Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T15:52:40.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Propagation and stability of vorticity–entropy waves in a non-uniform flow

Published online by Cambridge University Press:  07 March 2007

O. V. ATASSI
O. V. ATASSI*
Affiliation:
University of Seville, Seville, Spain41092 and Pratt & Whitney, East Hartford, CT 06108, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The evolution of disturbances in an annular duct with a non-isentropic radially varying mean flow is studied. Linear and nonlinear analyses are carried out to examine how the mean velocity and density gradients affect the stability and coupling between the disturbances. To isolate the effect of the mean-velocity gradients from that of the mean-density gradients two mean flows are considered, one with a Gaussian density profile and a uniform axial velocity and the other with Gaussian density and Gaussian axial-velocity distributions. For small-amplitude disturbances with the former mean flow profile, the vortical disturbances convect with the mean flow and density fluctuations grow linearly in space as a result of the interaction of the mean-density gradient with the disturbance radial velocity. Eigenmode analysis of the latter profile shows that unstable modes with exponential growth occur owing to the inflection point in the mean-velocity profile. These modes are almost independent of the mean-density profile and are most unstable for low azimuthal wavenumbers. Nonlinear solutions support the linear results and show an algebraic growth of the density for a range of azimuthal wavenumbers and both uniform and non-uniform mean-velocity profiles. The growth of the velocity fluctuations, however, is strongly dependent on the azimuthal wavenumber of the incident disturbance and the mean-velocity profile. The largest growth in the disturbance is observed at radial locations where the largest mean-flow gradients exist. Owing to the growth of the density fluctuations, coupled vorticity–entropy waves are observed downstream of a forced harmonic excitation in a non-isentropic flow. The forcing amplitudes of the incident waves were varied to see how the solutions change with amplitude. As the amplitude is increased, the waves continue to grow and a steepening of the gradients is observed as they propagate downstream until eventually very sharp density and velocity fronts form. These results show that the mean-flow and density profiles play an important role in the evolution of low-azimuthal-wavenumber disturbances which can couple strongly to the duct acoustic modes during combustion instabilities.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 575 , March 2007 , pp. 149 - 176
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Atassi, H. M., Ali, A. A., Atassi, O. V. & Vinogradov, I. V. 2004 Scattering of incident disturbances by an annular cascade in a swirling flow. J. Fluid Mech. 499, 111–138.CrossRefGoogle Scholar
Atassi, O. V. 2003 Computing the sound power in non-uniform flow. J. Sound Vib. 266, 75–92.CrossRefGoogle Scholar
Atassi, O. V. 2004 Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys. 197, 737–758.Google Scholar
Atassi, O. V. & Ali, A. 2002 Inflow/outflow conditions for internal time-harmonic Euler equations. J. Comput. Acoust. 10, 155–182.CrossRefGoogle Scholar
Atassi, O. V. & Galan, J. M. 2005 Nonreflecting boundary conditions for the euler equations. AIAA Paper 2005–1585.Google Scholar
Cooper, A. J. & Peake, N. 2001 Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow. J. Fluid Mech. 445, 207–234.CrossRefGoogle Scholar
Cooper, A. J. & Peake, N. 2005 Upstream-radiated rotor–stator interaction noise in mean swirling flow. J. Fluid Mech. 523, 219–250.Google Scholar
Drazin, P. & Reid, D. 1984 Hydrodynamic Stability. John Wiley & Sons.Google Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.CrossRefGoogle Scholar
Golubev, V. V. & Atassi, H. M. 1998 Acoustic-vorticity waves in swirling flows. J. Sound Vib. 209, 203–222.CrossRefGoogle Scholar
Hanifi, A. & Henningson, D. 1998 The compressible inviscid algebraic instability for streamwise independent disturbances. Phys. Fluids 10, 1784–1786.Google Scholar
Jameson, A., Schmidt, W. & Turkel, E. 1981 Numerical solution of the euler equations by finite volume methods using runge-kutta time stepping schemes. AIAA Paper 81–1259.Google Scholar
Kerrebrock, J. L. 1977 Small disturbances in turbomachine annuli with swirl. AIAA J. 15, 794–803.Google Scholar
Kovàsnznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20.Google Scholar
Landahl, M. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Lin, H. & Szeri, A. J. 2001 Shock formation in the presence of entropy gradients. J. Fluid Mech. 431, 161–188.Google Scholar
Marble, F. & Candel, S. 1977 Acoustic disturbance from gas non-uniformities convected through a nozzle. J. Sound Vib. 55, 225–243.Google Scholar
Rienstra, S. W. 1999 Sound transmission in slowly varying circular and annular lined ducts with flow. J. Fluid Mech. 380, 279–296.CrossRefGoogle Scholar
Rienstra, S. W. & Eversman, W. 2003 Sound propagation in slowly varying lined flow ducts with arbitrary cross section. J. Fluid Mech. 495, 157–173.CrossRefGoogle Scholar
Soukhomlinov, V., Kolosov, V., Sheverev, V. & Otugen, M. 2002 Formation and propagation of a shock wave in a gas with temperature gradients. J. Fluid Mech. 473, 245–264.Google Scholar
Stow, S. R., Dowling, A. P. & Hynes, T. P. 2002 Reflection of circumferential modes in a choked nozzle. J. Fluid Mech. 467, 215–239.Google Scholar
Tyagi, M. & Sujith, R. 2003 Nonlinear distortion of travelling waves in variable-area ducts with entropy gradients. J. Fluid Mech. 492, 1–22.Google Scholar
Vilenski, G. & Rienstra, S. W. 2005 Acoustic modes in a ducted shear flow. AIAA Paper 2005–3024.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar