Its effect is often obscured by that of the other factors mentioned above. However, since 

 the amplification of small disturbances does provide a definite mechanism of transition 

 in available experiments and since it is an inherent instability for which definite mathe- 

 matical theories can be developed, it is an important phase of the study of the mecha- 

 nism of transition. Indiscriminant application of the theory to practical problems should, 

 however, be avoided. 



From the theoretical point of view, the study of the stability of simple exact 

 solutions, such as the Couette flow and the plane Poiseuille flow, has merits over the 

 discussion of the boundary layer. (Indeed, Taylor [1] questioned Tollmien's theory [2] 

 of the instability of the laminar boundary layer, because the growth of its thickness in 

 the direction of the flow is neglected with only a mathematical justification.) Fortu- 

 nately, the two cases of exact solutions mentioned above are indeed typical cases illus- 

 trating the two basic types of instability. As is well known, the theory developed for 

 such flows can be — and has been — applied to study the stability of boundary layers. 



The first successful investigation of the instability of laminar flow was Taylor's 

 work on the motion between rotating cylinders [3]. He obtained experimental confirma- 

 tion of his theoretical calculations without great difficulty. The cause of instability is 

 also not hard to find. Indeed, when the inner cylinder is rotating and the outer cylinder 

 is at rest, the centrifugal forces tend to produce a secondary flow. The effect of viscosity 

 in this case is then to produce the unstable basic flow, and to provide some damping 

 effect on the secondary motion. On the other hand, when the outer cylinder is rotating 

 and the inner one is at rest, the flow is expected to be stable. 



A similar situation my be expected to hold in the case of the boundary layer 

 over a concave wall. Here, the fast moving particles are closer to the center of curva- 

 ture and the flow is expected to be unstable when the viscosity coefficient is too small 

 to provide sufficient damping. The instability of such flows was first predicted by 

 Gortler, [4] and Liepmann's experiments [5] generally support his conclusions. 



The more interesting case is that of parallel flows. Here, one is concerned with 

 wavy disturbances. Lord Rayleigh predicted the instability of the flow when the velocity 

 profile has a point of inflection. There has been ample evidence to support the theory. 

 However, in the case of flow through a channel, where the curvature of the profile is 

 all of the same sign, Heisenberg [6] also predicted instability of the flow caused by the 

 effect of viscosity. The theory was criticized by Noether [7] on mathematical grounds. 

 To some of the mathematical issues, we shall return in Section 4. Tollmien [2] studied 

 the case of the boundary layer and made the theory more convincing. But general 

 acceptance of the theory came only after the calculations of Tollmien and Schlichting 

 [8] were largely verified by the experimental measurements of Schubauer and Skram- 

 stad [9]. Figure 1 shows the comparison between theory and experiment. The dashed 

 curves were obtained by Schlichting following Tollmien's approach. The solid curves 

 were obtained by Shen [10] by using a method developed by Lin [11] on the basis of 

 Heisenberg's approach. It is seen that the latter results agree better with experiments. 

 Substantially the same neutral curve was earlier obtained by Tollmien and by Lin. 



What is the mechanism of the instability of parallel flows? When there is a point 

 of inflection in the velocity profile, the cause may be traced to a mechanism of vorticity 

 redistribution in a parallel flow. There is a maximum magnitude of vorticity at the 

 point of inflection. If we imagine a disturbance giving an exchange of particles on the 

 two sides of the maximum (remembering that such an exchange must be made without 

 change of the vorticity in a perfect fluid), we see that a small disturbance can exist 

 without an essential change of the vorticity distribution of the main flow. Thus, there 

 is no "restoring force" to resist the disturbance. On the other hand, with a monotone 

 increase or decrease of vorticity, an exchange of two fluid elements results in an excess 

 and a defect of vorticity; and it is known that such elements will tend to migrate back 

 to the layer where it came from. Thus, there is stability of the basic motion. 



Instability of a parallel flow without a point of inflection in the velocity profile 



354 



