Boundary layers laminar flow

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

Cylindrical Boundary Layer Laminar boundary layers on cylindrical surfaces, with flow parallel to the cylinder axis, are described by Glauert and LighthiU Proc. R. Soc. [London], 230A, 188-203 [1955]), Jaffe and Okamura (Z. Angety. Math. Phys., 19, 564—574 [1968]) and Stewartson ((J. Appl Math., 13, 113-122 [1955]). For a turbulent boundaiy layer, the total drag may be estimated as... 

Numerically determine the heat transfer rate variation with two-dimensional laminar boundary layer air flow over a fiat plate with a uniform heat flux at the surface. Compare the numerical results with those given by the similarity solution. 

Laminar and turbulent regions of the boundary layer during flow over aflat plate. 

As Re increases, the adjacent vortices are elongated and shedding of vortices occurs (Karman s vortices are formed). Finally, for Re > 1000, the remote wake becomes completely turbulent [117]. At the same time, the separation point moves toward the midsection and even a bit farther upstream. For such values of Re, we can speak about a pronounced boundary layer. In a large part of the boundary layer, the flow remains laminar [486], Strong turbulence within the boundary layer occurs for considerably higher Reynolds numbers (Re 2x 105), at which the cylinder drag drops rapidly [117], This phenomenon is called the drag crisis. 

Starting from Re = 0.5 x 103, one can speak of a developed hydrodynamic boundary layer. The flow remains laminar in a considerable part of this layer [486]. If the Reynolds number varies in the range 0.5 X 103 < Re < 0.5 x 105, the separation point 6S of the laminar boundary layer gradually moves from 71.2° to 95° [427,486],... 

Figures 6.2 and 6.3 indicate the order of magnitude of concentration polarization for laminar and turbulent flows through tubular membranes. The diagrams illustrate the dependence of the concentration boundary layer on flow conditions along the membrane (Re) and on the permeation flux (Pew).

It is essential for the rotating-disc that the flow remain laminar and, hence, the upper rotational speed of the disc will depend on the Reynolds number and experimental design, which typically is 1000 s or 10,000 rpm. On the lower lunit, 10 s or 100 rpm must be applied in order for the thickness of tlie boundary layer to be comparable to that of the radius of the disc. 

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

Fig. 5. Entrance flows in a tube or duct (a) separation at sharp edge (b) growth of a boundary layer (illustrated for laminar flow).

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. 

Continuous Flat Surface Boundaiy layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48 7 shows the continuous flat surface (Saldadis, AIChE J., 7, 26—28, 221-225, 467-472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. FluidMech., 26,145—161 [1966]). For a laminar boundary layer, the thickness is given by... 

For a gas in laminar flow over a condensed phase sample of length L, the mass transport across the boundary layer, in terms of the flux of molecules from the sample to die gas phase, is therefore... 

In a laminar boundary layer, no mixing takes place and the flow is parallel. In this case the heat transfer occurs mainly by conduction through the boundary layer. 

The preceding discussion has attempted to formulate the situation for laminar boundary layer flow as accurately as possible and to obtain precise correlation between the heat transfer and mass transfer factors. 

This obviously implies that the skin friction exerted on an airplane wing or body will depend on whether the boundary layer on the surface is laminar or turbulent, with laminar flow yielding the smaller skin friction drag. 

This form of attack, especially as affecting copper alloys in sea water, has been widely studied since the pioneer work of Bengough and May . Impingement attack of sea water pipe and heat exchanger systems is considered in Sections 1.6 and 4.2. In such engineering systems the water flow is invariably turbulent and the thickness of the laminar boundary layer is an important factor in controlling localised corrosion. 

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