Aims. The Vishniac instability is thought to explain the complex structure of radiative supernova remnants (SNRs) when a blast wave has propagated from a central explosion.Methods. In this paper, we present numerical studies with the two-dimensional (2D) code HADES. We compare simulations of non-cooling perturbed SNRs, with simulations of perturbed SNRs experiencing radiative losses. In the first case, a low adiabatic index involves a high compression rate that can mimic the effect of radiative losses, whereas a cooling function is used in the second case.Results. The development of the perturbation is analyzed with and without cooling. First, we show that with no cooling but with a low adiabatic index, the perturbation grows in agreement with the theory. Second, although in a first stage the initial Vishniac instability (VI) vanishes for SNR undergoing radiative losses and a large adiabatic index equal to 5/3, simulations show that at a later time a new and growing perturbation appears and the mode l′ of this new perturbation is twice the mode l of the initial one (l′ = 2 × l).Conclusions. Simulating SNR evolutions in similar conditions to theoretical conditions, that is, an adiabatic expansion and adiabatic index lower than 1.2, VI is found to occur in accordance with theoretical predictions. When cooling, instead of a low adiabatic index, which is included in the model, simulations demonstrate that in the late stage of SNR evolution, a doubled mode VI develops even for an adiabatic index equal to 5/3. These two phenomena, VI for high adiabatic index and the mode doubling process, are new and demonstrated in this paper.