Question 11 of 13: Why the Moody Friction Factor Jumps at the Laminar–Turbulent Transition

Question 11: Why the Moody Friction Factor Jumps at the Laminar–Turbulent Transition (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Approaching the critical Reynolds number (Re ≈ 2300) from below, laminar flow follows the smooth relation f = 64/Re; the friction factor is falling steadily as Re rises. The instant the flow becomes turbulent, however, f jumps to a value close to double what the laminar formula would have given at that same Re — then continues along the (much higher) turbulent curve. This is fundamentally different from the drag-coefficient chart on page 9, where C_D for a sphere or ellipsoid changes gradually with Reynolds number over most of its range (apart from its own drag-crisis dip at very high Re) because the flow pattern around a bluff body evolves continuously as the boundary layer thickens and separation shifts.

The physical reason for the pipe-flow discontinuity is a genuine change in the underlying momentum-transport mechanism, not a gradual reshaping of an existing one. In laminar flow, momentum is transferred across the pipe purely by viscous shear between adjacent fluid layers sliding smoothly past one another; wall shear stress, and hence f, depends only on the (smoothly varying) velocity gradient at the wall. Once the flow becomes turbulent, chaotic eddies begin transporting momentum radially across the pipe far more effectively than viscous diffusion alone — this new transport mechanism switches on abruptly once disturbances in the flow can no longer be damped out by viscosity, rather than growing in strength gradually from zero. The near-doubling of f reflects that a genuinely different, much more effective momentum-transfer process has taken over, not a small perturbation of the laminar one.

Implication for a pipe at constant flow rate. If the SAME volumetric flow rate is passed through the pipe under laminar conditions versus turbulent conditions, the head loss hf = f(L/D)(V²/2g) very nearly doubles the instant the flow becomes turbulent, even though V (and hence the flow rate) has not changed at all. In a real system this has two consequences: (1) if the driving head is fixed (e.g. a constant reservoir level), the pipe cannot actually deliver the same flow rate in both regimes — the flow rate itself must drop when the higher-f turbulent value applies, since a larger loss cannot be sustained by an unchanged available head; and (2) if a pump is instead sized to guarantee a fixed flow rate regardless of regime, that pump must supply roughly double the friction-loss head (and hence proportionally more power, since pumping power scales with H×Q) the moment the flow tips into turbulence. Operating a system right at the critical Reynolds number is therefore inherently unstable: small disturbances that nudge the flow between laminar and turbulent produce a disproportionate jump in energy dissipation, so pipeline designers deliberately avoid sizing a system to operate near Re ≈ 2300.

Energy accounting. No energy is gained anywhere in this process — only lost, and lost at a step-change rate. The "jump" in f is entirely a jump in how much of the flow's mechanical energy is irreversibly converted to heat via turbulent dissipation per unit length of pipe; total energy is still conserved overall (energy in = mechanical energy out + heat generated by friction), but the FRACTION dissipated as heat per metre of pipe rises abruptly at transition, which is why the EGL for a turbulent pipe flow falls noticeably faster per unit length than a laminar flow at the same Reynolds number would.