ScienceWeek

HISTORY OF PHYSICS: ON LUDWIG PRANDTL (1875-1953)

The following points are made by Roddam Narasimha (Nature 2004 432:807):

1) In August 1904 Ludwig Prandtl (1875-1953), a 29-year old professor of mechanics at the Technical University of Hanover, presented a remarkable paper at the Third International Mathematical Congress in Heidelberg. The paper was a scientific time bomb -- it made no great impact at the congress, and was not translated into English until 1928. But by the 1920s and 1930s, the powerful idea in that paper and the reputation of its author had spread across the world, helping to create modern fluid dynamics out of ancient hydraulics and 19th-century hydrodynamics.

2) In retrospect, the paper, with twelve photographs of water flow past bodies, ten other diagrams, and only eight equations, can be seen as full of understated daring. The heart of the paper is in one paragraph, sandwiched between some preliminary mathematics and a terse but wide-ranging exposition of the explanatory power of the idea. That exposition covered how vortices emerge in mixing layers, behind cylinders, at the edge of a plate moving normal to its plane and so on, and when and why flows separate from the solid surface they are supposed to follow.

3) The tone of that key paragraph is deceptively casual. The formal problem it tackled is the flow past an aligned flat plate -- a problem that is trivial in inviscid hydrodynamics, but was crucial to the new fluid dynamics that was being created. First, a small parameter, (e) , is identified to represent viscosity, or more precisely a reciprocal of the Reynolds number. Classical perturbation methods would not work here, for the limit (e) to 0 is singular: the limit of the full solution (for example, at the surface) is not the solution of the limiting equation. Prandtl's method was to divide the flow into different regions in which the dynamics are different -- an outer inviscid flow, and a thin inner viscous ("boundary") layer next to the surface. In each of these regions only the dominant terms in the Navier-Stokes equations were collected. The equations for each region were separately solved, but their boundary conditions were cleverly selected so that the solutions blended into each other to yield a "unified" approximation.

4) In that same paragraph, Prandtl also showed how the partial differential equations that governed the boundary layer could be reduced to one ordinary differential equation using a combination of the independent variables, and thus solved the resulting nonlinear equation (which was not even explicitly written out) numerically, and provided a rough value for the drag of the plate and a sketch of the solution. And he did all this using only roughly 25 lines.

5) This potentially precise calculation of laminar-flow flat-plate drag opened a door between the previously sealed chambers of hydraulics and hydrodynamics. Practitioners of these disciplines had long poured scorn on each other -- hydraulics was often called a science of variable constants, and hydrodynamics the mathematics of dry water. Reality, Prandtl demonstrated, was not only within the scope of Navier-Stokes equations, but even accessible to mathematics.[1-3]

References:

1. Birkhoff, G. Hydrodynamics: A Study in Logic, Fact and Similitude (Princeton Univ. Press, 1960)

2. van Dyke, M. Perturbation Methods in Fluid Mechanics (Academic Press, 1964)

3. Vogel-Prandtl, J. Ludwig Prandtl Mitt. 107, (Max-Planck-Institut für Strömungsforschung, 1993)

Nature http://www.nature.com/nature

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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)

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