Counterflow and wall stagnation flow with three-dimensional strain

Abstract

Three-dimensional (3D) viscous counterflows and wall stagnation flows are analyzed with differing normal strain rates in each of the three directions. Reduction of the equations to a similar form is obtained allowing for variations in density due to temperature and composition, heat conduction, and, for the counterflow, mass diffusion and the presence of a flame. Solutions to the Navier-Stokes equations are obtained without the boundary-layer approximation. For the steady and unsteady incompressible counterflows, analytical solutions are obtained for the flow field and the scalar fields subject to heat and mass transfer. In steady, variable-density configurations, a set of ordinary differential equations (ODEs) governs the two transverse velocity and the axial velocity profiles as well as the scalar-field variables. Diffusion rates for mass, momentum, and energy depend on the two normal strain rates parallel to the counterflow interface or the wall and thereby not merely on the sum of those two strain rates. For thin diffusion flames, the location, burning rate, and peak temperature are readily obtained. Solutions for planar flows and axisymmetric flows are obtained as limits here. Results for the velocity and scalar fields are found for a full range of the distribution of normal strain rates between the two transverse directions, various Prandtl number values, and various ambient (or wall) temperatures. For counterflows with flames and stagnation layers with hot walls, velocity overshoots are seen in the viscous layer, yielding an important correction of theories based on a constant-density assumption.