Boundary-layer theory separation

Several results may be derived from the use of boundary layer theory depending upon the velocity profile assumed to exist outside the boundary layer. Lee and Barrow (LIO) used the velocity profile of Tomotika (T3) which was, in turn, fitted to the surface pressure data of Fage (FI) at Re = 157,000. This profile predicts separation at d = 8L as noted in Section A.l. 

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... 

Peridier, V.J., Smith, F.T. and Walker, J.D.A. (1991a). Vortex-induced boundary-layer separation. Part 2. Unsteady interacting boundary-layer theory. J. Fluid Mech. 232, 133-165. 

The boundary layer theory has been widely accepted and used to describe the transport phenomena in CVD processes. High-performance CVD systems require designers to focus on the geometrical parameters of the reaction chamber, the orientation and arrangement of the preforms in the chamber, as well as some other important components, such as pipes, distributor, exit and so forth. Due to drag effects around the boundary layer of preforms, it is very important to design the preforms and the reaction chamber and aim to avoid the boundary layer separation such that they experience a minimum drag force. The details of these effects are discussed in Chapter 6. 

The reader may wonder why this change in flow structure invalidates the boundary-layer theory. Upstream of the separation point, the flow still divides into a boundary layer of the usual kind on the surface and an outer, potential flow. However, the potential-flow approximation applies only outside any region of nonzero vorticity. When these regions all... 

Re < 40, and the predicted pressure distribution from potential-flow theory. The ability of the boundary-layer theory to predict separation is probably its most important characteristic. 

This quantity was plotted in Fig. 10-3.21 Also shown is the measured pressure distribution for Re in the range 30 to 40. It is evident that the measured and predicted distributions are in close agreement over the front portion of the cylinder. Thus it is not surprising that boundary-layer theory, based on the potential-flow pressure distribution, should be quite accurate up to the vicinity of the separation point. It is this fact that explains the ability of boundary-layer theory to provide a reasonable estimate of the onset point for separation, as we shall demonstrate shortly. 

Before we turn to other topics, it is worth considering the physical events that lead to separation. There are two plausible ways to explain the phenomenon. A common feature of these two mechanisms is that viscous effects play a critical role. Experimentally, we find that separation (or at least the downstream recirculating wake that we associate with separation) usually occurs for Reynolds numbers larger than some critical value, and we infer from this that separation is basically a large-Reynolds-number phenomenon. Indeed, this point of view is consistent with the fact that separation can be predicted by boundary-layer theory, which is an asymptotic theory for Re -> oo. In spite of this, separation is a phenomenon that we can explain only by considering the consequences of viscous contributions to the motion of the fluid. 

Problem 10-12. Higher-Order Approximations for the Blasius Problem. The classical boundary-layer theory represents only the first term in an asymptotic approximation for Re 1. However, in cases involving separation, we do not seek additional corrections because the existence of a separation point signals the breakdown of the whole theory. When the flow does not separate, we can calculate higher-order corrections, and these provide useful insight and results. In this problem, we reconsider the familiar Blasius problem of streaming flow past a semi-infinite flat plate that is oriented parallel to a uniform flow. 

For many real flows, the streamlines separate from the body around which they are flowing. This results in the formation of eddying wakes, low pressure behind the body, and large drag forces. This is not predictable by perfect-fluid theory, but it can be approached through boundary-layer theory. 

In Sec. 10 8 we discussed separation. In almost every case, separation results in a great increase in drag, which is normally a very undesirable result. From boundary-layer theory it is possible to make some good estimates of when separation will or will not occur. In particular, it can be shown that... 

Anonymous (1966). Clark B. Millikan. Journal of Aeronautics and Astronautics 4(1) 14-15. P Karman von, T., Millikan, C.B. (1934). On the theory of laminar boundary layers involving separation. US Government Printing Washington DC. 

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). 

Figure 8.5 shows a Venturi meter. The theory is the same as for the orifice meter but a much higher proportion of the pressure drop is recoverable than is the case with orifice meters. The gradual approach to and the gradual exit from the orifice substantially eliminates boundary layer separation. Thus, form drag and eddy formation are reduced to a minimum. 

We also need to develop the theories for hquid film coefficient to use in the aforementioned equations. For drops that are close to spherical, without separation, Levich (1962) assumed that the concentration boundary layer developed as the bubble interface moved from the top to the bottom of a spherical bubble. Then, it is possible to use the concepts applied in Section 8.C and some relations for the streamlines around a bubble to determine Kl. ... 

There are some cases where this approach fails. One such case is that in which significant regions of separated flow exist. In this case, although the boundary layer equations are adequate to describe the flow upstream of the separation point, the presence of the separated region alters the effective body shape for the outer inviscid flow and the velocity outside the boundary layer will be different from that given by the inviscid flow solution over the solid surface involved. For example, consider flow over a circular cylinder as shown in Fig. 2.16. Potential theory gives the velocity, ui, on the surface of the cylinder as ... 

---