Turbulent flow eddy properties

For freely suspended bioparticles the most likely flow stresses are perceived to be either shear or normal (elongation) stresses caused by the local turbulent flow. In each case, there are a number of ways of describing mathematically the interactions between turbulent eddies and the suspended particles. Most methods however predict the same functional relationship between the prevailing turbulent flow stresses, material properties and equipment parameters, the only difference between them being the constant of proportionality in the equations. Typically, in the viscous dissipation subrange, theory suggests the following relationship for the mean stress [85] ... 

Most examples of flow in nature and many in industry are turbulent. Turbulence is an instability phenomenon caused, in most cases, by the shearing of the fluid. Turbulent flow is characterized by rapid, chaotic fluctuations of all properties including the velocity and pressure. This chaotic motion is often described as being made up of eddies but it is important to appreciate that eddies do not have a purely circular motion. 

It is the large scale eddies that are responsible for the very rapid transport of momentum, energy and mass across the whole flow field in turbulent flow, while the smallest eddies and their destruction by viscosity are responsible for the uniformity of properties on the fine scale. Although it is the fluctuations in the flow that promote these high transfer rates, it is... 

The work of Laufer (L3) indicates that eddy viscosity is not isotropic in shear flow. For this reason it is unlikely that eddy conductivity is isotropic in such flows. Therefore, uncertainties in the application of eddy conductivities must arise when it is assumed that this transport coefficient is isotropic. Until additional experimental information is available, it appears reasonable to consider eddy conductivity as isotropic except in circumstances when the vectorial nature (J4, R2) of the eddy viscosity may be estimated. Such an approximation appears acceptable since the measurements available described the conductivity normal to the axis of flow, which is the direction in which most detail is required in the prediction of temperature distribution in turbulently flowing streams. Throughout the remainder of this discussion all eddy properties will be treated as isotropic. Such a simplification is open to uncertainty, and further experimentation will be required in order to determine the error introduced by neglect of the vectorial characteristics of these macroscopic transport quantities. 

In Eq. (32) the subscript 1 refers to the location of the transition between the boundary flow and the turbulent stream. It appears that these analytical considerations by Deissler represent improvements over his earlier approach (D2). For the boundary flow close to the wall the foregoing expressions are based on the assumption that the eddy properties may be evaluated from... 

The physical mechanism of heat transfer in turbulent flow is quite similar to that in laminar flow the primary difference is that one must deal with the eddy properties instead of the ordinary thermal conductivity and viscosity. The main difficulty in an analytical treatment is that these eddy properties vary across the boundary layer, and the specific variation can be determined only from experimental data. This is an important point. All analyses of turbulent... 

When flow rates are high, a litjuid will not flow in a laminar fashion, but will become turbulent. The litjuid will start to flow in a chaotic way, forming large and small swirls and eddies. The flow rate at which this happens depends on the geometry of the machine, on the flow rate applied, and on the overall viscosity of the mixture. The transition from laminar to turbulent flow is characterized by the Reynolds number the critical Reynolds number depends on the geometry and the product properties (see Equations (15.9)-(15.10)). 

It will be useful to develop an understanding of the relationship between turbulent flow field and dispersion. Measurement techniques need to be developed for the measurement of turbulent flows in multiphase systems. The relationship between eddy diffusion and the dispersion coefficient needs to be brought out over a wide range of particle sizes, setthng velocities, column diameters, column heights, phase velocities, and physical properties. 

Here, is referred to as turbulent or eddy viscosity, which, in contrast to molecular viscosity, is not a fluid property but depends on the local state of flow or turbulence. It is assumed to be a scalar and may vary significantly within the flow domain, k is the turbulent kinetic energy (normal turbulent stresses) and can be expressed as... 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

A fundamental problem in performing a turbulent flow analysis involves determining the eddy diffusivities as a function of the mean properties of the flow. Unlike the molecular diffusivities, which are strictly fluid properties, the eddy diffusivities depend strongly on the nature of the flow they can vary from point to point in a boundary layer, and the specific variation can be determined only from experimental data. 

In turbulent flows, the transport of momentum, heat, and/or individual species within gradients of velocity, temperature, and concentration is caused predominantly by the chaotic motion of elements of fluid (eddies). This mixing process transports properties much more effectively than the molecular processes identified with viscosity, thermal conductivity, and diffusion. A rather complete description of these processes is given in Refs. 71-73. 

Currently there exist computers with sufficient storage capacity and speed to allow computation of these time-dependent motions for rather simple flows with finite difference meshes sufficiently fine to resolve the larger eddies of the motion. Even with such computations, however, it is necessary to model the effects of the eddies that are too small to be resolved. It is believed that since the transport of properties is governed by the larger eddies, the modeling process is less critical in these computations than where the entire turbulence is modeled. These turbulence simulations are still too costly for routine engineering computations and are used primarily to study the physics of particular turbulent flows. In fact, the results provide much more information than an engineer may ever want or need. 

The evaluation of these statistical second moments is the goal of turbulence models. These models fall into two categories. First are models in which the turbulent fluxes are expressed in the same functional form as their laminar counterparts, but in which the molecular properties of viscosity, thermal conductivity, and diffusion coefficient are supplemented by corresponding eddy viscosities, conductivities, and diffusivities. The primary distinction is the recognition that the eddy coefficients are properties of the turbulent flow field, not the... 

---