Comparison of scale analysis and numerical simulation for saturated zone convective mixing processes

Abstract

Scale analysis can be used to predict a variety of quantities arising from natural systems where processes are described by partial differential equations. For example, scale analysis can be applied to estimate the effectiveness of convective missing on the dilution of contaminants in groundwater. Scale analysis involves substituting simple quotients for partial derivatives and identifying and equating the dominant terms in an order-of-magnitude sense. For free convection due to sidewall heating of saturated porous media, scale analysis shows that vertical convective velocity in the thermal boundary layer region is proportional to the Rayleigh number, horizontal convective velocity is proportional to the square root of the Rayleigh number, and thermal boundary layer thickness is proportional to the inverse square root of the Rayleigh number. These scale analysis estimates are corroborated by numerical simulations of an idealized system. A scale analysis estimate of mixing time for a tracer mixing by hydrodynamic dispersion in a convection cell also agrees well with numerical simulation for two different Rayleigh numbers. Scale analysis for the heating-from-below scenario produces estimates of maximum velocity one-half as large as the sidewall case. At small values of the Rayleigh number, this estimate is confirmed by numerical simulation. For larger Rayleighmore » numbers, simulation results suggest maximum velocities are similar to the sidewall heating scenario. In general, agreement between scale analysis estimates and numerical simulation results serves to validate the method of scale analysis. Application is to radioactive repositories.« less