We study various cases of nonequilibrium gas flows with violation of the Fourier law of heat transfer, when the heat flux and the temperature gradient have the same signs. The kinetic description of flows far from equilibrium can lead to strong differences from the classical Chapman-Enskog formalism. The relationship of the effect with the second law of thermodynamics is discussed, and the validity of the H-theorem for all considered flows is confirmed. The given anomalous transport is complex and needs to be verified, therefore, calculations based on two numerical methods are compared, namely DMBE (Direct Method for the Boltzmann Equation) and DSMC (Direct Simulation Monte Carlo). Anomalous heat transfer has been found for a number of problems with a characteristic length of nonequilibrium flows on a scale of the order of the mean free path, such as spatial inhomogeneous relaxation, heat conduction between plates with nonequilibrium boundary conditions, flow around plates, and flows through membranes (grids) and others. The aim of this paper is to explore the various variants for anomalous heat transfer, keeping in mind also adequate experimental testing and possible applications of the effects. The production technologies of structured thin materials have reached a high level of development, which in turn leads to new problems. One of these problems is the passage of supersonic flows through membranes (grids) as a result of which nonequilibrium flows with special properties are formed. Behind the membranes, i.e. grid system nonequilibrium flows exhibit anomalous heat transfer properties. Our simulation is focused in particular on flows, which may be of interest given current trends in the development of new membrane technologies. The possible experimental tests are discussed. The proposals are also considered for a microrefregirator system in the nonequilibrium supersonic flow and for a cascade of microrefrigerators based on the anomalous heat between surfaces.