Aims. To form kilometre-sized planetesimals, the streaming instability is an efficient method for overcoming the barriers to planet formation in protoplanetary discs. The streaming instability has been extensively modelled by hydrodynamic simulations of gas and a single dust size. However, recent studies considering a more realistic case of a particle size distribution have shown that this will significantly decrease the growth rate of the instability. We follow up on these studies by evaluating the polydisperse streaming instability in the non-linear regime to see if clumping can occur in the same manner as the monodisperse streaming instability and determine the size distribution in the densest dust structures.

Methods. We employ 2D hydrodynamic simulations in an unstratified shearing box with multiple dust species representing an underlying continuous dust size spectrum using FARGO3D. We use the Gauss-Legendre quadrature in dust size space to calculate the drag force on the gas due to a continuous dust size distribution. These simulations are compared to previous analytical results of the polydisperse streaming instability in the linear phase. We then look at the saturated non-linear phase of the instability at the highest density regions and investigate the dust size distribution in the densest dust structures.

Results. When sampling the size distribution, the error in the growth rate converges significantly faster with the number of dust sizes using the Gauss-Legendre quadrature method than the usual uniform sampling method. In the non-linear regime, the maximum dust density reached in the polydisperse case is reduced compared to the monodisperse case. Larger dust particles are most abundant in the densest dust structure because they are less coupled to the gas and can therefore clump together more than the smaller dust grains. Contrary to expectations based solely on dust-gas coupling, our results reveal a distinct peak in the size distribution that arises from the size-dependent spatial segregation of the highest-density regions, where particles with the largest Stokes numbers are located just outside the densest areas of the combined dust species.

Conclusions. The 2D unstratified polydisperse streaming instability is less efficient than its monodisperse counterpart at generating dense clumps that may collapse into planetesimals, and in the densest regions, the distinct dust size distribution could be related to the size distribution that ends up in the planetesimal and can mimic the size distribution of dust growth.