What is Fluid Turbulence?

When people hear I study turbulence, they usually assume that I am referring to the phenomena that leaves airline passengers with a queazy feeling in their stomachs and a fear that the aircraft they are sitting in is about to go plummeting towards the earth. Of course, they are correct (well, not about the plummeting). The rapid fluctuations in wind velocity that causes an aircraft to suddenly lift or drop, are indeed associated with the same physical phenomena. Turbulent flows, such as these, are not restricted to the Earth’s atmosphere and actually have a far more significant effect on our everyday lives.

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Low speed Laminar flow.

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High speed Turbulent flow.

The easiest way to observe turbulence, and something you no doubt do on a regular basis, is simply to turn on a tap. When the faucet is open a small amount, the water exits the tap at a slow velocity and forms a transparent jet of relatively uniform velocity, referred to as a Laminar flow. As you continue to open the faucet the flow rate and the speed of the water increases, causing a sudden change to a frothy, opaque jet with random fluctuations its appearance and in the water velocity, typical of a Turbulent flow. These velocity fluctuations are made visible by the presence of air bubbles in the flow, which appear due to the entrainment of air into the jet. Similar behaviour can be observed when watching ocean waves, rivers and streams. In fact, attempts to visualise and understand turbulence can be traced back to sketches of streams by Da Vinci (see featured image at top of post). This drawing clearly shows the presence of multiple swirling motions, like miniature tornadoes, often referred to as turbulent eddies or coherent vortices or vortical structures. These eddies exist over a range of sizes or scales are present in flows ranging from atmospheric weather patterns to turbulent flames. In fact most of the flows encounter in both nature and industry are turbulent, including the flow through oil and gas pipe lines, the flow over cars, trains, ships, aircraft, wind and steam turbines, and even the blood flow through parts of the human heart.

Turbulent flows have been the focus of considerable study for over 130 years. Despite this, and contrary to popular belief, turbulence is still an immature science that lacks fundamental understanding of its complex dynamics. This lack of understanding is responsible for wasteful sub-optimal design and in some cases fatigue and failure of wind turbines and jet engines, greater air resistance (drag) over cars, ships, aircraft, etc. and increased fuel consumption.

It is estimated that close to half of the energy spent to move fluids through pipes and canals and to propel vehicles through air or water is dissipated by the turbulence that is generated in the thin layer of fluid at the surface of a solid body.

– J. Jimenez. Recent developments on wall-bounded turbulence. Rev. R. Acad. Cien. Serie A. Mat., 101(2):187–203, 2007.

“Turbulence remains the last great unsolved problem of classical Physics”
– Nobel laureate in Physics Richard Feynman

Surprisingly there is no clear definition of a turbulent flow, which I guess is a measure of how little we truly understand about this phenomena. Instead a flow is said to be turbulent when it possesses the following properties (n.b. more mathematic aspects like rotational flow and high Reynolds number have been neglected):

  1. Irregular / Random – quantities (e.g velocity) show a random variation in space and time, such that is impossible to predict the exact flow at a point and given instant in time;
  2. Wide range of scales – consists of eddies or coherent motions, some of which may be 1000s of times larger than others;
  3. Three dimensional – instantaneous eddies must be three-dimensional, even if the mean flow may be consider two-dimensional;
  4. Highly diffusive – exhibit rapid mixing of momentum, mass and temperature;
  5. Highly dissipative – kinetic energy in the flow will be quickly converted to internal energy in the fluid and must have a constant supply of energy to persist.

In a mathematical sense turbulence is the result of the non-linear and chaotic nature of the equations that govern the flow of fluids and as such much of the analysis and interpretation of turbulence can often becomes highly mathematical. Unfortunately the field of mathematics is not able to provided a general solution for these equations (with the exception of a few very basic cases), which means that in order to predict the turbulent flow over an aircraft we must simulate all scales of a turbulent flow from the size of the aircraft down to fractions of a millimetre. On present supercomputers this would require:

“several thousand years to compute the flow for one second of flight time!”

– Parviz Moin and John Kim (http://turb.seas.ucla.edu/~jkim/sciam/turbulence.html)

Obviously this is not possible. Instead engineers and designers are forced to use highly simplified turbulence models will model the effect of small scales and only require the simulation of larger scales on a coarser grid (as done in computational fluid dynamics or CFD). These models are typically based on empirical constants, which generally require significant fine tuning and validation via the use of carefully scaled wind tunnel experiments. This comes at significant expense and is partially responsible for the relatively slow evolution of commercial aircraft, whose shapes have remained relatively unchanged for the past 50 years.

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