Turbulence fully isotropic

While the fully isotropic assumption is not a good match to physical reality, the implications of isotropy are profound for turbulence modeling and measurements. Isotropy allows the entire turbulent spectrum to be defined from one component of fluctuating velocity, because the flow is perfectly without directional preference. It allows simplification of the equations to include only the normal stresses. It also allows one to make spectral arguments to simplify the measurement of the dissipation. This assumption is so powerful that it is often invoked in the hope that it will be good enough for a flrst approximation, despite the fact that it is a poor match for the full physical reality. 

Fig. 1. Dimensionless stress in fully developed turbulent flow given by the theory of isotropic turbulence...

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

A complete theory of turbulence is still lacking, so we must restrict our discussions to two general cases of interest to us (a) the diffusion of particles from point or line sources where the turbulence may be said to be isotropic, and (b) the behavior of particles near large land surfaces— as for example, dust storms. We shall begin our discussion with an explanation of the meaning of eddy-diffusion, which is characteristic of the conditions to be more fully discussed later. 

The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

Ten Cate A, Derksen JJ, Portela LM, Van den Akker HEA Fully resolved simulations of colliding monodisperse spheres in forced isotropic turbulence, J Fluid Mech 519 233-271, 2004. http //dx.doi.org/10.1017/s0022112004001326. 

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