Should any member of the fire department die and not leave a beneficiary entitled to a pension under this article, such funds as the member shall have paid into the fire department retirement system at the time of member's death shall be paid to such person as the member shall have nominated by written designation duly executed and filed with the board. In default of the designation of the beneficiary, all payments made by a member and on the member's behalf by the city shall accrue to the fire department retirement system. (Code 1979, § 15-135)