Turbulent inner scale

V is kinematics viscosity of carrying liquid. Pulsations with A L are referred to as large-scale pulsations. For them Re.i 1, therefore the flow of liquid caused by these pulsations, has a non-viscous character. Reduction of pulsation scale results in decrease of Reynolds number. At some value A = Ao, called the inner scale of turbulence, Reynolds number... 

Consider now motion of small particJes in turbulent flow of liquid. Assume that the volume concentration of particJes is small enough, so it is possible to neglect their influence on the flow of hquid. The large-scale pulsations transfer a particJe together with layers of hquid adjoining to it. Small-scale pulsations with A R, where R is particle radius, cannot involve the particJe in their motions -the particle behaves in this respect as a stationary body. Pulsations of intermediate scales do not completely involve the particle in their motion. Consider the case most interesting for apphcations, when respective densities of particle p and external liquid are only slightly different from one another, and radius of the particle is much less than inner scale of turbulence, that is R A . Thus, for water-oil emulsion pjp 1.1-1.5. Let Uq be the velocity of hquid at the particle s location, and Ui the velocity of particJe relative to hquid. At full entrainment of particle by the hquid, the same force would ad on the particle as on... 

First, consider the drops of size J > /lo, where A is the inner scale of turbulence. Then large-scale pulsations (A A. 1), which don t vary too much on distances of the order of the drop size, do not exert a noticeable influence on these drops. Hence, the deformation and breakage of such drops can be caused only by small-scale pulsations. For such pulsations, the change of pulsation velocity Ui on the distance equal to the drop size 2R is... 

Consider now the drops whose size is smaller than the inner scale of turbulence (R Ao). It is obvious that the breakage of such drops can be caused only by pulsations with scale A < Aq, i.e. pulsations whose motion is accompanied by large forces of viscous friction. Therefore only the force of viscous friction at the drop surface can function as the main mechanism causing drop deformation. The criterion of strong deformation of a drop is the equality of forces of viscous friction and surface tension... 

Consider the coalescence of drops with fiilly retarded (delayed) surfaces (which means they behave as rigid particles) in a developed turbulent flow of a lowconcentrated emulsion. We make the assumption that the size of drops is much smaller than the inner scale of turbulence R Ao), and that drops are non-deformed, and thus incapable of breakage. Under these conditions, and taking into account the hydrodynamic interaction of drops, the factor of mutual diffusion of drops is given by the expression (11.70). To determine the collision frequency of drops with radii Ri and Ri (Ri < Ri), it is necessary to solve the diffusion equation (11.36) with boundary conditions (11.39). Place the origin of a spherical system of coordinates (r, 0,0) into the center of the larger particle of radius i i. If interaction forces between drops are spherically symmetrical, Eq. (11.36) with boundary conditions (11.39) assumes the form... 

The factor of turbulent diffusion is given by (13.86) it was derived using the assumption that the size of drops is small in comparison with the inner scale of turbulence R Ao-... 

Let us find the collision frequency of conducting uncharged spherical drops in a turbulent fiow of a dielectric liquid in the presence of a uniform external electric field. Just as before, we assume a developed fiow, with drop sizes smaller than the inner scale of turbulence. We assume the drops to be undeformed, which is possible if the external electric field strength Eo does not exceed the critical value and the size of drops is sufficiently small. Under these conditions, the factor of mutual diffusion of drops of two types 1 and 2 with regard to hydrodynamic interaction is given by (13.86), while h and are given by the expressions (13.85) that apply to drops with a completely retarded surface. We must also take into account molecular and electric interaction forces acting on the drops. 

The inner scale of turbulence, 2q, deflnes the character of hydrodynamic and mass-exchange processes in areas in which size is greater or smaller than 2q. Since processes in the vicinity of drops are of greatest interest, the size of these regions is commensurable with drop sizes. Let be the average radius of an ensemble of drops under consideration. The character of the processes then depends on the ratio Rav/2o-... 

Xo on condensation growth of drops Inner scale of turbulence m... 

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

The periodicities are adimensionalized with the two scale parameters of the inner layer of the turbulent boundary, the kinematic viscosity v and the friction velocity uT, by the equation ... 

Sj r — r ) Dirac delta function which is zero everywhere except when r = r, infinite on interface, and integral unity Sy length scale characterizing the inner layer in turbulent boundary layers (m)... 

A survey concerning coherent structures which can play an important part in the phenomenon of drag reduction by polymer additives is given. Most of these structures have long been known. Recent measurements have shown that both the scaling of low speed streaks and the bursting frequency remain constant when made dimensionless with inner variables. Stream wise vortices have been visually verified and their intensities anemometrically determined. Various authors have independently shown that too little attention has been paid to the problem of small-scale structure in turbulence measurements. 

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