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Self-similarity of fluid residence time statistics in a turbulent round jet

  • Dong-hyuk Shin (a1), R. D. Sandberg (a2) and E. S. Richardson (a3)


Fluid residence time is a key concept in the understanding and design of chemically reacting flows. In order to investigate how turbulent mixing affects the residence time distribution within a flow, this study examines statistics of fluid residence time from a direct numerical simulation (DNS) of a statistically stationary turbulent round jet with a jet Reynolds number of 7290. The residence time distribution in the flow is characterised by solving transport equations for the residence time of the jet fluid and for the jet fluid mass fraction. The product of the jet fluid residence time and the jet fluid mass fraction, referred to as the mass-weighted stream age, gives a quantity that has stationary statistics in the turbulent jet. Based on the observation that the statistics of the mass fraction and velocity are self-similar downstream of an initial development region, the transport equation for the jet fluid residence time is used to derive a model describing a self-similar profile for the mean of the mass-weighted stream age. The self-similar profile predicted is dependent on, but different from, the self-similar profiles for the mass fraction and the axial velocity. The DNS data confirm that the first four moments and the shape of the one-point probability density function of mass-weighted stream age are indeed self-similar, and that the model derived for the mean mass-weighted stream-age profile provides a useful approximation. Using the self-similar form of the moments and probability density functions presented it is therefore possible to estimate the local residence time distribution in a wide range of practical situations in which fluid is introduced by a high-Reynolds-number jet of fluid.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Antonia, R. A., Satyaprakash, B. R. & Hussain, A. K. M. F. 1980 Measurements of dissipation rate and some other characteristics of turbulent plane and circular jets. Phys. Fluids 23 (4), 695700.
Balo, J. N. & Cloirec, P. L. 2000 Validating a prediction method of mean residence time spatial distributions. AIChE J. 46 (4), 675683.
Batterman, S.2004 Assessment of small-scale incinerators for health care waste. Tech. Rep. World Health Organization.
Bilger, R., Kim, S. & Martin, S. 2004 Direct numerical simulation of turbulent premixed flames with a marker field and application to RANS and LES. In Proceedings of the Summer Program, pp. 255267. Center for Turbulence Research.
Boussinesq, J. 1877 Essai sur la théorie des eaux courantes. Imprimerie nationale.
Costa, M., Silva, P. & Azevedo, J. L. T. 2003 Measurements of gas species, temperature, and char burnout in a low-NOx pulverized-coal-fired utility boiler. Combust. Sci. Technol. 175 (2), 271289.
Danckwerts, P. V. 1953 Continuous flow systems, distribution of residence times. Chem. Engng Sci. 2 (1), 113.
Dowling, D. R. & Dimotakis, P. E. 1990 Similarity of the concentration field of gas-phase turbulent jets. J. Fluid Mech. 218, 109141.
Enjalbert, N., Domingo, P. & Vervisch, L. 2012 Mixing time-history effects in large eddy simulation of non-premixed turbulent flames: flow-controlled chemistry tabulation. Combust. Flame 159 (1), 336352.
Fayolle, F., Belhamri, R. & Flick, D. 2013 Residence time distribution measurements and simulation of the flow pattern in a scraped surface heat exchanger during crystallisation of ice cream. J. Food Engng 116 (2), 390397.
Gampert, M., Narayanaswamy, V., Schaefer, P. & Peters, N. 2013 Conditional statistics of the turbulent/non-turbulent interface in a jet flow. J. Fluid Mech. 731, 615638.
Ghirelli, F. & Leckner, B. 2004 Transport equation for the local residence time of a fluid. Chem. Engng Sci. 59 (3), 513523.
Gomet, L., Robin, V. & Mura, A. 2012 Influence of residence and scalar mixing time scales in non-premixed combustion in supersonic turbulent flows. Combust. Sci. Technol. 184 (10–11), 14711501.
Grout, R. W. 2007 An age extended progress variable for conditioning reaction rates. Phys. Fluids 19 (10), 105107.
Haworth, D. C. & Pope, S. B. 1986 A generalized Langevin model for turbulent flows. Phys. Fluids 29 (2), 387405.
Hussein, H. J., Capp, S. P. & George, W. K. 1994 Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech. 258, 3175.
Kennedy, C. A., Carpenter, M. H. & Lewis, R. M. 2000 Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Maths 35 (3), 177219.
Kennedy, C. A. & Gruber, A. 2008 Reduced aliasing formulations of the convective terms within the Navier–Stokes equations for a compressible fluid. J. Comput. Phys. 227, 16761700.
Langevin, P. 1908 Sur la théorie du mouvement brownien. C. R. Hebd. Séances Acad. Sci. 146, 508533.
Langmuir, I. 1908 The velocity of reactions in gases moving through heated vessels and the effect of convection and diffusion. J. Am. Chem. Soc. 30 (11), 17421754.
Levenspiel, O. 1999 Chemical Reaction Engineering. Wiley.
Mackley, M. R. & Saraiva, R. M. C. N. 1999 The quantitative description of fluid mixing using Lagrangian- and concentration-based numerical approaches. Chem. Engng Sci. 54 (2), 159170.
Mi, J., Nathan, G. J. & Nobes, D. S. 2001 Influence of jet exit conditions on the passive scalar field of an axisymmetric free jet. J. Fluid Mech. 432 (4), 91125.
Mouangue, R., Obounou, M., Gomet, L. & Mura, A. 2014 Lagrangian intermittent modelling of a turbulent lifted methane-air jet flame stabilized in a vitiated air coflow. Flow Turbul. Combust. 92 (3), 731765.
Moullec, Y. L., Potier, O., Gentric, C. & Leclerc, J. P. 2008 Flow field and residence time distribution simulation of a cross-flow gas–liquid wastewater treatment reactor using CFD. Chem. Engng Sci. 63 (9), 24362449.
Mulenga, F. K. & Chimwani, N. 2013 Introduction to the use of the attainable region method in determining the optimal residence time of a ball mill. Intl J. Miner. Process. 125, 3950.
Nambully, S., Domingo, P., Moureau, V. & Vervisch, L. 2014 A filtered-laminar-flame PDF sub-grid scale closure for LES of premixed turbulent flames. Part I: formalism and application to a bluff-body burner with differential diffusion. Combust. Flame 161 (7), 17561774.
Nauman, E. B. 2008 Residence time theory. Ind. Engng Chem. Res. 47 (10), 37523766.
Nordström, J. & Carpenter, M. H.1998 Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier–Stokes equations. Tech. Rep. NASA/CR-1998-207681 ICASE Report No. 98-19. National Aeronautics and Space Administration.
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.
Poinsot, T. J. & Lele, S. K. 1992 Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101 (1), 104129.
Pope, S. B. 1985 PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11 (2), 119192.
Pope, S. B. 2000 Turbulent Flows, vol. 1. Cambridge University Press.
Prandtl, L. 1925 Bericht über untersuchungen zur ausgebildeten turbulenz. Z. Angew. Math. Mech. 5 (2), 136139.
Ricou, F. P. & Spalding, D. B. 1961 Measurements of entrainment by axisymmetrical turbulent jets. J. Fluid Mech. 11, 2132.
Sandberg, M. 1981 What is ventilation efficiency? Build. Environ. 16 (2), 123135.
Sandberg, M. & Sjöberg, M. 1983 The use of moments for assessing air quality in ventilated rooms. Build. Environ. 18 (4), 181197.
Sandberg, R. D. 2011 An axis treatment for flow equations in cylindrical coordinates based on parity conditions. Comput. Fluids 49, 166172.
Sandberg, R. D. & Sandham, N. D. 2006 Nonreflecting zonal characteristic boundary condition for direct numerical simulation of aerodynamic sound. AIAA J. 44 (2), 402405.
Sandberg, R. D., Sandham, N. D. & Suponitsky, V. 2012 DNS of compressible pipe flow exiting into a coflow. Intl J. Heat Fluid Flow 35, 3344.
Sandberg, R. D. & Tester, B. J. 2016 Mach-number scaling of individual azimuthal modes of subsonic co-flowing jets. J. Fluid Mech. 793, 209228.
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory. Springer.
von Smoluchowski, M. 1916 Drei vorträge über diffusion, Brownsche molekularbewegung und koagulation von kolloidteilchen part I. Physik. Z. 17, 557571.
Spalding, D. B. 1958 A note on mean residence-times in steady flows of arbitrary complexity. Chem. Engng Sci. 9 (1), 7477.
Sutherland, W. 1893 LII. The viscosity of gases and molecular force. Phil. Mag. Series 5 36 (223), 507531.
Tollmien, W. 1926 Brechnung turbulenter Ausbreitungsvogange. Z. Angew. Math. Mech. 6, 468478 [English translation 1945, NACA TM 1085].
Touber, E. & Sandham, N. D. 2009 Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble. Theor. Comput. Fluid Dyn. 23 (2), 79107.
Wygnanski, I. & Fiedler, H. 1969 Some measurements in the self-preserving jet. J. Fluid Mech. 38 (3), 577612.
Xu, G. & Antonia, R. 2002 Effect of different initial conditions on a turbulent round free jet. Exp. Fluids 33 (5), 677683.
Yeung, P. K. & Pope, S. B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531586.
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