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ScienceWeek
FLUID DYNAMICS: ON THE ONSET OF TURBULENCE
The following points are made by Friedrich H. Busse (Science 2004 305:1574):
1) The transition to turbulence in fluid flow is an everyday experience. As a faucet is slowly opened, the initially laminar flow of water changes into an irregular chaotic flow. As a result, friction is much increased and, for the same discharge, a higher pressure head must be applied than in the laminar case. This transition is of fundamental importance in engineering problems dealing with fluid flows.
2) The study of the onset of turbulence has a long history. In 1839, Hagen first noted the existence of two distinct flow regimes in the discharge from pipes (2). Some 50 years later, Reynolds (3) realized that the transition between these regimes depends only on a dimensionless number, Re = UD/v, where U denotes the mean velocity averaged over the circular cross section of the pipe, D is its diameter, and (v) is the kinematic viscosity of the fluid.
3) In pipe flows, disturbances of finite amplitude are responsible for the transition to turbulence. Reynolds noticed as much when he reported that the transition was delayed to higher values of Re when a particularly smooth entrance region of the pipe was used. However, theoretical studies can treat easily only infinitesimally small disturbances, and this is one reason why theoretical understanding of the transition to turbulence in shear flows has been slow to emerge. For laminar flow in a channel between parallel plates, such analysis suggests that laminar flow should become unstable at Re = 7696, but experiments indicate a much lower value of ~1500 for the transition (4). For flow between two parallel plates sliding relative to each other with speed U (plane Couette flow) and for flow through a circular pipe, the discrepancies are even larger: No growing infinitesimal disturbances could be found theoretically at any Reynolds number.
4) With today's powerful computers, it is not difficult to simulate turbulent fluid flows at Reynolds numbers of several thousands. Good agreement between statistical properties of turbulence in experiments and in numerical simulations has been found (5), but a detailed understanding of the transition process is still lacking.
5) For configurations other than plane parallel flow, theoretical studies have been more successful. For example, when the circularly symmetric flow between differentially rotating coaxial cylinders becomes unstable, axisymmetric vortices are formed, the amplitude of which increases smoothly with the Reynolds number. This is a typical example of a supercritical bifurcation(6), in contrast to the unstable subcritical bifurcations that occur in plane parallel shear flows in the absence of rotation.
6) For plane Couette flow and pipe flow, theoretical studies have not found evidence for bifurcation at finite values of Re. Nevertheless, the belief in the existence of relatively simple solutions describing states of fluid flow distinct from the basic states of plane Couette flow or pipe flow has persisted. These solutions must be expected to be unstable; therefore, numerical methods are usually not capable of producing them, just as experiments do not exhibit them.(1)
References (abridged):
1. B. Hof et al., Science 305, 1594 (2004)
2. G. H. L. Hagen, Pogg. Ann. 46, 423 (1839)
3. O. Reynolds, Proc. R. Soc. London A 35, 84 (1883).
4. Here the same definition of the Reynolds number is used as for pipe flow except that D now refers to the width of the channel.
5. J. G. M. Eggels et al., J. Fluid Mech. 268, 175 (1994)
6. "Bifurcation" is a mathematical term used when a secondary solution branches from a primary one.
Science http://www.sciencemag.org
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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)
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