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Wilson-Fisher renormalization of discrete gravity-capillary wave turbulence in viscous fluids

We report an experimental realization of Wilson-Fisher renormalization in driven surface-wave turbulence across Newtonian fluids spanning nearly six decades in Raynolds number. Discrete capillary and gravity turbulence define two universality classes selected by interaction topology: triadic resonances for capillary waves and effectively tetradic scattering for gravity waves. Navier-Stokes viscosity is the relevant perturbation that renormalizes spectral transfer and terminates the cascade. The resulting framework predicts the Kolmogorov cutoff from the balance of nonlinear transfer and viscous damping, and Reynolds scaling of the integrated inertial spectral weight. Laser Doppler Vibrometry quantitatively confirms these renormalized scaling laws, establishing discrete gravity-capillary turbulence as a tunable laboratory for nonequilibrium crossoever criticality.