Isotropy plays a central role in turbulence studies, both as an indicator of universality and as a simplification that permits estimates of crucial quantities from limited measurements.
An assumption underlying all of modern turbulence theory is that there exists a universal structure that is common to all turbulent flows. Although large scale motions are obviously affected by body forces and boundary effects, the small scales (on which mixing takes place) are presumed to be independent of such situation-specific factors (Kolmogorov 1941). A central component of this universality assumption is isotropy. Since the Navier-Stokes equations themselves are isotropic, anisotropy enters only through boundary conditions and body forces, and is therefore assumed to be confined to the large scales. In other words, if the small scales are independent of large-scale effects, they must be isotropic. In real flows, of course, the condition of small-scale isotropy cannot be satisfied exactly (as it requires complete separation of large and small scales). It is important that we understand the degree to which isotropy is violated in particular geophysical flows, so that we may assess the applicability of results based on the assumption of universality.
A very practical example occurs in attempts to measure dissipation rates of kinetic energy and scalar variances in laboratory and field experiments. The mathematical definitions of these quantities are composed of several terms, not all of which can be measured simultaneously. One then uses formulas based on the assumption of isotropy to estimate the missing terms. Most of what we know about these dissipation rates is based on measurements interpreted by assuming isotropy. It is therefore crucial to assess the degree to which the assumption is valid, and the corresponding effects of anisotropy on the accuracy of our measurements.
Computer simulations provide an ideal tool for the investigation of these issues, since they deliver a comprehensive, four-dimensional view of flow evolution. In this project, we analyze results from a sequence of simulations of mixing events typical of the ocean thermocline. These analyses allow us to quantify anisotropy as a function of the buoyancy Reynolds number,, and to assess the accuracy of many approximations that are used to estimate dissipation rates from field and laboratory data. Convergence with increasing buoyancy Reynolds number suggests that there is, in fact, an approximately-universal state at sufficiently high, and that the present results are therefore applicable to turbulent flows in general.
KELVIN-HELMHOLTZ BILLOW EVOLUTION
Flow evolution in a typical simulation is summarized in figure 1. At t=0, the flow consists of a stratified shear layer whose properties vary only in the vertical, i.e. the flow is one-dimensional. Flow is to the right in the upper layer, to the left in the lower layer, and the initial bulk Richardson number, Ri, is less than ¼. The primary Kelvin-Helmholtz instability forms a periodic train of vortices, two wavelengths of which are simulated here. Transverse secondary instability causes adjacent vortices to merge (figure 1a), after which three-dimensional secondary instabilities (figure 1b) trigger the transition to turbulence (figure 1c). Ri increases throughout this process, and eventually exceeds ¼. Beyond this point, turbulence gradually decays (figure 1d).
Figure 1: Stages in the evolution of a stratified shear layer as revealed by isosurfaces of the deformation rate. (a) Laminar rollup and pairing of 2D Kelvin-Helmholtz vortices. (b) 3D secondary instability triggers the transition to turbulence. (c) Fully developed turbulence. (d) Turbulence has decayed to form gravity waves and elongated pancake vortices.
The evolution of the large (energy-containing) motions from two- to three-dimensional form is illustrated further in figure 2a. The transition to turbulence occurs approximately 3 buoyancy periods into the simulation. Beyond this point, vertical motions (denoted by) decay rapidly. Streamwise motions dominate the final state of slow decay.
Throughout the simulation, the large scales are significantly anisotropic. This is, in fact, necessary for the growth of turbulence. As we will see, however, the small scales approach a nearly isotropic state after the transition to turbulence.
Figure 2: The evolution of turbulence in a stratified shear layers, illustrated by volume-averaged component kinetic energies. The upper axis gives the time in buoyancy periods, i.e. .
ANISOTROPY ON LARGE AND SMALL SCALES
In the Kolmogorov (1941) view of turbulence, kinetic energy is injected into the turbulent eddies at large scales, then cascaded down the spectrum to the smallest scales, where it is dissipated by viscosity. The production and dissipation ranges are separated by the inertial subrange. While the physics of the production range vary from flow to flow, the dissipation range is insulated from large-scale effects and is governed only by the intrinsic properties of the Navier-Stokes equations.
This idealized picture applies only in the
limit of large Reynolds numbers. Reynolds numbers found in the
ocean thermocline can be far from this limit, as the simulated
spectra shown in figure
3a illustrate. While there is a clear
distinction between production and dissipation ranges, there is
also significant overlap between them, and there is no intervening
inertial subrange. This raises the possibility that the dissipation
range could be influenced significantly by anisotropic large-scale
Figure 3: Spectral representations of the velocity field in fully developed turbulence. (a) TKE dissipation occurs mainly at scales smaller than 10 times the Kolmogorov scale. Production of TKE occurs primarily at larger scales. Dissipation range isotropy requires sufficient separation of these two spectral ranges. (b) Representative velocity gradient spectra reveal significant anisotropy at large scales, much less at small scales.
The velocity gradient spectra shown in figure 3b provide a first look at the degree of isotropy in the different spectral ranges. As we have seen in figure 2, the large scales are distinctly anisotropic. By the spectral peak (at which dissipation is a maximum), however, the spectra have converged and remain close down to the smallest resolved scales. This suggests that the small scales are significantly less anisotropic than the large scales.
DISSIPATION RANGE ANISOTROPY
Small-scale anisotropy is quantified using the dissipation anisotropy tensor:
in which subscripts following commas indicate partial differentiation. The properties of are summarized here by its first nonzero invariant: , a positive semidefinite quantity that is zero only when the flow is perfectly isotropic. Values of computed at selected times from all eight simulations are shown in figure 4. At both early and late times, is close to its maximum value, indicating that the small scales are strongly anisotropic. When turbulence is at its most energetic, however, drops to values near zero, showing that the dissipation range is nearly isotropic.
Figure 4: Invariants of the dissipation anisotropy tensor versus time in buoyancy periods (a) and versus buoyancy Reynolds number with time indicated by colors (b). In isotropic flow,
A remarkable result from the present analyses is the degree to which dissipation range anisotropy is determined by the buoyancy Reynolds number, (figure 4b). This is particularly evident after the transition to turbulence is complete (green and blue symbols), whence the dependence of on collapses to a well-defined curve.
ISOTROPIC ESTIMATES OF THE DISSIPATION RATE
Estimation of the turbulent kinetic energy dissipation rate, e , is often a crucial step in the analyses of data from both laboratory and field experiments. The exact expression for e is a lengthy sum of terms involving quadratic combinations of velocity derivatives. Since only a few of these derivatives can be measured simultaneously in most experiments, additional assumptions are needed in order to estimate the missing terms. The most common assumption is that the dissipation range is isotropic, in which case e can be computed from any velocity gradient.
For example, suppose that a standard shear probe executed a profile through a turbulent billow in the streamwise (x) direction. This would provide measurements of the transverse derivatives and , from which e could be estimated as
Accessible as it is, this estimate is only valid insofar as the dissipation range is isotropic. The results described above suggest that the accuracy of this approximation should depend on. The ratio of to e is shown as a function of in figure 5a. Note that e may be drastically underestimated if is much smaller than .
Estimates based on a spanwise profile with a shear probe as considerably more reliable (figure 5b). Data from a vertical profile tend to overestimate e quite significantly. (In the case of vertical profiles, the inaccuracy can be reduced by hi-pass filtering the data, as is done in standard profiling operations.)
Figure 5: Commonly used isotropic estimates of the TKE dissipation rate, scaled by the true dissipation rate and plotted as functions of the buoyancy Reynolds number.
More recent field experiments have employed Pitot tubes, which allow measurements of longitudinal velocity derivatives. In the present simulations, estimates of e based on longitudinal derivatives tend to be more resistant to anisotropy errors than those based on transverse derivatives. For example, the accuracy of the estimate based on a vertical profile,
is shown in figure 5d. The results shown in figure 5 are similar to results obtained by Itsweire et al (1993) for a uniform gradient model. This similarity provides further support for the universality of dissipation range physics.
Acknowledgement: This research was sponsored by the National Science Foundation.
Kolmogorov, A.N. 1941: "Dissipation of energy in locally isotropic turbulence", reprinted in Proc. R. Soc. Lond. A 434, p. 15 (1991).
Itsweire, E., J. Koseff, D. Briggs and J. Ferziger 1993: Turbulence in stratified shear flows: Implications for interpreting shear-induced mixing in the ocean, J. Phys. Oceanogr. 23, 1508.