6.1 Synopsis

The aim of the AcousticSolver is to predict acoustic wave propagation. Through the application of a splitting technique, the flow-induced acoustic field is totally decoupled from the underlying hydrodynamic field.

6.1.1 Linearized Euler Equations

The Linearized Euler Equations (LEE) are obtained by linearizing the Euler Equations about a mean flow state (  --   )
 ρ,c2, u. Hence, they describe the evolution of perturbations (pa, ρa, ρua) around this state. In conservative form, the LEE are given as:

∂U--+ ∂F-1 + ∂F-2+  ∂F-3+ CU   = W
∂t    ∂x1    ∂x2    ∂x3
(6.1)

with

                                 ⌊  a⌋
                                 | pa|
                                 ||-ρ ||
                            U =  ||ρua1|| ,
                                 ⌈ρua2⌉
                                  ρua3
      ⌊--a-2   -- a⌋         ⌊--a-2   -- a⌋         ⌊--a-2   -- a⌋
      |ρu1c  +-u1p |         |ρu2c  +-u2p |         |ρu3c  +-u3p |
      || ρua1-+ u1ρa ||         || ρua2-+-u2ρa ||         || ρua3-+-u3ρa ||
F 1 = || ρua1u1 + pa ||,  F 2 = ||   ρua1u2    ||,  F 3 = ||   ρua1u3    ||,
      ⌈   ρua2u1    ⌉         ⌈ ρua2u2 + pa ⌉         ⌈   ρua2u3    ⌉
          ρua3u1                  ρua3u2                ρua3u3 + pa
     ⌊       ∂u           1        ∂p  1        ∂p   1       ∂p ⌋
      (γ - 1)∂xkk     0    ρ (1 - γ)∂x1 ρ (1 - γ) ∂x2-  ρ (1- γ )∂x3
     ||     0         0-        0            0             0     ||
 C = ||     0      uk ∂u1      ∂u1          ∂u1           ∂u1    || .
     ||     0      u- ∂∂xuk2      ∂∂xu12          ∂∂xu22           ∂∂xu32    ||
     ⌈            --k∂∂xuk3      ∂∂xu13          ∂∂xu23           ∂∂xu33    ⌉
           0      uk ∂xk      ∂x1          ∂x2           ∂x3
(6.2)

(6.3)

(6.4)

By default, the source term vector W is zero and has to be specified by an appropriate forcing.

6.1.2 Acoustic Perturbation Equations

The acoustic perturbation equations (APE-1/APE-4) proposed by Ewert and Schroeder [14] assure stable aeroacoustic simulations. These equations are similar to the LEE, but account for acoustic perturbations exclusively. The AcousticSolver implements the APE-1/4 type operator:

∂pa-
 ∂t + c2∇⋅(           )
  --a   -pa
  ρu  + u c2- = ˙ωc (6.5a)
∂ua
----
 ∂t + ∇(u-⋅ua ) + ∇( pa)
  --
  ρ = ˙ωm , (6.5b)

where (u,c2,ρ) represents the base flow and (ua,pa) the acoustic perturbations. Similar to the LEE, the acoustic source terms ˙ωc and ˙ωm are by default zero and must be specified e.g. by an appropriate forcing. This way, e.g. the APE-1, APE-4 [14] or revised APE equations [17] can be obtained. Expressed as hyperbolic conservation law, the APE-1/4 operator reads:

∂U--+ ∂F-1 + ∂F-2 + ∂F-3 = W
 ∂t   ∂x1    ∂x2    ∂x3
(6.6)

with

                               ⌊  a⌋
                               | pa|
                           U = ||u 1||,
                               ⌈ua2⌉
                                ua3
     ⌊  --        ⌋        ⌊ --        ⌋        ⌊--2 a    a--⌋
       ρc2ua1 + pau1         ρc2ua2 + pau2        |ρc u3 + p u3|
     || ujua + pa∕ρ||        ||     0     ||        ||     0      ||
F1 = |⌈    j 0     |⌉ , F2 = |⌈u-ua + pa∕ρ|⌉ , F3 = ||--   0    --|| .
                             j j                ⌈ujuaj + pa∕ρ ⌉
            0                    0
(6.7)

(6.8)