Modeling turbulent flow over fractal trees using renormalized numerical simulation: Alternate formulations and numerical experiments
Authors: Graham Jason , Meneveau Charles 
Published: December 2012

Simulating turbulent flows over objects characterized by hierarchies of length-scales poses special challenges associated with the cost of resolving small-scale elements. If these are treated as subgrid-scale elements, their effects on the resolved scales must be captured realistically. Most importantly, the associated drag forces must be parameterized. Prior work [S. Chester, C. Meneveau, and M. B. Parlange, “Modeling turbulent flow over fractal trees with renormalized numerical simulation,” J. Comput. Phys. 225, 427–448 (2007)10.1016/j.jcp.2006.12.009] proposed a technique called renormalized numerical simulation (RNS), which is applicable to objects that display scale-invariant geometric (fractal) properties. The idea of RNS is similar to that of the dynamic model used in large eddy simulation to determine model parameters for the subgrid-stress tensor model in the bulk of the flow. In RNS, drag forces from the resolved elements that are obtained during the simulation are re-scaled appropriately by determining drag coefficients that are then applied to specify the drag forces associated with the subgrid-scale elements. The technique has already been applied to model turbulent flow over a canopy of fractal trees [S. Chester, C. Meneveau, and M. B. Parlange, “Modeling turbulent flow over fractal trees with renormalized numerical simulation,” J. Comput. Phys. 225, 427–448 (2007)10.1016/j.jcp.2006.12.009], using a particular set of assumptions in evaluating the drag coefficient. In the current work we introduce a generalized framework for describing and implementing the RNS methodology. Furthermore, we describe various other possible practical implementations of RNS that differ on important, technical aspects related to (1) time averaging, (2) spatial localization, and (3) numerical representation of the drag forces. As part of this study, several RNS formulations are presented and compared. The various models are first implemented and compared in simulations of a canopy consisting of fractal-like trees with planar cross section placed on a periodic lattice. The results indicate that the time averaged, local, and explicit formulation of RNS is superior. The advantages of time averaging can be understood based on the dynamic similarity of the time-averaged, rather than the instantaneous, forces, as well as from numerical stability considerations. Spatial localization is superior since it enables modeling spatially non-homogenous geometries, while the explicit formulation is found superior due to numerical issues. Using the time averaged, local, and explicit RNS formulation, a simulation of flow over a canopy with more complex, non-planar cross section, trees is performed, for which experimental data are available on the overall drag coefficient (Cd ≈ 0.35). The drag force on an entire tree (resolved plus subgrid-scale elements) obtained with RNS is found to be Cd = 0.32, i.e., close to, but with a 8% discrepancy to the measured value. In this flow, the contribution of the subgrid-scale elements to the total drag is dominant, nearly 75%, highlighting the importance of the model.