Mathematical tools based on algebraic topology (homology) provide new ways to describe complex patterns of fluid flow observed in laboratory experiments. First, we will discuss how homology can be used to quantify non-Oberbeck-Boussinesq (NOB) effects in weakly turbulent Rayleigh-Benard convection patterns. We then will describe homology-based methods to measure dynamical finite-size effects in spatiotemporally-chaotic convective flows. Finally, we will outline work in progress in which topological characteristics of fluid flows from both lab experiments and corresponding numerical simulations are encoded in persistence diagrams.