Boundary-Layer Flow and Turbulence

In Sections 3.8 and 3.9 the Navier-Stokes equations were used to find relations that described laminar flow between flat plates and inside circular tubes, flow of ideal fluids, and creeping flow. In this section the flow of fluids around objects will be considered in more detail, with particular attention being given to the region close to the solid surface, called the boundary layer. 

In the boundary-layer region near the solid, the fluid motion is greatly, affected by this solid surface. In the bulk of the fluid away from the boundary layer the flow can often be adequately described by the theory of ideal fluids with zero viscosity. However, in the thin boundary layer, viscosity is important. Since the region is thin, simplified solutions can be obtained for the boundary-layer region. Prandtl originally suggested this division of the problem into two parts, which has been used extensively in fluid dynamics. 

In order to help explain boundary layers, an example of boundary-layer formation in the steady-state flow of a fluid past a flat plate is given in Fig. 3.10-1. The velocity of the fluid upstream of the leading edge at x = 0 of the plate is uniform across the entire fluid stream and has the value. The velocity of the fluid at the interface is zero and the velocity in the x direction increases as one goes farther from the plate. The velocity, approaches asymptotically the velocity v of the bulk of the stream. 

The dashed line L is drawn so that the velocity at that point is 99% of the bulk velocity The layer or zone between the plate and the dashed line constitutes the boundary layer. When the flow is laminar, the thickness 5 of the boundary layer increases 

Reynolds number is less than 2 x 10 the flow is laminar, as shown in Fig. 3.10-1. 

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BOUNDARY-LAYER FLOW AND TURBULENCE IN HEAT TRANSFER... 

Figure 5.7-1. Laminar flow offluid past a flat plate and thermal boundary layer. Sec. 5.7 Boundary-Layer Flow and Turbulence in Heat Transfer...

Sec. 5.7 Boundary-Layer Flow and Turbulence in Heat Transfer... 

Considering the case of Eq. (4.244), it is normal to describe a real mass transfer case by taking into consideration the boundary layer flows and the turbulence by using a mass transfer factor which is defined by... 

In order to illustrate the main features of the analysis of turbulent flow, attention will be restricted to two-dimensional boundary layer flows and to axially symmetric pipe flows. It will also be assumed that the fluid properties are constant and that the mean flow is steady. 

In the present chapter and in the following two chapters, which are concerned with turbulent boundary layer flows and with turbulent duct flows, respectively, consideration will be restricted to forced flows, i.e., the effect of buoyancy forces on the mean flow and on the turbulence structure will be assumed to be negligible. Some discussion of the effect of buoyancy forces on turbulent flows will be given in Chapter 9. 

In order to utilize this equation it is necessary to use other equations to describe some of the terms in this equation and/or to model some of the terms in this equation. To illustrate how this is done, attention will be given to two-dimensional boundary layer flow. For two-dimensional boundary layer flows the turbulence kinetic energy equation, Eq. (5.S2), has the following form, some further rearrangement having been undertaken ... 

In 1968 a conference was held at Stanford on turbulent boundary layer prediction-method calibration (S3), where for the first time a large number of methods, totaling 29, were compared on a systematic basis. This comparison established the viability of prediction methods based on various closure models for the partial differential equations describing turbulent boundary layer flows, and has stimulated considerable more recent work on this approach. 

The separation efficiency of a hydrocyclone has a character of probability. This is to do with the probability of the position of the different particles in the entrance to the cyclone, their chances of separation into the boundary layer flow and the general probability character of turbulent flow. Coarse particles are always more likely to be separated than fine particles. Effectively, the hydrocyclone processes the feed solids by an efficiency curve called grade efficiency , which is a percentage increasing with particle size (see chapter 3 for more details about grade efficiency). Figure 6.6 shows the process schematically the solids in the feed enter the cyclone and are... 

Internal Flow. Depending on the atomizer type and operating conditions, the internal fluid flow can involve compHcated phenomena such as flow separation, boundary layer growth, cavitation, turbulence, vortex formation, and two-phase flow. The internal flow regime is often considered one of the most important stages of Hquid a tomiza tion because it determines the initial Hquid disturbances and conditions that affect the subsequent Hquid breakup and droplet dispersion. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Figures 4.34 and 4.35 represent two extreme cases. Drying processes represent the case shown in Fig. 4.34 and distillation processes represent Fig. 4.35. Neither case represents a convective mass transfer case while the gas flow is in the boundary layer, other flows are Stefan flow and turbulence. Thus Eqs. (4.243) and (4.244) can seldom be used in practice, but their forms are used in determining the mass transfer factor for different cases.

Chien, K. Y. Predictions of channel and boundary layer flows with a low-Reynolds-nuraber turbulence model. AIAA J., vol, 20, pp. 33-18, 1982. 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

This result makes it clear that particle stress is strongly dependent on the interaction between the particles and the interface, so that electrostatic and also hydrophobic and hydrophilic interactions with the phase boundary are particularly important. This means that the stress caused by gas sparging and also by boundary-layer flows, as opposed to reactors with free turbulent flow (reactors with impellers and baffles), may depend on the particle system and therefore applicability to other material systems is limited. 

The stress caused by gas sparging and also by boundary-layer flows, as opposed to reactors with free turbulent flow (reactors with impellers and bafQes), may depend on the particle system. 

For reactors with free turbulent flow without dominant boundary layer flows or gas/hquid interfaces (due to rising gas bubbles) such as stirred reactors with bafQes, all used model particle systems and also many biological systems produce similar results, and it may therefore be assumed that these results are also applicable to other particle systems. For stirred tanks in particular, the stress produced by impellers of various types can be predicted with the aid of a geometrical function (Eq. (20)) derived from the results of the measurements. Impellers with a large blade area in relation to the tank dimensions produce less shear, because of their uniform power input, in contrast to small and especially axial-flow impellers, such as propellers, and all kinds of inclined-blade impellers. 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

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