Governing Equations
The continuity equation derived for a control volume is given by
(3-13)
and the corresponding momentum balance equation is
(3-14)
where Z is the inner circumference of the pipe and A = A(x,  p,...) is the inner wetted cross-sectional area, u is the area-averaged mean velocity, which is also defined in the tangential direction u = uet, p is the mean pressure along the pipe, τw is the wall drag force, and F is a volume force. The gradient is taken in the tangential direction et. The term β is a flow profile correction factor relating the mean of the squared total velocity to the square of the mean velocity. Such that
(3-15)
where
and
are the local non-averaged parameters. Again p and u are the area-averaged dependent variables.
Linearization
The governing equations are now linearized; that is, all variables are expanded to first order assuming stationary zero-order values (steady-state background properties). The acoustic variations of the dependent variables are assumed small and on top of the background values. This is done according to the following scheme:
where A0 is often only function of x; however, A0 can be changed by external factors such as heating or structural deformation, thus the time dependency. The first-order terms represent small perturbations on top of the background values (zero order). They are valid for
Moreover, the perturbations for the fluid density and cross-sectional area are expanded to first order in p0 in a Taylor series such that
where the subscript s refers to constant entropy; that is, the processes are isentropic. The relations for the fluid compressibility and the cross-sectional area compressibility are
Here, β0 is the fluid compressibility at the given reference pressure p0, the isentropic bulk speed of sound is denoted cs, and ρ0 is the fluid density at the given reference temperature and reference pressure. βA is the effective compressibility of the pipe’s cross-sectional A0 due to changes in the inner fluid pressure. The bulk modulus K is equal to one over the compressibility.
Inserting the above expansions into the governing equations (Equation 3-13 and Equation 3-14) and retaining only first-order terms yield the pipe acoustics equations including background flow. These are:
(3-16)
where c is the effective speed of sound in the pipe (it includes the effect due to the elastic properties of the pipe defined through KA). The bulk modulus for the cross-sectional area KA is given by the pipe material properties according to the so-called Korteweg formula (see Ref. 2). For a system with rigid pipe walls cs = c as KA tends to infinity.
Using the fact that the velocity is taken along the tangential direction et, the governing equations are rewritten in terms of the scalar values u and p and projected onto the tangent. The 0 subscript is dropped on the density and area and the 1 subscript is also dropped on the dependent variables.
(3-17)
where is the tangential derivative, τw is the tangential wall drag force (SI unit: N/m2) and F is a volume force (SI unit: N/m3).
Governing Equations
Pipe Acoustics, Transient Interface
Finally, the expression for the time derivative of the pressure in the momentum equation is replaced by spatial derivatives using the continuity equation. This yields the equations solved in the Pipe Acoustics, Transient interface:
(3-18)
Pipe Acoustics, Frequency Domain Interface
In the frequency domain all variables are assumed to be time harmonic such that
(3-19)
inserting this into the governing Equation 3-18 (and dropping the tilde) yields the equations solved in the Pipe Acoustics, Frequency Domain interface:
(3-20)
where ω = 2π f is the angular frequency and f is the frequency.