Theoretical arguments predict that the distribution of cold dark matter in spiral galaxies has peaks in velocity space associated with nonthermalized flows of dark matter particles. We searched for the corresponding peaks in the spectrum of microwave photons from axion to photon conversion in a cavity detector for dark matter axions. We found none and place limits on the density of any local flow of axions as a function of the flow velocity dispersion over the axion mass range 1.98 to 2.17 microeV.