Transition to turbulence

A study of turbulence is incomplete without some mention of °uid stability and the transition to turbulence. We have previously noted that transition occurs when the °uid equilibrium is upset. Although not a topic that will be explored in this class, here is a brief overview of some ideas and methods used in studying °uid stability. The study of hydrodynamics stability is a complete area of study and specialization in itself. The main objectives of this branch of °uid mechanics are to understand the conditions under which a given laminar °ow becomes unstable, and give some information about the subsequent development of the instability. The development of instabilities in a laminar °ow is the ̄rst step towards the transition to turbulence. In most °uid mechanics applications the ability to control the transition would greatly increase engineering e±ciency and performance. For example, to achieve the lowest drag around aerodynamic bodies such as aircraft or cars, it is most desirable to delay transitionAlthough in the case of airfoil design in extreme operating conditions \vortex generators" are sometimes used to promote a turbulent boundary layer on certain areas of the wing. This is because a turbulent boundary layer will not separate as easily as a laminar boundary layer in an adverse pressure gradient. There is a tradeo® here to obtain the best overall performance.. It is obviously important to know when and where transition occurs. In combustion devices, high turbulence levels promote the mixing of fuel and oxidizer. In supersonic combustion devices, there are di±culties encountered in obtaining well mixed reactants, as a result of the stability of these °ows. A better understanding of the transition process is necessary to achieve better combustion performance. Numerous excellent text books are completely devoted to hydrodynamic stability. See for example, Drazin and Reed[1], Chandrsekar[2], or Lin [3], to mention only a few. Here we will brie°y touch upon some of the analytic methods, and the type of information that these analyses yield. From a historical perspective, the most well known experiment on hydrodynamic stability was carried out by Osborne Reynolds in 1888. He performed a set of experiments in which he carefully studied the behavior of °ow in a pipe by varying di®erent °ow conditions. Speci ̄cally, by varying the diameter of the pipe, the velocity of the °uid, and the viscosity of the °uid, Reynolds found that there was a relationship between these variables that indicated the transition from a smooth laminar °ow, to a complex turbulent °ow. Namely, when the value of V D=o (which we now know as the Reynolds number) exceeded a particular value, the perturbations began to grow, and the instantaneous °ow structure became very complex. The transition depended