Renormalization Group Analysis of the Velocity-Dependent Random Force Model of Fluid Flow through Random Media

We study a statistical mechanics model of fluid flow through random media. The effects of randomness in the media are renormalized into a velocity-dependent random force term in the Navier-Stokes equations. The asymptotic properties of the long-wavelength fluctuations are deduced by using the dynamic renormalization group method. Unlike the Foster, Nelson, and Stephen's random force model where the convection term is marginal, the convection term in our model is renormalized to vanish so that the model is well-defined within the context of the ε expansion. A Kolmogorov 5/ 3-energy spectrum is obtained when the force correlation is tuned for the Richardson separation law.