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This work describes both traveling and standing vorticoacoustic waves in circular tubes that are driven by axisymmetric headwall injection. In this process, perturbation tools, field decomposition, and boundary-layer theory are jointly used. First, perturbation expansions are initiated to linearize the Navier–Stokes equations. Second, a Helmholtz decomposition of the first-order disturbances is pursued to identify a suitable set of acoustic wave equations. The last step consists of solving for the vortical mode using boundary-layer theory and a viscous expansion of the unsteady rotational set. At the outset, an explicit formulation for arbitrary headwall injection is obtained and confirmed both numerically and through limiting process verifications; the latter take into account special cases involving uniform and bell-shaped injection profiles. The resulting formulation is then described using both laminar and turbulent injection patterns. Using four canonical cases, the characteristics of the evolving vorticoacoustic wave, including its penetration depth, spatial wavelength, and overshoot factor, are systematically explored and discussed. Several fundamental flow features are also unraveled including the radial, tangential, and axial velocities of the time-dependent vortical field. Most rotational flow features are found to depend on the penetration number, the Strouhal number, and the distance from the centerline. The corresponding standing modes are expressed in closed form and shown to be appreciable in view of their amplitudes that twice exceed those associated with strictly traveling waves. Finally, by extending the boundary-layer analysis from the headwall to the sidewall, a uniformly valid wave approximation is achieved, which remains observant of the no-slip requirement everywhere.
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On the compressible biglobal stability of the mean flow motion in porous tubes
In this work, a compressible biglobal stability approach is used to investigate the growth characteristics of hydrodynamic and vorticoacoustic waves in porous tubes with uniform wall injection. The retention of compressibility effects enables us to construct a physics-based formulation that is capable of predicting both hydrodynamic and vorticoacoustic wave motions simultaneously with no need for mode decomposition. At first, we show that, in the absence of a mean flow, the stability framework reproduces traditional Helmholtz frequencies and modal shapes. This confirms the embedment of the wave equation within the compressible Navier–Stokes framework. We then proceed to simulate the idealized motion in solid rocket motors, often modeled as porous tubes, where a mean flow expression is available. Specifically, using the compressible Taylor–Culick profile as a base flow, our solver produces a comprehensive frequency spectrum that returns both hydrodynamic and vorticoacoustic modes in one swoop with the added benefit of pinpointing the flow-induced longitudinal, radial, and mixed frequencies at user-prescribed tangential modes. Moreover, we find that increasing the flow Mach number leads to a slight reduction in the vorticoacoustic frequencies relative to their strictly acoustic counterparts. Similar results are obtained while increasing the Reynolds number and aspect ratio, thus affirming the origin of frequency shifts often observed in motor firings. Finally, the vorticoacoustic velocity fluctuations are shown to resemble those obtained asymptotically. Particularly, their depths of penetration appear to be controlled by the penetration number, a dimensionless parameter that combines the effects of sidewall injection, oscillatory frequency, viscosity, and chamber radius.
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- Award ID(s):
- 1761675
- PAR ID:
- 10589354
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 33
- Issue:
- 8
- ISSN:
- 1070-6631
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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