Dynamic Similarity and the Reynolds Number

Mathematics deals with pure numbers: 1, 2, 3, fractions and more exotic numbers like π. Since π depends on lengths, we might think its value depends on our units. But it is the ratio of the circumference of a circle to its diameter and, as long as both are measured in the same units — centimetres, inches or whatever — the ratio is always the same, a pure dimensionless number [TM247 or search for “thatsmaths” at irishtimes.com].

In applied mathematics, engineering, astronomy, and so on, we deal with dimensional quantities: the mass of the Earth, the power of an engine, the lifetime of an atom, the distance to a star. All of these change their numerical value when the units of measurement change, just as prices change when we convert from dollars to Euros.

The Navier-Stokes Equations

Suppose we are designing a racing car or a jet aircraft. A scale model can be built and tested in a wind tunnel at a fraction of the cost of testing a full-scale prototype but, to guarantee that the behaviour of the model faithfully simulates the real system, we must ensure that they are dynamically similar, that the same mathematical equations with the same parameters describe both.

The equations of fluid flow were formulated by French engineer Claude-Louis Navier and, more systematically, by the Irish scientist George Gabriel Stokes. The Navier-Stokes equations have wide application, to aircraft, train and car design, air pollution analysis, ocean currents and weather prediction. Stokes is honoured by a commemorative plaque in the townland of Screen, Co. Sligo. Navier’s name is inscribed on the Eiffel Tower.

Transition to Turbulence

A quantity in the equations called the coefficient of viscosity represents fluid friction. It is a dimensional number: when we apply the equations to a small-scale model, we must use a different value for this coefficient to maintain dynamic similarity. The ratio that we need to conserve is called the Reynolds number.

In 1883, Belfast-born Osborne Reynolds presented to the Royal Society his experiments with fluids, noting two distinct types of motion: laminar flow, where the fluid elements follow one another smoothly and steadily along clear lines, and turbulent flow in which they move in sinuous, eddying paths that are chaotic and unpredictable.

Reynolds had found that the transition from laminar to turbulent flow depends on the ratio between inertial and viscous forces — the Reynolds number. When this number reaches a critical value, the motion changes suddenly from smooth and orderly flow, becoming irregular and unpredictable.

The Reynolds number occurs in diverse problems: fish swimming in the sea, burning gas in a jet turbine, blood being pumped through our arteries, dispersal of pollutants in the atmosphere and hurricanes threatening the coastline. It is thanks to this number that we can design wind farms and aircraft using small-scale models, confident that our predictions will be valid at full scale.

Other Applications. Other Numbers.

The Reynolds number is also proving valuable in non-traditional fields. The flow of money through an economic system is modelled by equations with velocity representing the speed of cash-flow and transaction costs acting like viscosity. Information in computer networks can also be simulated in this way, with velocity related to bandwidth and viscosity arising from limitations of processing speed. In both cases, a ratio analogous to the Reynolds number allows us to characterise different regimes of behaviour and predict transitions between them.

Many other dimensionless numbers are valuable in fluid dynamics: The Froude number, the Richardson number, the Rossby number, the Ekman number, the Mach number and the Prandtl number. But, viscosity being ubiquitous and playing such a fundamental role in determining the nature of fluid flows, the Reynolds number is the most widely applicable of these.