## 1 Introduction  
The exponential growth of e-commerce and the rise of on-demand services have placed increasing pressure on urban logistics systems, particularly in the last-mile segment [1, 2]. Traditional delivery vehicles contribute significantly to road congestion, noise, and environmental pollution [3, 4], prompting a growing interest in alternative delivery solutions. In this context, unmanned aerial vehicles (UAVs) have emerged as a promising approach to reduce delivery times, bypass road traffic, and lower the carbon footprint of parcel distribution in dense urban areas [5, 6]. Despite their advantages, UAV-based delivery systems face major challenges, especially regarding their limited energy autonomy, which restricts their delivery range and operational flexibility [7, 8]. In response, hybrid approaches have been explored that couple UAVs with ground vehicles to extend their effective range. A particularly promising concept involves integrating drones with public transportation systems (PTSs) such as buses or trains [9; 10] enabling drones to \"ride\" on these vehicles and deploy near delivery locations, thereby reducing the need for long-range flights and eliminating frequent returns to depots. However, existing works on UAV-PTS integration tend to focus primarily on routing strategies or high-level delivery schemes [10; 11; 12] while neglecting the accurate modeling of energy consumption. Most rely on idealized assumptions such as constant power draw, negligible drag forces, or perfect flight trajectories [8; 13]. These simplifications lead to significant errors in battery sizing and mission feasibility assessment, especially in dynamic and constrained urban environments. This leads to the core scientific challenge addressed in this paper: \"How can we accurately predict the energy consumption of UAVs operating in realistic multimodal delivery systems, accounting for physical constraints and environmental disturbances, to enable robust battery sizing and mission planning?\" [7]. To address this gap, we propose a physically-grounded, scenario-aware energy modeling framework tailored for UAVs operating in conjunction with public transport networks [12; 14]. Our model combines Newtonian dynamics, aerodynamic forces, and mission constraints with computational fluid dynamics (CFD) simulations to correct for the influence of surrounding structures and moving vehicles on drone thrust [7; 15]. We define four delivery scenarios of increasing complexity, including ideal flight, delivery time pressure, thrust disturbances due to urban obstacles, and early return maneuvers to synchronize with PTS schedules. Our method is applied to a real-world case study using a graph-based digital twin of the bus network of Belfort, France, with UAV and bus data provided by the public transport operator. Results show that failing to consider delivery constraints and aerodynamic disturbances can lead to energy underestimation of up to 16%, which is critical for energy-aware mission planning and fleet reliability. The contributions of this work are as follows:  
* We develop a complete and transferable energy estimation methodology for UAV-based parcel delivery systems integrated with public transport networks. * We integrate CFD-derived corrections into the physical model to account for real-world aerodynamic effects. * We define and analyze four mission scenarios, each reflecting common operational constraints in urban delivery. * We validate the approach on a real urban network, showing its applicability for battery sizing and sustainable aerial delivery planning. The rest of this paper is organized as follows. Section 2 reviews the related literature on last-mile delivery, UAV-vehicle integration, and energy consumption modeling. Section 3 describes the system architecture and modeling assumptions. Section 4 presents the energy estimation framework, including dynamic modeling and CFD integration. Section 5 introduces the experimental setup and simulation parameters.