Two scaling transformations for the numerical computation of multidimensional unsteady laminar flames

Numerical computations of the unsteady propagation of multidimensional laminar flames are difficult to perform because the structure of the flame must be resolved and the thickness of the flame is often several orders of magnitude smaller than the characteristic dimension of the field. Two scaling transformations are discussed which make possible such computations by reducing the computation time to acceptable levels. The α-transformation is associated with the spatial uniformity of the time-varying pressure and is shown to be valid for low-Mach-number flows. It reduces computational time by either reducing the total number of time steps or, in the RICE code, the number of iterations required each time step. The β-transformation is related to the small thickness of the flame with respect to the characteristic dimension of the field and is shown to be useful for high Reynolds number flows. It reduces computation time by reducing the number of necessary grid points which, in turn, is achieved by broadening the flame front without altering the flame speed. The β-transformation was introduced by T. D. Butler and P. J. O'Rourke (''Proceedings, Sixteenth Symposium on Combustion, August 1976,'' pp. 1403--1516, The Combustion Institute, Pittsburgh, 1977). The validity of the two transformations is shown theoretically and through a series of computer calculations