Motivated by cancer cell capture and sorting applications, we investigate the scalar velocity field of two-dimensional (2D) laminar flows through configurations of fixed congruent circular disks with a disordered hyperuniform (HU) distribution of the disk centers. Disordered HU many-particle systems suppress large-scale density fluctuations like perfect crystals and yet possess no Bragg peaks, and are endowed with many novel physical properties. Here, we generate 2D HU configurations of congruent nonoverlapping (hard) circular disks with various packing densities via Monte Carlo simulations and obtain the corresponding scalar velocity fields of laminar flows via the lattice Boltzmann method. Using numerical spectral analysis, we find that hyperuniform configurations of the disks can lead to hyperuniform flow fields for a certain range of disk packing fraction φ and normalized flow rate q over porosity ϵ=1-φ. The degree of hyperuniformity of the flow fields, as quantified via the spectral density value at the zero-wave-number limit, reduces as φ and q increase and eventually the fields become nonhyperuniform. The transition from HU flow fields to non-HU fields as φ and q vary is captured using a "phase diagram." For purpose of comparison, disordered nonhyperuniform configurations of circular disks are generated via the random sequential addition process and the resulting flow field is found to be nonhyperuniform. To complement our numerical analysis, we show analytically that HU configurations of circular disks, in the dilute (low-packing-density) limit, lead to HU flow field. Our findings have implications in the design and optimization of microfluidic chips for capture and differentiation of subpopulations in circulating tumor cells.