Identification of Swirl Waves using Local Stability Analysis Alp Albayrak, Wolfgang Polifke 15.09.15

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

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

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 + 𝜕 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

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 ⋮

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

Oscillatory modes are damped faster. ( 𝑢 𝜃 )

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

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 2 + 1 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)

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

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

Identification of Swirl Waves using Local Stability Analysis Alp Albayrak, Wolfgang Polifke 11.09.15

Oscillatory modes are damped faster.