The present work focuses on structure functions in homogeneous isotropic turbulence. Structure functions are statistics (more precisely, higher-order moments) of the velocity difference evaluated at two points in space, separated by some distance r. While most of the work found in the literature is based on phenomenology and thus requires additional assumptions besides homogeneity and continuity, the present work aims at examining structure functions based on the Navier-Stokes equations, the governing equations of motion for incompressible fluids. For that reason, firstly the system of structure function equations is discussed and analysed, with emphasis on their dissipative and pressure source terms. It is found that the dissipative source terms and equations derived thereof contain the higher moments of the (pseudo-)dissipation. Next, the viscous range is examined more closely. It is found that there are exact solutions for even-order longitudinal structure functions, which are determined by the higher moments of the dissipation and the viscosity. These findings are then used to define exact order-dependent dissipative cut-off scales, which reduce to the well-known Kolmogorov scales for the second order. Furthermore, the inertial range scaling exponents are examined and contributions by the viscous and unsteady terms to the second-order balances are analysed. Finally, streamline segment statistics are briefly considered, because the higher conditional moments are conceptually similar to the longitudinal structure functions.
