In this paper we study the statistical laws of relative dispersion in two-dimensional turbulence by deriving an exact equation governing its evolution in time, then evaluating the magnitude of its various terms in numerical experiments, which allows us to check the validity of the classical dispersion laws: the equivalent to the Richardson-Obukhov t3 law in the energy cascade range, and the Kraichnan-Lin exponential law in the enstrophy cascade range. We examine theoretically and experimentally the conditions of validity of both laws. It is found that the t3 law is obtained in the energy inertial range provided the separation scale of the particles is smaller by an order of magnitude than the injection scale. When the t3 law is reached, the relative acceleration correlations are observed to have reached a statistical quasistationary stage: this would tend to justify in the energy inertial range of two-dimensional turbulence a working hypothesis formulated by Lin & Reid (1963); also, the necessity of starting from very small initial separations to get the t3 law may be explained by the time necessary for relative acceleration correlations to reach the statistical quasi-stationary regime. On the other hand, the Kraichnan-Lin exponential law is, strictly speaking, never observed; it is in fact reduced to a very short transient stage when the relative dispersion characteristic time reaches its minimum value, as predicted by Batchelor.

Two-dimensional turbulence is investigated experimentally in thin liquid films. This study shows the spontaneous formation of couples of opposite-sign vortices in von Kármán wakes. The structure of these couples, their behaviour and their role in turbulent flows is then studied using both a numerical simulation and laboratory results.

In a previous article we introduced a dissipative circular geometry in which stationary states of the shear flow instability were obtained. We show here that the dynamical behaviour of this flow depends strongly on the aspect ratio of the cell. In large cells, where the number of vortices is large, transitions from a mode with m vortices to a mode with (m−1) vortices occur through localized processes. In contrast to that situation, in small cells, transition takes place after a series of bifurcations which correspond to the successive breaking of all the symmetries of the flow.

We show that, provided an adequate forcing term is introduced, a two-dimensional numerical simulation of this flow is sufficient to recover all the dynamical processes which characterize the experimental flow.

The dynamics of vorticity in two-dimensional turbulence is studied by means of semi-direct numerical simulations, in parallel with passive-scalar dynamics. It is shown that a passive scalar forced and dissipated in the same conditions as vorticity, has a quite different behaviour. The passive scalar obeys the similarity theory à la Kolmogorov, while the enstrophy spectrum is much steeper, owing to a hierarchy of strong coherent vortices. The condensation of vorticity into such vortices depends critically both on the existence of an energy invariant (intimately related to the feedback of vorticity transport on velocity, absent in passive-scalar dynamics, and neglected in the Kolmogorov theory of the enstrophy inertial range); and on the localness of flow dynamics in physical space (again not considered by the Kolmogorov theory, and not accessible to closure model simulations). When space localness is artificially destroyed, the enstrophy spectrum again obeys a k−1 law like a passive scalar. In the wavenumber range accessible to our experiments, two-dimensional turbulence can be described as a hierarchy of strong coherent vortices superimposed on a weak vorticity continuum which behaves like a passive scalar.

The equilibrium spectra of two-dimensional numerical model flows are studied from the viewpoint of microcanonical ensemble averages. The method leads to accurate numerical verification of the ergodic, or mixing, hypothesis in the case of systems constrained to a finite number of degrees of freedom.

Kraichnan's (1967) predictions concerning a simultaneous direct enstrophy cascade and inverse energy cascade for high Reynolds number two-dimensional turbulence are tested numerically using a variant of the eddy-damped quasi-normal approximation. For the initial-value problem, an analytic study using this theory shows that, in the zero-viscosity limit, energy and enstrophy are conserved for arbitrarily long times, contrary to the three-dimensional case, where the energy is conserved for only a finite time, after which it is dissipated. Non-local effects in the enstrophy inertial range, which are difficult to treat by conventional numerical schemes (Leith 1971; Leith & Kraichnan 1972), are shown to be representable by an additional diffusion term in the spectral equation. The resulting equation, including non-local effects, is integrated numerically. When enstrophy and energy are continuously injected at a fixed wavenumber, it is shown numerically that a quasi-steady regime is obtained where enstrophy cascades to large wavenumbers across a k−3 inertial range with zero energy transfer while energy flows indefinitely to small wavenumbers across a $k^{-\frac{5}{3}}$ inertial range with zero enstrophy transfer.

The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws.The mathematical model is a set of six coupled differentialequations governing the time evolution of both the tumour cell population(cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunalenterocyte population, to be shielded from unwanted side effectsduring a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent freeplatinum Pt (the active drug) infusion law being given,the mathematical model allows to compute the time evolution of both tumour andvilli populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicityrhythm. The model has been derived using, as far as possible, experimental data.We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L ∞ criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.

Energy transfers between modes obtained from the proper orthogonal decomposition (POD) of a turbulent flow past a backward-facing step are analysed with the aim of providing guidelines for modelling unresolved modes in truncated POD–Galerkin models. It is observed that energy transfers are local in the POD basis, and that the Fourier-decomposition-based concepts of forward and backward energy cascades are also valid in the POD basis, the net effect being a forward energy cascade. General features of the eddy-viscosity representation of kinetic energy transfers are investigated through a priori tests. It is observed that the ideal eddy-viscosity model should exhibit a cusp behaviour near the cutoff mode.