Accuracy in Numerical Solution of the Particle Concentration Field in Laminar Wall-Bounded Flows with Thermophoresis and Diffusion

The accuracy of numerical finite-difference solutions to the one-way-coupled Eulerian partial differential equations for particle concentration in the presence of thermophoresis and diffusion is explored at different Schmidt numbers in laminar boundary-layer flow of a hot gas over a cold wall. Crank-Nicolson and MacCormack space-marching solutions to the coupled partial differential equations are compared with essentially exact solutions to the self-similar ordinary differential equation problem to determine the requirements for achieving accuracy in numerical solutions. When the diffusion sublayer at the wall is to be resolved, in flows laden with nanometer particles, the cell "Peclet" number referenced to the thermophoretic velocity and grid spacing in the wall-normal direction, and particle diffusion coefficient, serves as a criterion for the accuracy of space-marching solutions and determines the required number of wall-normal grid points, which is proportional to the particle Schmidt number. This criterion should be a useful guide in computations of other wall-bounded flows with thermophoresis, for which no accuracy criterion exists. When the diffusion sublayer at the wall is too thin to be resolved, as in flows laden with micron-size or larger particles, outer solutions to the particle concentration equation with no Brownian particle diffusion give excellent predictions of both the particle concentration profile and the flux of particles to the wall.