"second year of" is correct and usable in written English.
You could use it, for example, to describe something that happened in a certain year: "She began her second year of college in the fall."
She's only in her second year of coursework.
His charity, water.org, is in its second year of operation.
This was my second year of teaching the course.
This is the second year of the two-year experiment.
My grandmother died in the second year of the famine.
The first and second year of the BSc.
Fruit quality characteristics were affected by nutrient rates only after the second year of fertilizer application.
Bimatoprost sustained an excellent safety profile during the second year of treatment.
In Vietnam, however, no recommendations for pertussis boosters beyond the second year of life exist.
This paper reports results after the second year of this ongoing project, and has four objectives.
The analysis showed that a significant variation between clones begins in the second year of growth.
