Abstract
Asymptotic methods are applied to study a plane turbulent jet of a viscous incompressible fluid flowing out through a narrow slot into a space filled with a fluid. The complete Navier—Stokes equations are considered. The characteristic Reynolds number is assumed to be large, while the jet thickness is small. In analyzing the problem the method of many scales is used, which makes it possible to determine and investigate a steady secondary flow within the turbulent jet. The secondary steady solutions are analytically obtained for the normal and longitudinal velocity components and the pressure. It is shown that the self-induced fluid outflow of the jet core toward the jet periphery is the main secondary flow ensuring the supply of the kinetic energy from a maximum velocity zone into the turbulence production zone. The solutions obtained are in good agreement with the available experimental data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. R. Miller and E. W. Comings, “Static Pressure Distribution in the Free Turbulent Jet,” J. Fluid Mech. 3(1), 1–16 (1957).
L. J. S. Bradbury, “The Structure of a Self-Preserving Turbulent Plane Jet,” J. Fluid Mech. 23(1), 31–64 (1965).
W. Rodi, “A Review of Experim en tal Data on Uniform Density, Free Turbu lent Boundary Layers,” in: B. E. Launder (ed.), Studies in Convection, Vol. 1 (Acad. Press, New York, 1975).
E. Gutmark and I. Wygnanski, “The Planar Turbulent Jet,” J. Fluid Mech. 3(3), 465–495 (1976).
R. C. Deo, J. Mi, and G. J. Nathan, “The Influence of Nozzle Aspect Ratio on Plane Jets,” Exp. Thermal Fluid Sci. 31, 825–838 (2007)).
A. S. Ginevskii, Theory of Turbulent Jets and Wakes (Mashinostroenie, Moscow, 1969) [in Russian].
G. N. Abramovich, T. A. Girshovich, S. Yu. Krasheninnikov, A. N. Sekundov, and I. P. Smirnova, Theory of Turbulent Jets (Nauka, Moscow, 1984) [in Russian].
W. Schneider, “Boundary-Layer Theory of Free Turbulent Shear Flows,” Z. Flugwiss. Weltraumforschung 15(3), 143–158 (1991).
I. I. Vigdorovich, “Turbulent Thermal Boundary Layer on a Plate. Reynolds Analogy and Heat Transfer Law over the Entire Range of Prandtl Numbers,” Fluid Dynamics 52(5), 631–645 (2017).
B. Scheichl, “Time-Mean Turbulent Shear Flows: Classical Modelling-Asymptotic Analysis-New Perspectives,” in: S. Bullett, T. Fearn, and F. Smith (eds.), Fluid and Solid Mechanics (World Sci., 2016), pp. 71–108.
V. B. Zametaev and A. R. Gorbushin, “Evolution of Vortices in 2D Boundary Layer and in the Couette Flow,” AIP Conf. Proc. 1770, 030044 (2016).
A. R. Gorbushin and V. B. Zametaev, “Asymptotic Analysis of Viscous Fluctuations in Turbulent Boundary Layers,” Fluid Dynamics 53(1) (2018).
H. Schlichting, “Berechnung ebener periodischer Grenzschichtströmungen,” Physikal. Z. 33, 327–335 (1932)).
M. Tatsuno, “Secondary Flow Induced by a Circular Cylinder Performing Unharmonic Oscillations,” J. Phys. Soc. Japan 50, 330–337 (1981)).
C. C. Lin, “Motion in the Boundary Layer with a Rapidly Oscillating External Flow,” in: Proc. 9th Intern. Congress Appl. Mech., 1957, Brussels, Vol. 4 (1956), pp. 155–167.
T. J. Pedley, “Two-Dimensional Boundary Layers in a Free Stream which Oscillates without Reversing,” J. Fluid Mech. 55, 359–383 (1972)).
V. I. Yudovich, “An Example of the Birth of a Secondary Stationary or Periodic Flow upon the Loss of Stability of Laminar Flow of a Viscous Incompressible Fluid,” Prikl. Mat. Mekh. 29(3), 453–467 (1965).
K. Stewartson, “The Theory of Unsteady Laminar Boundary Layers,” Adv. Appl. Mech. 6, 1–37 (1960)).
N. Riley, “Unsteady Laminar Boundary Layers,” SIAM Review 17, 274–297 (1975)).
O. Reynolds, “On the Dynamic Theory of Incompressible Viscous Fluids and the Determination of the Criterion,” Phil. Trans. Roy. Soc. A 186, 123 (1895).
A. N. Kolmogorov, “Local Structure of Turbulence in an Incompressible Fluid at Very Large Reynolds Numbers,” Dokl. Akad. Nauk SSSR 30(4), 299–303 (1941).
Lord Rayleigh, “On the Stability, or Instability, of Certain Fluid Motions,” Sci. Papers. Cambridge Univ. Press 1, 474–487 (1880)).
P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, 2004).
M. R. Head, “Entrainment in the Turbulent Boundary Layer,” R. & M. No. 3152 (1960). ARC Technical Report.
G. L. Brown and A. Roshko, “On Density Effects and Large Structure in Turbulent Mixing Layers,” J. Fluid Mech. 64(4), 775–816 (1974).
Acknowledgments
The authors wish to thank Vik.V. Sychev and G.N. Dudin for active discussion during the investigations and T.V. Khokhlova for assistance in preparing the plots.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2019, No. 2, pp. 99–111.
Rights and permissions
About this article
Cite this article
Gorbushin, A.R., Zametaev, V.B. & Lipatov, I.I. Steady Secondary Flow in a Plane Turbulent Free Jet. Fluid Dyn 54, 244–256 (2019). https://doi.org/10.1134/S0015462819020071
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0015462819020071