where u = u(
r,
t) is the particle velocity associated with the acoustic wave motion. The
total particle velocity is given by
where V denotes the local mean velocity for the fluid motion (the mean flow is labeled
u0 in the linearized Euler and Navier-Stokes interfaces). The dynamic equations for this mean-flow field are described in the next subsection. For now, just assume
V to be a given
irrotational background velocity field; hence, also the mean-flow velocity can be defined in terms of a potential field
Φ, by
V = ∇Φ.
In deriving this equation, all variables appearing in the full nonlinear fluid-dynamics equations were first split in time-independent and acoustic parts, in the manner of Equation 5-9. Then, linearizing the resulting equations in the acoustic perturbation and eliminating all acoustic variables except the velocity potential gives
Equation 5-10. Thus, the density
ρ in this equation is the time-independent part. The corresponding acoustic part is
ρ(r,
t) = p(r,
t)/c02 where
p is the acoustic pressure, given by
Hence, once Equation 5-10 has been solved for the velocity potential, the acoustic pressure can easily be calculated.
the out-of-plane wave number kz enters the equations when the
∇ operators are expanded:
The default value of the out-of-plane wave number is 0, that is, no wave propagation perpendicular to the 2D plane. In a mode analysis the equations are solved for
kz.
the azimuthal wave number m similarly appears in the equation as a parameter:
Here ρ0 (SI unit: kg/m
3) is the background mean flow density,
V (SI unit: m/s) denotes the background mean velocity, and
c0 (SI unit: m/s) refers to the speed of sound. The software solves the equation for the velocity potential
, with SI unit m
2/s. The validity of this equation relies on the assumption that
ρ0,
V, and
c0 are approximately constant in time, while they can be functions of the spatial coordinates.
for the eigenmodes, , and eigenvalues, λ = −ikz, on a bounded two-dimensional domain,
Ω, given well-posed edge conditions on
∂Ω. In this equation,
is the velocity potential,
ρ0 is the background mean flow density,
c0 is the speed of sound,
ω is the angular frequency, and
kz is the out-of-plane wave number or propagation constant. Furthermore,
Vt denotes the background mean velocity in the tangential plane while
Vn is the background mean velocity component in the normal direction.