The distribution of the heat flux on the walls in a closed cylinder during the compression of a non-reactive hydrogen-air mixture is investigated using data from a direct numerical simulation (DNS) of the compression stroke under engine relevant conditions. It is found that the relation of the temperature field with the local heat flux distribution depends strongly on the distance from the wall. The strong correlation of the two quantities within the viscous sublayer deteriorates with increasing distance from the wall; in the outer layer they become uncorrelated. In contrast the flow in the wall-normal direction transports hot gases towards and cold gases away from the wall and the velocity in the wall-normal direction is correlated to the heat flux distribution also further away from the wall. A local flow away from the wall results in a lower and more uniform heat flux while flow towards the wall result in significantly higher and strongly fluctuating heat fluxes. Ejection streams localized in the near-wall region are found to be responsible for the highly fluctuating heat flux distributions. The joint PDF distributions of wall-normal velocity and temperature with the heat flux collapse when scaled by density-weighted wall-normal units, which can be used to model the heat flux distribution within coarser RANS and LES cells and may therefore provide a promising basis for future engine wall heat transfer models. (C) 2015 Elsevier Ltd. All rights reserved.