Spectrum of turbulence

The three models were calculated with the same chemical and physical inputs with the only exception of convection, for which we adopted the Full Spectrum of Turbulence convective model (FST, Canuto Mazzitelli 1991), and the MLT model (Vitense 1953) with two values of the free parameter connected to the mixing length a = 1.7 (the standard value, used to reproduce the evolution of the Sun) and a = 2.1. 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

Fig. 2. Kinetic-energy spectrum of turbulence in shear flow.

Microturbulence seldom occurs alone it is generally driven by larger-scale motions microturbulence is the high-wavenumber part of the atmospheric spectrum of turbulence. For the stars discussed here the origin of the motion field may be found in pulsations or in convective motions. Such motions have been discovered in a Cyg (Boer et al., 1987 ) they have up- and downward velocities of 14 km s-1 and the... 

Prudman, I. (1951). A comparison of Heisenberg s spectrum of turbulence with experiment. Proc. Camb. Phil. Soc., 47, 158-176. 

Taylor GI (1938) The Spectrum of Turbulence. Proc. Roy. Soc. London Series A 164 476-490... 

In other cases the application of this concept has been further extended simulating faster turbulent fluctuations that are within the turbulence spectrum. For such dynamic simulations, using Reynolds averaged models, the Ic-quantity represents the turbulent kinetic energy accumulated on the fraction of the spectrum that is represented by the modeled scales. Therefore, to compare the simulated results obtained with this type of models with experimental data, that is averaged over a sufficient time period to give steady-state data (representing the whole spectrum of turbulence), both the modeled and the resolved scales have to be considered [68]. 

For large Reynolds number the kinetic energy spectrum of turbulence has the form E k) oc k-S/3... 

It can also be shown [8] that there exists a relatively simple mathematical relationship between the spectrum of turbulence and the correlation coefficient, so that a detailed, accurate measurement of one enables the calculation... 

In turbulent flow of liquid, random pulsation motions characterized by a set of pulsation velocities are imposed on average movement with velocity 1/ in a certain direction. Turbulent pulsations are characterized not only by velocities, but also by distances at which these velocities undergo noticeable change. These distances are referred to as pulsation scales and are denoted as 2. The set of values X represents a spectrum of turbulent pulsations varying from 0 up to a maximal value, having the order of linear scale of cross-sectional area of current flow. So, at motion in a pipe of diameter I the greatest value X is equal to I. Every pulsation movement is characterized by its Reynolds number Re = Xuxjv, where... 

Since for all dispersion organs in the ranges which are important for the practice the gas dispersion occurs due to turbulence mechanisms it is to be expected that there is a relationship between the turbulence properties and the efficiency of gas dispersion. It has already been pointed out that efficient gas dispersion is only possible, if the microeddy size is smaller than de cind if the percentage of microturbulence is high enough. Therefore the power spectrum of turbulence should influence the efficiency of gas dispersion. Fig. 10 shows one-dimensional power spectra in stirred teink reactors of different sizes according to van der Molen and van Maanen (22). As usual, the energy content of turbulent eddies was plotted as a function of the wave number, K, which is inversely proportional to the diameter of the... 

It is often not sufficient to characterize the magnitude of small-scale fluctuations, but also to understand the scale of the fluctuations or, more precisely, the distribution of the energy of the fluctuations over different frequencies or wave numbers. This is known as the spectrum of turbulence. For example, in the case of shaking of towers or bridges, only relatively high frequencies are important in contrast, high frequencies are relatively inefficient for mixing air masses with different properties. 

G. I. Taylor, The spectrum of turbulence, Proc. Royal Society, London, A164, 476, 1938. 

Fluctuations of the velocity field in a turbulent flow vary both in space and time. It is this time-space information that is characterized using the energy spectrum of turbulence. If the variations of one turbulent velocity component u x, f) are known over a length interval [0, L] in a direction Ox at a given time t, the variations of u x, f) may also be expanded into a Fourier series with respect to the space coordinate x Thus, [11.2] is extrapolated into ... 

Classical computational fluid dynamics (CFD) deals with a well-established system of equations. Typically, transport parameters and kinetic constants (if any) are also well defined. Scientific CFD problems are complicated by the presence of turbulence in the system. The spectrum of turbulence covers many orders of magnitude, which requires exhausting computing resources to resolve all the space scale and timescales. 

In RANS-based simulations, the focus is on the average fluid flow as the complete spectrum of turbulent eddies is modeled and remains unresolved. When nevertheless the turbulent motion of the particles is of interest, this can only be estimated by invoking a stochastic tracking method mimicking the instantaneous turbulent velocity fluctuations. Various particle dispersion models are available, such as discrete random walk models (among which the eddy lifetime or eddy interaction model) and continuous random walk models usually based on the Langevin equation (see, e.g.. Decker and... 

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