Problem 5.19: Spark ignited spherical combustion
A combustible mixture of air and fuel is initially at rest at a density r1 and uniform pressure p∞. At t = 0, the mixture is ignited at the origin by a spark and a flame front begins to move radially outward from the origin at a constant speed Vf. As the flame front overtakes a gas particle, it converts the combustible mixture of density r1 to a hotter combustion product of lower density r2. This occurs so fast that the flame front can be modeled as an infinitesimally thin density discontinuity. Because the volume of a fluid particle increases as it is enveloped by the flame front, the combustible mixture ahead of the expanding flame front is pushed radially outward by the front. The combustion products of density r2 that are left behind the flame must, however, be stationary relative to the reference frame fixed in the origin.
(a) In a reference frame fixed in the origin, determine the radial outflow velocity v(r,t) of the combustible mixture. There will be different expressions for the unburned gas ahead of the flame front (r > Vft) and the burned gas left behind (r < Vft). See also Prob. 3.17. Show that the velocity distribution can be expressed in the dimensionless form , where is either a dimensionless radial distance at fixed time or a dimensionless inverse time at fixed r, and is a dimensionless density difference between the unburned and burned gases.