Stability of a two-dimensional jet

Calculations are presented for the stability of a two-dimensional jet. Since the jet spreads, initially very rapidly, a parallel-flow approximation may not be used. The method of multiple scales is used to account for the flow divergence. The growth of axial velocity fluctuations in the jet is examined as a function of axial and transverse location. Neutral curves are presented on the basis of various definitions of the neutral points. The results are compared with parallel flow calculations and inclusion of the flow divergence is found to predict increased growth rates. Various features of the calculations indicate that the multiple scales technique has only a limited range of validity, and an analysis is presented that determines this range. N spite of the fact that many flows of practical interest, such as boundary layers, jets, and wakes, are nonparallel, linear stability analyses historically have used a quasiparallel mean flow approximation. Although this approximation might at first glance appear reasonable for bounded flows where the critical Reynolds number is high, it is clearly less justifiable for free shear flows where experiments show the flow to be unstable at very low Reynolds numbers where the basic flow is strongly divergent. Various intuitive approaches to account for the effects of flow divergence, e.g., Cheng1 and Ko and Lessen,2 have been adopted. Formal expansions of the linear stability problem about some axial location have been obtained by Lanchon and Eckhaus3 and by Ling and Reynolds.4 In the latter case numerical calculations were performed for the problems of the Blasius boundary layer, the two-dimensional jet, and the two-dimensional flat-plate wake. However, their expansion in powers of e, a small parameter depending on the flow under study, e.g., e = (x0U ^lv)~l/2 for the Blasius boundary layer, wherex0 was some axial location, breaks down when (x—x0) - 0(e ~ } ). Alternative expansion schemes have been introduced by Bouthier,5'6 Gaster,7 Saric and Nayfeh,8 and Eagles and Weissman9 which eliminate these difficulties and describe the cumulative effects of divergence of the flow. The method used in the present work makes use of the method of multiple scales that is described in detail by Nayfeh.10 In the present work the linear stability of a two-dimensional jet flow is examined. A similar analysis has been performed by Garg and Round,11 however, as will be described later in this paper, there are several omissions in their work. It is recognized that since the rate of spread of the jet in the region of instability is relatively high it is questionable whether the use of an expansion scheme that uses the spread rate as the expansion parameter is justified without the inclusion of many terms in the expansion. Thus, it is to be expected that the numerical results obtained in this paper will only provide a qualitative assessment of the effects of flow divergence. However, it should be noted that in another use of the method of multiple scales where the expansion parameter was not very small, Kaiser and Nayfeh12 have shown good agreement between the multiple scales solution and wave-envelope and weighted-residual techniques for sound propagation in nonuniform ducts up to a wall slope of 0.2 with good qualitative agreement up to a slope of 0.4.