Turbulence velocity

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. MacroscopicaUy, this energy dissipation requires pressure drop, or velocity decrease. The ener dissipation rate per unit mass is usually denoted . For steady ffow in a pipe, the average energy dissipation rate per unit mass is given by... 

The universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the eddy viscosity,... 

The neglect of a low turbulence effect and a laminar flow is not justified in regions close to solid surfaces where the turbulent velocity fluctuations... 

Rielly and Marquis (2001) present a review of crystallizer fluid mechanics and draw attention to the inconsistency between the dependence of crystallization kinetic rates on local mean and turbulent velocity fields and the averaging assumptions of conventional well-mixed crystallizer models. 

Warholic MD, Schmidt GM, Hanratty TJ (1999) The influence of a drag-reducing surfactant on a turbulent velocity field. J Fluid Mech 388 1-20... 

The Kolmogorov velocity field mixes packets of air with different passive scalars a passive scalar being one which does not exchange energy with the turbulent velocity flow. (Potential) temperature is such a passive scalar and the temperature fluctuations also follow the Kolmogorov law with a different proportionality constant... 

In this study, the flame can be classified as a wrinkled flame throughout most of the flow field. The main findings of [25] are related to both (1) the question of how the turbulent velocity field is affected by the chemical reaction and induced expansion phenomena and (2) the measurements of mean flame surface density and the... 

A very fine space resolution is required to measure the gradient of turbulent velocity fluctuations and calculate turbulent dissipation directly from the definition [5, 6]. 

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

The quantity G of the effective mixing mass flux is determined by the turbulent velocity fluctuations at the bubble-layer edge. The distance of the edge of the bubble layer from the wall is taken as the distance at which the size of the turbulent eddies is k times the average bubble diameter. Weisman and Pei have determined empirically that k equals 2.28. Only a fraction of the turbulent velocity fluctuations produced are assumed to be effective in reaching the wall. The effective velocity fluctuations are those in which the velocity exceeds the average velocity away from the wall produced by evaporation heat flux q"b. At the bubble layer-core interface, the effective mass flux to the wall is computed as... 

A more realistic approach to quantify the pressure field is to consider the effect of turbulence [6]. For a pipe flow, the turbulent pressure fluctuations are due to velocity perturbations as a result of the formation of eddies. The instantaneous turbulent velocity can be calculated by assuming a sinusoidal velocity variation in... 

In real life, the parcels or blobs are also subjected to the turbulent fluctuations not resolved in the simulation. Depending on the type of simulation (DNS, LES, or RANS), the wide range of eddies of the turbulent-fluid-flow field is not necessarily calculated completely. Parcels released in a LES flow field feel both the resolved part of the fluid motion and the unresolved SGS part that, at best, is known in statistical terms only. It is desirable that the forces exerted by the fluid flow on the particles are dominated by the known, resolved part of the flow field. This issue is discussed in greater detail in the next section in the context of tracking real particles. With a RANS simulation, the turbulent velocity fluctuations remaining unresolved completely, the effect of the turbulence on the tracks is to be mimicked by some stochastic model. As a result, particle tracking in a RANS context produces less realistic results than in an LES-based flow field. 

Note that the correction terms are proportional to fT and result from turbulent velocity fluctuations (represented by a gradient-diffusion model). For the multi-environment model the composition vector is defined by... 

Plot laminar and turbulent velocity profiles for steady state flow in a cylindrical pipe for a maximum velocity gm = 5 m/s using the radial positions 2r d - 0, 0.2, 0.4, 0.6 and 0.8. 

Chigiefl211 found that turbulence in a liquid jet has important disturbing influences throughout the liquid flow. At the liquid surface, turbulent velocity fluctuations directly cause protuberances and roughness that result in direct stripping by surrounding air flow. Large eddy structures in the air flow penetrate into the liquid and... 

This term represents the spatial transport of e by turbulent velocity fluctuations. 

The transported PDF models discussed so far in this chapter involve the velocity and/or compositions as random variables. In order to include additional physics, other random variables such as acceleration, turbulence dissipation, scalar dissipation, etc., can be added. Examples of higher-order models developed to describe the turbulent velocity field can be found in Pope (2000), Pope (2002a), and Pope (2003). Here, we will limit our discussion to higher-order models that affect the scalar fields. 

Gibson, C. H., G. R. Stegen, and R. B. Williams (1970). Statistics of the fine structure of turbulent velocity and temperature fields measured at high Reynolds numbers. Journal of Fluid Mechanics 41, 153-167. 

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined the integral scale l0, which characterizes... 

Above and to the left of the criterion line is the region in which 4 < 4.-According to the Klimov-Williams criterion, the turbulent velocity gradients in this region, or perhaps in a region defined with respect to any of the characteristic lengths, are sufficiently intense that they may destroy a laminar flame. 

The length scale for describing the spatial variation of the pollutant species, L, is large when compared with the turbulent mean free path, /, a distance traveled by a particle in time Ti. Assuming the root-mean-square (rms) turbulent velocity of the particle is , this condition can be expressed as L > / = mJl. 

The elements of Taylor s treatment are as follows. Consider a particle of fluid with turbulent velocity w in a homogeneous turbulent field. The distance that the fluid particle travels in time t is,... 

Again, any turbulence velocity and length scale are sufficient to use in these dimensionless parameters, as long as they are used in all parameters. Tamburrino and Gulliver used the bottom shear velocity and channel depth. Equation (8.69) provides a measure of surface renewal rate ... 

Another important function of the impeller in three-phase systems is to generate finely dispersed and homogeneously distributed bubbles throughout the vessel. At the same time, the turbulent velocities should be sufficiently high to prevent coalescence of gas bubbles. The most important variable concerning gas dispersion is the gas holdup in the vessel. 

Turbulent flow means that, superimposed on the large-scale flow field (e.g., the Gulf Stream), we find random velocity components along the flow (longitudinal turbulence) as well as perpendicular to the flow (transversal turbulence). The effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick s first law (Eq. 18-6), where the molecular diffusion coefficient is replaced by the so-called turbulent or eddy diffusion coefficient, E. For instance, for transport along the x-axis ... 

Recall our short discussion in Section 18.5 where we learned that turbulence is kind of an analytical trick introduced into the theory of fluid flow to separate the large-scale motion called advection from the small-scale fluctuations called turbulence. Since the turbulent velocities are deviations from the mean, their average size is zero, but not their kinetic energy. The kinetic energy is proportional to the mean value of the squared turbulent velocities, Mt2urb, that is, of the variance of the turbulent velocity (see Box 18.2). The square root of this quantity (the standard deviation of the turbulent velocities) has the dimension of a velocity. Thus, we can express the turbulent kinetic energy content of a fluid by a quantity with the dimension of a velocity. In the boundary layer theory, which is used to describe wind-induced turbulence, this quantity is called friction velocity and denoted by u. In contrast, in river hydraulics turbulence is mainly caused by the friction at the... 

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