<p>The law of the wall (LoW) provides a universal description of the mean velocity profile in wall-bounded flows at sufficiently high Reynolds numbers. However, deviations from the LoW are observed at low Reynolds numbers, often vaguely attributed to the so-called low-Reynolds-number effect. These deviations pose significant challenges for near-wall turbulence modeling, as real-world engineering flows span both low and high Reynolds number regimes. Moreover, the absence of a universal scaling for the mean flow underscores distinct flow physics between these regimes. This study introduces a velocity transformation that aligns velocity profiles at both low and high Reynolds numbers with the LoW, via involving the effect of the variation of the total shear stress. The proposed transformation is validated against a wide range of wall-bounded flows, including channel, pipe, Couette, Poiseuille-Couette, and zero-pressure-gradient boundary layer flows, with the Reynolds number ranges from 100 to 10000. Furthermore, a new near-wall model is developed based on this mean flow transformation, offering improved accuracy over conventional LoW-based models.</p>