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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 ∂t + ∂x1 + ∂x2 + ∂x3 + CU = W
(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 [13] 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 = ω˙c (6.5a)
∂ua ---- ∂t + ∇(u-⋅ua) + ∇( pa) -- ρ = ω˙m, (6.5b)

where (u,c2,ρ) represents the base flow and (ua,pa) the acoustic perturbations. Similar to the LEE, the acoustic source terms ω˙c and ω˙m are by default zero and must be specified e.g. by an appropriate forcing. This way, e.g. the APE-1, APE-4 [13] or revised APE equations [15] can be obtained. Expressed as hyperbolic conservation law, the APE-1/4 operator reads:

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

with

⌊ a⌋ | pa| U = ||u 1||, ⌈ua2⌉ ua3 ⌊--2 a a--⌋ ⌊--2 a a--⌋ ⌊ --2 a a--⌋ |ρc-u1a + pa u1| |ρc u2 + p u2| | ρc u3 + p u3| F1 = || ujuj + p ∕ρ || ,F2 = ||-- 0 --||,F3 = || 0 || . ⌈ 0 ⌉ ⌈ujuaj + pa∕ρ ⌉ ⌈ -- 0 -⌉ 0 0 ujuaj + pa∕ρ
(6.7)

(6.8)

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