## 1 Introduction  
Over past decades, studies on velocity distribution and its accuracy in open-channel flow have been conducted [1], which are useful for numerous research fields, including river restoration, sediment transport, river morphology, power plant design, and hydrometry [1, 2]. The logarithmic trend has been followed by the vertical velocity profile above aquatic vegetation, as shown by various experimental and theoretical studies on open-channel vegetated flows [3, 4, 5]. According to [PERSON] et al. [4], a vegetated layer submerged within the flow increases flow resistance and can be treated as exhibiting a similar effect to a roughness layer. In addition, [PERSON] et al. [6] and [PERSON] [7] proposed water-worked bedforms, which usually occur in natural channels, to be another factor that alters velocity distribution in near-bed flow regions. Overall, the non-vegetated flow velocity profile is determined by the bed's shear stress and physical flow parameters, while drag and friction parameters with respect to vegetation are the key determinants for vegetated flow, i.e., for submerged and emerged cases.  
Excessive sediment transport induced by propeller-equipped big ships is common in harbours and waterways [8, 9]. Studies by [PERSON] et al. [10] and [PERSON] et al. [11] discussed the impact of propeller-induced jets on the velocity profile and bed scouring. According to [PERSON] et al. [12], the characteristics of velocity downstream from jet diffusion are similar to a submerged 3D jet. Besides imitating real-world cases, the bed scouring causedby the 3D propeller-induced flow has also been studied experimentally by [PERSON] et al. [13], and their findings further emphasised the impact of propeller flow, particularly in shallow flow fields. As identified by [PERSON] et al. [10] and [PERSON] et al. [11], the axial velocity along the propeller axis of rotation is the most important component for altering the flow's characteristic as it releases forward thrust to push the ship in the desired direction; hence, more research is required to understand its behaviour in various real-world flow conditions to achieve a functional guideline that can maximise ship operations.  
Up until now, there has been a lack of analytical modelling investigations of the combined flow field from propeller-induced jet and vegetated flow (the combined effect is shown in the schematic diagram of Figure 1). The existing method to investigate this complex engineering problem is by using numerical models, which can be computationally costly. In view of this research field's importance to reduce ship malfunction in shallow flow conditions, this study researched an analytical approach to understand and, hence, mitigate this problem. To the author's knowledge, this study is the first to theoretically combine and study the propeller-induced jet and vegetated flow coherently. To systematically study the mentioned complex flow, this paper will first investigate the flow model with submerged vegetation to obtain a suitable representation of the vegetated velocity profile. After that, the impact of the ship's propeller will be studied and included into the vegetated flow model to achieve a calculation of the combined velocity characteristic of propeller jet-induced vegetated flow. This study will also validate the presented modelling approach through separate vegetated and propeller-induced flow experiments reported in the literature.