Volumetric flow rate ($Q$)

The volumetric flow rate measures volume crossing an oriented surface per unit time:For a planar cross-section of area $A$, define the mean normal velocity $\bar{U}=A^{-1}{\int }_A\mathbf{v}\cdot \mathbf{n}dA$; then $Q=A\bar{U}$ exactly. The units are ${\mathrm{m}}^3{\mathrm{s}}^{-1}$. A nonuniform velocity profile changes the mean, rather than invalidating this definition.

In steady incompressible flow without leaks or branches, mass conservation makes $Q$ the same at every pipe section. In a compressible stream the conserved quantity is mass flow rate, and $Q$ can change as mass density changes. Collecting discharged liquid for a measured interval gives an elementary flow-rate measurement, provided the collection captures the full flow and the volume is interpreted at the appropriate conditions.

River discharge is volume flux across a channel section. For approximately uniform cross-sectional velocity it equals flow area times mean velocity. Its time dependence at a fixed station is a hydrograph.

A hydrograph records discharge against time at a fixed location. A nonlinear kinematic wave can steepen an upstream increase into a sharp front; longitudinal diffusion or additional storage rounds the resulting rise.

The sverdrup, symbol $\mathrm{Sv}$, is a unit of volume flux used for ocean circulation:It expresses the integrated transport across a section, rather than the velocity at one point. A spatially uniform current of $0.1\mathrm{m}{\mathrm{s}}^{-1}$ through a section $100\mathrm{km}$ wide and $1\mathrm{km}$ deep would transport $10\mathrm{Sv}$ in this deliberately idealized calculation.

Real sections have variable velocity and depth, so transport is obtained by integrating the normal velocity over the section. Opposite flows in different parts can partly cancel in the signed net transport, even when the local speeds are large. A quoted value in sverdrups therefore needs the section and direction convention before it can be compared with a local current-meter reading.

The mass flow rate through an oriented cross-section is the integral of mass density times normal velocity. For constant density it is density times the volumetric flow rate. In a steady pipe without leaks, conservation of mass makes this rate independent of cross-section.

Coolant mass flow rate per open flow area, measured in $\mathrm{kg}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. A decrease in total flow does not establish a decrease in mass flux when the open area also changes.

For a flow invariant in one horizontal direction, the volume flux per unit width is the velocity integrated across the flow depth. It has dimensions of area per unit time.

A discharge coefficient multiplies an ideal inviscid opening-flow rate to represent contraction and dissipative losses. Its value depends on the opening geometry and flow regime.