UNSTABLE BURSTS IN THE NEAR REGION OF AN INTERNAL FREE SHEAR LAYER J.C.Crepeau* and L.K.Isaacson** Department of Mechanical and Industrial Engineering University of Utah Salt Lake City, UT 84112 Abstract A deterministic approach is used to analyze the flow in the near region of an internal free shear layer. Measurements of the time between bursts in the near region identify the possibility of the existence of chaotic attractors in the flow. The experimental data give some insight into the existence of these attractors. The turbulent Navier-Stokes equations along with a velocity profile derived by Stuart are used to model the velocity in this flow regime. Computer solutions of the linearized form of the rate of change of the Fourier components of the turbulent (deterministic) flow field reveal the existence of a region of instability. I. Introduction Originally, investigators believed turbulent fluid flow to be a completely random process, and they employed statistical approaches to explain and model the flow. However, within the past few decades, deterministic methods and models have been incorporated into the study of the transition to turbulence. By using the deterministic method, this paper attempts to describe unstable bursts in the near region of an internal free shear layer. The current studies evolved from research done on free shear flows in internal cavities, which simulates the flow in segmented solid propellant rocket motors. By understanding the flow in this type of configuration, the engineer can avoid hazardous acoustic oscillations, and motor performance can be improved. ~orenz' in 1963 presented three nonlinear, first-order ordinary differential equations of the form, LORENZ BIFURCATION DIAGRAM -20 1-u- I I 5 1 A 0 10 15 20 25 30 35 COEFFICIENT R Figure 1 Plot of x vs.r of the Lorenz equations. Lorenz derived Equation (1) by making severe approximations on a set of equations which were derived by saltzman2 to study finite amplitude convection. The profound behavior of the resulting trajectories justified Lorenz's approximation. Equation (1) is called a chaotic or strange attractor because of its behavior at certain values of their coefficients. For example, the bifurcation diagram3 in fig. 1 gives the steady state values of x for different values of the coefficient r. Equation (1) is solved numerically with a fourth-order Runge-Kutta routine through 1000 iterations, keeping o=10, b=8/3 and r varies from r=0.5 to r=35 in intervals of 0.5. Notice an abrupt change near the value r=27. This change is the transition point from a stable attractor (Fig. 2a) to an unstable attractor (Fig. 2b). The plots of Figs. 2a and 2b are typical for values of r in the stable and unstable regimes respectively. As will be shown later, the equations used to model internal free shear layers exhibit some of the same types of stability transitions. Packard et al.'l explained how the existence of low- dimensional chaotic dynamical systems can be experimentally Figure 2a Figure 2b x vs.z plot of Lorenz equations, stable and unstable regimes. 'Graduate student. Student member, AIAA **Professor, Mechanical Engineering. Associate Fellow,AIAA Copyright O 1988 bv the American Institute of Aeronautics -- - and Astronautics, Inc. All rights reserved. 853 determined. They indicate that to specify the state of a three-dimensional system at any time, the coordinates can be based on any three independent quantities that uniquely and smoothly label the states of the attractor (italics included in original paper). For example, the state-space coordinates x(t), y(t), z(t) are used in fig. 2b. The delayed-time phase space, ~(9, ~(t-z),x(~-22) is another possible configuration. Aliabadi et al. performed experiments where the data are presented in the delayed-time phase space. Crutchfield et performed a novel experiment by measuring the time between drops of a randomly dripping faucet, and plotting the results in t , t,,+2 phase space (Fig. 3). They found that the resulting p ? ots compared favorably with a variant of Henon's rule. h en on's' rule is a q& Plots from Crutchfield et al. paper describing time between drops from a dripping faucet experiment.