Turbulent Nonlinear Dynamics Of The Shell Model

The intermittent nature of turbulence is the essence character of turbulence and one of central problems in theoretical studies of turbulence.Since the seventies in last century,the nonlinear dynamical approaches have provided a new view on turbulence. Turbulence connects closely with the bifurcation,chaos and strange attractor in nonlinear dynamical systems.Through the studies on the low dimension of dynamical systems,many kinds of transition routes to the chaotic attractors have been found out. The three famous routes are Hopf bifurcation brought forward by Ruelle and Takens, doubling periodic bifurcation brought forward by Feigenbaum and intermittent transition found by Pomean and Menneville.Nowadays one has believed that turbulence occurrence has close relations with the dynamical chaotic attractor appearance in nonlinear systems.This thesis is about the studies on the nonlinear dynamics and scaling in shell model of turbulence,called Gledzer-Ohkitani-Yamada shell model.In the numerical simulation for the 22 shells of GOY model,We find by varying parameterδ:For each shell in the inertial range,the phases of intermittent orbit parts in velocity phase space display clockwise rotation randomly,counter clockwise rotation randomly and oscillation randomly which have period three with shell number.The directions of unstable periodic orbit parts of each shell in inertial range form the rotational,reversal and locked cascade which also have period three with shell number.Through calculating statistical average of unstable periodic time series length,we obtain critical scaling of parameterδabout the average by fitting.We calculate relative scaling exponents of the whole time series of intermittent turbulence which is very similar to the kolmogorov linear scaling,and besides,we calculate the relative scaling exponents of structure function about unstable periodic time series parts and intermittent time series parts.The results show that intermittent turbulence also has the character of nonlinear scaling of velocity structure function of well-developed turbulence,which is contributed by the strongest and lowest intermittency parts of the intermittent time series.Through variation of parameter f₀ andδ,the dynamical behaviors on f_{0—δ} parameter surface is investigated for GOY model.We determine the type of intermittency transitions to chaotic attractor is saddle node bifurcation by calculating the scaling exponent,the pairs of stable and unstable periodic orbits and Liyapunov exponents.We plot phase diagram on f_{0—δ} parameter surface which is divided into periodic,quasi-periodic and intermittent chaos area.By means of varying Taylor-microscale Reynolds number,we calculate the relative scaling exponents of velocity structure function which change a little with Taylor-microscale Reynolds number,and we indicate f_{0—δ} the parameter range of nonlinear the relative scaling of velocity structure function.