Boundary layer transition length

Air flows over a wide t-m long flat plate which has a uniform surface temperature of 80°C, the temperature of the air ahead of the plate being 20°C. The air velocity is such that the Reynolds number bas J on the length of the plate is 5 x 106. Derive an expression for the local wall heat flux variation along the plate. Use the Reynolds analogy and assume the boundary layer transition occurs at a Reynolds number of 10 ... 

For the inlet length of a pipe in which the boundary layers are forming, the equations in the previous section will give an approximate value for the heat transfer coefficient. It should be remembered, however, that the flow in the boundary layer at the entrance to the pipe may be streamline and the point of transition to turbulent flow is not easily defined. The results therefore are, at best, approximate. 

The cause of this initial smooth zone and the subsequent fairly sudden transition to wavy flow is not completely clear. Working on a much larger scale, with mostly turbulent flow of the liquid layers on dam spillways, Bauer (Bl) has shown that the length of the smooth initial region is the same as the distance required for the turbulent boundary layer, which... 

Adams, J.C., Jr and Hodge, B.K., The Calculation of Compressible, Transitional, T ir-bulent, and Relaminarizational Boundary Layers over Smooth and Rough Surfaces Using an Extended Mixing Length Hypothesis , AIAA Paper 77-682, Albuquerque, NM, 1977. 

Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48fe is applicable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE J., 7, 26-28, 221-225, 467- 72 [1961]). The critical-length Reynolds number for transition is Re, = 200,000. The laminar boundary layer thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49 the drag and entrainment rate are obtained from the momentum area 0 and displacement area A evaluated at x = L. 

Consider the parallel flow of a fluid over a flat plate of length L in the flow direction, as shown in Fig. 7-6. The. t-coordinate is measured along the plate surface from the leading edge in the direction of the flow. The fluid approaches the plate in the, T-direction with a uniform velocity V and temperature T. The flow in the velocity boundary layers starts out as laminar, but if the plate is sufficiently long, the flow become.s turbulent at a distance. r r from the leading edge where the Reynolds number reaches its criiical value for transition. 

Transition length for laminar and turbulent flow. The length of the entrance region of the tube necessary for the boundary layer to reach the center of the tube and for fully developed flow to be established is called the transition length. Since the velocity varies not only with length of tube but with radial distance from the center of the tube, flow in the entrance region is two dimensional. 

If the laminar and transitional zones are short, the turbulent boundary layer can be assumed to begin at the leading edge of the plate. In this case, for surface length Reynolds number of a few million, Eq. 6.163 can be employed in Eq. 6.166 to establish the following simple relationship ... 

Now, to ascertain how large an error we made by assuming that the entire boundary layer jwas turbulent, we assume that transition from laminar to turbulent flow takes place at an of 10. From the above, this corresponds to a distance of of the length of the plate so the boundary layer over the first 0.2 ft presumably is laminar. For this area the drag due to a laminar boundary layer is given byiEq. 11.19 as... 

If the velocity profile at the entrance region of a tube is flat, a certain length of the tube is necessary for the velocity profile to be fully established. This length for the establishment of fully developed flow is called the transition length or entry length. This is shown in Fig. 2.10-6 for laminar flow. At the entrance the velocity profile is flat i.e., the velocity is the same at all positions. As the fluid progresses down the tube, the boundary-layer thickness increases until finally they meet at the center of the pipe and the parabolic velocity profile is fully established. 

The velocity profile is dictated by a boundary layer that moves out from the wall as illustrated in figure 6. The form of the equation governing this velocity-profile development means that just below turbulent flow at a Reynolds number of say 2300, a length of 138 diameters is needed to achieve fully-developed flow. (However for turbulent flow—see next section—lengths are much shorter because the relationship is given by L/d = 4.4 so that just above the transition to turbulence, we only need 16 diameters to obtain fuUy developed flow [7].)... 

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