412 are geometrically similar led to the notion of aself similar boundary layer

412 are geometrically similar led to the notion of

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4.12 are “geometrically similar” led to the notion of a self-similar boundary layer and use of a mathematical technique known as a similarity transformation to reduce the above PDEs to a single ordinary differential equation boundary-value problem. This resulting equation, however, does not possess an exact solution, but it is so easily solved to arbitrarily high accuracy on a computer that the boundary-layer solution that results is often considered exact. Further Discussion of Pipe Flow We can now return to our discussion of flow in a circular pipe with a better understanding of what is occurring in the entrance region of the pipe. The preceding description of boundary-layer flow over a flat plate is applicable in a basic sense to this entrance flow until the boundary layers extend an appreciable distance out from the pipe walls. In particular, it is reasonable to expect that as long as the boundary-layer thickness satisfies δ R the boundary-layer approximations given above will be valid. This, of course, holds only very near the pipe entrance; nevertheless, flow development farther from the entrance is still strongly influenced by boundary-layer growth, but now velocity profiles outside the boundary layer are also adjusting—unlike the situation with the flat plate boundary layer discussed above. In any case, we can now describe the entrance length in pipe flow as the required distance in the flow direction for the “boundary layers” from opposite sides of the pipe to merge. This distance
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4.5. PIPE FLOW 129 cannot be predicted exactly, but many experiments have been conducted to determine it under various flow conditions. There are two important results to emphasize. First, for laminar flow it is known that L e 0 . 06 DRe , (4.33) where D is the pipe diameter, and for turbulent flow the entrance length can be anywhere in the range of 20 to 100 pipe diameters, with the following formula sometimes given: L e = 4 . 4 DRe 1 / 6 . We observe that in both laminar and turbulent flows one can define a dimensionless entrance length, L e /D , that is simply a power of the Reynolds number. Beyond this distance the flow is said to be fully developed . Fully-developed flow can be charac- terized by three main physical attributes: i ) “boundary layers” from opposite sides of the pipe have merged (and, hence, can no longer continue to grow); ii ) the streamwise velocity component satisfies u z = 0; iii ) the radial (or in the case of, e.g. , square ducts, the wall-normal) component of velocity is zero, i.e. , v = 0. These flow properties, especially the latter two, will be very important in the sequel as we attempt to solve the N.–S. equations for the problem of pipe flow. 4.5.2 The Hagen–Poiseuille solution We now derive another exact solution to the N.–S. equations, subject to the following physical assumptions: steady, incompressible, axisymmetric, fully-developed, laminar flow. We will use the result to arrive at formulas useful in practical pipe flow analyses and then employ these, with some modifications based on experimental observations, to treat flows in pipes having irregularly-shaped
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