Spatial instability of flow in a semiinfinite cylinder with fluid injection through its porous walls

Griffond, Jérôme; Casalis, Grégoire; Pineau, Jean-Philippe

doi:10.1016/s0997-7546(00)00105-9

Cited by 86 publications

(33 citation statements)

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“…They demonstrated that the flow becomes unstable downstream of a critical abscissa. Casalis, Avalon & Pineau (1998) showed that such an analysis compared well with experimental data, while Griffond, Casalis & Pineau (2000) extended the local analysis to cylindrical geometries. The stability of the Taylor-Culick flow was finally addressed by Chedevergne, Casalis & Féraille (2006) using global stability theory, which accounts for the non-parallel effects.…”

Section: Introductionmentioning

confidence: 92%

Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection

Cerqueira

Sipp

2014

J. Fluid Mech.

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We have performed linearized direct numerical simulations (DNS) of flow induced by radial wall injection forced by white-noise Gaussian forcings. We have shown that the frequency spectrum of the flow exhibits low-frequency discrete peaks in the case of a spatial structure of the forcing that is large scale. On the other hand, we observed that the spectrum becomes smooth (with no discrete peaks) if the spatial structure of the forcing is of a smaller extent. We have then tried to analyse these results in the light of global stability analyses. We have first computed the eigenvalue spectrum of the Jacobian and shown that the computed eigenvalues in the frequency range of interest were strongly damped and extremely sensitive to numerical discretization choices if large domains in the axial direction were considered. If shorter domains are used, then the eigenvalues are more robust but still extremely sensitive to the location of the upstream and downstream boundaries. Analysis of the -pseudo-spectrum showed that eigenvalues located in a region displaying 'background' values of below 10 −12 were extremely sensitive and confirmed that all values in this region of the spectrum were actually quasi-eigenvalues. The eigenvalues are therefore ill-behaved and cannot be invoked to explain the observed discrete frequency selection mechanism. We have then performed a singular value decomposition of the global resolvent matrix to compute the leading optimal gains, optimal forcings and optimal responses, which are robust quantities, insensitive to numerical discretization details. We showed that the frequency response of the flow with the large-scale forcing can accurately be reproduced by an approximation based on the leading optimal gain/forcing/response. Analysis of this approximation showed that it is the projection coefficient of the forcing onto the leading optimal forcing that is responsible for the discrete frequency selection mechanism in the case of the large-scale forcing. From a more physical point of view, such a discrete behaviour stems from the streamwise oscillations of the leading optimal forcings, whose wavelengths vary with frequency, in combination with finite extent forcings (which start or end at locations where the leading optimal forcings are strong). Experimental results in the literature are finally discussed in light of these findings.

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Section: Introductionmentioning

confidence: 92%

Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection

Cerqueira

Sipp

2014

J. Fluid Mech.

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“…The latter are formulated along the lines of the local non-parallel (LNP) approach, in which all of the non-zero components of the basic flow are retained in the Navier-Stokes equations. In this vein, Casalis et al (1998), Griffond, Casalis & Pineau (2000) and , 2001 have applied the LNP approach to injection-driven fluid motions in porous channels and tubes using the planar and axisymmetric steady flow profiles of Taylor (1956) and Culick (1966), respectively. These are used to mimic the bulk gaseous motion in slab and circular-port rocket motors.…”

Section: Introductionmentioning

confidence: 99%

Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

2012

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In this article, a biglobal stability approach is used in conjunction with direct numerical simulation (DNS) to identify the instability mode coupling that may be responsible for triggering large thrust oscillations in segmented solid rocket motors (SRMs). These motors are idealized as long porous cylinders in which a Taylor-Culick type of motion may be engendered. In addition to the analytically available steadystate solution, a computed mean flow is obtained that is capable of securing all of the boundary conditions in this problem, most notably, the no-slip requirement at the chamber headwall. Two sets of unsteady simulations are performed, static and dynamic, in which the injection velocity at the chamber sidewall is either held fixed or permitted to vary with time. In these runs, both DNS and biglobal stability solutions converge in predicting the same modal dependence on the size of the domain. We find that increasing the chamber length gives rise to less stable eigenmodes. We also realize that introducing an eigenmode whose frequency is sufficiently spaced from the acoustic modes leads to a conventional linear evolution of disturbances that can be accurately predicted by the biglobal stability framework. While undergoing spatial amplification in the streamwise direction, these disturbances will tend to decay as time elapses so long as their temporal growth rate remains negative. By seeding the computations with the real part of a specific eigenfunction, the DNS outcome reproduces not only the imaginary part of the disturbance, but also the circular frequency and temporal growth rate associated with its eigenmode. For radial fluctuations in which the vorticoacoustic wave contribution is negligible in relation to the hydrodynamic stability part, excellent agreement between DNS and biglobal stability predictions is ubiquitously achieved. For axial fluctuations, however, the DNS velocity will match the corresponding stability eigenfunction only when properly augmented by the vorticoacoustic solution for axially travelling waves associated with the Taylor-Culick profile. This analytical approximation of the vorticoacoustic mode is found to be quite accurate, especially when modified using a viscous dissipation function that captures the decaying envelope of the inviscid acoustic wave amplitude. In contrast, pursuant to both static and dynamic test cases, we find that when the frequency of the introduced eigenmode falls close to (or crosses over) an acoustic mode, a nonlinear mechanism is triggered that leads to the emergence of a † Instability waves in solid rocket motors 191 secondary eigenmode. Unlike the original eigenmode, the latter materializes naturally in the computed flow without being artificially seeded. This natural occurrence may be ascribed to a nonlinear modal interplay in the form of internal, eigenmode-toeigenmode coupling instead of an external, eigenmode pairing with acoustic modes. As a result of these interactions, large amplitude oscillations are induced.

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“…This may be owed to its association with several studies involving hydrodynamic instability [23][24][25][26][27][28], acoustic instability [29][30][31][32][33][34][35], wave propagation [36][37][38][39], particle-mean flow interactions [40], and rocket performance measurements [41][42][43]. The Taylor-Culick solution was originally verified to be an adequate representation of the expected flowfield in SRMs both numerically by Sabnis et al [44] and experimentally by Dunlap et al [45,46], thereby confirming its character in a nonreactive chamber environment.…”

Section: Doi: 102514/1j055949mentioning

confidence: 95%

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Saad

Majdalani

2017

AIAA Journal

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The mean gaseous motion in solid rocket motors has been traditionally described using an inviscid solution in a porous tube of fixed radius and uniform wall injection. This model, usually referred to as the Taylor-Culick profile, consists of a rotational solution that captures the bulk gaseous motion in a frictionless rocket chamber. In practice, however, the port radius increases as the propellant burns, thus leading to time-dependent effects on the mean flow. This work considers the related problem in the context of viscous motion in a porous tube and allows the radius to be time dependent. By implementing a similarity transformation in space and time, the incompressible Navier-Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions. This equation is then solved both numerically and asymptotically, using the injection Reynolds number Re and the dimensionless wall regression ratio α as primary and secondary perturbation parameters. In this manner, closedform analytical solutions are obtained for both large and small Reynolds number with small-to-moderate α. The resulting approximations are then compared with the numerical solution obtained for an equivalent third-order ordinary differential equation in which both shooting and the irregular limit that affects the fourth-order formulation are circumvented. This code is found to be capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios.

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Spatial instability of flow in a semiinfinite cylinder with fluid injection through its porous walls

Cited by 86 publications

References 0 publications

Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection

Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection

Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

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