Abstract
Turbulent flows are highly intermittent—for example, they exhibit intense bursts of vorticity and strain. Kolmogorov theory1, 2 describes such behaviour in the form of energy cascades from large to small spatial and temporal scales, where energy is dissipated as heat. But the causes of high intermittency in turbulence, which show non-gaussian statistics3, 4, 5, are not well understood. Such intermittency can be important, for example, for enhancing the mixing of chemicals6, 7, by producing sharp drops in local pressure that can induce cavitation (damaging mechanical components and biological organisms)8, and by causing intense vortices in atmospheric flows. Here we present observations of the three components of velocity and all nine velocity gradients within a small volume, which allow us to determine simultaneously the dissipation (a measure of strain) and enstrophy (a measure of rotational energy) of a turbulent flow. Combining the statistics of all measurements and the evolution of individual bursts, we find that a typical sequence for intense events begins with rapid strain growth, followed by rising vorticity and a final sudden decline in stretching. We suggest two mechanisms which can produce these characteristics, depending whether they are due to the advection of coherent structures through our observed volume or caused locally.
The problem of turbulence has been notoriously difficult to resolve, because it involves the interaction of flow length and timescales over many orders of magnitude. The intense, localized events that are characteristic of turbulence can cause difficulty in numerical simulations, via the generation of small scales and large local gradients. Simulations have some advantage in that they provide access to the entire field; alternatively, experiments can yield long observational records that are computationally impractical. Turbulence experiments also have limitations. Many measurements have relied on invasive techniques9 that can affect the flow being studied. Usually, measurements do not capture the full three-dimensional nature of the flow; instead, they rely on one- or two-component approximations of velocities and velocity gradients. A fully three-dimensional (3D), spatially resolved technique is needed to measure important quantities such as the energy dissipation and helicity within the flow. Good temporal resolution is also required to study the rapid development of intense events.
A few recent experiments have captured many of these features. Tao et al.10, 11 used holographic particle image velocimetry (HPIV) to take elegant 3D snapshots of turbulent channel flow, though the technique is not time-resolved. Using two very different technologies, La Porta et al.12 and Mordant et al.13 tracked individual particles (a lagrangian measurement) in highly turbulent flows. This method provides direct access to the velocity and acceleration of the flow, but cannot yield velocity gradients.
We have designed and implemented a 3D particle image velocimetry technique14 that obtains time series of the three components of velocity and all nine velocity gradients measured within a small volume. That is, this instrument allows us not only to study the statistics of intense, intermittent events, but also to observe and characterize their dynamics. In the following, we describe this technique and the results from a turbulent flow.
To visualize the 3D velocity field locally, we align three laser sheets such that each sheet illuminates one face of a small, cubic test volume, with the three sheets crossing at one vertex of the cube. The sheets illuminate 1 µm spherical polystyrene fluorescent particles that track the motion of the fluid. Three high-speed video cameras, recording at 125 frames per second, focus on the sheets, such that each camera sees only those particles in its own sheet. The key to reconstructing the 3D flow from these three slices is that the test volume be sufficiently small so that the flow within is, to good approximation, locally linear:
where M = ∂vi/∂xj is the matrix of gradients. We use the two-dimensional velocity vectors from all three faces to estimate b and M for the whole cube. The dissipation ε and enstrophy Ω can be calculated at a given time step from the measured gradients: ε = νΣij(Mij + Mji)2/2 and Ω = Σij(Mij - Mji)2/4 = ω2/2, where ω = ▿ × v is the vorticity and ν is the kinematic viscosity. In a turbulent flow, for equation (1) to hold, the test region must be of the order of, or smaller than, the Kolmogorov length η, the length scale at which viscosity smooths out the flow. We use three long-distance microscopes to image a small region (of size 1.8η), from a working distance of 43.3 cm.
Our flow is produced in a transparent cubic container of side l = 22 cm. Two square grids with a mesh size of lg = 0.85 cm are rigidly connected to each other with a vertical spacing of 6.0 cm and are vertically oscillated with a position y = y0 sin (ωgt) to drive the flow. The measurement volume at the centre of the tank is horizontally and vertically centred between the two grids.
The Reynolds number, which gives the approximate ratio of inertial to viscous forces in the flow, can be defined in terms of the tank size and the velocity amplitude of the grid, Re = lωgy0/ν; we present data for a flow with Re = 48,200 and Rλ = v2x,rms/ν
(∂xvx)2
1/2 = 54. The time average of the dissipation is ε¯ = 0.0336 cm2 s-3, which can be used to estimate the Kolmogorov length scale for this flow, η = (ν3/ε¯)1/4 = 0.074 cm.
To gather sufficient statistics, 5 × 106 sets of vectors and matrices for grid turbulence have been gathered, corresponding to 0.8 terabytes of video data. Video sequences of the flow reveal periods of unidirectional flow occasionally interrupted by visually striking vortices and hyperbolic flow patterns. Traces of ε and Ω (Fig. 1) reflect the intermittent nature of the flow. These distributions are qualitatively comparable to those observed in numerical simulations15. The probability distributions Pr(ε) and Pr(Ω) (Fig. 2) both display long tails with high values more common for Ω than ε, indicating that vorticity is more intermittent than strain. Each component of the gradient matrix M assumes positive and negative values and, by continuity, must occasionally pass through zero. A one-dimensional approximation ε ≈ 15ν(∂xux)2 of these quantities, as used in some previous experiments3, implies that very low values of ε occur with much higher probability than we observe. In reality, it is unlikely that all nine gradient components simultaneously drop to zero.
Figure 1: Time traces of the dissipation ε and enstrophy Ω.
![Figure 1 : Time traces of the dissipation |[epsiv]| and enstrophy |[Omega]|. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com](nature01334-f1.0.jpg)
The two traces above show intense events occurring in a turbulent flow. Dissipation curves are marked with crosses, and enstrophy curves are shown as continuous curves. The time averages are ε¯ = 0.0336 cm2 s-3 and Ω¯ = 4.13 s-2 with intermittent bursts reaching up to 40 times those values.
High resolution image and legend (46K)Figure 2: Distributions of the dissipation and enstrophy.
The long tails of the probability distribution functions extend toward high values of ε (crosses) and Ω (circles), indicating that these are highly intermittent quantities. For values larger than their means, the distributions are well approximated by stretched exponentials, Pr(ε) ≈ exp[ε-0.62 ± 0.08] and Pr(Ω) ≈ exp[Ω-0.45 ± 0.07], as indicated by the solid lines. Inset, log–log plots show significant deviations from log-normality, which would yield parabolas here.
High resolution image and legend (84K)Taken over the entire data set, the measure εΩ = 1.699ε¯Ω¯ indicates that these two quantities are not statistically independent. A scatter plot of ε versus Ω (Fig. 3) highlights the interdependence of these quantities. In particular, especially high values of ε coincide with high values of Ω sixteen times more frequently than if they were independent.
Figure 3: A scatter plot of dissipation versus enstrophy.
The interdependence of these two quantities is clearly evident. The shading Pr(ε,Ω)/Pr(ε)Pr(Ω) would be 1 if ε and Ω are independent. While values close to their averages appear uncorrelated, high values of ε and Ω occur simultaneously about sixteen times more often than would be expected. Particularly low values appear similarly correlated. Only points with Pr(ε,Ω) > 10-5 were plotted.
High resolution image and legend (58K)Our observations of the fluctuating gradients permit a statistical analysis of the interdependencies of the stretching and rotating parts of the field. It is useful to split the gradient matrix M into symmetric (representing strain) and antisymmetric (representing rotation) parts: M = S + A. Note that the dissipation is proportional to the sum of the squared components of S and that the enstrophy is proportional to the sum of the squared components of A. We observe that the time derivatives ε˙ and Ω˙ are systematically affected by the local conditions of ε (and the three eigenvalues of S, which we denote as λi) and Ω. In particular, the dissipation on average is rising when the largest eigenvalue of S, λ1, is large (see Fig. 4a), and falling when the enstrophy is large (see Fig. 4b).
Figure 4: Conditional averages of the growth rate of dissipation ε˙.
![Figure 4 : Conditional averages of the growth rate of dissipation |[epsiv]||[dot]|. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com](nature01334-f4.0.jpg)
a, The average of ε˙ conditioned on the largest eigenvalue of the strain matrix λ1 being larger than a value λc. Gradients, represented by λ1, are subject to a self-steepening mechanism, causing the dissipation to rise. Note that ε = 2νΣiλi2. b, The average of ε˙ conditional on the enstrophy Ω being larger than a value Ω0. The observations show an unambiguous decline in the dissipation when rotations are strong. We interpret this as a vortex ‘squelch’, whereby the vortices disrupt by mixing the self-steepening of the straining motions. c, The average of ε˙ conditional on the ratio of the two largest eigenvalues of the strain matrix. Arrows indicate local flow stretching directions. The dissipation, on average, grows when there are two contracting directions, while for λ2/λ1 > 0.25 the dissipation, on average, falls. As the contracting directions are the ones unstable in growth by a Burger's mechanism, this may indicate that having two contracting directions enhances that mechanism.
High resolution image and legend (91K)The time dependence of the dissipation can be due to several mechanisms. Although we will also advance the idea of local kinematic causes, another explanation invokes coherent structures being advected into our measurement volume. Because our observations are probably an admixture of the two phenomena, we will discuss interpretations relating to both.
Previous observations of vortex tubes with nearby sheets of high dissipation16 offer one possible explanation to the behaviour seen in Fig. 1. An advecting velocity correlated with these structures is consistent with the observed statistics in the following way. Figure 4b shows the dissipation falling as the enstrophy increases. This is consistent with a dissipation decrease as the centre of a vortex core is approached, and that the high dissipation sheet precedes but does not trail the vortex. Figure 4c shows that the front of this preceding dissipation sheet would typically be compressive in two directions, while the back would have one compressive direction.
We can learn about gradient fluctuations by studying our data and by examining the Navier–Stokes equations. If we expand the velocity field locally as a first-order Taylor series in x, exactly as in equation (1), and substitute it into the Navier–Stokes equations, we obtain the following equation for M:
= -M2 - Π/ρ, where Π is the Hessian matrix for the pressure field17. This expansion neglects several factors, all quadratic in x, including the effect of viscosity and the effect of advection in convecting gradients into our local region. This expression retains gradient steepening by the nonlinear term (M2) and the nonlocal effect of pressure to enforce incompressibility. Note that Π is undetermined except by boundary conditions and incompressibility, Tr(M2 + Π) = 0; this is due to the nonlocality of the pressure. Finally, splitting M into symmetric and antisymmetric parts yields:
These form a local basis for the interaction of rotation (A) and strain (S). Because the matrix A contains only the components of the vorticity, equation (3) gives ω˙ = Sω, indicating how the strains stretch and amplify vorticity17.
The kinematics of equations (2) and (3) can be related to our observed statistics and by observations of events such as those displayed in Fig. 1.
First, the gradients show self-steepening instabilities, causing ε˙ > 0. These instabilities are probably of the same type that causes the gradients to grow in the Burger's equation. The conditional statistic shown in Fig. 4a substantiates this effect. Second, the presence of large gradients causes the vorticity to grow (Ω˙ > 0) owing to the well-known vortex stretching mechanism. Finally, once the vorticity has grown, it causes the dissipation to fall, that is, Ω
Ω¯ implies that ε˙ > 0, as is seen in Fig. 4b and anticipated in equation (2) (although the effect of the -A2 term depends in a nontrivial way on the directions involved).
The events shown in Fig. 1 were chosen for their particularly large amplitudes of ε, but they show a sequence bearing out these three features. The gradients (as indicated by ε) first grow, in the absence of a large vorticity. Once the gradients are large, the vorticity (through Ω) shows a sudden, rapid increase. The peaking of the enstrophy is then accompanied by the rapid decline of the dissipation. We see no clear evidence, through conditional statistics, for a mechanism by which the vortices decline. It may be that slower viscous forces are required to damp the vortices, because kinematic mechanisms, as represented in equations (2) and (3), are ineffectual at damping vorticity.
Using this optical technique, we present these experiments to directly observe the development, interdependence and interaction of turbulent strain and vorticity. We have directly quantified intermittent events in the dissipation and enstrophy and discussed their relation to advecting structures. Our analysis includes a characterization of self-steepening and the first statistical evidence for vortex ‘squelch’ (see Fig. 4) of the dissipation. Our observations highlight how an understanding of the temporal interactions between stretching and vorticity is crucial to the science of extreme events in turbulence.
Methods
Each image frame was broken into four 108 × 128 pixel subsections (9.7% overlap). A two-dimensional cross-correlation is performed between each subsection in subsequent images, a process which identifies the best two-dimensional pixel shift to particle images from one frame to the next. A weighted averaging technique is used to identify the cross-correlation peak to sub-pixel resolution, estimated at 0.25 pixels. A linear least-squares method fits the calculated velocity vectors and three-vector locations from each camera to the linear shear model of equation (1) to estimate b and M. Vector validation and bootstrapping18 are used to reduce the effect of spurious vectors caused by too few particles. The least-squares calculation is constrained to be divergence-free, that is, Tr(M) = 0.
