In case of a sudden reduction of pipe diameter, without streamlining, the flow is not able to follow the sharp bend into the narrower pipe. As a result, there is flow separation, creating recirculating separation zones at the entrance of the narrower pipe. The main flow is contracted between the separated flow areas, and later on expands again to cover the full pipe area.
There is not much head loss between cross section 1, before the contraction, and cross section 3, the vena contracta at which the main flow is contracted most. But there are substantial losses in the flow expansion from cross section 3 to 2. These head losses can be expressed by using the Borda–Carnot equation, through the use of the coefficient of contraction μ:
with A the cross-sectional area at the location of strongest main flow contraction 3, and A the cross-sectional area of the narrower part of the pipe. Since A ≤ A, the coefficient of contraction is less than one: μ ≤ 1. Again there is conservation of mass, so the volume fluxes in the three cross sections are a constant (for constant fluid density ρ):
with v, v and v the mean flow velocity in the associated cross sections. Then, according to the Borda–Carnot equation (with loss coefficient ξ=1), the energy loss ΔE per unit of fluid volume and due to the pipe contraction is:
The corresponding loss of total head ΔH can be computed as ΔH = ΔE/(ρg).
According to measurements by Weisbach, the contraction coefficient for a sharp-edged contraction is approximately: