Contraction and Expansion Losses

Losses due to contraction and expansion of flow between cross sections are determined during the standard step profile calculations. Manning's equation is used to calculate friction losses, and all other losses are described in terms of a coefficient times the absolute value of the change in velocity head between adjacent cross sections. When the velocity head increases in the downstream direction, a contraction coefficient is used; and when the velocity head decreases, an expansion coefficient is used.

As shown in the figure above, the flow contraction occurs between cross sections 4 and 3, while the flow expansion occurs between sections 2 and 1. The contraction and expansion coefficients are used to compute energy losses associated with changes in the shape of river cross-sections (or effective flow areas). The loss due to expansion of flow is usually larger than the contraction loss, and losses from short abrupt transitions are larger than losses from gradual transitions. Typical values for contraction and expansion coefficients under subcritical flow conditions are shown in Table 5-2 below:

Table 5-2 Subcritical Flow Contraction and Expansion Coefficients

The maximum value for the contraction and expansion coefficient is 1.0. As mentioned previously, a detailed study was completed by the Hydrologic Engineering Center entitled "Flow Transitions in Bridge Backwater Analysis" (HEC, 1995). A summary of this research, as well as recommendations for contraction and expansion coefficients, can be found in "Flow Transitions in Bridge Backwater Analysis".

In general, contraction and expansion coefficients for supercritical flow should be lower than subcritical flow. For typical bridges that are under class C flow conditions (totally supercritical flow), the contraction and expansion coefficients should be around 0.03 and 0.05 respectively. For abrupt bridge transitions under class C flow, values of 0.05 and 0.1 may be more appropriate.