We introduce and validate a novel mathematical model for computing the radial profiles of both axial and circumferential velocity components, respectively, of the swirling flow exiting the runner of hydraulic turbines within the full operating range. We assume an incompressible, inviscid, axisymmetrical, and steady swirling flow, with vanishing radial velocity at runner outlet. First we find the correlation between the flux of moment of momentum downstream the turbine runner and the operating regime given by turbine's discharge and head. Second, we express the relationship between the axial and circumferential velocity components, corresponding to the fixed pitch runner blades, using the swirl-free velocity instead of the traditional relative flow angle at runner outlet. It is shown that the swirl-free velocity profile practically does not change with the operating regime. Third, we introduce a constrained variational problem corresponding to the minimization of the flow force while maintaining the prescribed discharge and flux of moment of momentum. This formulation also accounts for a possible central stagnant region to develop when operating the turbine far from the best efficiency point. Fourth, we show that by representing the unknown axial velocity profile with a suitable Fourier-Bessel series, the discharge constraint can be automatically satisfied. The resulting numerical algorithm is robust and produces results in good agreement with available data for both axial and circumferential velocity profiles measured on a model Francis turbine at several operating regimes. Our mathematical model is suitable for the early optimization stages of the runner design, as it provides the swirling flow configuration at runner outlet without actually computing the runner. By optimizing the parameterized swirl-free velocity profile one can achieve through the inverse design approaches the most suitable runner blades configuration at the trailing edge. (C) 2011 Elsevier Inc. All rights reserved.