A method of inactivating microorganisms such as viruses within a fluid such as a biological fluid is disclosed. The method includes the steps of providing a UV reactor, which may take the form of an elongated generally annular reaction chamber surrounding at least one elongated UV lamp, moving the fluid within the reaction chamber in a primary flow directed along the length of the UV lamp, and inducing a circulating secondary flow within the fluid with the secondary flow being superimposed on the primary flow. As the fluid moves through the reaction chamber in the primary flow, it is circulated repeatedly toward and away from the UV lamp in the circulating secondary flow to provide uniform and controllable exposure of the entire volume of fluid to ultraviolet radiation. Microorganisms such as viruses are thus inactivated while desirable components in the fluid, such as proteins, are preserved without the use of a free radical scavenger.