ScienceWeek

APPLIED PHYSICS: ON TURBULENCE IN PIPE FLOW

The following points are made by Richard Fitzgerald (Physics Today 2003 February):

1) In 1883, Osborne Reynolds (1842-1912) published his landmark paper on the transition from smooth, laminar flow to turbulent flow in cylindrical pipes. Drawing water through a horizontal glass pipe, Reynolds injected a narrow stream of dye and looked for the onset of eddies as he varied the flow velocity and the water viscosity (dependent on water temperature). He found that the transition to turbulence was very sensitive to disturbances and typically occurred above a critical value of about 2000 for the ratio of UD/v, where U is the average (or bulk) velocity, D is the pipe diameter, and v is the kinematic viscosity.(1) This ratio, which parameterizes the relative strengths of inertial and viscous forces, is now known as the Reynolds number, Re.

2) Understanding the nature of the transition to turbulence has been an ongoing quest ever since Reynolds's first experiments (and was the subject of Werner Heisenberg's PhD thesis in 1923). For pipe flow, the underlying Navier-Stokes equations, which describe the fluid dynamics of a system, have a laminar solution that has been found numerically to be stable for all Reynolds numbers. Indeed, in exquisitely controlled experiments, laminar flow at Reynolds numbers up to 100,000 has been observed.

3) And yet in practice, most pipe flows -- at least for Re above about 2000, a typical value for a moderate flow of water from a faucet -- are turbulent. Because laminar flow is linearly stable -- that is, stable against infinitesimal perturbations -- a finite-amplitude perturbation must be required to kick pipe flow out of that state and into a turbulent mode.

4) The nature of that transition is of more than academic interest. For a given pressure drop along a pipe, turbulent flow will result in a flow rate an order of magnitude smaller than laminar flow. To avoid large pressure and flow fluctuations associated with the turbulence transition, oil and gas pipelines are usually operated in the less-efficient turbulent regime. The ability to predict -- and perhaps eventually control -- the transition to turbulence could be a real boon.

5) In recent experiments, B. Hof et al (2) have measured how the threshold amplitude of turbulence-producing perturbations in pipe flow scales with Re. They have apparently unambiguously determined the thresholds in controlled experiments for the first time.(3-5)

References (abridged):

1. O. Reynolds, Proc. R. Soc. London 35, 84 (1883)

2. B. Hof, A. Juel, T. Mullin, Phys. Rev. Lett. 91, 244502 (2003)

3. See, for example, the discussion and references in P. J. Schmid, D. S. Henningson, Stability and Transition in Shear Flows, Springer-Verlag, New York (2001)

4. S. J. Chapman, J. Fluid Mech. 451, 35 (2002)

5. F. Waleffe, Phys. Fluids 9, 883 (1997); 15, 1517 (2003)

Physics Today http://www.physicstoday.org

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ON TURBULENCE

The following points are made by Roberto Benzi (Science 2003 301:605):

1) Turbulence -- the chaotic behavior of fluid flows -- occurs in a wide variety of flows, from the dispersal of pollutants in the atmosphere to the flow of air around automobiles and airplanes, and new approaches (1) have been introduced that facilitate numerical simulations of these complex processes.

2) All turbulent flows can be described by a set of nonlinear partial differential equations, which were first introduced almost 200 years ago by C.L. Navier (1785-1836) and George Stokes (1819-1903). The degree of turbulence can be quantified by the Reynolds number Re = UL/v, where (U) and (L) are the typical velocity and scale of the flow, respectively, and (v) is the kinematic viscosity due to molecular forces. For a car moving at 100 km/hour, Re is about 10^(7).

3) In 1941, Andrei Kolmogorov (1903-1987) (2) proposed that the statistical properties of turbulence are universal at large values of Re, that is, they depend only on the rate of energy dissipation. Among the many consequences of his theory, one can easily show that velocity gradients scale as Re^(1/2) at large Reynolds number. This means that the smallest scale at which one can observe chaotic or turbulent behavior is N = Re^(-3/4)L, where (N) is the Kolmogorov length.

4) Since then, researchers have combined new theoretical ideas (3,4) with numerical simulations and data analysis (5) to show, and in some cases rigorously demonstrate, that the statistical properties of turbulence are indeed universal, although not in the way originally proposed by Kolmogorov: At sufficiently small scales, the probability distribution of turbulence and its strong intermittent fluctuations do not depend on the forcing mechanism or other large-scale properties. The conceptual and practical consequences of these results are still to be explored, and researchers are facing a new period of scientific excitement in this field.

5) Direct numerical simulations of the Navier-Stokes equations have played a major role in studies of turbulent flows. However, such simulations are computationally very demanding. A modern supercomputer can perform accurate direct numerical simulations for Re <= 10^(4) -- far below what is needed to simulate a car or aircraft in engineering applications. Moreover, the geometry of turbulent flows in real-life applications can be complex, requiring a complex grid to resolve the dynamics of the flow. Direct numerical simulations thus face two problems: complex geometry and limited computer power.

References (abridged):

1. H. Chen et al., Science 301, 633 (2003)

2. A. N. Kolmogorov, Dokl. Akad. Nauk. SSSR 30, 9 (1941)

3. U. Frisch, Turbulence, the Legacy of A. Kolmogorov (Cambridge Univ. Press, Cambridge, 1996)

4. G. Falkovich, K. Gawedzki, M. Vergassola, Rev. Mod. Phys. 73, 913 (2001)

5. R. Benzi et al., Physica D 96, 162 (1996)

Science http://www.sciencemag.org

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ON DEFECT TURBULENCE

The "Prandtl number" is a dimensionless number occurring in the dimensional analysis of convection in a fluid due the presence of a hot body, the number given by Cn/Kr, where (C) is the heat capacity per unit volume of the fluid, (n) is the viscosity of the fluid, (K) is the thermal conductivity of the fluid, and (r) is the density of the fluid.

The following points are made by K.E. Daniels and E. Bodenschatz (Phys. Rev. Lett. 2002 88:034501):

1) Weakly driven, dissipative pattern-forming systems often exhibit the spatiotemporally chaotic state of defect turbulence, where the dynamics of a pattern is dominated by the perpetual nucleation, motion, and annihilation of point defects (or dislocations) [1]. Examples can be found within striped patterns in wind driven sand, electroconvection in liquid crystals [2], nonlinear optics [3], fluid convection [4,5], autocatalytic chemical reactions, and Langmuir circulation in the oceans. Researchers hope that the dynamics of these very different systems can be characterized by a universal description which is based only on the defect dynamics.

2) A first description of defect turbulence was given by Gil et al (1990) for a spatiotemporally chaotic state of the complex Ginzburg-Landau equation (CGLE). They postulated that the nucleation rate for defect pairs is independent of the number of pairs M, and based on the topological nature of defects the annihilation rate is proportional to M^(2). Through detailed balance, they showed that these assumptions lead to a squared Poisson distribution for the probability distribution function of M. They also found agreement with this probability distribution function in simulations of the CGLE with periodic boundary conditions. Rehberg et al. [2] measured probability distribution function of M for defect turbulence in electroconvection of nematic liquid crystals and found it to be consistent with the predicted squared Poisson distribution. Later, Ramazza et al [3] investigated a defect turbulent state in optical patterns and found that their data were not conclusive. To date, studies in both simulation and experiment have relied purely on comparisons of the probability distribution functions. The gain and loss rates, fundamental to the universal description of defect turbulence, have not been measured. In addition, effects due to boundaries have not been considered.

3) The authors report experimental results on the defect turbulent state of undulation chaos in inclined layer convection of a fluid of Prandtl number of approximately 1. By tracking all defects in a finite area of the convection cell, the authors measured, for the first time, defect creation, annihilation, leaving, and entering rates for a defect turbulent state. The observed pair creation and annihilation rates agree with predictions.

References (abridged):

I. P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).

2. I. Rehberg, S. Rasenat, and V. Steinberg, Phys. Rev. Lett. 62, 756 (1989).

3. P. Ramazza, S. Residori, G. Giacomelli, and F. Arecchi, Europhys. Lett. 19, 475 (1992).

4. S. W. Morris, E. Bodenschatz, D. S. Cannell, and G. Ahlers, Phys. Rev. Lett. 71, 2026 (1993).

5. A. L. Porta and C. M. Surko, Physica (Amsterdam) 139D, 177 (2000).

Phys. Rev. Lett. http://prl.aps.org

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