(a) Linear regime. (b) Nonlinear regime.
Figure 2.2: Snapshots of streamlines in the vicinity of a Helmholtz resonator neck gen-erated with the CFD environment introduced in Chp. 4.
instead, which are finally dissipated via the turbulent cascade process. The energy which the jet carries—and thus also the acoustic loss—depends on the particle velocity in a nonlinear manner. Moreover, the level of forcing influences the induced flow at the opening of the resonator noticeably. The change of the flow characteristics impacts not only the acoustic losses as described above, but also the fluid mass taking part in the resonant flow motion. This corresponds to a variation of mass in the mass-spring-damper model and, thus, changes the eigenfrequency of the resonator. It is common to distinguish between the linear and the nonlinear regime. In the linear regime, where the acoustic forcing is sufficiently low, only the linear effects are present. The resonator is said to be operated in the nonlinear regime when the forcing is high enough to trigger nonlinear effects. Typical streamlines for both regimes are illustrated for an Helmholtz resonator in Fig. 2.2. The flow separation at the edges can clearly be seen in the nonlinear regime presented in Fig. 2.2b. In this thesis, these nonlinear effects are studied in detail (see Sec. 5) and a data-based reduced-order model for their description is developed (see Sec. 6).
2.2 Governing Equations
This thesis investigates the acoustic damping of resonators, i. e., the losses of sound propa-gating in a fluid when it interacts with those devices. All relevant effects are fully included in the Navier-Stokes equations for Newtonian, compressible fluids without external forces (see for instance the manuscript by Davidson [8]). This set of equations consists of the continuity, themomentum, and the energy equation. The relation between the density ρ and the velocity of the fluid ui in space xi and timet can be written in Einstein notation for the mass and momentum equation as:
dρ
dt +ρ∂ui
∂xi
= 0 , and (2.1)
ρdui
dt =−∂p
∂xi
+ ∂
∂xj
2µ Sij −2 3µ∂uk
∂xk
δij
. (2.2)
Above, the operator d·/dt denotes the material derivative d·/dt=∂·/∂t+ui∂ ·/∂xi. The pressure, the viscosity, and the strain rate tensor are referred to as p, µ, and Sij, respectively. The rate tensor is defined as Sij = 1/2 (∂ui/∂xj +∂uj/∂xi). The transport equation for the internal energy e reads as:
ρde
Above, the variables T and kT stand for the temperature and the thermal conductivity, respectively. In addition to the partial differential equations (PDEs) above, two equations of state are required for a complete problem description. Here, the ideal gas law can be applied for all cases considered:
p=ρRsT , (2.4)
where Rs is the specific gas constant. For an ideal gas, the specific internal energy e and the temperature T are linked by the relation:
e=
where cv and cp denote the specific heat capacities at constant specific volume and at constant pressure, respectively.
The set of equations above describes the whole physics involved. However, it is often meaningful to consider some simplification of them. Disregarding viscosity, the momentum equation (2.2) reduces to the Euler equation
ρdui
dt + ∂p
∂xi
= 0 . (2.6)
If additionally the flow is assumed to be incompressible and irrotational, i. e.
ωi ≡ǫijk∂uk/∂xj = 01, it further reduces to the incompressible Bernoulli equation
∂φ
∂t +1
2uiui+p
ρ = const. . (2.7)
This equation is valid along a streamline and the variable φ stands for the potential of the irrotational flow ∂φ/∂xi = ui. It is often starting point for the analysis of so-called acoustically compact elements, see Sec. 2.5.
In acoustics, isentropic disturbances of flow variables are considered [9]. Hence, any varying quantity q is decomposed into its mean q0 and its fluctuating parts q′, i. e. q =q0+q′. If the fluctuating parts are small in comparison to a suitable reference value, it is valid with only minor loss of generality to neglect higher-order products of fluctuating quantities.
The continuity (2.1) and the Euler equation (2.6) for the fluctuating quantities read as d0ρ′
1ǫijk denotes the permutation tensor defined asǫijk=
2.2 Governing Equations where d0 ·/dt stands for the material derivative with respect to the mean flow u0, i. For an isentropic compression, the fluctuating pressure p′ and density ρ′ are related in first order approximation as
p′ =c20ρ′ , (2.10)
where the constantc0 is defined asc20 = (∂p/∂ρ)swith the indexsindicating the isentropic relation. The variable c0 is named speed of sound, since acoustic perturbations propagate in space with that speed, as it can be seen below in Eq. (2.12). For an ideal gas, the speed of sound is given by c0 = √
γRsT, where γ denotes the adiabatic index γ = cp/cv. By combination of Eqs. (2.8), (2.9) and (2.10), the so-calledwave equationcan be formulated:
d20p′
dt2 −c20 ∂2p′
∂xi∂xi
= 0 . (2.11)
When the acoustic pressure p′ is know, the particle velocityu′i can be deduced from it by applying Eqs. (2.8) and (2.10).
In many applications, the sound propagates in a 1-D manner. This is, e. g., valid for plane waves in a duct with constant cross section area. Hence, a 1-D configuration is considered in the following paragraph.
The wave equation (2.11) can be factorized as d0
dt +c0 ∂
∂x d0
dt −c0 ∂
∂x
p′ = 0 . (2.12)
Each factor represents an operator known from the convection equation with the con-vective speeds ±(c0∓u0). In this form, it can be seen that the solution of the 1-D wave equation consists of two perturbations traveling upstream and downstream with the speed c0 relative to the mean fluid motion. These two characteristic waves, also known as Rie-mann invariants, are defined as
f = 1 2
p′ ρ0c0
+u′
and g = 1 2
p′ ρ0c0 −u′
. (2.13)
Accordingly, the acoustic velocity and pressure is given in terms of f and g as u′ =f−g and p′
ρ0c0 =f+g , (2.14)
respectively.
In linear acoustics, equations are often transformed into frequency domain by the Fourier transform F. Quantities in the frequency domain are indicated by ˆ· and the angular frequency is denoted as ω in the following. The pressure in frequency domain ˆp is given by
ˆ
p(ω) =F{p}(ω) = Z ∞
−∞
p(t) e−iωtdt . (2.15)
When the pressure is determined in the frequency domain, the inverse Fourier transform F−1 yields the corresponding value in the time domain
p(t) =F−1{pˆ}(t) = 1 2π
Z ∞
−∞
ˆ
peiωtdω . (2.16)
A quantity gained by the inverse Fourier transform is complex-valued. Without loss of generality, the real part of this quantities can be considered as the physical quantity. The 3-D wave equation (2.11) without mean flow is transformed to
∂2pˆ
∂xi∂xi
+k2pˆ= 0 (2.17)
and is named Helmholtz equation. The parameter k is called wavenumber and is defined as k ≡ ω/c0. In a 1-D setup, the propagation of a f-wave in a duct with length l can be written in terms of the wavenumber as: fout = exp(−ilk)fin, where fin/out denote the f-wave entering and leaving the duct, respectively.