Rayleigh's problem at low Mach numbers based on kinetic theory.

Abstract

we have detP J^ " detP' Rewriting Eq. (12), = I -KH T P'HKT =/•-/)" (32) (33) ;)Applying Eq. (27) and rearranging terms,From Eq. (12) detK(R (detK) 2 = (36) (37) Combining Eqs. (32, 35, and 37) gives the final result, Eq.CD. SummaryAn expression has been derived [Eq.(1) ] which determines how to update the determinant of the covariance matrix when a measurement is incorporated. For the usual case of the measurement being of smaller dimension than the state, this result offers computational advantages because of the reduction of the order of the determinants to be evaluated. Even greater computational savings are possible for stationary measurement statistics or for one-dimensional measurements. The results found by Cercignani et al. are exact analytical solutions, and the case considered by Chu is nonlinear. The purpose of this paper is twofold: first, to present a solution to the Rayleigh problem; and second, to demonstrate the accuracy and utility of the discrete ordinate method for linearized, time-dependent problems. Since many practical situations will not readily yield analytical solutions, the results presented in this study indicate that the method of discrete ordinates offers a promising tool for the numerical solution of transient problems. This method has been sue-