Internal Flows

Lecture 27, March 12, 2004 • Reminder Assignment 3 now due Monday Today • Overview of ch7 • Begin chapter 8 – Internal Flows Internal flows are constrained by walls entirely, instead of being a surface exposed to a free stream flow. We have a lot of the background to understand the ramifications of this, but we need to consider a few key differences. The Reynolds number still governs the flow regime (laminar/turbulent), but it is based on a different length scale, the pipe diameter (or hydraulic diameter for non-circular sections). This results in a very different number for transition to turbulence.

Turbulence can be expected for Re greater than about 2300 using this length scale, as compared to 500,000 when we used the distance along the plate as a length scale. For constant density flow, conservation of mass dictates that Um must remain constant.

Clearly the no slip condition at the wall means that umax > Um. Consider a developing flow, as in the figure on the board. Once the boundary layers from opposite walls meet, the flow is considered fully developed and the velocity profiles no longer change in the flow direction. There is still a pressure gradient though, as the pressure forces must balance the friction forces on the pipe walls (there are no other significant forces acting). If the velocity profiles in and out are the same, then the momentum flux in is the same as the momentum flux out, and conservation of momentum (Newton’s Second Law) reduces to a force balance.