Boundary-layer theory scaling

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. 

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... 

T temperature scale in turbulent boundary layer theory (—)... 

Abstract. Interaction between a current and surface-active material is considered. Some simple cases where the substrate motion is steady and 2D is analysed using standard boundary layer theory. Questions like how is the transversal dimension of a slick related to the film pressure and the substrate convergence and how strong substrate motion does it take to break up a surface film , are addressed. It is pointed out that the answers depend on whether the film can be considered stagnant, or develops self-organised motion. It is further pointed out how small scale thermal convection at the ocean surface is easily suppressed by a slick. 

Hence, an equivalent form of the previous scaling law, 5c A.mix ", is Sc l/(Re Sc) where m = for boundary layer theory adjacent to a solid-liquid interface in the creeping flow regime, and m = for gas-liquid interfaces. As expected, the boundary layer thickness at any position along the interface decreases at higher flow rates and increases when the diffusivity is larger. Since... 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

Generalized kinetic energy in Lagrangian mechanics Temperature scale in turbulent boundary layer theory (—) Dimensionless temperature in turbulent boundary layer theory (—)... 

Two other historical asides about this result are interesting. First, the dimensionless quantities b and b suggested by Reynolds were renamed y-factors by Chilton and Colburn. These factors are common in the older literature, especially as Jd and Jh. Second, the exponent of on the Schmidt and Prandtl number is frequently subjected to theoretical rationalization, especially using boundary-layer theory. Chilton is said to have cheerfully conceded that the value of was not even equal to the best fit of the data, but was chosen because the slide rules in those days had square-root and cube-root scales, but no other easy way to take exponents. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, "separation of variables (not the classical concept) and the use of "continuous transformation groups. The basic theory is available in Ames (1965). 

The analyses of Hunt, Liebovich and Richards, 1988 [287] and of Finnigan and Belcher, 2004 [189] divide the flow in the canopy and in the free boundary layer above into a series of layers with essentially different dynamics. The dominant terms in the momentum balance in each layer are determined by a scale analysis and the eventual solution to the flow held is achieved by asymptotically matching solutions for the flow in each layer. The model apphes in the limit that H/L 1. By adopting this limit, Hunt, Liebovich and Richards [287] were able to make the important simplification of calculating the leading order perturbation to the pressure held using potential how theory. This perturbation to the mean pressure, A p x, z), can then be taken to drive the leading order (i.e. 0(II/I.) ]) velocity and shear stress perturbations over the hill. 

In spite of this, we shall see that potential-flow theory plays an important role in the development of asymptotic solutions for Re i>> 1. Indeed, if we compare the assumptions and analysis leading to (10-9) and then to (10-12) with the early steps in analysis of heat transfer at high Peclet number, it is clear that the solution to = 0 is a valid first approximation lor Re y> 1 everywhere except in the immediate vicinity of the body surface. There the body dimension, a, that was used to nondimensionalize (10-1) is not a relevant characteristic length scale. In this region, we shall see that the flow develops a boundary layer in which viscous forces remain important even as Re i>> 1, and this allows the no-shp condition to be satisfied. 

The conclusion to be drawn from the preceding discussion is that the potential-flow theory (10-9) [or, equivalently, (10 12) and (10 13)] does not provide a uniformly valid first approximation to the solution of the Navier Stokes and continuity equations (10-1) and (10 2) for Re 1. Furthermore, our experience in Chap. 9 with the thermal boundary-layer structure for large Peclet number would lead us to believe that this is because the velocity field near the body surface is characterized by a length scale 0(aRe n), instead of the body dimension a that was used to nondimensionalize (10-2). As a consequence, the terms V2co and u V >, in (10 6), which are nondimensionalized by use of a, are not 0(1) and independent of Re everywhere in the domain, as was assumed in deriving (10-7), but instead are increasing fimctions of Re in the region very close to the body surface. Thus in... 

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