The shear stress at the bed of a channel influences important benthic processes such as sediment transport. Several methods exist to estimate the bed shear stress in bare channels without vegetation, but most of these are not appropriate for vegetated channels due to the impact of vegetation on the velocity profile and turbulence production. This study proposes a new model to estimate the bed shear stress in both vegetated and bare channels with smooth beds. The model, which is supported by measurements, indicates that for both bare and vegetated channels with smooth beds, within a viscous sublayer at the bed, the viscous stress decreases linearly with increasing distance from the bed, resulting in a parabolic velocity profile at the bed. For bare channels, the model describes the velocity profile in the overlap region of the Law of the Wall. For emergent canopies of sufficient density (frontal area per unit canopy volume a ! 4:3 m 21 ), the thickness of the linear-stress layer is set by the stem diameter, leading to a simple estimate for bed shear stress.

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IntroductionOne dimensional numerical simulation of free-surface and pressurized flows is a useful engineering tool for a wide range of practical applications in civil engineering. The method can be used as long as no 2D and 3D hydraulic effects are predominant and must be thus taken into account. For instance, large rivers networks are often managed and developed by means of 1D models [1,2]. Similarly, simulation of pressurized flow in pipes networks such as water supply or sewer systems relies traditionally on such models [3,4]. Finally, 1D models can be reliably considered in the design process of many hydraulic structures such as water intake, bottom outlet tailrace tunnel, flushing galleries in dams [5]. On account of the large number of practical applications concerned, an efficient prediction of 1D flow features is an obvious need. Developing a unified 1D model for all the situations of interest in civil engineering remains however challenging. Various flow patterns may indeed coexist in actual situations:1. Free surface flows, where supercritical, subcritical and transcritical conditions could co-exist [2], are usually modelled, including the discontinuities (hydraulic jump), on the basis of the conservative Saint-Venant equations [6,7]. 2. Pressurized flows are traditionally described by the water hammer equations [4]. 3. Mixed flows, characterized by a simultaneous occurrence of free-surface and pressurized flow, are still nowadays an issue of research [8][9][10][11] for its mathematical description and its numerical solution. To achieve our purpose to develop a universal solver handling free-surface, pressurized and mixed flow, it is then required 1. to establish a unified mathematical model which overcomes the dissimilarity between the sets of equations describing pressurized and free-surface flows; 2. to set an efficient resolution scheme for this model. As previously mentioned, different mathematical approaches to describe free-surface, pressurized and mixed flow in a unified framework have been developed to date and are still subject to many research. Shock-tracking methods consists in solving separately free-surface and pressurized flows through different sets of equations [12,13]. Rigid Water Column Approach treats each phase separately on the basis of a specific set of equations in focusing on the air behaviour [14]. Nevertheless, such algorithms are very complicated and casespecific. Finally, the so-called shock-capturing approach is a family of method which computes pressurized and free-surface flows by using a single set of equations [8][9][10][11] . In this paper, such an approach is used, based on the model of the Preissmann slot [15]. In particular, this paper focuses on steady state flows which are of great interest for engineers. Design guidelines for many hydraulic structures specify indeed that specific critical steady states have to be addressed. Practitioners should then rely on robust and efficient 1D solvers suitable for each flow pattern (free-surface, pressurized and mixed) in o...

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a b s t r a c tHydraulic models available in literature are unsuccessful in simulating accurately and efficiently environmental flows characterized by the presence of both air-water interactions and free-surface/pressurized transitions (aka mixed flows). The purpose of this paper is thus to fill this knowledge gap by developing a unified one-dimensional mathematical model describing free-surface, pressurized and mixed flows with air-water interactions. This work is part of a general research project which aims at establishing a unified mathematical model suitable to describe the vast majority of flows likely to appear in civil and environmental engineering (pure water flows, sediment transport, pollutant transport, aerated flows. . .). In order to tackle this problem, our original methodology consists in both time-and spaceaveraging the Local Instant Formulation, which includes field equations for each phase taken separately and jump conditions, over a flow cross-section involving a free-surface. Subsequently, applicability of the model is extended to pressurized flows as well. The first key result is an original 1D Homogeneous Equilibrium Model which describes two-phase free-surface flows. It is proven to be fundamentally multiphase, to take into account scale heterogeneities of environmental flow and to be very easy to solve. Next, applicability of this free-surface model is extended to pressurized flows by using the classical Preissmann slot concept. A second key result here is the introduction of an original negative Preissmann slot to simulate sub-atmospheric pressurized flows. The model is then closed by using constitutive equations suitable for air-water flows. Finally, this mathematical model is discretised by means of a finite volume scheme and validated by comparison with experimental results from a physical model in the case of a steady flow in a large scale gallery.