Cédric Beaume

Applied Mathematician

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Turbulence

As opposed to laminar flows, which can be observed when fluid (macro-)particles move along parallel undisturbed layers and does not yield any macroscopic mixing, turbulent flows are characterized by their ability to produce mixing on the macroscopic scale. Turbulence occurs in a large variety of flow configurations but no additional effects than shear are necessary to observe it. As a result, it can be studied in very simple physical systems.

Plane Couette flow

One of the simplest flow configurations that exhibit turbulence is plane Couette flow, i.e., the three-dimensional incompressible flow confined between two parallel walls and driven by their differential motion. This flow is solved for using the Navier–Stokes equation and the incompressibility constraint with no external forcing term and is solely driven by the imposition of the wall motion via the boundary conditions. The resulting system involves a single parameter, the Reynolds number (hereafter Re), which quantifies the ratio between inertia forces and viscous dissipation. On the one hand, for small Reynolds numbers, only laminar flow in the form of a constant shear (or, equivalently, a linear velocity distribution) between the walls can be observed. On the other hand, turbulence prevails at large Reynolds numbers.

To illustrate turbulence, a representative simulation at Re = 500 in a small domain (half the wall distance being used as the unit of space, the domain has size 2 in the wall-normal direction, is 4π-periodic in the streamwise direction and 2π-periodic in the spanwise direction) was run. The resulting turbulent flow is shown in terms of the streamwise velocity at a constant streamwise location in Video 1 and in the mid-plane between the walls in Video 2. The sheets of constant streamwise velocity visible in Video 1 are undulated in the spanwise direction to produce streaks. They are maintained by rolls which advect some of the negative (resp. positive) streamwise velocity up (resp. down) from the bottom (resp. top) wall. These structures have a non-trivial dependence on the streamwise direction, as shown in Video 2. Importantly, the flow is chaotic, as the "plume dance" in Video 1 and the "billow highway" in Video 2 indicate. It is this temporal complexity that leads to macroscopic mixing.

Video 1: Turbulent plane Couette flow in a domain of streamwise period 4π and spanwise period 2π at Re=500 displayed using the streamwise velocity at a constant streamwise location in the spanwise (x-axis) and wall-normal (y-axis) plane. The top (resp. bottom) wall has velocity 1 (resp. -1) so that it moves into the screen (resp. toward you). Each second of the video corresponds to 10 advective time units.

Video 2: Same as Video 1 except that the plane of representation is the mid-plane between the walls. The streamwise (resp. spanwise) direction is represented along the x- (resp. y-)axis.

The small domain simulation presented above allows to make two critical observations: (i) the flow displays temporal complexity and (ii) it is supported by spatial structures, namely streaks and rolls. Videos 3 and 4 show similar flow conditions to Videos 1 and 2 but within a larger domain: the streamwise and spanwise periods of the flow are here both 4 times larger than in the smaller domain. This less constraining domain allows complex spatial dynamics to develop. Many plumes of various sizes coexist and interact with each other (Video 3), while billows propagate along irregular paths and give rise to coalescence and splitting events (Video 4). The flow is thus constituted of a multitude of structures of different size and amplitude, each of which describing a behavior similar to that identified in Videos 1 and 2. While a given structure may be out of phase with its neighbors, interaction between neighboring structures is a strong dynamical mechanism that enhances flow complexity. The resulting spatio-temporal dynamics is a feature shared by most turbulent flows and is what makes them instantly recognizable.

Video 3: Same as Video 1 but in a domain of streamwise period 16π and of spanwise period 8π.

Video 4: Same as Video 2 but in a domain of streamwise period 16π and of spanwise period 8π.

The turbulence presented above is typical of the transitional regime, i.e., of Reynolds numbers which are barely large enough to sustain turbulence. We can get more insight into turbulence by increasing the Reynolds number. A similar simulation to that in Videos 3 and 4, except for a Reynolds number three times larger, is reported in Videos 5 and 6. Video 5 shows that, for Re=1500, the dynamics occurs over shorter time-scales. The plumes are finer and do not extend as far into the domain as they did for Re=500 (Video 3). As shear builds at the wall, the boundary layers become thinner and momentum becomes better mixed over a larger fraction of the domain. Video 6 highlights the fact that the structures constituting the flow become finer as the Reynolds number is increased. As a consequence, larger Reynolds number flows display a larger range of spatial scales, each of which undergoing dynamics on its own time-scale: the smaller the structure, the faster the dynamics. This accounts for the ever-growing complexity of turbulent flows as the Reynolds number is increased.

Video 5: Same as Video 3 but for Re=1500.

Video 6: Same as Video 4 but for Re=1500.

Research supported by the Leverhulme Trust under Grant RPG-2018-311.

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