Scaling laws for optimal turbulent flow in tree-like networks with smooth and rough tubes and power-law fluids

Abstract

In this paper, we establish optimized scaling laws for fully-developed turbulent flows of incompressible power-law fluids with index n within tree-like branching networks. Our analysis considers turbulence in both smooth and rough channels while accounting for constraints on the network’s volume and surface area. To characterize flow conditions, we introduce a dimensionless conductance parameter, E, and examine its dependence on the diameter ratio \(\beta \), branch splitting N, length ratio \(\gamma \), and the number of branching generations m. The results indicate that E decreases as \(\gamma \), N, and m increase, underscoring the impact of these factors on flow conductance. Furthermore, under volume constraints, we determine the optimal conditions for flow in both smooth and rough tube fractal networks, leading to distinct scaling relationships as \(\displaystyle \frac{D_{k+1}}{D_{k}} = \beta ^* = N^{-(10n+1)/(24n+3)} \), and \(\displaystyle \frac{D_{k+1}}{D_{k}} = N^{-3/7} \) (or \(\dot{m}_k \propto D_k^{(24n+3)/(10n+1)}\) and \(\dot{m}_k \propto D_k^{7/3}\) ), respectively, where \( D_k \) represents the tube diameter and \( \dot{m}_k \) denotes the mass flow rate at the \( k \)th branching level. Interestingly, in rough tube networks, the scaling behavior remains unaffected by the fluid index \( n \), whereas in smooth tube networks, it exhibits a dependence on \( n \). Likewise, when constrained by surface area, the optimal flow conditions vary between smooth and rough tube networks, each following distinct scaling laws as \(\displaystyle \frac{D_{k+1}}{D_{k}} = \beta ^* = N^{-(10n+1)/(21n+2)} \), and \(\displaystyle \frac{D_{k+1}}{D_{k}} = N^{-1/2} \) (or \(\dot{m}_k \propto D_k^{(21n+2)/(10n+1)}\) and \(\dot{m}_k \propto D_k^{2}\) ), respectively. The smooth tube network once again exhibits dependence on the fluid index \( n \). Furthermore, our analysis reveals that in volume-constrained networks, the scaling exponent slope decreases as \( n \) increases, whereas in surface-area-constrained networks, the trend is reversed. We find that the optimal flow condition aligns with the uniform distribution of pressure drop across each branching level within the network. This applies under both constraints for turbulent flow behavior in both smooth and rough-walled tubes. We validated our results against various previous theoretical predictions under limiting conditions. Additionally, this study explores the interplay between geometric and flow properties of parent and daughter tubes in branching networks. Furthermore, we also establish extended scaling-laws under optimal conditions, addressing critical parameters such as length ratios, average flow velocities, tube-volume, and surface-area within the fractal network. In summary, our research significantly extends the relevance of Murray’s Law in understanding and optimizing branching networks under diverse constraints and fluid characteristics. By addressing the interplay between non-Newtonian fluid behaviors and wall properties, our findings offer valuable strategies to enhance the performance and efficiency of engineering systems reliant on fluid dynamics.