Fluid ideas

Sophie's first experience of gravity-wave-generated turbulence. Painted by Rosie Reid
Sophie’s first experience of gravity-wave-generated turbulence. Painted by Rosie Reid


Fluid Ideas

John Reid

Over the last four centuries the Newtonian description of the physical world in terms of differential equations has been brilliantly successful. It began with precise and accurate descriptions of the motion of the planets and their satellites and found further application in the design of the machinery which became the basis of industrial civilization. At first General Relativity appeared to undermine Newton’s ideas, but in effect it amounted to only a minor refinement of the Newtonian system.

A more profound threat to the Newtonian hegemony was the quantum theory which followed the “Ultraviolet Catastrophe”: the realization that some physical quantities are granular and require a radically different approach. At about the same time, the Second Law of Thermodynamics was proposed leading to the concept of entropy which describes how matter and energy are ordered and which requires the Universe be granular rather than continuous.

Newton’s ideas were successful when applied to the dynamics of planetary motions because friction and turbulence are negligibly small at planetary scales. Planetary motions are, to all intents and purposes, deterministic and amenable to a description in terms of differential calculus. Machinery is intentionally designed to minimise friction and turbulence and to be amenable to a deterministic description. This even applies to semi-conductor design where so called “race conditions” are eliminated in order to preclude any possibility of stochastic behaviour in electronic components.

But it is not true of fluids. Stochastic behaviour in the form of turbulence clearly plays a major role in the dynamics of fluids. The Navier–Stokes equations, which describe fluids in purely Newtonian terms, fail in high Reynolds Number regimes where turbulence is generated. This is the “Fluid Catastrophe”. In effect the Quantum Revolution bypassed Fluid Dynamics, whose practitioners still cling to the 19th century idea of the continuum.

The belief that any real fluid can only be dealt with as a deterministic, Newtonian continuum has had a stifling effect on development. Fluid Dynamics has become the province of Applied Mathematicians who are skilled in the manipulation of partial differential equations but in very little else. They are not trained to perform experiments. They do not have an empirical, “Popperian” outlook. They are mathematical Rationalists who only pay lip service to the scientific method. They are not really scientists at all but they they think they are.

Fluid dynamicists might argue that there is no alternative, that only a Newtonian approach to fluids will work. The way to break the stranglehold of determinism is to regard the mathematical expressions of physical laws, such as the Navier–Stokes equations, not as immutable, deterministic Laws but, rather, as constraints. The future state of a fluid is then predicted as that state which maximises the entropy within constraints determined by conservation of mass, momentum and energy. Because of our imperfect knowledge of the initial boundary conditions and the existence of bifurcation points in the evolving trajectory of the system, future states can never be known with certainty but, rather, a most likely state is predicted and its variance estimated. Russian physicists and mathematicians such as Kolmogorov, Monin, Obukhov and Kitaigorodskii laid the groundwork for this approach, but their work has been overshadowed by the advent of the digital computer which favoured fully deterministic numerical models. These now dominate the discipline. The current fad for running “ensembles” of deterministic models with randomized boundary conditions is naive and simplistic. Internally such models are kept stable only by means of unrealistic parameter values and smoothed topography and bathymetry. Garbage in, garbage out.

Science is not a collection of given“truths” like religion; it is a work in progress. This applies as much to Fluid Dynamics as to any other branch of science. What is needed is for Fluid Dynamics to become, once again, the empirical science that flowered in the 19th century. A simple experient with a cubic metre wax block could test the reality of the liquid-in-solid convection. If successful, new insights into Mantle dynamics would be gained. The wave-wave interaction algorithms, presently used in numerical wave models, were initially borrowed, second-hand, from Quantum Mechanics. This accounts for the poor performance of these models. What is needed are empirical studies of those wave-wave interactions which really do occur among gravity waves on a fluid surface and which involve entropy increases due to wave-breaking. Initially such studies could be carried out with the wave tanks and hydraulic basins presently used by naval architects. This would open a new field similar to the field of sub-atomic particle interactions which followed the development of the cyclotron. These experiments are not expensive; they are within the reach of a small university department.

It is time for Fluid Dynamics to recognize its “catastrophe”, to put its house in order and to become an empirical science once again. Until that happens, its predictions of future states of the Earth’s atmosphere must be taken with a large grain of salt.

Contemporary Fluid Dynamics is based on a myth:  that a fluid is a continuum which can be fully described by the deterministic equations of differential calculus. This idea is at odds with the stochastic assumption underlying statistical inference and with the post-quantum conception of entropy. It is a relic of a bygone era when the natural world was regarded as a perfect machine. Once the world was Paley’s timepiece, created and wound up by God at the Creation and left ticking steadily for us to examine and to marvel at. Perhaps this was appropriate in the early 19th Century but there has since been a vast increase in the range of phenomena accessible to science. Wherever we look we see a Universe that is chaotic and unpredictable; more like stock market than timepiece.

Furthermore it is not the pristine perfection of a Laplacian universe that matters, but rather its imperfections. Without occasional, random imperfections in nucleic acids, life could never have evolved beyond the virus. Turbulence is everywhere, in cumulus clouds, in breaking waves, in the sound of a clarinet and yet it is inaccessible to the elegant equations of 19th century physics. By presupposing an underlying perfection, we blind ourselves to the amazing realities of the world around us.

Only by experiment and observation can we truly see.

The above is an extract from the Author’s forthcoming book, “The Fluid Catastrophe” (Cambridge Scholars Publishing, Newcastle-on-Tyne).