Turbulent Flows Generated by Multi-scale Structures - from a Fractal Tree to a Fractal Canopy

The vegetation canopy has a significant impact on various physical and biological processes. Most scaled laboratory experimental studies have used canopy element models that consist of rigid vertical strips or cylindrical rods that can be typically represented through only one or a few characteristic length scales while, most natural vegetation is highly multi-scale with branches and sub-branches, covering a wide range of length scales. Since fractals provide a convenient idealization of multi-scale objects. We study the flows generated by multi-scale, fractal structures. We first present results of the momentum transport and energy flux across scales in the near-wake flow of a single fractal tree-like object. Detailed mean velocity and turbulence stress profiles are documented, as well as their downstream development. The transverse momentum fluxes can be quantified well by the Boussinesq eddy-viscosity model and associated Prandtl mixing length-scale. We find that the measured mixing length increases with increasing streamwise locations. Conversely, the measured eddy viscosity and mixing length decrease with increasing elevation, which differs from eddy viscosity and mixing length behaviors of traditional boundary layers or canopies studied before. In order to find an appropriate length for the flow, several models are proposed, based on the notion of superposition of scales. These proposed models agree well with the measured mixing length. The subgrid-scale kinetic energy fluxes, of relevance to Large Eddy Simulations, are obtained by spatially filtering the velocity field at various scales. The subgrid-scale fluxes exhibit strong scale-dependence which is different from the behavior in canonical flows. Using the data, we measure the scale-dependent injection of kinetic energy and find that the length-scale associated with momentum transport identified before successfully collapses the measured trends of kinetic energy injection. The results confirm that information about multi-scale clustering of branches as it occurs in fractals has to be incorporated into subgrid-scale models of flows through canopies with multiple scales. Beyond studies of flows behind a single fractal tree, we also report the turbulence inside and above an entire canopy consisting of an array of fractal tree models. An index-matching technique is utilized based on matching the refractive indices of the tree elements and the working fluid. Such index matching provides optical paths for both light illuminations and camera recordings. As a result, the flow fields between each individual branch can be well resolved, which in prior work was very challenging or impossible due to optical interference. We observe distinct spatial partitioning of flows into wake and non-wake regions. The local velocities deviate significantly from spatially averaged values and the streamwise normal dispersive flux inside the canopy has larger magnitude (up to four times) than the corresponding Reynolds normal stress. The eddy viscosity and mixing length are deduced from the data and the mixing length can be described by models proposed for flows behind a single tree. Above the fractal canopy, the flow structures are quite different from that in typical boundary layer flows. From the instantaneous velocity field, we observe jet-like motions which are directly ejected from the region inside the canopy. Based on the linear stochastic estimation and the lead-lag correlation from two-point correlation, we identify the structure inclination angle of about 36° which is much steeper than the angle in typical boundary layer flows. We find that close to the canopy, in addition to large scale structures, the flows contain small-scale coherent structures which could contribute, at small scale, to momentum transport. A physical space filtering technique is then applied to the velocity field to study such structures