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Turbulent convection is thought to act as an effective viscosity ($ν_E$) in damping tidal flows in stars and giant planets. However, the efficiency of this mechanism has long been debated, particularly in the regime of fast tides, when the tidal frequency ($ω$) exceeds the turnover frequency of the dominant convective eddies ($ω_c$). We present the results of hydrodynamical simulations to study the interaction between tidal flows and convection in a small patch of a convection zone. These simulations build upon our prior work by simulating more turbulent convection in larger horizontal boxes, and here we explore a wider range of parameters. We obtain several new results: 1) $ν_E$ is frequency-dependent, scaling as $ω^{-0.5}$ when $ω/ω_c \lesssim 1$, and appears to attain its maximum constant value only for very small frequencies ($ω/ω_c \lesssim 10^{-2}$). This frequency-reduction for low frequency tidal forcing has never been observed previously. 2) The frequency-dependence of $ν_E$ appears to follow the same scaling as the frequency spectrum of the energy (or Reynolds stress) for low and intermediate frequencies. 3) For high frequencies ($ω/ω_c\gtrsim 1-5$), $ν_E\propto ω^{-2}$. 4) The energetically-dominant convective modes always appear to contribute the most to $ν_E$, rather than the resonant eddies in a Kolmogorov cascade. These results have important implications for tidal dissipation in convection zones of stars and planets, and indicate that the classical tidal theory of the equilibrium tide in stars and giant planets should be revisited. We briefly touch upon the implications for planetary orbital decay around evolving stars.