Direct numerical simulations of the κ-mechanism -: I.: Radial modes in the purely radiative case

Context. We present a purely-radiative hydrodynamical model of the kappa-mechanism that sustains radial oscillations in Cepheid variables. Aims. We determine the physical conditions favourable for the kappa-mechanism to occur inside a layer, with a configurable conductivity-hollow. We complete nonlinear direct numerical simulations ( DNS) that initiate from these most favourable conditions. Methods. We compare the results of a linear-stability analysis, applied to radial modes using a spectral solver, and a DNS, which is developed from a high-order finite difference code. Results. We find that by changing the location and shape of the hollow, we can generate well-defined instability strips. For a given position in the layer, the amplitude and width of the hollow appear to be key parameters to vary to attain unstable modes driven by the kappa-mechanism. The DNS, starting from the favourable conditions, confirm both the growth rates and the structures of linearly-unstable modes. Nonlinear saturation is produced by intricate couplings between excited fundamental mode and higher damped overtones.