Volume Velocity of a Gas

The volume velocity $U$ of a gas flow is defined as particle velocity $u$ times the cross-sectional area $A$ of the flow, or

$$U(t,x) = u(t,x) A(x)$$

where $x$ denotes position along the flow, and $t$ denotes time in seconds. Volume velocity is thus in physical units of volume per second (m$\null^3$/s).

When a flow is confined within an enclosed channel, as it is in an acoustic tube, volume velocity is conserved when the tube changes cross-sectional area, assuming the density $\rho$ remains constant. This follows directly from conservation of mass in a flow: The total mass passing a given point $x$ along the flow is given by the mass density $\rho$ times the integral of the volume volume velocity at that point, or

$$M(t_0:t_f,x) = \rho\int_{t_0}^{t_f} U(t,x) dt.$$

As a simple example, consider a constant flow through two cylindrical acoustic tube sections having cross-sectional areas $A_1$ and $A_2$, respectively. If the particle velocity in cylinder 1 is $u_1$, then the particle velocity in cylinder 2 may be found by solving

$$u_1 A_1 = u_2 A_2$$

for $u_2$.

It is common in the field of acoustics to denote volume velocity by an upper-case $U$. Thus, for the two-cylinder acoustic tube example above, we would define $U_1\isdeftext u_1A_1$ and $U_2\isdeftext u_2A_2$, so that

$$U_1 = U_2$$

would express the conservation of volume velocity from one tube segment to the next.