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A body moving on a curved path has a constant speed. It has a varying velocity. Velocity varies in direction. So the difference in the vectors of velocities give rise to the acceleration.  As speed V is constant ,  the body turns equal angles Ф in equal intervals of time .  so angular velocity ω is constant.

The change in magnitude of velocity is V2-V1. The magnitude is radius times the angle through which the body turns.  
       V2 - V1 = radius * angle turned = R * Ф = R * ω * time duration

    acceleration = ΔV/Δt = R ω   since R and ω are constant, a is constant in magnitude. 

acceleration varies in direction as it is always pointing towards the center of circular path or  curved path.