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Rapid distortion theory and the ‘problems’ of turbulence

Published online by Cambridge University Press:  26 April 2006

J. C. R. Hunt  and
D. J. Carruthers
J. C. R. Hunt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CBS 9EW, UK
D. J. Carruthers
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CBS 9EW, UK
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Abstract

The ‘problems’ associated with analysing different kinds of turbulent flow and different methods of solution are classified and discussed with reference to how the turbulent structure in a flow domain depends on the scale and geometry of the domain's boundary, and on the information provided in the boundary conditions. Rapid distortion theory (RDT) is a method, based on linear analysis, for calculating ‘rapidly changing turbulent’ (RCT) flows under the action of different kinds of distortion, such as large-scale velocity gradients, the effects of bounding surfaces, body forces, etc. Recent developments of the theory are reviewed, including the criteria for its validity, and new solutions allowing for the effects of inhomogeneities and boundaries.

We then consider the contribution of RDT to understanding the fundamental problems of ‘slowly changing turbulent’ (SCT) flows, such as why are similar and persistent features of the local eddy structure found in different kinds of shear flow, and what are the general features of turbulent flows near boundaries. These features, which can be defined in terms of certain statistical quantities and flow patterns in individual flow realizations, are found to correspond to the form of particular solutions of RDT which change slowly over the time of the distortion. The most general, features are insensitive to the energy spectrum and to the initial anisotropy of the turbulence. A new RDT analysis of the energy spectra E(k) indicates why, in shear flows at moderate Reynolds number, the turbulence tends to have similar forms of spectra for eddies on a local scale, despite the Reynolds number not being large enough for the existence of a nonlinear cascade and there being no universal forms of spectra for unsheared turbulence; for this situation, the action of shear dU1/dx2 changes the form of the spectrum, so that, as β = (tdU1/dx2 increases, over an increasing part of the spectrum defined in terms of the integral scale L by β−1 [Gt ] kL, E(k) ∝ k−2, whatever the form of initial spectrum of E0(k) (provided E(k) = o(k−2) for kL [Gt ] 1).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 212 , March 1990 , pp. 497 - 532
Copyright
© 1990 Cambridge University Press

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References

Adrian, R. J. & Moin, P., 1988 Stochastic estimation of organized turbulent structure: homogeneous shear flow. J. Fluid Mech. 190, 531–559.Google Scholar
Auton, T. R., Hunt, J. C. R. & Prud'homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.Google Scholar
Batchelor, G. K.: 1953 The Theory of Homogeneous Turbulence, Cambridge University Press, 197 pp.
Batchelor, G. K.: 1955 The effective pressure exerted by a gas in turbulent motion. In Vistas in Astronomy (ed. A. Beer), vol. 1, pp. 290–295. Pergamon.
Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.
Batchelor, G. K. & Proudman, I., 1954 The effects of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7, 83–103.Google Scholar
Batchelor, G. K. & Proudman, I., 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.Google Scholar
Bradshaw, P.: 1967 The turbulence structure of equilibrium boundary layers. J. Fluid Mech. 29, 625–645.Google Scholar
Britter, R. E., Hunt, J. C. R. & Mumford, J. C. 1979 The distortion of turbulence by a circular cylinder. J. Fluid Mech. 92, 269–301.Google Scholar
Britter, R. E., Hunt, J. C. R. & Richards, K. J. 1981 Analysis and wind-tunnel studies of speed-up, roughness effects and turbulence over a two-dimensional hill. Q. J. R. Met. Soc. 107, 91–110.Google Scholar
Cambon, C. & Jacquin, L., 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–318.Google Scholar
Carruthers, D. J., Fung, J. C. H. & Hunt, J. C. R. 1989 The emergence of characteristic eddy motion in turbulent shear flows. Proc. Organized Structures and Turbulence in Fluid Mechanics (ed. M. Lesieur & O. Métais).
Carruthers, D. J. & Hunt, J. C. R. 1986 Velocity fluctuations near an interface between a turbulent region and a stably stratified layer. J. Fluid Mech. 165, 475–501.Google Scholar
Champagne, F. H., Harris, V. G. & Corrsin, S., 1970 Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech. 41, 81–139.Google Scholar
Comte-Bellot, G. & Mathieu, J., 1987 (eds.) Advances in Turbulence. Proc. First European Turbulence Conference, Lyon, July 1986. Springer.
Craya, A.: 1958 Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P. S. T. Ministère de l'Air, vol. 345.Google Scholar
Crow, S. C.: 1968 Turbulent Rayleigh shear flow. J. Fluid Mech. 32, 113–120.Google Scholar
Davidson, P. A., Hunt, J. C. R. & Moros, A. 1968 Turbulent recirculating flows in liquid metal magnetohydrodynamics. Prog. Astronaut. Aeronaut. 111, 400–420.Google Scholar
Deissler, R. G.: 1968 Effects of combined two-dimensional shear and normal strain on weak locally homogeneous turbulence and heat transfer. J. Math. Phys. 47, 320–331.Google Scholar
Dritschel, D.: 1989 Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 77–146.Google Scholar
Durbin, P. A.: 1981 Distorted turbulence in axisymmetric flow. Q. J. Mech. Appl. Maths 34, 489–500.Google Scholar
Durbin, P. A. & Hunt, J. C. R. 1980 On surface pressure fluctuations beneath turbulent flow round bluff bodies. J. Fluid Mech. 100, 161–184.Google Scholar
Dussage, J. P. & Gaviglio, J., 1981 Bulk dilatation effects on Reynolds stresses in the rapid expansion of a turbulent boundary layer at supersonic speeds. Proc. Symp. Turbulent Shear Flows, Davis, pp. 2.33–2.38.Google Scholar
Ferré, J. A. & Giralt, F. 1989 Pattern-recognition analysis of the velocity field in plane turbulent wakes. J. Fluid Mech. 198, 27–64.Google Scholar
Gartshore, I. S., Durbin, P. A. & Hunt, J. C. R. 1983 The production of turbulent stress in a shear flow by irrotational fluctuations. J. Fluid Mech. 137, 307–329.Google Scholar
Gaster, M., Kit, E. & Wygnanski, I., 1985 Large-scale structures in a forced turbulent mixing layer. J. Fluid Mech. 150, 23–39.Google Scholar
Gilbert, A.: 1988 Spiral structures and spectra in two-dimensional turbulence. J. Fluid Mech. 193, 475–497.Google Scholar
Goldstein, M. E.: 1978 Unsteady vortical and entropic distortion of potential flow round arbitrary obstacles. J. Fluid Mech. 89, 433–468.Google Scholar
Goldstein, M. E. & Atassi, H., 1976 A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust. J. Fluid Mech. 74, 741–765 (and Corrigendum. 91. 1979, 788).Google Scholar
Goldstein, M. E. & Durbin, P. A., 1980 The effect of finite turbulence spatial scale on the amplification of turbulence by a contracting stream. J. Fluid Mech. 98, 473–508.Google Scholar
Graham, J. M. R.: 1976 Turbulent flow past a porous plate. J. Fluid Mech. 73, 565–591.Google Scholar
Hayakawa, M. & Hussain, F., 1989 Three-dimensionality of organized structures in a plane turbulent wake. J. Fluid Mech. 206, 375–404.Google Scholar
Ho, C. M. & Huerre, P., 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Hunt, J. C. R.: 1973 A theory of turbulent flow round two-dimensional bluff bodes. J. Fluid Mech. 61, 625–706.Google Scholar
Hunt, J. C. R.: 1978 A review of the theory of rapidly distorted turbulent flow and its applications. Proc. XIII Biennial Fluid Dynamics Symp., Kortowo, Poland, Fluid Dyn. Trans. 9, 121–152.Google Scholar
Hunt, J. C. R.: 1984 Turbulence structure in thermal convection and shear-free boundary layers. J. Fluid Mech. 138, 161–184.Google Scholar
Hunt, J. C. R.: 1987 Vorticity and vortex dynamics in complex turbulence flow. Proc. CANCAM (Canadian Congress of Applied Mechanics, London, Ontario). Trans. Can. Soc. Mech. Engng 11, 21–35.Google Scholar
Hunt, J. C. R.: 1988 Studying turbulence using direct numerical simulation: 1987 Center for Turbulence Research NASA Ames/Stanford Summer Programme. J. Fluid Mech. 190, 375–392.Google Scholar
Hunt, J. C. R. & Graham, J. M. R. 1978 Free-stream turbulence near plane boundaries. J. Fluid Mech. 84, 209–235.Google Scholar
Hunt, J. C. R., Newley, T. M. J. & Weng, W.-S. 1989 Analysis and computation of turbulent boundary layers with varying pressure gradients. Joint IMA-SMAI Conf. on Computational Methods in Aeronautical Fluid Dynamics, University of Reading, April 1987 (ed. P. Stow). Oxford University Press.
Hussain, A. K. M. F.: 1983 Coherent structures – reality and myth. Phys. Fluids 26, 2816–2850.Google Scholar
Hussain, A. K. M. F.: 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.Google Scholar
Jeandel, D., Brison, J. F. & Mathieu, J., 1978 Modelling methods in physical and spectral space. Phys. Fluids 21, 169–182.Google Scholar
Kawai, H.: 1990 Pressure fluctuations on an upwind surface of two-dimensional square prisms in a turbulent flow. J. Fluid Mech. (submitted).Google Scholar
Kida, S. & Hunt, J. C. R. 1989 Interaction between turbulence of different scales over short times. J. Fluid Mech. 201, 411–445.Google Scholar
Komori, S., Ueda, H., Ogino, F. & Mizushina, T., 1983 Turbulence structure in stably stratified open channel flow. J. Fluid Mech. 130, 13–26.Google Scholar
Landahl, M. T.: 1984 Coherent structures in turbulence and Prandtl's mixing length theory. Z. Flugwiss. Weltraumforsch. 8, 233.Google Scholar
Landahl, M. T.: 1990 Theoretical model for VITA-educed coherent structures in the wall region of a turbulent boundary layer. Proc. 2nd Stanford Summer School, pp. 209–220.Google Scholar
Launder, B. E., Reece, G. J. & Rodi, W., 1975 Progress in the development of a Reynolds stress turbulence closure. J. Fluid Mech. 68, 537–566.Google Scholar
Launder, B. E. & Spalding, D. B., 1972 Mathematical Models of Turbulence. Academic.
Lee, M. J.: 1985 Ph.D. thesis. Mech. Eng. Stanford University.
Lee, M. J. & Hunt, J. C. R. 1989 The structure of sheared turbulence near a boundary. Center for Turbulence Research, Stanford, Rep. CTR-S88, pp. 221–242.Google Scholar
Lee, M. J., Kim, J. & Moin, P., 1987 Turbulent structure at high shear rate. Turbulent Shear Flows, 6, pp. 22.6.1–22.6.6. Springer.
Leith, C. E.: 1978 Objective methods for weather prediction. Ann. Rev. Fluid Mech. 10, 107–128.Google Scholar
Lesieur, M.: 1987 Turbulence in Fluids. Martinus Nijhoff.
Lesieur, M. & Métais, O. 1989 Proc. Pole European Pilote de Turbulence (PEPIT). ERCOFTAC Summer School, Lyon. July 1989.
Leslie, D. C.: 1973 Developments in the Theory of Turbulence. Clarendon.
Lessen, M.: 1979 On the power laws for turbulent jets, wakes and shearing layers and their relationship to the principle of marginal instability. J. Fluid Mech. 88, 535–540.Google Scholar
Liu, J. T. C.: 1989 Contributions to the understanding of large-scale coherent structures in developing free turbulent shear flows. Adv. Appl. Mech. 26, 183–309.Google Scholar
Lumley, J.: 1965 The structure of inhomogeneous turbulent flows. Proc. Intl Coll. on Radio Wave Propagation (ed. A. M. Yaglom & V. I. Takasky), Dokl. Akad. Nauk. SSSR. 166–178.
Lumley, J. L.: 1978 Computational modelling of turbulent flows. Adv. Appl. Mech. 18, 126–176.Google Scholar
Mason, P. J. & King, J. C., 1985 Measurements and predictions of flow and turbulence over an isolated hill of moderate slope. Q. J. R. Met. Soc. 111, 617–640.Google Scholar
Maxey, M. R.: 1978 Aspects of unsteady turbulent shear flow. Ph.D. dissertation. University of Cambridge.
Maxey, M. R.: 1982 Distortion of turbulence in flows with parallel streamlines. J. Fluid Mech. 124, 261–282.Google Scholar
Moffatt, H. K.: 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28, 571–592.Google Scholar
Moffatt, H. K.: 1984 Simple topological aspects of turbulent vorticity dynamics. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), pp. 223–230. Elsevier.
Monin, A. S. & Yaglom, A. M., 1971 Statistical Theory of Turbulence, vol. II. MIT Press.
Mumford, J. C.: 1982 The structure of the large eddies in fully developed turbulent shear flows. Part 1. The plane jet. J. Fluid Mech. 118, 241–268.Google Scholar
Murakami, S. & Mochida, A., 1988 3D numerical simulation of airflow around a cubic model by means of a k–ε model. J. Wind. Engng Indust. Aerodyn. 31, No. 2.Google Scholar
Pearson, J. R. A.: 1959 The effect of uniform distortion on weak homogeneous turbulence. J. Fluid Mech. 5, 274–288.Google Scholar
Phillips, O. M.: 1955 The irrotational motion outside a free turbulent boundary. Proc. Camb. Phil. Soc. 51, 220–229.Google Scholar
Rodi, W.: 1988 Recent developments in turbulence modelling. Proc. 3rd Intl Symp. on Refined Flow Modelling and Turbulence Measurements, Tokyo, July.Google Scholar
Rogallo, R. S.: 1981 Numerical experiments in homogeneous turbulence. NASA Tech. Memo. 81315.Google Scholar
Rogers, M. M.: 1990 The response of a passive scalar field to a mean scalar gradient in rapidly sheared turbulent flow. Phys. Fluids (submitted).Google Scholar
Rogers, M. M. & Moin, P., 1987 The structure of the vorticity field in homogeneous turbulent flows. J. Fluid Mech. 176, 33–66.Google Scholar
Ruderich, R. & Fernholz, H. H., 1986 An experimental investigation of a turbulent shear flow with separation, reverse flow and reattachment. J. Fluid Mech. 163, 283–322.Google Scholar
Saffman, P. G.: 1967 Note on decay of homogeneous turbulence. Phys. Fluids 10, 1349.Google Scholar
Savill, A. M.: 1987 Recent developments in rapid-distortion theory. Ann. Rev. Fluid Mech. 19, 531–570.Google Scholar
Sreenivasan, K. R.: 1985 The effect of a contraction on a homogeneous turbulent shear flow. J. Fluid Mech. 154, 187–213.Google Scholar
Sreenivasan, K. R. & Narasimha, R., 1978 Rapid distortion of axisymmetric turbulence. J. Fluid Mech. 84, 497–516.Google Scholar
Taylor, G. I. & Batchelor, G. K., 1949 The effect of wire gauze on small disturbances in a uniform stream. Q. J. Mech. Appl. Maths 2, 1–29.Google Scholar
Tennekes, H.: 1988 Numerical weather prediction: illusions of security, tales of imperfection. Weather 43, 165–169.Google Scholar
Tennekes, H. & Lumley, J. L., 1971 A First Course in Turbulence. MIT Press.
Thomas, N. H. & Hancock, P. E., 1977 Grid turbulence near a moving wall. J. Fluid Mech. 82, 481–496.Google Scholar
Townsend, A. A.: 1961 Equilibrium layer and wall turbulence. J. Fluid Mech. 11, 87–120.Google Scholar
Townsend, A. A.: 1970 Entrainment and the structure of turbulent flow. J. Fluid Mech. 41, 13–46.Google Scholar
Townsend, A. A.: 1976 Structure of Turbulent Shear Flow. Cambridge University Press.
Townsend, A. A.: 1980 The response of sheared turbulence to additional distortion. J. Fluid Mech. 98, 171–191.Google Scholar
Tucker, H. J. & Reynolds, A. J., 1968 The distortion of turbulence by irrotational plane strain. J. Fluid Mech. 32, 657–673.Google Scholar
Uzkan, T. & Reynolds, W. C., 1967 A shear-free turbulent boundary layer. J. Fluid Mech. 28, 803–821.Google Scholar
Weber, W.: 1868 Über eine Transformation der hydrodynamischen Gleichungen. J. Reine Angew. Math. 68, 286.Google Scholar
Wray, A. & Hunt, J. C. R. 1989 Algorithms for classification of turbulent structures. In Proc. IUTAM Symp. on Topological Fluid Mechanics. Cambridge University Press.
Wyngaard, J. C. & Cote, O. R., 1972 Modelling buoyancy driven mixed layers. J. Atmos. Sci. 33, 1974–1988.Google Scholar
Zeman, O. & Jensen, N. O., 1987 Modification of turbulence characteristics in flow over hills. Q. J. R. Met. Soc. 113, 55–80.Google Scholar