^{1,a)}, Graham J. Nathan

^{2}and Jianchun Mi

^{3}

### Abstract

A similarity analysis is presented of the momentum field of a subsonic, plane air jet over the range of the jet-exit Reynolds number Re h (≡ U b h/υ where U b is the area-averaged exit velocity, h the slot height, and υ the kinematic viscosity) = 1500 − 16 500. In accordance with similarity principles, the mass flow rates, shear-layer momentum thicknesses, and integral length scales corresponding to the size of large-scale coherent eddy structures are found to increase linearly with the downstream distance from the nozzle exit (x) for all Re h . The autocorrelation measurements performed in the near jet confirmed reduced scale of the larger coherent eddies for increased Re h . The mean local Reynolds number, measured on the centerline and turbulent local Reynolds number measured in the shear-layer increases non-linearly following x 1/2, and so does the Taylor microscale local Reynolds number that scales as x 1/4. Consequently, the comparatively larger local Reynolds number for jets produced at higher Re h causes self-preservation of the fluctuating velocity closer to the nozzle exit plane. The near-field region characterized by over-shoots in turbulent kinetic energy spectra confirms the presence of large-scale eddy structures in the energy production zone. However, the faster rate of increase of the local Reynolds number with increasing x for jets measured at larger Re h is found to be associated with a wider inertial sub-range of the compensated energy spectra, where the −5/3 power law is noted. The downstream region corresponding to the production zone persists for longer x/h for jets measured at lower Re h . As Re h is increased, the larger width of the sub-range confirms the narrower dissipative range within the energy spectra. The variations of the dissipation rate (ɛ) of turbulent kinetic energy and the Kolmogorov (η) and Taylor (λ) microscales all obey similarity relationships, , η/h ∼ Re h −3/4, and λ/h ∼ Re h −1/2. Finally, the underlying physical mechanisms related to discernible self-similar states and flow structures due to disparities in Re h and local Reynolds number is discussed.

R.C.D. acknowledges the Endeavour International Research Award, the Adelaide Achiever's Research Award, and ARC Linkage Grant for his Ph.D. studies. It is expressly stated that all experiments were undertaken at the School of Mechanical Engineering, The University of Adelaide, Australia. The analysis and completion of the final manuscript was undertaken by R.C.D. at the University of Southern Queensland (Springfield) that has been kindly supported by the Department of Mathematics and Computing. Finally, we thank both reviewers for their insightful comments which have strengthened the manuscript.

I. INTRODUCTION

II. ANALYTICAL OVERVIEW

III. EXPERIMENTAL DESIGN

A. Plane jet facility

B. Hot wire anemometry

C. Jet exit conditions

IV. RESULTS AND GENERAL DISCUSSION

A. Mean and turbulent statistics

B. Isotropic turbulence statistics

C. Explanation of flow structures by spectral analysis

D. Further analysis of flow structure

V. FURTHER DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Eddies
- 65.0
- Reynolds stress modeling
- 64.0
- Turbulent flows
- 44.0
- Turbulent jets
- 40.0
- Viscosity
- 15.0

## Figures

(a) A schematic of the plane jet nozzle, (b) time-averaged flow, and (c) laboratory dimensions. Note: U c ≡ centerline mean velocity, y 0.5 ≡ half-width (y-value where U(x, y) = 1/2U c(x)).

(a) A schematic of the plane jet nozzle, (b) time-averaged flow, and (c) laboratory dimensions. Note: U c ≡ centerline mean velocity, y 0.5 ≡ half-width (y-value where U(x, y) = 1/2U c(x)).

The jet-exit conditions defined by (a) displacement thickness (δm), momentum thickness (θm) and shape factors (H) at x/h = 0.5 reported by Deo et al. 17 (b) Momentum-based Reynolds number (Re θ) and shear layer-peak turbulent Reynolds number ( ) measured at x/h = 0.25. In part (a), δm and θm at x/h = 1 for high-AR rectangular jet of Namar and Otugen 12 are shown.

The jet-exit conditions defined by (a) displacement thickness (δm), momentum thickness (θm) and shape factors (H) at x/h = 0.5 reported by Deo et al. 17 (b) Momentum-based Reynolds number (Re θ) and shear layer-peak turbulent Reynolds number ( ) measured at x/h = 0.25. In part (a), δm and θm at x/h = 1 for high-AR rectangular jet of Namar and Otugen 12 are shown.

Streamwise evolution of mass flow rates m(x) normalized by bulk mass flow rate (m 0) at the exit plane.

Streamwise evolution of mass flow rates m(x) normalized by bulk mass flow rate (m 0) at the exit plane.

Dependence of entrainment rates (K m ) and virtual origin (x 04/h) on Re h .

Dependence of entrainment rates (K m ) and virtual origin (x 04/h) on Re h .

Streamwise evolutions of momentum thickness θ m(x) for various Re h .

Streamwise evolutions of momentum thickness θ m(x) for various Re h .

The Re h -dependence of the shear-layer growth rate (K θ) and mixing length (l m) based on Prandtl's mixing length hypothesis where l m is proportional to K y as embodied in Eq. (13) .

The Re h -dependence of the shear-layer growth rate (K θ) and mixing length (l m) based on Prandtl's mixing length hypothesis where l m is proportional to K y as embodied in Eq. (13) .

Streamwise evolutions of the integral length scale, Λ (x) /h for various cases of Re h . The asymptotic decrease of K Λ with increasing Re h is shown in the inset.

Streamwise evolutions of the integral length scale, Λ (x) /h for various cases of Re h . The asymptotic decrease of K Λ with increasing Re h is shown in the inset.

Streamwise evolutions of (a) local Reynolds number (Re θ) based on mean momentum thickness θ m(x), (b) local Reynolds number (Re y0.5) based on jet half-widths y 0.5(x). Note that part (b) has been modified after Deo 30 and Deo et al. 17

Streamwise evolution of the one-dimensional turbulent kinetic energy E k normalized by U 2 c(x) together with the data of Namar and Otugen. 12

Streamwise evolution of the one-dimensional turbulent kinetic energy E k normalized by U 2 c(x) together with the data of Namar and Otugen. 12

Lateral profiles of turbulence intensity u n = <[u(x, y)]>1/2/U c(x) measured across the jet for Re h = (a) 1500, (b) 3000, (c) 7000, (d) 10 000, and (e) 16 500.

Lateral profiles of turbulence intensity u n = <[u(x, y)]>1/2/U c(x) measured across the jet for Re h = (a) 1500, (b) 3000, (c) 7000, (d) 10 000, and (e) 16 500.

Dependence of streamwise turbulence intensity on Re θ(x) for Re h = 3000–16 500.

Dependence of streamwise turbulence intensity on Re θ(x) for Re h = 3000–16 500.

Streamwise evolutions of the local turbulent Reynolds number . Note: <u 2 p(x) > 1/2 ≡ peak value of shear layer turbulence intensity y 0.5 (x) ≡ jet half-width and <u 2 p(x) > 1/2 is deduced from Fig. 10 .

Streamwise evolutions of the local turbulent Reynolds number . Note: <u 2 p(x) > 1/2 ≡ peak value of shear layer turbulence intensity y 0.5 (x) ≡ jet half-width and <u 2 p(x) > 1/2 is deduced from Fig. 10 .

Streamwise evolutions of the peak shear-layer turbulence intensity, <u 2 p(x) > 1/2/U c(x) in %. Symbols are identical to Fig. 12 .

Streamwise evolutions of the peak shear-layer turbulence intensity, <u 2 p(x) > 1/2/U c(x) in %. Symbols are identical to Fig. 12 .

Lateral profiles of (a) the skewness factors (Su) and (b) the flatness factors (F u) of the fluctuating velocity measured across the jet at x/h = 20. The dashed (- - -) line shows the maximum skewness in the outer boundary layer.

Lateral profiles of (a) the skewness factors (Su) and (b) the flatness factors (F u) of the fluctuating velocity measured across the jet at x/h = 20. The dashed (- - -) line shows the maximum skewness in the outer boundary layer.

Streamwise evolutions of the Kolmogorov microscales (η/h) as per Deo et al. 17

Streamwise evolutions of the Taylor microscales (λ/h).

Streamwise evolutions of the Taylor microscales (λ/h).

Centerline variation of the local Taylor microscale-based Reynolds number, λ(x)/υ. The plane jet data of Mi et al. 42 is included.

Centerline variation of the local Taylor microscale-based Reynolds number, λ(x)/υ. The plane jet data of Mi et al. 42 is included.

The streamwise evolutions of the ratio of sampling frequency, f cut-off against (a) Kolmogorov frequency, f k = U c (x)/ [2π η (x)], (b) Eddy turnover Taylor frequency, f inertial. Note: dashed line in (a) shows over-filtering for higher Re h cases.

The streamwise evolutions of the ratio of sampling frequency, f cut-off against (a) Kolmogorov frequency, f k = U c (x)/ [2π η (x)], (b) Eddy turnover Taylor frequency, f inertial. Note: dashed line in (a) shows over-filtering for higher Re h cases.

The Re h -dependence of (a) Kolmogorov scales (η), (b) Taylor microscales (λ) in the self-preserving field measured between x/h = 50 and 160.

The Re h -dependence of (a) Kolmogorov scales (η), (b) Taylor microscales (λ) in the self-preserving field measured between x/h = 50 and 160.

The lateral distributions of (a) Kolmogorov scales (η), (b) Taylor microscales (λ) measured at x/h = 20.

The lateral distributions of (a) Kolmogorov scales (η), (b) Taylor microscales (λ) measured at x/h = 20.

The lateral distributions of (a) the dissipation of turbulent kinetic energy (ɛh/U 3 b), (b) Kolmogorov frequency, f k (x, y) = U(x, y)/2πη at x/h = 20. The sampling frequency (f c) is shown with the dashed line.

The lateral distributions of (a) the dissipation of turbulent kinetic energy (ɛh/U 3 b), (b) Kolmogorov frequency, f k (x, y) = U(x, y)/2πη at x/h = 20. The sampling frequency (f c) is shown with the dashed line.

Streamwise evolutions of turbulent kinetic energy spectra, on Re h between x/h = 0 and 10. The dashed lines show energy production zones and nth spectra have been transformed in coordinate-axis by a factor of 10n.

Streamwise evolutions of turbulent kinetic energy spectra, on Re h between x/h = 0 and 10. The dashed lines show energy production zones and nth spectra have been transformed in coordinate-axis by a factor of 10n.

Streamwise evolutions of compensated spectra of the turbulent kinetic energy, on Re h and Re y0.5 measured between x/h = 20 and 160. The width of the inertial sub-range where ∼ k −5/3 for is characterized by the flat region on the ordinate axis.

Streamwise evolutions of compensated spectra of the turbulent kinetic energy, on Re h and Re y0.5 measured between x/h = 20 and 160. The width of the inertial sub-range where ∼ k −5/3 for is characterized by the flat region on the ordinate axis.

The probability density functions PDF of centerline velocity fluctuations u c(x). The Gaussian distribution PDF is shown.

The probability density functions PDF of centerline velocity fluctuations u c(x). The Gaussian distribution PDF is shown.

(a) Lateral distribution of the intermittency factor, γ estimated indirectly from the flatness factor of the fluctuating velocity at x/h = 20 compared with Gutmark and Wygnanski's 47 direct measurements. (b) Autocorrelation functions, R uu on the jet centerline for Re h = 1500 and 16 500.

(a) Lateral distribution of the intermittency factor, γ estimated indirectly from the flatness factor of the fluctuating velocity at x/h = 20 compared with Gutmark and Wygnanski's 47 direct measurements. (b) Autocorrelation functions, R uu on the jet centerline for Re h = 1500 and 16 500.

Comparing the probability density function (PDF) of the fluctuating velocity (u) at x/h = 20 between the jet centerline (blue dash line) and edge (red full line). The y axis is normalised by p(u n) at u n = 0 and Gaussian PDF is .

Comparing the probability density function (PDF) of the fluctuating velocity (u) at x/h = 20 between the jet centerline (blue dash line) and edge (red full line). The y axis is normalised by p(u n) at u n = 0 and Gaussian PDF is .

The Re h -dependence of turbulent viscosity (υT) and turbulent Reynolds number (Re T = U b h /υ T).

The Re h -dependence of turbulent viscosity (υT) and turbulent Reynolds number (Re T = U b h /υ T).

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