Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-10T12:39:08.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Secondary flows in a plane channel: their relationship and comparison with turbulent flows

Published online by Cambridge University Press:  20 April 2006

B. L. Rozhdestvensky  and
I. N. Simakin
B. L. Rozhdestvensky
Affiliation:
Keldysh Institute of Applied Mathematics, Academy of Sciences of the USSR, Miusskaja Sq. 4, Moscow 125047, USSR
I. N. Simakin
Affiliation:
Keldysh Institute of Applied Mathematics, Academy of Sciences of the USSR, Miusskaja Sq. 4, Moscow 125047, USSR
Get access Rights & Permissions [Opens in a new window]

Abstract

Two- and three-dimensional non-stationary viscous-fluid flows in a plane channel are considered. By means of efficient computational algorithms for direct integration of the incompressible Navier-Stokes equations the evolution of these flows over large time intervals is simulated. Classes of two- and three-dimensional non-stationary flows with stationary integral characteristics (the flow rate, mean pressure gradient, total energy of pulsations etc.) were found. Such flows are called secondary flows. Two-dimensional secondary flows have only qualitative similarity to turbulent flows observed in experiments. Three-dimensional secondary flows agree very well, even quantitatively, with turbulent flows. The principal characteristics of turbulent flows such as drag coefficient, mean-velocity profile, the distributions of the pulsation velocity components and some others are reproduced in three-dimensional secondary flows with good accuracy.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 147 , October 1984 , pp. 261 - 289
Copyright
© 1984 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flow. Academic Press.
Davies, S. J. & White, C. M. 1928 An experimental study of the flow of water in pipes of rectangular section. Proc. R. Soc. Lond. A 119, 92–107.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Fox, L. 1962 Chebyshev methods for ordinary differential equations. Comp. J. 4, 318–331.Google Scholar
George, W. D., Hellums, J. D. & Martin, B. 1974 Finite-amplitude neutral disturbances in plane Poiseuille flow. J. Fluid Mech. 63, 765–771.Google Scholar
Goldshtik, M. A. & Stern, V. N. 1977 Hydrodynamic Stability and Turbulence. Novosibirsk, Nauka.
Gottlieb, D. & Orszag, S. A. 1977 Numerical analysis of spectral methods: theory and applications. SIAM, NSF—CBMS Monograph 29.Google Scholar
Herbert, T. 1977 Finite amplitude stability of plane parallel flows. In Proc. AGARD Symp. on Laminar—Turbulent Transition no. 3.
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I, II. Springer.
Kao, T. W. & Park, C. 1970 Experimental investigation of the stability of channel flows. Part 1. Flow of a single liquid in rectangular channel. J. Fluid Mech. 43, 145–164.Google Scholar
Kaporin, I. E. 1980 New fast Fourier transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 20, 1054–1058.Google Scholar
Kreplin, H. & Eckelmann, M. 1979 Behaviour of the three fluctuating velocity components in the wall region of a turbulent channel flow. Phys. Fluids 22, 1233–1239.Google Scholar
Lanczos, C. 1956 Applied Analysis. Prentice-Hall.
Moin, P. & Kim, J. 1980 On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries. J. Comp. Phys. 35, 381–392.Google Scholar
Orszag, S. A. 1971 Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations. Stud. Appl. Maths 50, 293–327.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flows. J. Fluid Mech. 96, 159–205.Google Scholar
Orszag, S. A. & Patera, A. T. 1980 Subcritical transition to turbulence in plane channel flows. Phys. Rev. Lett. 45, 989–993.Google Scholar
Orszag, S. A. & Patera, A. T. 1981 Calculation of von Kármán's constant for turbulent channel flow. Phys. Rev. Lett. 47, 832–835.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Patel, V. C. & Head, M. R. 1969 Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech. 38, 181–201.Google Scholar
Petrov, G. I. 1940 The application of the Galerkin method to the problem of stability of viscous fluid flow. Prikl. Mat. i Mekh. 4, 3–12.Google Scholar
Rozhdestvensky, B. L. 1973 On the applicability of finite-difference methods to the solution of Navier—Stokes equations at large Reynolds numbers. Dokl. Akad. Nauk SSSR 211, 308–311.Google Scholar
Rozhdestvensky, B. L., Ermakova, E. I. & Priymak, V. G. 1977 A study of the stability of difference schemes of higher order accuracy. Keldysh Inst. Appl. Maths Preprint 14.Google Scholar
Rozhdestvensky, B. L. & Priymak, V. G. 1982 Numerical simulation of two-dimensional turbulence in a plane channel. Comp. Fluids 10, 117–126.Google Scholar
Rozhdestvensky, B. L. & Simakin, I. N. 1982a Methods of numerical simulation of unsteady flows of incompressible viscous fluid in a plane channel. Keldysh Inst. Appl. Maths Preprint 191.Google Scholar
Rozhdestvensky, B. L. & Simakin, I. N. 1982b Nonstationary secondary flows in a plane channel and stability of the Poiseuille flow with respect to finite disturbances. Dokl. Akad. Nauk SSSR 266, 1337–1340.Google Scholar
Rozhdestvensky, B. L. & Simakin, I. N. 1982c Stability of the Poiseuille flow with respect to finite disturbances. Chisl. Metody Mekh. Sploshnoy Sredy, Novosibirsk 13, 89–128.Google Scholar
Rozhdestvensky, B. L. & Simakin, I. N. 1983 Two- and three-dimensional secondary flows in a plane channel, their relationship and comparison with turbulent flows. Dokl. Akad. Nauk SSSR 273, 553–558.Google Scholar
Rozhdestvensky, B. L. & Yanenko, N. N. 1978 Systems of Quasi-Linear Equations. Moscow, Nauka [English transl. Am. Math. Soc. Monograph 55 (1983)].
Samarsky, A. A. 1977 Theory of Difference Schemes. Moscow, Nauka.
Serrin, J. 1959 Mathematical principles of classical fluid mechanics. In Handbuch der Physik, Bd. VIII/1 (ed. S. Flügge & C. Truesdell). Springer.
Shkadov, V. YA. 1973a Some methods and problems of the hydrodynamic stability theory. Nauch. Trudy Inst. Mekh. Moscow Univ. 25.Google Scholar
Shkadov, V. YA. 1973b On nonlinear evolution of the disturbances in plane-parallel Poiseuille flow. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza no. 2, pp. 49–57.Google Scholar
Simakin, I. N. 1983a Investigation of the stability and accuracy of the methods of numerical simulation of nonstationary flows in a plane channel. Keldysh Inst. Appl. Maths Preprint 49.Google Scholar
Simakin, I. N. 1983b Numerical investigation of the accuracy of simulation of unsteady flows in a plane channel. Keldysh Inst. Appl. Maths Preprint 60.Google Scholar
Stewartson, K. 1974 Some aspects of nonlinear stability theory. Fluid Dyn. Trans. 7, 101–128.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part I. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.Google Scholar
Stuart, J. T. 1971 Non-linear stability theory. Ann. Rev. Fluid Mech. 3, 347–370.Google Scholar
Thomas, T. Y. 1942 Qualitative analysis of the flow of fluids in pipes. Am. J. Maths 64, 754.Google Scholar
Whan, G. A. & Rothfus, R. R. 1959 Characteristics of transition flow between parallel plates. A.I.Ch.E J. 5, 204–208.Google Scholar
Yanenko, N. N. 1967 Fractional Steps Method of the Solution of Many-dimensional Problems in Mathematical Physics. Novosibirsk, Nauka.
Zahn, J.-P., Toomre, J., Spiegel, E. A. & Gough, D. O. 1974 Nonlinear cellular motions in Poiseuille channel flow. J. Fluid Mech. 64, 319–345.Google Scholar