On the Use of the Squire-Long Equation to Estimate Radial Velocities in Swirling Flows

A method to estimate the radial velocity in swirling flows from experimental values of the axial and tangential velocities is presented. The study is motivated by the experimental difficulties to obtain this component in a draft tube model as evidenced in the Turbine-99 IAHR∕ERCOFTAC Workshop. The method uses a two-dimensional nonviscous description of the flow. Such a flow is described by the Squire-Long equation for the stream function, which depends on the boundary conditions. Experimental values of the axial velocities at the inlet and outlet of the domain are used to obtain the boundary conditions on the bounded domain. The method consists of obtaining the equation related to the domain with an iterative process. The radial velocity profile is then obtained. The method may be applied to flows with a swirl number up to about $Sw=0.25$⁠. The critical value of the swirl number depends on the velocity profiles and the geometry of the domain. The applicability of the methodology is first performed on a swirling flow in a diffuser with a half angle of $3deg$ at various swirl numbers, where three-dimensional (3D) laser Doppler velocimeter (LDV) velocity measurements are available. The method is then applied to the Turbine-99 test case, which consists in a model draft tube flow where the radial inlet velocity was undetermined. The swirl number is equal to $Sw=0.21$⁠. The stability and the convergence of the approach is investigated in this case. The results of the pressure recovery are then compared to the experiments for validation.