An important open question in fluid dynamics concerns the effect of small scales in structuring a fluid flow. In oceanic or atmospheric flows, this is aptly captured in wave–current interactions through the study of the well-known Langmuir secondary circulation. Such wave–current interactions are described by the Craik–Leibovich system, in which the action of a wave-induced velocity, the Stokes drift, produces a so-called “vortex force” that causes streaking in the flow. In this work, we show that these results can be generalized as a generic effect of the spatial inhomogeneity of the statistical properties of the small-scale flow components. As demonstrated, this is well captured through a stochastic representation of the flow.
We present a variational assimilation technique (4D-Var) to reconstruct time resolved incompressible turbulent flows from measurements on two orthogonal 2D planes. The proposed technique incorporates an error term associated to the flow dynamics. It is therefore a compromise between a strong constraint assimilation procedure (for which the dynamical model is assumed to be perfectly known) and a weak constraint variational assimilation which considers a model enriched by an additive Gaussian forcing. The first solution would require either an unaffordable direct numerical simulation (DNS) of the model at the finest scale or an inaccurate and numerically unstable large scale simulation without parametrisation of the unresolved scales. The second option, the weakly constrained assimilation, relies on a blind error model that needs to be estimated from the data. This latter option is also computationally impractical for turbulent flow models as it requires to augment the state variable by an error variable of the same dimension. The proposed 4D-Var algorithm is successfully applied on a 3D turbulent wake flow in the transitional regime without specifying the obstacle geometry. The algorithm is validated on a synthetic 3D data-set with full-scale information. The performance of the algorithm is further analysed on data emulating large-scale experimental PIV observations.
A stochastic representation based on a physical transport principle is proposed to account for mesoscale eddy effects on the evolution of the large-scale flow. This framework arises from a decomposition of the Lagrangian velocity into a smooth in time component and a highly oscillating term. One important characteristic of this random model is that it conserves the energy of any transported scalar. Such an energy-preserving representation is tested for the coarse simulation of a barotropic circulation in a shallow ocean basin, driven by a symmetric double-gyres wind forcing. The empirical spatial correlation of the random small-scale velocity is estimated from data of an eddy-resolving simulation. After reaching a turbulent equilibrium state, a statistical analysis of tracers shows that the proposed random model enables us to reproduce accurately, on a coarse mesh, the local structures of the first four statistical moments (mean, variance, skewness and kurtosis) of the high-resolution eddy-resolved data.
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