Entrance Length

Figure 7.4: Flow at  the entrance to a pipe

Consider a flow entering a pipe. Let us think of the entering flow being uniform, so inviscid. As soon as the flow 'hits' the pipe many changes take place. The most important of these is that viscosity imposes itself on the flow and the "No Slip" condition at the wall of the pipe comes into effect. Consequently the velocity components are each zero on the wall, ie., u = v = 0. The flow adjacent to the wall decelerates continuously. We have a layer close to the body where the velocity builds up slowly from zero at wall to a uniform velocity towards the center of the pipe. This layer is what is called the Boundary Layer. Viscous effects are dominant within the boundary layer. Outside of this layer is the inviscid core where viscous effects are negligible or absent.

The boundary layer is not a static phenomenon; it is dynamic. it grows meaning that its thickness increases as we move downstream. From Fig. it is seen that the boundary layer from the walls grows to such an extent that they all merge on the centreline of the pipe. Once this takes place, inviscid core terminates and the flow is all viscous. The flow is now called a Fully Developed Flow. The velocity profile becomes parabolic. Once the flow is fully developed the velocity profile does not vary in the flow direction. In fact in this region the pressure gradient and the shear stress in the flow are in balance. The length of the pipe between the start and the point where the fully developed flow begins is called the Entrance Length. Denoted by L_e, the entrance length is a function of the Reynolds Number of the flow. In general,

$${L_e \over d} \approx 0.06 Re_d,    \texttt{for a Laminar Flow}.$$ (7.1)

$${L_e \over d} \approx 4.4 Re_d^{1/6},    \texttt{for a Turbulent Flow}.$$ (7.2)

At critical condition, i.e., Re_d =2300, the L_e/d for a laminar flow is 138. Under turbulent conditions it ranges from 18(at Re_d = 4000) to 95 (at Re_d=10⁸)

Application decides whether a long enough entrance length is required or a shorter one is required. Take wind tunnel for instance. Here the aim is to have a uniform flow over the model in the test section. So one desires to have a longer entrance length.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney