To better understand the dynamics of Kelvin–Helmholtz instabilities in environmental flows, their evolution is investigated using direct numerical simulations (DNS). Two-dimensional DNS is used to examine the large-scale and small-scale structures of the instability at high Reynolds and Prandtl numbers that represent real environmental flows. The semi-analytical model of Corcos and Sherman (J Fluid Mech 73:241–264, 1976) is used to explain the physics of these simulations prior to saturation of the KH billow, and also provide a computationally efficient prediction of the vortex dynamics of the instability. The DNS results show that the large-scale structure of the billow does not depend on the Reynolds number for sufficiently high Reynolds numbers. The billow structure reveals a less straightforward dependence on the Prandtl number. Predictions of the model of Corcos and Sherman (J Fluid Mech 73:241–264, 1976) improve as Reynolds number and Prandtl number increase. The small-scale structure of the vorticity and density fields vary with both Reynolds and Prandtl numbers. Three-dimensional DNS of KH flows and their transition to turbulence are used to study small length scales. Based on the thickness of the braid, a simple method is introduced to estimate the Batchelor scale, which can be used as a guide for the resolution required for the direct numerical simulation of two and three-dimensional Kelvin–Helmholtz flow fields.