Boundary layer thickness turbulent flow

Even when the flow in the body of the boundary layer is turbulent, flow remains laminar in the thin layer close to the solid smface, the so-called laminar sub-layer. Indeed, the bulk of the resistance to momentum, heat and mass transfer hes in this thin film and therefore interphase heat and mass transfer rates may be increased by decreasing its thickness. As in pipe flow, the laminar sub-layer and the turbulent region are separated by a buffer layer in which viscous and inertial effects are of comparable magnitudes, as shown schematically in Figure 7.1. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

The boundary layer thickness gradually increases until a critical point is reached at which there is a sudden thickening of the boundary layer this reflects the transition from a laminar boundary layer to a turbulent boundary layer. For both types, the flow outside the boundary layer is completely turbulent. In that part of the boundary layer near the leading edge of the plate the flow is laminar and consequently this is known as a... 

In terms of hydrodynamics, the boundary layer thickness is measured from the solid surface (in the direction perpendicular to a particle s surface, for instance) to an arbitrarily chosen point, e.g., where the velocity is 90-99% of the stream velocity or the bulk flow ((590 or (599, respectively). Thus, the breadth of the boundary layer depends ad definitionem on the selection of the reference point and includes the laminar boundary layer as well as possibly a portion of a turbulent boundary layer. 

Apart from the nature of the bulk flow, the hydrodynamic scenario close to the surfaces of drug particles has to be considered. The nature of the hydrodynamic boundary layer generated at a particle s surface may be laminar or turbulent regardless of the bulk flow characteristics. The turbulent boundary layer is considered to be thicker than the laminar layer. Nevertheless, mass transfer rates are usually increased with turbulence due to the presence of the viscous turbulent sub-layer. This is the part of the (total) turbulent boundary layer that constitutes the main resistance to the overall mass transfer in the case of turbulence. The development of a viscous turbulent sub-layer reduces the overall resistance to mass transfer since this viscous sub-layer is much narrower than the (total) laminar boundary layer. Thus, mass transfer from turbulent boundary layers is greater than would be calculated according to the total boundary layer thickness. 

Boundary Layer. There exists a quiescent boundary layer through which the organotin species must diffuse before being carried by the sea water flow past the surface of the coating. The boundary layer under laminar flow conditions and under turbulent flow conditions are quantitatively defined (9) the thickness of the layer (L) decreases as the fluid velocity increases. 

Equation (4) states that the linear deposition rate vj is a diffusion controlled boundary layer effect. The quantity Ac is the difference in foulant concentration between the film and that in the bulk flow and c is an appropriate average concentration across the diffusion layer. The last term approximately characterizes the "concentration polarization" effect for a developing concentration boundary layer in either a laminar or turbulent pipe or channel flow. Here, Vq is the permeate flux through the unfouled membrane, 6 the foulant concentration boundary layer thickness and D the diffusion coefficient. 

Other factors do intervene. Significant solar heating of the soil surface, so that the soil becomes warmer than the air, causes vertical thermal convection currents to develop within the boundary layers. This introduces turbulence or instability that acts to move the chemical signature up into the free air. When the molecules are moved into the free flow of the air, the effect is to reduce the concentration by dilution. Conversely, when the soil surface is cooler than the air, thermal convection is inhibited, with the result that the molecules are effectively trapped in the boundary layer. This effect is strengthened by the cooling of the air adjacent to the surface, which increases its viscosity. Higher viscosity lowers the Reynold s number, thus decreasing boundary layer thickness. 

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 solid 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 boundary layer thickness is conventionally taken 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 simplified by scaling arguments. Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

As with the program for laminar boundary layer flow discussed in Chapter 3, the turbulent boundary flow program calculates the velocity and temperature boundary layer thicknesses using, as discussed above, the assumption that A is the value of Y at which U = 0.99U and At is the value of Y at which 0 = 0.010. If either A or At is greater than the number of grid points is increased by ten. 

For sufficiently large electrodes with a small vibration amplitude, aid < 1, a solution of the hydrodynamic problem is possible [58, 59]. As well as the periodic flow pattern, a steady secondary flow is induced as a consequence of the interaction of viscous and inertial effects in the boundary layer [13] as shown in Fig. 10.10. It is this flow which causes the enhancement of mass-transfer. The theory developed by Schlichting [13] and Jameson [58] applies when the time of oscillation, w l is small in comparison with the time taken for a species to diffuse across the hydrodynamic boundary layer (thickness SH= (v/a>)ln diffusion timescale 8h/D), i.e., when v/D t> 1. Re needs to be sufficiently high for the calculation to converge but sufficiently low such that the flow does not become turbulent. Experiment shows that, for large diameter wires (radius, r, — 1 cm), the condition is Re 2000. The solution Sh = 0.746Re1/2 Sc1/3(a/r)1/6, where Sh (the Sherwood number) = kmr/D and km is the mass-transfer coefficient,... 

Figure 7-6. Schematic illustration of originally nonturbulent air (straight anrows in upwind side on left) flowing over the top of a flat leaf, indicating the laminar sublayer (shorter straight anrows), the turbulent region (curved arrows), and the effective boundary layer thickness, 5bl. The length of an arrow indicates the relative speed, and the curvature indicates the local direction of air movement. A similar airflow pattern occurs on the lower leaf surface.

The simulations were performed assuming that the flow is laminar. Additionally, the contact angle is assumed to be known. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, 5j, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, Jj. =1. 4(vfiCil ... 

Eq. (3.150) represents the wall law for turbulent flow, first formulated by Prandtl in 1925. The functions f(y+) and g(y+) are of a universal nature, because they are independent of external dimensions such as the height of a channel and are valid for all stratified flows independent of the boundary layer thickness. 

We have seen how heat transfer and thus dry deposition of SO2 is reduced on large surfaces, due to the buildup of boundary layer thickness (which reduces the local gradients). However, there are economically important structural objects composed of many elements of small dimension which show the opposite effect. These include fence wire and fittings, towers made of structural shapes (pipe, angle iron, etc.), flagpoles, columns and the like. Haynie (11) considered different damage functions for different structural elements such as these, but only from the standpoint of their effect on the potential flow in the atmospheric boundary layer. The influence of shape and size act in addition to these effects, and could also change the velocity coefficients developed by Haynie (11), which were for turbulent flow. Fence wire, for example, as shown below, is more likely to have a laminar boundary layer. 

In practice, however there could be differences between the observed and estimated flux. The mass transfer coefficient is strongly dependent on diffusion coefficient and boundary layer thickness. Under turbulent flow conditions particle shear effects induce hydrodynamic diffusion of particles. Thus, for microfiltration, shear-induced difflisivity values correlate better with the observed filtration rates compared to Brownian difflisivity calculations.Further, concentration polarization effeets are more reliably predicted for MF than UF due to the fact diat macrosolutes diffusivities in gels are much lower than the Brownian difflisivity of micron-sized particles. As a result, the predicted flux for ultrafiltration is much lower than observed, whereas observed flux for microfilters may be eloser to the predicted value. 

Structure of the flow. Velocity profile. The flow in the boundary layer on a flat plate is laminar until Rex = U[X/v 3 x 105. On a longer plate, the boundary layer becomes turbulent, that is, its thickness increases sharply and the longitudinal velocity profile alters. 

Concentration polarization cannot be eliminated, but it can be minimized by decreasing boundary layer thickness. This is done by increasing the flow rate across the membrane surface or introducing turbulence promoters into the feed/reject stream. In order to achieve optimum performance, most membrane manufacturers will recommend a minimum feed rate to or from their elements and a maximum recovery in order to minimize the effects of concentration polarization. 

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