The role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov-Zakharov spectrum

The Weak Turbulence Theory has been applied to waves in thin elastic plates obeying the F\"oppl-Von K\'arm\'an dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the Weak Turbulence Theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the F\"oppl-Von K\'arm\'an equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves $\gamma_\mathbf{k}= a + bk^2$ is used. This confirms that the F\"oppl-Von K\'arm\'an equations are a valid theoretical framework to describe our experiments. When we progressively tune the dissipation so that to localize it at the smallest scales, we observe a gradual transition between the experimental spectrum and the Kolmogorov-Zakharov prediction. Thus it is shown dissipation has a major influence on the scaling properties stationary solutions of weakly non linear wave turbulence.