Summary: We analyze the properties of solutions to initial-boundary value problems for the time-dependent Navier-Stokes equations with periodic rapidly oscillating data having a zero mean. These problems are stated in bounded domains, for example, in three-dimensional ones. The oscillation period of data is determined by a small positive parameter \(\varepsilon\). The viscosity \(\nu\) in the equations can also be treated as a parameter. We present estimates for solutions to such problems that depend on the ratios of some powers of \(\varepsilon\) and \(\nu\). In the general case, the estimates presented hold when \(\nu\) is not too small compared with \(\varepsilon^2\). Under this assumption, the solutions to the problems are asymptotically small in the energy norm, which characterizes the smoothing property of the solutions. When the viscosity has the order of \(\varepsilon^2\), we can derive suitable estimates only under the assumption that the nonlinearity in the equations of the problems is small. Under these conditions, the asymptotics of the velocity may contain rapidly oscillating terms.