Forced isotropic turbulence: not so random after all?

Physical Review Letters has published a paper on turbulence by PhD student Moritz Linkmann and Alexander Morozov, both of the School of Physics & Astronomy.

Paper summary

Chaotic flows of liquids or gases far away from any boundary, like many atmospheric and oceanic flows, are often viewed by scientists as real-life realisations of the so-called forced isotropic turbulence - a classical idealised description of turbulent motion that dates back to the beginning of the 20th century. In this framework turbulence is driven by a vigorous large-scale stirring and was thought to be featureless at smaller scales: if one would look at the flow through a magnifying glass, one should see the same picture independent of the degree of magnification. This is in contrast with turbulence between boundaries, like flow in a pipe, where recent research showed that the flow is organised by spatially regular, unstable structures. 

Surprisingly, computer simulations performed in the Letter suggest that forced isotropic turbulence is not featureless and is much more similar to wall-bounded flows than was thought previously. We demonstrate that at moderate Reynolds number isotropic turbulence is always metastable and the probability of its sudden disappearance obeys an exponential law, previously found in experiments and simulations on pipe flows. 

The similarities between the two systems suggest a universal scenario in which turbulence is always organized around unstable, spatially regular structures. 

"It was very interesting to find that isotropic turbulence, which is an idealised system that allows the study of fundamental turbulent dynamics without having to consider external influences such as the geometry (and walls) of a container, is connected to real-world flows such as flow through a pipe.Moritz Linkmann, PhD student, School of Physics & Astronomy​

Below, Moritz Linkmann explains the research behind the paper.

Dynamical systems and the transition to turbulence in parallel shear flows

In parallel wall-bounded shear flows (such as flow through a pipe or counter-rotating cylinders) the transition to turbulence does not occur due to a linear instability of the laminar profile. The state space of the system, where each point corresponds to a flow state, is organised by a complicated collection of unstable flow states and the linearly stable laminar profile. Turbulence is then characterised as the system revolving around these unstable flow states (so-called `exact coherent structures'). The important point is that the laminar profile and the turbulent states remain dynamically connected and a sudden `escape' from the turbulent region of state space can occur. Localised turbulence in a flow can therefore suddenly relaminarise and this has been observed in many experiments of parallel shear flows. This relaminarisation is a memoryless process, that is, it does not depend on the amount of time the system has spent in the turbulent region of the state space. A characteristic timescale can be associated with this process which increases with Reynolds number as a double exponential. This implies that there is always a finite probability of relaminarisation, even at high Reynolds numbers: Localised turbulence in wall-bounded shear flows is transient.   

The transition to sustained turbulence then occurs due to a competing process,that is the splitting of a locally turbulent region into two. This process also has a characteristic timescale which now decreases with Reynolds number as a double exponential. The critical Reynolds number for sustained turbulence is then defined at the point where the two timescales are equal.

Collapse of isotropic turbulence

Isotropic turbulence and a parallel wall-bounded shear flow are a priori very different systems, isotropic turbulence being a simplified system studied in order to establish fundamental properties of turbulent flows. It is often thought of as high Reynolds number limit of wall-bounded flows, where the walls have negligible influence on the turbulent dynamics. It is traditionally studied in statistical terms and in numerical simulations, where the decay of turbulence due to viscous dissipation is balanced by an external energy input at the large scales. The emphasis in the simulations is usually on achieving high Reynolds numbers. The dynamics of forced isotropic turbulence has been thought to be as simpler than that of parallel shear flows. In particular, isotropic turbulence is not expected to show transitional behaviour and nothing is known about its phase space structure.

In this study we investigated the same system at low Reynolds numbers and observe a sudden collapse of turbulence in favour of a large-scale ordered flow. This can be seen in the figures, which show streamlines of the flow before and after the collapse of turbulence. A transition from a 'disordered' to an 'ordered' flow is clearly visible. This collapse of turbulence happens despite the continuous stirring of the flow at the large scales and had been observed in a previous study already. Our analysis, which is new in the sense that it is the first time that a dynamical systems approach has been applied to isotropic turbulence, now shows that this collapse has very similar features to relaminarisation events in pipe flow: It is a memoryless process with a characteristic timescale that increases with Reynolds number in much the same way as in wall-bounded shear flows. Furthermore, the base flow appears to be linearly stable. This shows that isotropic turbulence at low Reynolds numbers is transient and it suggests that the phase space dynamics of wall-bounded shear flows and isotropic turbulence may be very similar.

Outlook: the universality class of the transition to turbulence

Recent research suggests that the transition to turbulence in parallel shear flows constitutes a second-order nonequilibrium phase transition belonging to the Directed Percolation universality class. That is, the transition to turbulence in these flows behaves like, for example, the spread of diseases or wildfires. The similarities between relaminarisation in pipe flow and the collapse of isotropic turbulence reported here suggest that this behaviour may in fact be a universal feature of turbulence and plans are under way to investigate this further. 

This work has made use of the resources provided by the Edinburgh Compute and Data Facility ( Moritz Linkmann and Alexander Morozov acknowledge support from the UK Engineering and Physical Sciences Research Council (EP/K503034/1 and EP/I004262/1).

Reference: M. Linkmann and A. Morozov, Sudden Relaminarization and Lifetimes in Forced Isotropic Turbulence, Phys. Rev. Lett. 115, 134502