Analysis of the time scales in time periodic Darcy flows

Abstract:

We investigate unsteady flow in a porous medium under time - periodic (sinusoidal) pressure gradient. DNS were performed to benchmark the analytical solution of the unsteady Darcy equation with two different expressions of the time scale : one given by a consistent volume averaging of the Navier - Stokes equation [1] with a steady state closure for the flow resistance term, another given by volume averaging of the kinetic energy equation [2] with a closure for the dissipation rate . For small and medium frequencies, the analytical solutions with the time scale obtained by the energy approach compare well with the DNS results in terms of amplitude and phase lag. For large frequencies (f > 100 [Hz]) we observe a slightly smaller damping of the amplitude. This study supports the use of the unsteady form of Darcy's equation with constant coefficients to solve time - periodic Darcy flows at low and medium frequencies. Our DNS simulations, however, indicate that the time scale predicted by the VANS approach together with a steady - state closure for the flow resistance term is too small. The one obtained by the energy approach matches the DNS results well. At large frequencies, the amplitudes deviate slightly from the analytical solution of the unsteady Darcy equation. Note that at those high frequencies, the flow amplitudes remain below 1% of those of steady state flow. This result indicates that unsteady porous media flow can approximately be described by the unsteady Darcy equation with constant coefficients for a large range of frequencies, provided, the proper time scale has been found.