In spite of the ubiquity of free-surface flows across both nature and industry, theoretical and numerical modelling remains challenging. Continuum flows, whether Newtonian or Non-Newtonian, are governed by the Navier–Stokes equations, whilst the particle-particle interactions within a granular flow can be modelled by simple collision mechanics. However, the computational effort required to solve these equations can often render numerical modelling intractable as modellers must either resolve all scales of the velocity field or model every particle-particle interaction. This issue is compounded by the fact that in many cases, modellers are not interested in the fine detail of the flow, but instead wish to study the bulk features of the flow. These bulk features include: the development of a free-surface due to topographic changes or the spatial distribution of the coarse-grained stress field within a granular flow, to name the two examples explored in this thesis. A common approach to model bulk features, without resolving the smallest scale features, is to perform a filtering operation on the governing Navier–Stokes equations or equations of collision mechanics. Unfortunately, the resulting equations often rely on a heuristic closure model to capture the interaction between the sub-filter scale features and the bulk features. In this thesis we explore two methods for deriving closure laws for filtered models – information entropy/energy dissipation maximisation and rheological inference – and apply these to two free-surface flows. In our first problem, we theoretically and numerically examine the interaction between a Blasius boundary layer and free-surface within a shallow Newtonian fluid. Depth-averaging the Navier Stokes equations yields an infinite system of equations that describes how the shape-factors (depth-wise moments of the stream-wise velocity profile) of the flow evolve. We apply an entropy maximisation method that: (i) produces a first-order accurate closure model for our equations; (ii) predicts analytical, steady-state solutions to the free-surface flow and (iii) provides valuable, new insight into the relationship between the rheology and information entropy of a flow. In our second problem, we experimentally and numerically investigate the stress/deformation-rate relationship (granular rheology) in the statistically steady flow of a refractive-index-matched, granular suspension within a rotating drum. Coarse-graining the governing equations yields the Cauchy momentum equations, which describe the relationship between coarse-grained velocity, pressure and stress. We develop and apply an inference algorithm to estimate the latent coarse-grained stress and pressure fields from granular velocimetry data and examine the relationship between inferred stress and shear rate within the context of existing rheology models. We demonstrate that our inferred data follows many of the previously observed stress/shear-rate trends and discuss where our data deviates away from theory. Finally, we posit how our method could be used to develop and validate more accurate models of granular rheology.