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The effects of the heliospheric current sheet (HCS) on the evolution of Alfvénic turbulence in the solar wind are studied using MHD simulations incorporating the expanding-box model. The simulations show that, near the HCS, the Alfvénicity of the turbulence decreases as manifested by lower normalized cross-helicity and larger excess of magnetic energy. The numerical results are supported by a superposed-epoch analysis using OMNI data, which shows that the normalized cross-helicity decreases inside the plasma sheet surrounding HCS, and the excess of magnetic energy is significantly enhanced at the center of HCS. Our simulation results indicate that the decrease of Alfvénicity around the HCS is due to the weakening of radial magnetic field and the effects of the transverse gradient in the background magnetic field. The magnetic energy excess in the turbulence may be a result of the loss of Alfvénic correlation between velocity and magnetic field and the faster decay of transverse kinetic energy with respect to magnetic energy in a spherically expanding solar wind.