Abstract
The initial stage of the onset of turbulence in a three-dimensional compressible inviscid shear flow is studied. An initial deterministic velocity perturbation in the form of one or several Fourier modes leads to the development of a cascade of instabilities, which is numerically simulated. The influence exerted on the formation of the cascade of instabilities and the transition to turbulence by the size of the computational domain, the shear layer width, and the initial conditions is analyzed. It is shown that the mechanism of turbulence onset is essentially three-dimensional. The influence of various flow parameters and initial conditions on the formation of the turbulence cascade is studied numerically.
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References
O. M. Belotserkovskii, Karman’s Lecture (Von Karman Institute for Fluid Dynamics, 1978).
O. M. Belotserkovskii, Turbulence and Instabilities (MZpress, Moscow, 2003).
O. M. Belotserkovskii, A. M. Oparin, and V. M. Chechetkin, Turbulence: New Approaches (Nauka, Moscow, 2003) [in Russian].
O. M. Belotserkovskii, A. M. Oparin, and V. M. Chechetkin, “Physical Processes Underlying the Development of Shear Turbulence,” Zh. Eksp. Teor. Fiz. 126, 577–584 (2004) [J. Exp. Theor. Phys. 99, 504-509 (2004)].
W. Thomson, “On the Propagation of Laminar Motion through a Turbulently Moving Inviscid Liquid,” Philos. Mag. 4 (47), 342 (1887).
Lord F. R. S. Rayleigh, “On the Stability, or Instability, of Certain Fluid Motions,” Proc. London Soc. S1-11, 57–72 (1880).
L. F. Richardson, “Atmospheric Diffusion Shown on a Distance-Neighbor Graph,” Proc. R. Soc. London 110, 709–737 (1926).
A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Viscous Fluid at Very High Reynolds Numbers,” Dokl. Akad. Nauk SSSR 30, 476–481 (1940).
O. V. Troshkin, Nontraditional Methods in Mathematical Hydrodynamics (Am. Math. Soc. Providence, RI, 1995).
O. M. Belotserkovskii and A. M. Oparin, Numerical Experiment: From Order to Chaos (Nauka, Moscow, 2000) [in Russian].
O. M. Belotserkovskii, V. M. Chechetkin, S. V. Fortova, and A. M. Oparin, “The Turbulence of Free Shear Flows,” Proceedings of International Workshop on Hot Points in Astrophysics and Cosmology (Dubna, 2005), pp. 191–209.
O. Belotserkovskii, S. Fortova, A. Oparin, et al., “The Turbulence in Free Shear Flows and in Accretion Disks,” Investigations of Hydrodynamic Instability and Turbulence in Fundamental and Technological Problems by Means of Mathematical Modeling with Supercomputers (Nagoya Univ., Nagoya, Japan, 2005), pp. 229–241.
E. E. Meshkov, “Instability of a Gas Interface Accelerated by a Shock Wave,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 5, 151–158 (1969).
L. G. Loitsyanskii, Mechanics of Liquids and Gases (Nauka, Moscow, 1978; Begell House, New York, 1996).
S. Yanase and Y. Kaga, “Zero-Absolute-Vorticity-State of Rotating Flows Between Plates,” Japan-Russia Seminar on Turbulence and Instabilities, September 29-30, 2003 (Tokyo, 2003).
J. Hoffman and C. Johnson, “On Transition To Turbulence in Shear Flow,” Preprint (R. Inst. Technol. Kunliga Tekniska Hogskolan, 2002).
S. V. Fortova, “The Cascade Mechanism in Free Shear Flows,” 24th International Conference on Interaction of Intense Energy Fluxes with Matter Poster Session, (Elbrus, Russia, 2009).
S. V. Fortova, “On Vortex Cascades in Shear Flow Instabilities,” 2nd International Conference and Advanced School on Turbulent Mixing and Beyond (Trieste, Italy, 2009).
O. M. Belotserkovskii, A. M. Oparin, O. V. Troshkin, and V. M. Chechetkin, “Constructive Modeling of Structural Turbulence: Computational Experiment,” Phys. Scripta 132 (2008)
A. M. Oparin, “Numerical Simulation of Problems Related to Intense Development of Hydrodynamic Instabilities,” in Advances in Numerical Simulation: Algorithms, Numerical Experiments, and Results (Nauka, Moscow, 2000), pp. 63–90 [in Russian].
A. M. Oparin, “Numerical Study of Hydrodynamic Instabilities,” Comput. Fluid Dyn. J. 10 (3), Special Issue, 327–332 (2001).
O. M. Belotserkovskii, L. M. Kraginskii, and A. M. Oparin, “Numerical Simulation of Three-Dimensional Flows in a Stratified Atmosphere Caused by Strong Large-Scale Disturbances,” Zh. Vychisl. Mat. Mat. Fiz. 43, 1722–1736 (2003) [Comput. Math. Math. Phys. 43, 1657-1670 (2003)].
A. Sakurai and K. Takayama, “Molecular Kinetic Approach to the Problem of Compressible Turbulence,” Phys. Fluids 15, 1282–1294 (2003).
O. M. Belotserkovskii, V. A. Gushchin, and V. N. Kon’shin, “Splitting Method for Analysis of Stratified Fluid Flows with Free Surface,” Zh. Vychisl. Mat. Mat. Fiz. 27, 594–609 (1987).
G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Univ. Press, Cambridge, 1953; Inostrannaya Literatura, Moscow, 1955).
V. M. Kovenya and N. N. Yanenko, Splitting Method in Gas Dynamics Computations (Nauka, Novosibirsk, 1981) [in Russian].
R. Courant, E. Isaacon, and M. Rees, “On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences,” Commun. Pure Appl. Math. 5, 124–130 (1952).
P. L. Roe, “Characteristic-Based Schemes for the Euler Equations,” Ann. Rev. Fluid Mech. 18, 248–254 (1986).
A. Harten, “High Resolution Schemes for Hyperbolic Conservation Laws,” J. Comput. Phys. 49, 151 (1983).
A. S. Monin, Topics in Turbulence (Nauka, Moscow, 1994) [in Russian].
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Original Russian Text © O.M. Belotserkovskii, S.V. Fortova, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 6, pp. 1126–1139.
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Belotserkovskii, O.M., Fortova, S.V. Macroscopic parameters of three-dimensional flows in free shear turbulence. Comput. Math. and Math. Phys. 50, 1071–1084 (2010). https://doi.org/10.1134/S0965542510060126
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DOI: https://doi.org/10.1134/S0965542510060126