Identifying universal laws within fluid mechanics is notoriously difficult. So it has come as a disappointment that recent experimental and computational results have cast doubt on one of the very few relations that was thought to hold true across all turbulent, bounded fluid flows. But now a physicist in Italy reckons that by tweaking the formula in question – which stipulates that such flows have a logarithmic velocity profile – it does indeed prove to be universal. All that is needed, he says, is a simple extra term to account for the effect of pressure variation along the structure through which the fluid flows. One turbulence expert, however, points out that this is not necessarily a new idea.

Turbulence is a ubiquitous phenomenon in nature and is seen by many as one of the greatest unsolved problems of classical physics. The challenge is understanding how a fluid, such as water flowing from a tap, makes the transition from a smooth laminar flow to a disordered turbulent flow as the velocity of flow increases. This characterized by a dimensionless ratio known as the Reynolds number, and this usually occurs when the number is around 3000.

Physicists have no doubt that the basic formula describing the temporal evolution of a flowing fluid – the Navier–Stokes equation – is correct. Applying the formula to turbulent fluids, however, is problematic. The equation can only be solved numerically in simple geometries – even water travelling along a uniform, circular pipe can only be described for very modest Reynolds numbers.

The best that can be done in more complex scenarios, says Paolo Luchini of the University of Salerno, Italy, is to calculate time-averaged local quantities using "somewhat ad-hoc" turbulence models. The values that emerge can then be used to work out the global quantities – such as total flow rate through a given pipe – which are of most interest to engineers.

Holy grail

Developing robust turbulence models has been a "holy grail" for physicists, according to Luchini, but to date the only one that has, he says, been "generally accepted" is one describing a fluid's velocity profile. This cross-sectional variation in the velocity of a confined fluid, such as one flowing through a pipe, arises because while the fluid generally flows quite freely down the centre of a pipe it is held back by friction at the edges. Indeed, at the very edge its velocity is generally zero.

Working out this profile for laminar flow is relatively straightforward. It just involves calculating the shear stress across the pipe caused by the variation in the fluid's velocity, resulting in a parabolic distribution. But when the flow is turbulent, eddies introduce a second source of stress. Unfortunately, this cannot – yet, at least – be worked out from first principles.

Logarithmic profile

Despite these difficulties, German engineer Ludwig Prandtl showed as far back as 1925 that the turbulent profile should be logarithmic. However, experiments and computer simulations done over the past 20 years or so have cast doubt on this idea. The experiments measured the velocity profile of a fluid flowing down a pipe with a variety of Reynolds numbers. If the logarithmic law indeed holds true then the different profiles, when plotted, should overlap one another once the axes are suitably adjusted. But the overlap wasn't perfect. Furthermore, the experimenters found that the shape of the profile also changed when fluids were sent along a different kind of channel – between parallel plane walls, for example.

This is a topic that can easily start up a fight at a specialist conference
Paolo Luchini, University of Salerno

Researchers have responded to this problem in various ways. Grigory Barenblatt of the University of California, Berkeley, US, proposed that the logarithmic law be replaced with a multi-parameter power law, while Hassan Nagib of the Illinois Institute of Technology, US, and colleagues said the law should remain logarithmic but that the coefficient used to multiply the logarithm, known as von Kármán's constant (ᴋ), should vary. Others, meanwhile, have continued to back the existing logarithmic law. "This is a topic that can easily start up a fight at a specialist conference," says Luchini.

Luchini's own solution involves the addition of an extra mathematical term. He arrived at this conclusion by chance while surveying the literature as part of a study of how modulations in the flow of a stream that are dictated by the shape of the stream bed affect the formation of sand dunes. Having devised, and then discarded, a series of candidate formulae, he found one that he says "had a physical basis and satisfied me more than those I had read about". The new term he introduces varies linearly with the pressure gradient along a pipe, which, he says, accounts for the variation in velocity profile with channel type while preserving the expression's "asymptotic" logarithmic nature.

Common correction

Not everyone, however, is convinced by the importance of the latest work. Peter Davidson of the University of Cambridge, UK, argues that it "has long been known" how to correct for pressure gradients when calculating turbulent velocity profiles. "There is not much new here," he says.

Luchini acknowledges that the effect of the pressure gradient on a velocity profile is "not in itself a new idea". But what is new, he maintains, is the addition of a new term to account for the effect, rather than modifying the value of ᴋ. The omission of such a term to date, he writes, "justifies the doubts that have arisen in the literature, whereas including it definitely shows that the logarithmic law is valid and the value of ᴋ is universal".

The research is reported in Physical Review Letters.