Why is there no centripetal acceleration?

A body that moves in a circle has a speed V and an acceleration a. In fact, the speed and acceleration are directed at a certain angle to the circle, however, for the convenience of calculations, it is customary to decompose them into 2 parts: one is directed tangentially to the circle, and the other is directed to the center. Acceleration directed towards the center is called centripetal acceleration, although in fact it is not a separate type of acceleration, but just a projection onto a coordinate system.