Entrance effect and entrance Length

Principle

Fig. 1 Flow at the entrance to a tube

Consider a flow entering a tube and suppose that the entering flow is uniform, so inviscid. As soon as the flow 'hits' the tube many changes take place. The most important of these is that viscosity imposes itself on the flow and the friction at the wall of the tube comes into effect. Consequently the velocity along the wall becomes zero (also in tangential direction). 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 tube. 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. 1 it is seen that the boundary layer from the walls grows to such an extent that they all merge on the centre line of the tube. 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 is 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 tube 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 Re of the flow.

The distance needed to restore a laminar flow to a parabolic Poiseuille flow (see Poiseuille's Law) is called the entrance length, being:

L_e,laminar = 0.06D∙Re,

where D is the tube diameter.

To restore a turbulent flow to parabolic flow, the entrance length is by approximation:

L_e,turbulent = 4.4D∙Re^1/6.

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=4000) to 95 (at Re=10⁸).

Application

Hemodynamics and flow in the airways system (see Flow through a stenosis, Flow in curvatures and Flow in bifurcations).

Numerical example Supposing that the aorta diameter is 3 cm and Re is 4000, then L_e,turbulent = 53 cm. This is much more than the first large bifurcations after the aortic arch. Also when we take (conceptually) into account the bifurcations in the aorta bend (which will disturb the process of restoration), the pulsatile character of the aorta flow and the compliance of the wall.

Source: http://www.aeromech.usyd.edu.au/aero/fprops/pipeflow/node9.html