Flow pipe, boundary layer theory

BOUNDARY LAYER THEORY APPLIED TO PIPE FLOW 11.5.1. Entry conditions... 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

Hughmark employed this u to derive a correlation for Son and Hanratty (1967) and Hughmark (1971,1974) correlated wall to fluid heat transfer in pipe flow based on the relatively simple and well-established boundary layer theory. In the case of pipe flow, momentum transfer is solely by skin friction because of the geometry involved. Nonetheless, this approach was extended to particle-fluid mass transfer in turbulent flow. The correlation proposed was of the following form ... 

Prandtl, [43] using boundary layer theory, mixing length hypothesis and law of the wall, developed the following theoretical law of friction for smooth pipes in turbulent flow... 

Adler measured the velocity distribution in a coiled pipe for laminar flow [86]. He found that the velocity profile differed considerably from the parabolic one which was due to the existence of secondary flow field. Using the boundary layer theory he derived the following relationship for relatively high flow rate but laminar range. 

The friction factor for single phase flows is presented in terms of boundary layer theory. Losses in pipes are summarized in terms of fittings and conduits commonly used on large engineering projects. Numerous books have been written for single-phase flows. This chapter limits Itself to a brief introduction. 

Boundary layer theory has been extensively covered by a number of authors. A book by Schlichting (1968) is considered one of the classical references on this subject. When a uniform flow approaches a plate, the particles at the wall of the plate are slowed down by the dynamic viscosity of the fluid. A layer called the boundary layer develops. When the flow enters a pipe, effects develop at the entrance until the flow is uniform. 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

Flow around single cylinders is the elementary model for (he fibrous filter and is the geometry of interest for deposition on pipes, wires, and other such objects in an air flow (Chapter 3). The flow patterns at low and high Reynolds numbers differ significantly, and thi.s affects impaction efficiencies. For Re > 100. the velocity distribution outside the velocity boundary layer can be approximated by inviscid flow theory. This approximates the velocity distribution best over the front end of the cylinder which controls the impaction efficiency. The components of the velocity in the direction of the mainstream flow, x, and normal to the main flow, y, are... 

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