The von Kármán (1921) rotating disk problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution. Exact representations of the non-steady velocity field and pressure are given by suitable power-series expansions in the angle of rotation, Ωt, with coefficients that are functions of a similarity variable. The first four equations for velocity coefficient functions are solved exactly in closed form, and the next six by numerical integration. This gives four terms in the series for the primary flow and three terms in each series for the secondary flow.The results indicate that the asymptotic steady state is approached after about 2 radians of the disk's motion and that it can be approximately obtained from the initial-value, time-dependent analysis. Furthermore, the non-steady flow has three phases, the first two of which are accurately and fully described with the terms computed. During the first-half radian (phase 1), the velocity field is essentially similar in time, with boundary-layer thickening the only significant effect. For 0·5 [lsim ] Ωt [lsim ] 1·5 (phase 2), boundary-layer growth continues at a slower rate, but simultaneously the velocity profiles adjust towards the shape of the ultimate steady-state profiles. At about Ωt = 1·5, some flow quantities overshoot the steady-state values by small amounts. In analogy with the ‘Greenspan-Howard problem’ (1963) it is believed that the third phase (Ωt > 1·5) consists of a small amplitude decaying oscillation about the steady-state solution.
Abstract.The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.Introduction and survey of literature. The quasilinear parabolic equation now known as the "one-dimensional Burgers equation,"first appeared in a paper by Bateman [4], who derived two of the essentially steady solutions (1.3 and 1.5 of our Table). It is a special case of some mathematical models of turbulence introduced about thirty years ago by J. M. Burgers [10], [11]. The distinctive feature of (1) is that it is the simplest mathematical formulation of the competition between convection and diffusion. It thus offers a relatively convenient means of studying not only turbulence but also the distortion caused by laminar transport of momentum in an otherwise symmetric disturbance and the decay of dissipation layers formed thereby. Moreover, the transformation u = -(2v/6)(dd/dx)relates u{x, t) and d(x, t) so that if 6 is a solution of the linear diffusion equationthen u is a solution of the quasilinear Burgers equation (1). Conversely, if u is a solution of (1) then 9 from (2) is a solution of (3), apart from an arbitrary time-dependent multiplicative factor which is irrelevant in (2).In connection with the Burgers equation, transformation (2) appears first in a technical report by Lagerstrom, Cole, and Trilling [38, especially Appendix B], and was published by Cole [21]. At about the same time it was discovered independently by Hopf [30] and also-in the context of the similarity solution u = t~l/2S(z), z = (4vt)~1/2x-by Burgers [14, p. 250]. The similarity form of the Burgers equation-the quasilinear ordinary differential equation for S(z)-is a Riccati equation [51], and can thus be regarded as a basis for motivating transformation (2) inasmuch as (2) is a standard means of linearizing the Riccati equation. More general hydrodynamical applications of this transformation have been discussed by Ames [1, chapter 2], Chu [20], and Shvets and Meleshko [55].
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