Dynamical-systems-theory approach to the wall region

Objective analysis of experimental measurements indicates that there are recurrent streamwise rolls present in the wall region, at least in the quadratic mean sense. Representation theorems permit optimal expansion of the instantaneous velocity field in the wall region in terms of these streamwise rolls. Without involving ourselves in the question of the source of these rolls, we ask how they will behave dynamically. Severely truncating our system, and using Galerkin projection, we obtain a closed set of non-linear ordinary differential equations with ten degrees of freedom. The methods of dynamical systems theory are applied to these equations. Loss to unresolved modes is represented by a Heisenberg parameter. We find that for large values of the Heisenberg parameter (large loss), we obtain stable streamwise rolls having the experimentally observed spacing. For smaller values of the parameter, we have traveling waves (corresponding to cross-stream drift of the rolls); we also find a heteroclinic attracting orbit giving rise to intermittency; and finally a chaotic state showing ghosts of all of the above. The intermittent jump from one attracting point to the other resembles in many respects the bursts observed in experiments. Specifically, the time between jumps, and the duration of the jumps, is approximately that observed in a burst; the jump begins with the formation of a narrowed and intensified updraft, like the ejection phase of a burst, and is followed by a gentle, diffuse downdraft, like the sweep phase of a burst. During the jump a spike of Reynolds stress is produced, as is observed in a burst, although the magnitude is limited in our model by the truncation of the high wavenumber components. The behavior is quite robust, much of it being due to the symmetries present (Aubry’s group has examined dimensions up to 128 with persistence of the global behavior). We have examined eigenvalues and coefficients obtained from experiment, and from exact simulation, which differ in magnitude. Similar behavior is obtained in both cases: in the latter case, the heteroclinic orbits connect limit cycles instead of fixed points, corresponding to cross-stream waving of the streamwise rolls. The bifurcation diagram remains structurally similar, but somewhat distorted. The role of the pressure term is made clear - it triggers the intermittent jumps, which otherwise would occur at longer and longer intervals, as the system trajectory is attracted closer and closer to the heteroclinic cycle. The pressure term results in the jumps occurring at essentially random times, and the magnitude of the signal determines the average timing. This clarifies the question of whether bursting scales with wall variables or with outer variables - evidently the structure of a burst scales with wall variables, while the time between bursts should scale in a complex way with both inner and outer variables. Stretching of the wall region shows that the model is consistent with observations of polymer drag reduction. Change of the third order coefficients, corresponding to acceleration or deceleration of thc mean flow, changes the hetcroclinic cycles from attracting to repelling, increasing or decreasing the stability, in agreement with observations. The existence of fixed points is an artifact introduced by the projection; however, a decoupled model still displays the rich dynamics. This sort of relatively simple model could be used as a “black box” in feed-back systems to control the boundary layer, as well as being used to predict pressure and stress fluctuations at the wall, and the effect of various drag reduction schemes. Feeding back eigenfunctions with the proper phase can delay the bursting, (the heteroclinic jump to the other fixed point), decreasing the drag. It is also possible to speed up the bursting, increasing mixing to control separation.