number St = f p D e /U j on Re. Note that the values of the St are those of f * that correspond to peaks of Φ u between x/D e = 2 and x/D e = 5 as indicated in Fig. 14 by the dashed lines.

Evolution of the centreline Φ u obtained at x/D e = 20, 30 and 40 in the form of fΦ u vs log (fY 1/2/U c ) for (a) Re = 8000, (b) Re = 12 000, (c) Re = 20 000, (d) Re = 30 000, (e) Re = 50 000.

Evolution of the centreline Φ u obtained at x/D e = 20, 30 and 40 in the form of fΦ u vs log (fY 1/2/U c ) for (a) Re = 8000, (b) Re = 12 000, (c) Re = 20 000, (d) Re = 30 000, (e) Re = 50 000.

Dependence of f * p = f p Y 1/2/U c on Re.

Dependence of f * p = f p Y 1/2/U c on Re.

Reynolds number dependent spectra of the centreline u measured at x/D e = 30.

Reynolds number dependent spectra of the centreline u measured at x/D e = 30.

Compensated velocity spectra f m Φ u of the centreline u measured at x/D e = 30 for (a) m = 5/3, and (b) m = 1.31–1.56.

Compensated velocity spectra f m Φ u of the centreline u measured at x/D e = 30 for (a) m = 5/3, and (b) m = 1.31–1.56.

The R λ dependence of the power-law exponent of Φ u , all for x/De = 30.

The R λ dependence of the power-law exponent of Φ u , all for x/De = 30.

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