## 1 Introduction  
Although modern turbomachinery design practice counts on CFD techniques to develop three-dimensional blade sections, simple methods with an empirical basis are still needed for the mean-line design. This preliminary design phase is the starting point and should provide a solid foundation for the final design, or else adding more complex flow analysis techniques would not contribute to a good result [1]. Furthermore, the empiric loss relations are used before the full 3D check, in the so-called axisymmetric through-flow calculation, usually run in the inverse mode in order to generate blade profiles from a prescribed pressure distribution [2; 3].  
The axial turbine performance simulation at a preliminary design phase is traditionally achieved from the determination of flow velocity diagrams. This task involves the use of aero-thermal relations as well the assessment of a flow loss model. At this phase the performance estimation shall not be dependent on detailed knowledge of blade profile, but only on the upstream and downstream velocity vectors and on some few geometrical parameters, such as the cascade solidity, the mean-line curvature length to chord ratio, the aspect ratio and the blade flare angle. The velocity vectors are then calculated at stations located upstream and downstream of each blade row, where the flow is supposed to be circumferentially uniform. For a known upstream velocity, the downstream velocity is determined taking into consideration the amount of thermal energy converted into kinetic energy, the mass conservation through a variable section annulus and the enthalpy or kinetic energy loss coefficient for that specific blade row [4]. On the other hand, the inlet to exit cascade density ratio will depend on the velocities, as given by the aero-thermal relations, making the velocity diagram problem an intricately coupled system of equations that need to be solved by an iterative procedure.  
The velocity diagrams are usually evaluated at a characteristic radius, which depicts the mean conditions found along the blade span. The mean radius is often used, but an alternative choice is that which divides the annulus section into two equal-area halves. If the blade span is high enough, it may be necessary to extend this analysis to a number of different positions, at least the hub, mean and tip radius. In fact, the circumferential flow needs an inward pressure force to be sustained, so that a radial pressure gradient is built. In this case, a radial equilibrium equation shall be included and a swirl distribution exponent is used to represent a free or forced vortex design [5]. In this situation the velocity components at different radial positions will be related each other, further increasing the problem complexity.  
The flow loss through the axial blade cascade is investigated in this work, from the wind tunnel cascade tests available in the literature. An accurate performance prediction is obtained from the presented method, taking into account the fluid dynamic grounding and the experimental data substantiation. A further motivation is to have an analytical method prone to computational implementation, in opposition to the traditional methods from purely empirical nature and graphical presentation. The traditional methods used as references are those attributed to [PERSON] [6], [PERSON] [7] and their later modifications [8; 9], [PERSON] [10] and [PERSON] [11].