Modeling turbulent flow over fractal trees with renormalized numerical simulation

High-Reynolds number flow over tree-like fractals is considered, with emphasis on the drag forces produced. Fractal objects display large scale-disparity and complexity while being amenable to a simple and standardized description. Hence, they offer an elegant idealization of the actual boundaries in practical applications where turbulence interacts with boundaries that are characterized by multiple length-scales. First, using large-eddy-simulation of flow over prefractal shapes with increasing numbers of branch generations, the dependence of the tree drag on the inner cutoff-scale of the fractal is studied. It is found that the convergence of the drag coefficient towards a value that is independent of inner cutoff-scale is very slow. In order to address this fundamental difficulty and avoid the need to resolve all the small-scale branches of the fractal, a new numerical modeling technique called renormalized numerical simulation (RNS) is introduced. RNS models the drag of the unresolved branches using drag coefficients measured from both resolved branches and unresolved branches as modeled in previous iterations of the procedure. The RNS technique and its convergence properties are tested by means of a series of simulations using different levels of resolution. Then, RNS is used to investigate the influence of the tree fractal dimension on the drag coefficient. The increase of the drag with fractal dimension is quantified for two types of tree geometry, in two flow configurations. Results illustrate that RNS enables numerical modeling of physical processes associated with fractal geometries using affordable computational resolution.