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Motivated by plane wave ultrasound image velocimetry (a.k.a. Echo PIV), we introduce a novel method to globally estimate the velocity field of a laminar and steady flow from the motion of tracer particles. Our approach exploits the fact that the equation of motion for pipe flows of constant circular cross-section is governed by a single global but unknown parameter. We connect this parameter to the Fourier spectrum of the input image sequence of the tracer particles and formulate our motion estimation problem as a spectral support estimation problem. The approach is validated on both simulated and experimental real data. It returns accurate velocity estimates after few seconds runtime and effectively copes with speckle patterns and noise.