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...