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It is proposed that the statistics of the inertial range of fully developed turbulence can be described by a quantized random multiplicative process. We then show that (i) the cascade process must be a loginfinitely divisible stochastic process (i.e., stationary independent log-increments); (ii) the inertial-range statistics of turbulent fluctuations, such as the coarse-grained energy dissipation, are log-Poisson; and (iii) a recently proposed scaling model [Z.-S. She and E. Leveque 72, 336 (1994)] of fully developed turbulence can be derived. A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems. PACS numbers: 47.27.Gs, 47.27.TeThe statistics of fully developed turbulence exhibit certain universal features as a result of strong nonlinear interactions. One set of intriguing quantities characterizing universal behavior of turbulent flows is a set of scaling exponents for two-point correlation functions, e.g. , g"of the velocity structure functions defined by an expressionset of quantities is an exponent~p for pth-order moment of locally averaged energy dissipation e over a ball of size 4: (et") -8". The range of the length scale 8 for the above power-law behavior to be valid is called an inertial range. Kolmogorov's refined similarity hypothesis[1] provides a relation between these two sets of quantities: g"= p/3 + r"t3. This Letter addresses a predictive model [2] of g"and r". Note that g2 characterizes the scaling for the kinetic energy fluctuations and is directly related to the exponent for the kinetic energy spectrum E(k) -k:n =1+ j2.During the past half century since Kolmogorov's 1941 seminal work [3] that predicts r"= 0 and g"= p/3, there has been a continual effort to experimentally determine the values of g"or r". There is now strong evidence [4 -7] that g"W p/3, which is usually referred to as the intermittency effects or anomalous scaling exponents. Many theoretical models have been proposed to address this phenomenon. Some approaches start with an ansatz of probability density function (PDF) of et such as the log-normal model [2] or a log-stable model [8], for example. Others [9 -12] propose discrete random multiplicative processes (RMP) modeling the energy cascade. The stochastic process that generates the energy cascade is furnished by random multiplicative factors Wt, t, relating the fluctuations of e~at two different length scales 8]and Zz. et, = Wq, t, et, . By construction, the multiplicative factor is independent of ez, . The probability distribution of Wt, t, determines r", since it is required that log(Wt", t, )/ log(82/Zi) = r". Existing cascade models of random multiplicative processes [10 -12] are obtained by making an ad hoc ansatz for the PDF of lVt, t, being composed of a certain number of discrete atoms described by one or more adjustable parameters. The parameters in the models are difficult to determine by plausible phys...

The solutions to Burgers equation, in the limit of vanishing viscosity, are investigated when the initial velocity is a Brownian motion (or fractional Brownian motion) function, i.e. a Gaussian process with scaling exponent 0 < h < 1 (type A) or the derivative thereof, with scaling exponent -1 < h < 0 (type B). Largesize numerical experiments are performed, helped by the fact that the solution is essentially obtained by performing a Legendre transform. The main result is obtained for type A and concerns the Lagrangian function x(a) which gives the location at time t = 1 of the fluid particle which started at the location a. It is found to be a complete Devil's staircase. The cumulative probability of Lagrangian shock intervals Δa (also the distribution of shock amplitudes) follows a (Δa)~h law for small Δa. The remaining (regular) Lagrangian locations form a Cantor set of dimension h. In Eulerian coordinates, the shock locations are everywhere dense. The scaling properties of various statistical quantities are also found. Heuristic interpretations are provided for some of these results. Rigorous results for the case of Brownian motion are established in a companion paper by Ya. Sinai. For type B initial velocities (e.g. white noise), there are very few small shocks and shock locations appear to be isolated. Finally, it is shown that there are universality classes of random but smooth (nonscaling) initial velocities such that the long-time large-scale behavior is, after rescaling, the same as for type A or B.

A generalized Reynolds analogy (GRA) is proposed for compressible wall-bounded turbulent flows (CWTFs) and validated by direct numerical simulations. By introducing a general recovery factor, a similarity between the Reynolds-averaged momentum and energy equations is established for the canonical CWTFs (i.e. pipes, channels, and flat-plate boundary layers that meet the quasi-one-dimensional flow approximation), independent of Prandtl number, wall temperature, Mach number, Reynolds number, and pressure gradient. This similarity and the relationships between temperature and velocity fields constitute the GRA. The GRA relationship between the mean temperature and the mean velocity takes the same quadratic form as Walz’s equation, with the adiabatic recovery factor replaced by the general recovery factor, and extends the validity of the latter to diabatic compressible turbulent boundary layers and channel/pipe flows. It also derives Duan & Martín’s (J. Fluid Mech., vol. 684, 2011, pp. 25–59) empirical relation for flows at different physical conditions (wall temperature, Mach number, enthalpy condition, surface catalysis, etc.). Several key parameters besides the general recovery factor emerge in the GRA. An effective turbulent Prandtl number is shown to be the reason for the parabolic profile of mean temperature versus mean velocity, and it approximates unity in the fully turbulent region. A dimensionless wall temperature, that we call the diabatic parameter, characterizes the wall-temperature effects in diabatic flows. The GRA also extends the analysis to the fluctuation fields. It recovers the modified strong Reynolds analogy proposed by Huang, Coleman & Bradshaw (J. Fluid Mech., vol. 305, 1995, pp. 185–218) and explains the variation of the temperature–velocity correlation coefficient with wall temperature. Thus, the GRA unveils a generalized similarity principle behind the complex nonlinear coupling between the thermal and velocity fields of CWTFs.