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Unified description of turbulent entrainment

Published online by Cambridge University Press:  03 December 2020

Maarten van Reeuwijk Open the ORCID record for Maarten van Reeuwijk [Opens in a new window] ,
J. Christos Vassilicos Open the ORCID record for J. Christos Vassilicos [Opens in a new window]  and
John Craske Open the ORCID record for John Craske [Opens in a new window]
Maarten van Reeuwijk*
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, LondonSW7 2AZ, UK
J. Christos Vassilicos
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Metiers ParisTech, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Feriet, F-59000Lille, France
John Craske
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We present a mathematical description of turbulent entrainment that is applicable to free-shear problems that evolve in space, time or both. Defining the global entrainment velocity $\bar {V}_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\bar {V}_g=\bar {u}_\zeta - \bar {\textrm {D}}h_t/\bar {\textrm {D}}t$, where $\bar {\textrm {D}}/\bar {\textrm {D}} t$ is the material derivative of the average flow field and $\bar {u}_\zeta$ is the average velocity perpendicular to the flow direction across the interface located at $\zeta =h_t$. The description is shown to reproduce well-known results for the axisymmetric jet, the planar wake and the temporal jet, and provides a clear link between the local (small scale) and global (integral) descriptions of turbulent entrainment. Application to unsteady jets/plumes demonstrates that, under unsteady conditions, the entrainment coefficient $\alpha$ no longer only captures entrainment of ambient fluid, but also time-dependency effects due to the loss of self-similarity.

JFM classification

Boundary Layers: Free shear layers Convection: Plumes/thermals Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A12
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Bisset, D. K., Hunt, J. C. R. & Rogers, M. M. 2002 The turbulent/non-turbulent interface bounding a far wake. J. Fluid Mech. 451, 383–410.CrossRefGoogle Scholar
Burridge, H. C., Parker, D. A., Kruger, E. S., Partridge, J. L. & Linden, P. F. 2017 Conditional sampling of a high Péclet number turbulent plume and the implications for entrainment. J. Fluid Mech. 823, 26–56.Google Scholar
Cafiero, G. & Vassilicos, J. C. 2019 Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets. Proc. R. Soc. Lond. A 475, 20190038.Google ScholarPubMed
Cantwell, B. & Coles, D. 1983 An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136, 321–374.CrossRefGoogle Scholar
Chauhan, K., Philip, J., de Silva, C. M., Hutchins, N. & Marusic, I. 2014 The turbulent/non-turbulent interface and entrainment in a boundary layer. J. Fluid Mech. 742, 119–151.CrossRefGoogle Scholar
Ching, C. Y., Fernando, H. J. S. & Robles, A. 1995 Break-down of line plumes in turbulent environments. J. Geophys. Res.: Oceans 100, 4707–4713.CrossRefGoogle Scholar
Corrsin, S. & Kistler, A. L. 1955 Free stream boundaries of turbulent flows. NACA Tech. Rep. 1244.Google Scholar
Craske, J. & van Reeuwijk, M. 2015 Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500–537.CrossRefGoogle Scholar
Craske, J. & van Reeuwijk, M. 2016 Generalised unsteady plume theory. J. Fluid Mech. 792, 1013–1052.CrossRefGoogle Scholar
Da Silva, C. B., Hunt, J. C. R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46, 567–590.CrossRefGoogle Scholar
Da Silva, C. B. & Métais, O. 2002 On the influence of coherent structures upon interscale interactions in turbulent plane jets. J. Fluid Mech. 473, 103–145.CrossRefGoogle Scholar
Da Silva, C. B. & Pereira, J. C. F. 2008 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids 20 (5), 055101.CrossRefGoogle Scholar
Davidson, G. A. 1986 A discussion of Schatzmann's integral plume nodel from a control volume viewpoint. J. Clim. Appl. Meteorol. 25, 858–866.2.0.CO;2>CrossRefGoogle Scholar
De Wit, L., Van Rhee, C. & Keetels, G. 2014 Turbulent interaction of a buoyant jet with cross-flow. J. Hydraul. Engng ASCE 140 (12), 04014060.Google Scholar
Deardorff, J. W., Willis, G. E. & Stockton, B. H. 1980 Laboratory studies of the entrainment zone of a convectively mixed layer. J. Fluid Mech. 100, 41–64.CrossRefGoogle Scholar
Devenish, B. J., Rooney, G. G. & Thomson, D. J. 2010 Large-eddy simulation of a buoyant plume in uniform and stably stratified environments. J. Fluid Mech. 652, 75–103.CrossRefGoogle Scholar
Dopazo, C., Martín, J. & Hierro, J. 2007 Local geometry of isoscalar surfaces. Phys. Rev. E 76 (5), 056316.CrossRefGoogle ScholarPubMed
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.CrossRefGoogle Scholar
Flanders, Harley 1973 Differentiation under the integral sign. Am. Math. Mon. 80 (6), 615–627.CrossRefGoogle Scholar
Gad-El-Hak, M. & Bushnell, D. M. 1991 Separation control: review. Trans. ASME: J. Fluids Engng 115, 5–30.Google Scholar
Garcia, J. R. & Mellado, J. P. 2014 The two-layer structure of the entrainment zone in the convective boundary layer. J. Atmos. Sci. 71, 1935–1955.CrossRefGoogle Scholar
Gaskin, S. J., McKernan, M. & Xue, F. 2004 The effect of background turbulence on jet entrainment: an experimental study of a plane jet in a shallow coflow. J. Hydraul. Res. 42 (5), 533–542.CrossRefGoogle Scholar
Head, M. R. 1958 Entrainment in the turbulent boundary layer. Reports & Memoranda 3152. Ministry of Aviation.Google Scholar
Holzner, M. & Luethi, B. 2011 Laminar superlayer at the turbulence boundary. Phys. Rev. Lett. 106 (13), 134503.CrossRefGoogle ScholarPubMed
Holzner, M. & van Reeuwijk, M. 2017 The turbulent/nonturbulent interface in penetrative convection. J. Turbul. 18, 260–270.CrossRefGoogle Scholar
Hunt, G. R. & Burridge, H. C. 2015 Fountains in industry and nature. Annu. Rev. Fluid Mech. 47, 195–220.CrossRefGoogle Scholar
Hunt, J. C. R., Rottman, J. W. & Britter, R. E. 1983 Some physical processes involved in the dispersion of dense gases. In Proc. UITAM Symposium on Atmospheric Dispersion of Heavy Gases and Small Particles (ed. G. Ooms & H. Tennekes), pp. 361–395. Springer.CrossRefGoogle Scholar
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, 31–75.CrossRefGoogle Scholar
Jahanbakhshi, R. & Madnia, C. 2018 The effect of heat release on the entrainment in a turbulent mixing layer. J. Fluid Mech. 844, 92–126.CrossRefGoogle Scholar
Jonker, H. J. J., Van Reeuwijk, M., Sullivan, P. & Patton, E. 2013 On the scaling of shear-driven entrainment: a DNS study. J. Fluid Mech. 732, 150–165.CrossRefGoogle Scholar
Kankanwadi, K. & Buxton, O. 2020 Turbulent entrainment from a turbulent background. J. Fluid Mech. 905, A35.CrossRefGoogle Scholar
Kato, H. & Phillips, O. M. 1969 On the penetration of a turbulent layer into stratified fluid. J. Fluid Mech. 37, 643–655.CrossRefGoogle Scholar
Kotsovinos, N. E. 1978 A note on the conservation of the volume flux in free turbulence. J. Fluid Mech. 86, 201–203.CrossRefGoogle Scholar
Krug, D., Chung, D., Philip, J. & Marusic, I. 2017 Global and local aspects of entrainment in temporal plumes. J. Fluid Mech. 812, 222–250.CrossRefGoogle Scholar
Krug, D., Holzner, M., Luethi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2015 The turbulent/non-turbulent interface in an inclined dense gravity current. J. Fluid Mech. 765, 303–324.CrossRefGoogle Scholar
List, E. J. 1982 Turbulent jets and plumes. Annu. Rev. Fluid Mech. 14 (1), 189–212.CrossRefGoogle Scholar
Mahesh, K. 2013 The interaction of jets with cross-flow. Annu. Rev. Fluid Mech. 45, 379–407.CrossRefGoogle Scholar
Mellado, J. P. 2012 Direct numerical simulation of free convection over a heated plate. J. Fluid Mech. 712, 418–450.CrossRefGoogle Scholar
Mellado, J. P. 2017 Cloud-top entrainment in stratocumulus clouds. Annu. Rev. Fluid Mech. 49 (1), 145–169.CrossRefGoogle Scholar
Neamtu-Halic, M. M., Krug, D., Mollicone, J. P., van Reeuwijk, M., Haller, G. & Holzner, M. 2020 Connecting the time evolution of the turbulence interface to coherent structures. J. Fluid Mech. 898, A3.CrossRefGoogle Scholar
Obligado, M., Dairay, T. & Vassilicos, J. C. 2016 Nonequilibrium scalings of turbulent wakes. Phys. Rev. Fluids 1 (4), 044409.CrossRefGoogle Scholar
Odier, P., Chen, J. & Ecke, R. E. 2014 Entrainment and mixing in a laboratory model of oceanic overflow. J. Fluid Mech. 746, 498–535.CrossRefGoogle Scholar
Philip, J., Meneveau, C., de Silva, C. M. & Marusic, I. 2014 Multiscale analysis of fluxes at the turbulent/non-turbulent interface in high Reynolds number boundary layers. Phys. Fluids 26, 015105.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Rajaratnam, N. 1976 Turbulent Jets. Developments in Water Science, vol. 5. Elsevier.CrossRefGoogle Scholar
Redford, J. A., Castro, I. P. & Coleman, G. N. 2012 On the universality of turbulent axisymmetric wakes. J. Fluid Mech. 710, 419–452.CrossRefGoogle Scholar
van Reeuwijk, M. & Craske, J. 2015 Energy-consistent entrainment relations for jets and plumes. J. Fluid Mech. 782, 333–355.CrossRefGoogle Scholar
van Reeuwijk, M. & Holzner, M. 2014 The turbulence boundary of a temporal jet. J. Fluid Mech. 739, 254–275.Google Scholar
van Reeuwijk, M., Holzner, M. & Caulfield, C. P. 2019 Mixing and entrainment are suppressed in inclined gravity currents. J. Fluid Mech. 873, 786–815.CrossRefGoogle Scholar
van Reeuwijk, M., Krug, D. & Holzner, M. 2018 Small-scale entrainment in inclined gravity currents. Environ. Fluid Mech. 18 (1), 225–239.CrossRefGoogle ScholarPubMed
van Reeuwijk, M., Salizzoni, P., Hunt, G. R. & Craske, J. 2016 Turbulent transport and entrainment in jets and plumes: a DNS study. Phys. Rev. Fluids 1, 074301.Google Scholar
de Rooy, W. C., Bechtold, P., Fröhlich, K., Hohenegger, C., Jonker, H. J. J., Mironov, D., Siebesma, A. P., Teixeira, J. & Yano, J.-I. 2013 Entrainment and detrainment in cumulus convection: an overview. Q. J. R. Meteorol. Soc. 139 (670), 1–19.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P., Dalziel, S. B. & Hunt, J. C. R. 2006 Time-dependent plumes and jets with decreasing source strengths. J. Fluid Mech. 563, 443–461.CrossRefGoogle Scholar
Schatzman, M. 1978 The integral equations for round buoyant jets in stratified flows. Z. Angew. Math. Phys. 29, 608–630.CrossRefGoogle Scholar
Sillero, J. A., Jimenez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to $\delta ^+\approx 2000$. Phys. Fluids 25, 105102.CrossRefGoogle Scholar
Silva, T. S., Zecchetto, M. & da Silva, C. B. 2018 The scaling of the turbulent/non-turbulent interface at high Reynolds numbers. J. Fluid Mech. 843, 156–179.CrossRefGoogle Scholar
Sreenivasan, K. R., Ramshankar, R. & Meneveau, C. 1989 Mixing, entrainment and fractal dimensions of surfaces in turbulent flows. Proc. R. Soc. Lond. A 421 (1860), 79–107.Google Scholar
Sullivan, P. P., Moeng, C. H., Stevens, B., Lenschow, D. H. & Mayor, S. D. 1998 Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci. 55, 3042–3064.2.0.CO;2>CrossRefGoogle Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Turner, J. S. 1962 The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13 (03), 356–368.CrossRefGoogle Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.CrossRefGoogle Scholar
Watanabe, T., Riley, J., De Bruyn Kops, S., Diamessis, P. & Zhou, Q. 2016 Turbulent/non-turbulent interfaces in wakes in stably stratified fluids. J. Fluid Mech. 797, R1.CrossRefGoogle Scholar
Watanabe, T., Riley, J., Nagata, K., Onishi, R. & Matsuda, K. 2018 a A localized turbulent mixing layer in a uniformly stratified environment. J. Fluid Mech. 849, 245–276.CrossRefGoogle Scholar
Watanabe, T., Sakai, Y., Nagata, K., Ito, Y. & Hayase, T. 2014 Enstrophy and passive scalar transport near the turbulent/non-turbulent interface in a turbulent planar jet flow. Phys. Fluids 26 (10), 105103.CrossRefGoogle Scholar
Watanabe, T., Zhang, X. & Nagata, K. 2018 b Turbulent/non-turbulent interfaces detected in DNS of incompressible turbulent boundary layers. Phys. Fluids 30 (3), 035102.CrossRefGoogle Scholar
Wells, M., Cenedese, C. & Caulfield, C. P. 2010 The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr. 40 (12), 2713–2727.CrossRefGoogle Scholar
Westerweel, J., Fukushima, C., Pedersen, J. M. & Hunt, J. C. R. 2005 Mechanics of the turbulent-nonturbulent interface of a jet. Phys. Rev. Lett. 95 (17), 174501.CrossRefGoogle ScholarPubMed
Whitney, H. 2005 Geometric Integration Theory. Dover.Google Scholar
Woodhouse, M. J., Phillips, J. C. & Hogg, A. J. 2016 Unsteady turbulent buoyant plumes. J. Fluid Mech. 794, 595–638.CrossRefGoogle Scholar
Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391–412.CrossRefGoogle Scholar
Xu, Y., Fernando, H. J. S. & Boyer, D. L. 1995 Turbulent wakes of stratified flow past a cylinder. Phys. Fluids 7 (9), 2243–2255.CrossRefGoogle Scholar
Zhou, Y. & Vassilicos, J. C. 2017 Related self-similar statistics of the turbulent/non-turbulent interface and the turbulence dissipation. J. Fluid Mech. 821, 440–457.CrossRefGoogle Scholar