Isotropic homogeneous turbulence

While the theoretical value (based on homogeneous, isotropic turbulence) of the Smagorinsky coefficient cs amounts to 0.165 (Mason and Callen, 1986), in many simulation studies lower values for cs proved to result in a better reproduction of experimental data. This may have to do with the abundant presence of shear flows in process equipment. Derksen (2003) reported that varying cs values in the range 0.08-0.14 does not have a large impact on the simulation results. A value of 0.12 is recommended. 

In homogeneous isotropic turbulence, the two-point velocity correlation function can be expressed (Pope 2000) in terms of the longitudinal (/) and transverse (g) auto-correlation functions ... 

Hughes, T. J. R., L. Mazzei, A. A. Oberai, and A. A. Wray (2001b). The multiscale formulation of large eddy simulation Decay of homogeneous isotropic turbulence. 

In contrast, most models of turbulent flame speeds assume homogeneous isotropic turbulence with no mean flow or strain. This leads to substantial differences between model predictions and experimental observations [4]. 

Among experimental studies of chemical reactions in turbulent media, fast reactions in tubular reactors with multijet injection of reactants are very popular, since the first experiments of Mao and Toor (34) and Vassiliatos and Toor (35). Their data have been (and are still) extensively exploited for testing theoretical models, although one may ask if homogeneous isotropic turbulence was perfectly controlled in these experiments. In order to rule out this objection, a new series of experiments was recently performed by Bennani et al. (28, 29, 30, 36) in a 0.29 m i.d. tube eliminating the influence of boundary layers. Turbulence was created by a grid and carefully controlled by velocity fluctuation measurements. Previous studies (2) had confirmed that the decrease of Ig with a non-reacting species (passive scalar) obeys Corrsin s equation ... 

Check for homogeneous isotropic turbulence in test experiments (influence of boundary layers, perturbations close to impellers). ... 

Design of experiments where hydrodynamic conditions are perfectly controlled small stirred reactors with high power input (no macromixing effects, circulation time tc << t, no spatial variation of e), multijet tubular reactors with a diameter large enough for pure homogeneous isotropic turbulence to be achieved. 

If one considers the decay of homogeneous isotropic turbulence with zero strain for large turbulence Reynolds numbers, while modeling the dissipation by... 

Homogeneous, isotropic turbulence cannot be assumed in the free jets. The authors in [541], therefore, utilized the Phoenics program (CHAM Ltd.) in connection with the slower diazotization reaction. The constants of the /c/e-turbulence model are adapted to well-known pictures of flow patterns and the turbulent Schmidt number determined to be Scturb = 1. It thereby succeeds in achieving the best description of the decoloration length. 

Homogeneous isotropic turbulence is a mathematical idealization of real turbulence, that was introduced by [159], allowing us to simplify the analysis of turbulence considerably, and thus gain insight into its behavior. Nevertheless, real turbulent flows do rarely approach homogeneity and isotropy. However, by using the homogeneous and isotropic turbulence concept the mathematical problem simplifies considerably and it is possible to obtain many specific mathematical results which explain several aspects of turbulent flows. 

In the following we will consider some basic results of the statistics of such a homogeneous isotropic turbulent field. The consequences of homogeneity and isotropy for the correlation functions were worked out by von Karman and Howarth [179] and the full derivations are available in classical books like [66, 8, 112, 113]. 

Zaichik, L. L, Simonin, O. Aupchenkov, V. M. 2003 Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence. Physics of Fluids 15(10), 2995-3005. 

According to this theory, the only parameter needed to describe the probability distribution of the relative velocity field K . in a homogeneous isotropic turbulent fluid is the average energy dissipation g i.e. 

A two dimensional configuration was considered (Fig. 4), where an initially planar laminar premixed flame evolves in a turbulent flow field. The chemicd scheme of [4] (9 species, 19 reactions) has been retained to prescribe chemical reactions. The initial homogeneous isotropic turbulence field was given according to the energy spectrum proposed by Von Karmann-Pao [5]. Figure 5 presents an example of DNS - results are for a turbulent Reynolds number mT/v = 289 in the fresh gas, a ratio of velocity fluctuations to laminar flame speed uYsi = 30 and an integral... 

Mixing of newtonian fluids in conventional stirred tanks is now reasonably well understood, although most models rely on the assumption of homogeneous isotropic turbulence. This is only a rough approximation. For instance, the fluctuating components u - and u 0 of the radial and tangential velocity were recently measured by Laser-Doppler Anemometry in a 6.3 dm reactor,. Values of the Reynolds Tension... 

Figure 2-13 Spectrum of velocity [E(n)] and temperature or concentration [r(n)] fluctuation wavenumbers (m ) in the equilibrium range of homogeneous isotropic turbulence for the case of large Sc or Pr (modified from Batchelor, 1959). In the Batchelor scale, k is either the thermal diffusivity (k/pCp) or the molecular diffusivity (Dab)-...

So far, our discussion of length scales has focused on the smallest scales of motion. At the larger scales of motion, Taylor (1921) considered the turbulent dispersion of fluid particles by homogeneous isotropic turbulence in the absence of molecular diffusion. In his model, each fluid particle leaving a point source in a uniform velocity field is expected to deviate from the linear mean path in a random manner, depending on the local nature of the turbulence. The RMS deviation of the particle paths is observed as a continued divergence, spread, or dispersion as the particles are carried downstream. This eddy motion occurs even... 

Hanratty. T.J. 1956. "Heat transfer through a homogeneous, isotropic turbulent field. AIChE Journal. 2, 359-362. 

Derksen JJ (2012) Direct numerical simulations of aggregation of monosized spherical particles in homogeneous isotropic turbulence. AIChE J 58 2589-2600... 

Ernst M, Sommerfeld M (2012) On the volume fraction effects of inertial colliding particles in homogeneous isotropic turbulence. J Fluids Eng 134 031302... 

Sommerfeld M (2001) Validation of a stochastic lagrangian modeUing approach for interparticle collisions in homogeneous isotropic turbulence. Int J Multiph Flow 27 1829-1858 Soo SL (1967) Fluid dynamics of multiphase systems. BlaisdeU Publishing Company, Waltham, Massachusetts... 

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