A Useful Approximation to the Error Function: Applications to Mass, Momentum, and Energy Transport in Shear Layers

Abstract

Specific mass, momentum, and energy flux transport integrals are given by the sequence I1 = ∫erfc(x)dx, I2 = ∫erfc2 (x)dx, and I3 = ∫erfc3 (x)dx for the error function velocity distribution typical of some laminar and turbulent shear layers. In this report, the Gaussian function A exp(−b(x−x0 )2 ) is used to approximate the error function within 0.21 percent, allowing a direct approximate closed-form evaluation of these transport flux integrals. Mass, momentum, and energy flux are accurate to within 0.22, 0.15, and 0.11 percent, respectively, over the entire shear layer. To achieve this same degree of accuracy with a Taylor series requires in excess of 10 terms. Each of the sequence of approximate functions can be inverted to yield the inverse function, i.e., erfc−1 (x), etc., also in closed-form. The results presented here are also applicable to the error function as it appears in heat transfer and probability and statistics type problems. A coefficient table is included to allow evaluation of the error function and its various integrals.