For boundary layers

The most widely used approach to channel flow calculations assumes a steady qua si-one-dimensional flow in the channel core, modified to account for boundary layers on the channel walls. Electrode wall and sidewall boundary layers may be treated differently, and the core flow may contain nonuniformities. 

Special Tubes A variety of special forms of the pitot tube have been evolved. Folsom (loc. cit.) gives a description of many of these special types together with a comprehensive bibliography. Included are the impact tube for boundary-layer measurements and shielded total-pressure tubes. The latter are insensitive to angle of attack up to 40°. 

Fig. C.l Exact solutions for the mathematical prototype for boundary-layer behavior. The solution shown is for varying values of the parameter e and for a = 0.4. Also shown is the solution to the reduced outer equation that does not satisfy the boundary condition at x=0.

Fig. C.2 Comparison of the inner solution with the exact solution for the mathematical prototype equation for boundary-layer behavior.

Since terms of the order of magnitude of (8/L)2 and smaller are being neglected, it follows from this equation that the y-momentum equation for boundary layer flow... 

Coordinate system used for boundary layer flow over a curved surface. 

The boundary layer integral equations have been derived above without recourse to the partial differential equations for boundary layer flow. They can, however, be determined directly from these equations. Consider, for example, the laminar momentum equation (2.140). Integrating this equation across the boundary layer to some distance from the wall, i being greater than the boundary layer thickness, gives because du/dy is zero outside the boundary layer and because dp/dx is independent of y ... 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

For boundary layer flows on bodies of other shape, the transition Reynolds number based on the distance around the surface from the leading edge of the body is usually increased if the pressure is decreasing, i.e., if there is a favorable pressure gradient, and is usually decreased if the pressure is increasing, i.e., if there is an unfavorable pressure gradient. 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

In the numerical solution for boundary layer flow given in this chapter it was assumed that transition occurred at a point i.e., the eddy viscosity was set equal to zero up to the transition point and then the full value given by the turbulence model was used. Show how this numerical method and the program based on it can be modified to allow for a transition zone in which the eddy viscosity increases linearly from zero at the beginning of the zone to the full value given by the turbulence model at the end of the zone. 

The ideas used in the previous chapter to derive analogy solutions for boundary layer flows can easily be extended to obtain such analogy solutions for turbulent 304... 

The same assumptions as were used in deriving the Reynolds analogy for boundary layer flow are now introduced, i.e., it is assumed that ... 

This is the basic Reynolds analogy equation for pipe flow. It is identical to that for boundary layer flow except that the mean velocity and temperature occur in place of the free-stream values. 

A consideration of the orders of magnitude of the terms in the momentum equation for boundary layer flow indicates that if u = o(u ), where u is characteristic free-stream velocity, then the buoyancy force term will be important if ... 

There have been significant contributions initially made by Helmholtz, Kelvin and Rayleigh (1880, 1887) using inviscid analysis. In their quest to justify their inviscid analysis, an assumption was made that viscous action due to its dissipative nature can be only stabilizing. Such was the impact of this observation that when Heisenberg (1924) submitted his dissertation solving perturbation equations including viscous terms for boundary layer... 

Q = [a +iaRe —c)Y/ . For boundary layer instability problems, i e —> 00 and then Q >> laj. This is the source of stiffness that makes obtaining the numerical solution of (2.3.21) a daunting task. This causes the fundamental solutions of the Orr-Sommerfeld equation to vary by different orders of magnitude near and far away from the wall. This type of behaviour makes the governing equation a stiff differential equation that suffers from the growth of parasitic error, while numerically solving it. 

Sengupta, T.K. and Sinha, A.P. (1995). Surface mass transfer A receptivity mechanism for boundary-layers. In Proc. 6 Asian Cong. Fluid Mech., (Eds. Y.T. Chew and C.P. Tso), 1242-1245. 

To enter or leave a leaf, the molecules must diffuse across an air boundary layer at the leaf surface (boundary layers are discussed in Chapter 7, Section 7.2 also see Fig. 7-6). As a starting point for our discussion of gas fluxes across such air boundary layers, let us consider the one-dimensional form of Fick s first law of diffusion, Jj = —Djdcj/dx (Eq. 1.1). As in Chapter 1 (Section 1.4B), we will replace the concentration gradient by the difference in concentration across some distance. In effect, we are considering cases that are not too far from equilibrium, so the flux density depends linearly on the force, and the force can be represented by the difference in concentration. The distance is across the air boundary layer adjacent to the surface of a leaf, 5bl (Chapter 7, Section 7.2, presents equations for boundary layer thickness). Consequently, Fick s first law assumes the following form for the diffusion of species across the boundary layer ... 

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