Nonlinear Energy Transfer Applied to Data Analysis in Combustion Instability

Jacob, Eric J.; Flandro, Gary A.; Gloyer, Paul

doi:10.2514/6.2011-5641

Cited by 3 publications

(1 citation statement)

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“…To do so requires consideration of nonlinear energy transfer pathways between the fundamental mode and its harmonics. 62,63 However, quantifying linear energy transfer in this way facilitates the distinct advantage of correlating driving and damping of periodic, unsteady flowfields with specific flow phenomena. This leads to identifying the influence of geometric features on the mean flow and ultimately the emergence of acoustic waves.…”

Section: Quantifying Unsteady Energy Transfermentioning

confidence: 99%

Transverse Vortico-Acoustic Waves in the Presence of Strong Mean Flow Shear Layers

Kovacic¹,

Batterson²,

Majdalani³

2015

51st AIAA/SAE/ASEE Joint Propulsion Conference

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Several analytical models describing the formation of resonant waves in simple geometric configurations are available in the open literature. Their solutions are predicated on several limiting assumptions, namely, that the mean flow velocity is uniform and that the wall injection Mach number is sufficiently small to justify asymptotic reductions to the governing equations. When considering acoustic waves in a convecting medium where 1 M  , the effect of mean flow motion appears at 2 ( ) M which, from an asymptotic standpoint, represents a higher-order correction that is traditionally dismissed with no afterthought. Naturally, the ensuing truncation leads to a simple Helmholtz-type wave equation. In practice though, the influence of the mean flow appears at leading order when accounting for acoustic boundary layers. Acoustic boundary layers, when strongly coupled with the mean flow, give rise to the so-called vorticity (or vortical) waves. When combined with the acoustic wave motion, the resulting pair constitutes what is typically referred to as a vortico-acoustic wave. By strictly enforcing the 1 M  condition, analytical solutions can be recovered for vortico-acoustic waves in simple planar and axisymmetric chambers. The present study focuses on a numerical approach that resolves the unsteady field equations computationally for the threefold purpose of verifying existing analytical solutions, providing practical bounds for their applicability, and extending their predictive capabilities to more complex configurations. In particular, the interaction between acoustic waves and a strong mean flow shear layer is investigated by computationally resolving the vortico-acoustic field developing behind a backward facing step. In this context, deviations from the analytical solutions are reported both at high injection Mach numbers and when considering the emergence of acoustically-synchronized vortex cells around a shear layer. These vortex cells exhibit a spatial instability that continues to grow as the flow is further accelerated. Lastly, the temporal stability of the flow is quantified by computing the rate of energy transfer into the unsteady field. Similarly to the spatial instability, the increased vorticity at higher injection velocities tends to drive temporal instability as well. Nomenclature a = chamber radius or half-height [m] s a = speed of sound [m/s] j = mode number M = injection Mach number, / inj s U a p = pressure [Pa] R = specific gas constant [J/kg-K] j R = pressure amplitude for mode j [Pa] r = spatial coordinate, , , r z  T = temperature [K] inj U = injection velocity [m/s] u = velocity vector [m/s] j u = velocity associated with mode j [m/s] Symbols  = acoustic velocity potential [m 2 /s]  = dynamic viscosity [N-m/s]  = density [kg/m 3 ] Ω = steady vorticity [1/s] ω = unsteady vorticity [1/s] j  = circular frequency of mode j [rad/s] Superscripts and Subscripts 0 = denotes a mean flow variable 1= denotes an unsteady flow variable , , m n l = denotes the tangential, radial, and axial acoust...

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Section: Quantifying Unsteady Energy Transfermentioning

confidence: 99%