Approach to the origin of turbulence on the basis of two-point kinetic theory

Abstract

Equations for the fluctuation correlation in an incompressible shear flow are derived on the basis of kinetic theory, utilizing the two-point distribution function which obeys the BBGKY hierarchy equation truncated with the hypothesis of “ternary” molecular chaos. The step from the molecular to the hydrodynamic description is accomplished by a moment expansion which is a two-point version of the thirteen-moment method, and which leads to a series of correlation equations, viz., the two-point counterparts of the continuity equation, the Navier-Stokes equation, etc. For almost parallel shearing flows the two-point equation is separable and reduces to two Orr-Sommerfeld equations with different physical implications. Solution of an eigenvalue problem for the Blasius boundary layer is obtained in a certain parallelism to the classical stability theory, and is used for predicting the transition Reynolds number of a “quiescent” Blasius flow in which thermodynamic fluctuations alone are the initiating mechanism. Also, the calculated spatial growth rate of fluctuation agrees with the Schubauer-Klebanoff experiment, which gives an account of unexplained experimental evidence that the fluctuation complex (turbulence bursts plus the Tollmien-Schlichting wave), as a whole, obeys a certain linear theory.