  Tissue growth underpins a wide array of biological and developmental
processes, and numerical modeling of growing systems has been shown to be a
useful tool for understanding these processes. However, the phenomena that can
be captured are often limited by the size of systems that can be modeled. Here,
we address this limitation by introducing a Lattice-Boltzmann method (LBM) for
a growing system that is able to efficiently model hydrodynamic length-scales.
The model incorporates a novel approach to describing the growing front of a
tissue, which we use to investigate the dynamics of the interface of growing
model tissues. We find that the interface grows with scaling in agreement with
the Kardar-Parisi-Zhang (KPZ) universality class when growth in the system is
bulk driven. Interestingly, we also find the emergence of a previously
unreported hydrodynamic instability when proliferation is restricted to the
tissue edge. We then develop an analytical theory to show that the instability
arises due to a coupling between the number of cells actively proliferating and
the position of the interface.
