1. Identification of Swirl Waves using Local Stability Analysis
Alp Albayrak,
Wolfgang Polifke

2. 2 distinct propagation scales are present
Acoustic wave propagation
𝑢 𝑝 = 𝑢 𝑥 ±𝑐
Swirl wave propagation
𝑢 𝑝 = 𝑢 𝑥
𝑢 θ ′
𝑢 𝑧 ′
𝑢 𝑧
𝑢 θ
Swirler

3. Recap from ICSV22: Swirl waves are not convective
𝑢 θ ′
𝑢 θ
𝑢 θ 𝑥
Output Plane
Constant Speed Model
Input Plane
Current Model
CFD Result

4. Modal decomposition can easily describe the propagation speeds.
Linearized NS Eq: 𝑢 ′ 𝑡,𝑧,𝑟 =𝑢 𝑡,𝑧,𝑟 − 𝑢 𝑧,𝑟
𝜕 𝑢 𝑟 ′ 𝜕𝑡 + 𝑢 𝑧 𝜕 𝑢 𝑟 ′ 𝜕𝑧 − 2 𝑢 𝜃 𝑢 𝜃 ′ 𝑟 =− 1 𝜌 𝜕 𝑝 ′ 𝜕𝑟 +𝜐 1 𝑟 𝜕 𝑢 𝑟 ′ 𝜕𝑟 + 𝜕 2 𝑢 𝑟 ′ 𝜕 𝑟 𝜕 2 𝑢 𝑟 ′ 𝜕 𝑧 2 − 𝑢 𝑟 ′ 𝑟 2
Normal Modes: 𝑢 𝑟 = 𝑢 ′ 𝑡,𝑧,𝑟 𝑒𝑥𝑝 𝑖 −𝜔𝑡+𝑘𝑧
−𝑖𝜔 𝒖 𝒓 +𝑖𝑘 𝑢 𝑧 𝒖 𝒓 − 2 𝑢 𝜃 𝑟 𝒖 𝜽 =− 1 𝜌 𝑑 𝒑 𝑑𝑟 +𝜐 1 𝑟 𝑑 𝒖 𝒓 𝑑𝑟 + 𝑑 2 𝒖 𝒓 𝑑 𝑟 2 − 𝑘 2 𝒖 𝒓 − 𝒖 𝒓 𝑟 2
Temporal analysis: fix 𝑘∈ℝ, find ω 𝑛 ∈ℂ and 𝑋 𝑛 𝑟 ∈ℂ
𝑢 𝑝 𝑛 = 𝑅𝑒 𝜔 𝑛 𝑘

5. Eigenvalue problem is constructed from modal decomposition and is easy to solve.
𝜐𝑘 2 + 𝜐 𝑟 2 +𝑖𝑘 𝑢 𝑧 − 𝜐𝐷 𝑟 −𝜐 𝐷 2 𝑢 𝑟 − 2 𝑢 𝜃 𝑟 𝑢 𝜃 + 𝐷 𝜌 𝑝 =𝑖𝜔 𝑢 𝑟 𝐴 𝑘 𝑢 =𝜔𝐵 𝑢 𝐴 11 𝐴 𝐴 ⋯ 𝐴 1𝑛 𝐴 𝐴 ⋮ ⋱ ⋮ 𝐴 𝐴 𝐴 𝑚1 ⋯ 𝐴 𝐴 𝐴 𝑚𝑛 𝑢 𝑧 ⋮ 𝑢 𝑟 ⋮ 𝑢 𝜃 ⋮ 𝑝 ⋮ =𝜔 𝐵 11 𝐴 𝐴 ⋯ 𝐵 1𝑛 𝐴 𝐴 ⋮ ⋱ ⋮ 𝐴 𝐴 𝐵 𝑚1 ⋯ 𝐴 𝐴 𝐵 𝑚𝑛 𝑢 𝑧 ⋮ 𝑢 𝑟 ⋮ 𝑢 𝜃 ⋮ 𝑝 ⋮

6. Each mode has its own growth rate 𝜔 𝑖 and propagation speed 𝜔 𝑟 𝑘 Propagation speeds are clustered around convective speed.
Convection speed
Analytical Model:
(𝑢 𝑝 ) 𝑚𝑎𝑥 =1.46
(𝑢 𝑝 ) 𝑚𝑖𝑛 =0.54
Growth Rate
Propagation speed

7. Oscillatory modes are damped faster.
( 𝑢 𝜃 )

8. Quantitative comparison is missing: Construct IR
Next steps
Quantitative comparison is missing: Construct IR

9. Analytical approach reveals important criteria for swirl waves.
𝐶 𝑢 𝜃 +𝑩 𝑢 𝑟 =0 2 𝑢 𝜃 𝐶𝑟 𝑢 𝜃 − 𝑢 𝑟 + 1 𝑘 2 𝑑 2 𝑢 𝑟 𝑑 𝑟 𝑟 𝑑 𝑢 𝑟 𝑑𝑟 − 𝑢 𝑟 𝑟 2 =0 𝐶=𝑖 𝑘 𝑢 𝑧 −𝜔 is convective operator. 𝑩= 𝑢 𝜃 𝑟 + 𝑑 𝑢 𝜃 𝑑𝑟 ,
𝑢 𝜃 =𝐾 𝑟 𝑛 → 𝑛<−1→𝐵<0 𝑛=−1→𝐵=0 𝑛>−1→𝐵>0 (free vortex)

10. Non-convective waves are not present for free vortex
𝑢 𝜃 =𝐶
𝑢 𝜃 =𝐶𝑟
𝑢 𝜃 =𝐶/ 𝑟 2
𝑢 𝜃 =𝐶/𝑟
𝐵<0
𝐵=0
𝐵>0

11. Further research Construct IR for quantitative comparison.
Comparison with global stability analysis (more realistic profiles?)
Build a model for 𝑢 𝜃 ′ → 𝑄 ′

12. Identification of Swirl Waves using Local Stability Analysis
Alp Albayrak,
Wolfgang Polifke

13. Oscillatory modes are damped faster.