Skip to main content Accessibility help

Equilibrium similarity solution of the turbulent transport equation along the centreline of a round jet

  • H. Sadeghi (a1), P. Lavoie (a1) and A. Pollard (a2)


A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of $\langle ({\it\delta}q)^{2}\rangle$ along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of $\langle q^{2}\rangle$ and the decay rate constants of $\langle u^{2}\rangle$ and $\langle v^{2}\rangle$ are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity $U$ and the streamwise distance $x-x_{0}$ from the virtual origin (i.e.  $U\propto (x-x_{0})^{-1}$ ), is sufficient to predict a power-law decay for the turbulence kinetic energy ( $\langle q^{2}\rangle \propto (x-x_{0})^{m}$ ), rather than requiring a power-law decay ( $m=-2$ ) as an additional ad hoc assumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate, $\langle {\it\epsilon}\rangle$ , varies as $(x-x_{0})^{m-2}$ . These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for $11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$ over the range $10\leqslant x/D\leqslant 90$ . For the present experiments, $\langle q^{2}\rangle$ exhibits power-law behaviour with $m=-1.83$ . The validity of this solution is confirmed using other experimental data where $\langle q^{2}\rangle$ follows a power law with $-1.89\leqslant m\leqslant -1.78$ . The present similarity form of the transport equation for $\langle ({\it\delta}q)^{2}\rangle$ is also shown to be closely satisfied by the experimental data.


Corresponding author

Email address for correspondence:


Hide All
Abdel-Rahman, A. A., Chakroun, W. & Al-Fahed, S. F. 1997 LDA measurements in the turbulent round jet. Mech. Res. Commun. 24, 277288.
Amielh, M., Djeridane, T., Anselmet, F. & Fulachier, L. 1996 Velocity near-field of variable density turbulent jets. Intl J. Heat Mass Transfer 39, 21492164.
Antonia, R. A. & Burattini, P. 2006 Approach to the 4/5 law in homogeneous isotropic turbuelnce. J. Fluid Mech. 550, 175184.
Antonia, R. A., Satyaprakash, B. A. & Hussain, A. K. F. M. 1980 Measurements of dissipation rate and some other characteristics of turbulent plane and circular jets. Phys. Fluids 23, 695700.
Antonia, R. A., Smalley, R. J., Zhou, T., Anselmet, F. & Danaila, L. 2003 Similarity of energy structure functions in decaying homogeneous isotropic turbulence. J. Fluid Mech. 487, 245269.
Burattini, P. & Antonia, R. A. 2005 The effect of different X-wire calibration schemes on some turbulence statistics. Exp. Fluids 38, 8089.
Burattini, P., Antonia, R. A. & Danaila, L. 2005a Scale-by-scale energy budget on the axis of a turbulent round jet. J. Turbul. 6, 111.
Burattini, P., Antonia, R. A. & Danaila, L. 2005b Similarity in the far field of a turbulent round jet. Phys. Fluids 17, 025101.
Champagne, F. H. 1978 The fine-scale structure of the turbulent velocity field. J. Fluid Mech. 86, 67108.
Corrsin, S. 1963 Turbulence: experimental methods. In Handbuch der Physik (ed. Flügge, S. & Truesdell, C. A.), pp. 524589. Springer.
Danaila, L., Anselmet, F. & Antonia, R. A. 2002 An overview of the effect of large-scale inhomogeneities on small-scale turbulence. Phys. Fluids 14, 24752484.
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R. A. 2001 Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87109.
Dimotakis, P. E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 6998.
Fellouah, H., Ball, C. G. & Pollard, A. 2009 Reynolds number effects within the development region of a turbulent round free jet. Intl J. Heat Mass Transfer 52, 39433954.
Fellouah, H. & Pollard, A. 2010 The velocity spectra and turbulence length scale distributions in the near to intermediate regions of a round free turbulent jet. Phys. Fluids 21, 115101.
Ferdman, E., Otugen, M. V. & Kim, S. 2000 Effect of initial velocity profile on the development of the round jet. J. Propul. Power 4, 676686.
Fulachier, L. & Antonia, R. A. 1983 Turbulent Reynolds and Péclet number re-defined. Intl Commun. Heat Mass Transfer 10 (5), 435439.
George, W. K. 1989 The self-preservation of turbulent flows and its relation to initial conditions and coherent structures. In Advances in Turbulence (ed. George, W. K. & Arndt, R.), Springer.
George, W. K. 1992 The decay of homogeneous isotropic turbulence. Phys. Fluids 4 (7), 14921509.
George, W. K. & Hussein, H. J. 1991 Locally axisymmetric turbulence. J. Fluid Mech. 233, 123.
Hearst, R. J., Buxton, O. R. H., Ganapathisubramani, B. & Lavoie, P. 2012 Experimental estimation of fluctuating velocity and scalar gradients in turbulence. Exp. Fluids 53, 925942.
Hinze 1975 Turbulence. McGraw-Hill.
Hussein, H. J., Capp, S. & George, W. K. 1994 Velocity measurements in a high-Reynolds number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech. 258, 3175.
Lavoie, P., Burattini, P., Djenidi, P. & Antonia, R. A. 2005 Effect of initial conditions on decaying grid turbulence at low $R_{{\it\lambda}}$ . Exp. Fluids 39 (5), 865874.
Lumley, J. L. 1965 Interpretation of time spectra measured in high-intensity shear flows. Phys. Fluids 8, 10561062.
Malmstrom, T. G., Kirkpatrick, A. T., Christensen, B. & Knappmiller, K. D. 1997 Centreline velocity decay measurements in low-velocity axisymmetric jet. J. Fluid Mech. 246, 363377.
Mi, J., Xu, M. & Zhou, T. 2013 Reynolds number influence on statistical behaviors of turbulence in a circular free jet. Phys. Fluids 25, 075101.
Panchapakesan, N. R. & Lumley, J. L. 1993 Turbulence measurements in axisymmetric jets of air and helium. Part 1. Air jet. J. Fluid Mech. 246, 197223.
Pollard, A. & Uddin, M. 2007 Self-similarity of coflowing jets: the virtual origin. Phys. Fluids 19, 068103.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Quinn, W. R. 2006 Upstream nozzle shaping effects on near field flow in round turbulent free jets. Eur. J. Mech. (B/Fluids) 25, 279301.
Ruffin, E., Schiestel, R., Anselmet, F., Amielh, M. & Fulachier, L. 1994 Investigation of characteristic scales in variable density turbulent jets using a secondorder model. Phys. Fluids 6, 27852799.
Sadeghi, H., Lavoie, P. & Pollard, A. 2014 The effect of Reynolds number on the scaling range along the centreline of a round turbulent jet. J. Turbul. 15, 335349.
Sadeghi, H. & Pollard, A. 2012 Effects of passive control rings positioned in the shear layer and potential core of a turbulent round jet. Phys. Fluids 24, 115103.
Taub, G. N., Lee, H., Balachandar, S. & Sherif, S. A. 2013 A direct numerical simulation study of higher order statistics in a turbulent round jet. Phys. Fluids 25, 115102.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Tong, C. & Warhaft, Z. 1994 Turbulence suppression in a jet by means of a fine ring. Phys. Fluids 6, 328333.
Weisgraber, T. H. & Liepman, D. 1998 Turbulent structure during transition to self-similarity in a round jet. Exp. Fluids 24, 210224.
Wygnanski, I. & Fiedler, H. 1969 Some measurements in the self-preserving jet. J. Fluid Mech. 38, 577612.
Xu, G. & Antonia, R. A. 2002 Effect of different initial conditions on a turbulent round free jet. Exp. Fluids 33, 677683.
Xu, M., Pollard, A., Mi, J., Secretain, F. & Sadeghi, H. 2013 Effects of Reynolds number on some properties of a turbulent jet from a long square pipe. Phys. Fluids 25, 035102.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed