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The linearized Navier-Stokes equations play a central role in describing the unsteady motion of a viscous fluid inside a porous tube. Asymptotic solutions of these equations have been found and here we extend the class of known solutions by solving the problem for an arbitrary mean-flow function of the Berman type. In the process, we show how not only do we recover, confirm, or correct some of the previously known solutions, but also find some completely new forms. It is interesting that, for sufficiently small injection, the Sexl profile can be restored from ours. Furthermore, we find that analytical, numerical and experimental results obtained by other investigators compare favourably with ours. The methods we apply provide accurate expressions for the main flow variables and help describe the ensuing oscillatory field. By appealing to a space-reductive multiple-scale technique, the problem's underlying length-scale is rigorously derived. Our results indicate that, irrespective of the mean-flow details, the unsteady component of vorticity initiated by small pressure disturbances can be more intense than its mean counterpart. No vortical study in porous tubes can therefore be complete unless it incorporates the unsteady field contribution.

s t AIANASMEJSAUASEE Joint Propulsion Conference and ExhmitJuly 10=12,1995/San Diego, CA For penniaalon to copy QI rmpubllah, tontrci tha Amorlean tnrHM. ol Aoro~utlla and Aalronautlca 370 CEnfrnt Promonado, S.W., Washington, D.C. 20024 Downloaded by KUNGLIGA TEKNISKA HOGSKOLEN KTH on July 31, 2015 | http://arc.aiaa.org | Abstract NomenclatureThe flow-turning effect in rocket stability prediction has been the focus of controversy since it was first introduced several decades ago. Because it arose in a strictly one-dimensional context, its incorporation into three-dimensional motor instability computations has been questioned. A complete three-dimensional unsteady viscous analytical solution for thc flow in a rocket chamber with a realistic mean flow field is used in this paper to establish the three-dimensional flow-turning stability integral. The key to a physical understanding of flow-turning is recognizing the role played by vorticity in acousticlmean flow interactions. Crocco's theorem clarifies the origin of unsteady vorticity in the presence of axial acoustic waves. The transfer of energy from acoustic pressure oscillations to vortical waves provides the mechanism by which gas particles produced in the combustion process acquire the axial motion of the acoustic wave. However, Culick's original analysis is incomplete because it did not account for the effect of these processes on the unsteady normal flow field near the propellant surface. The corrected radial velocity fluctuation leads to an additional driving effect. Hence the net stabilizing influence is significantly smaller than predicted by the classical flow-turning model..J

High-amplitude pressure oscillations in solid propellant rocket motor combustion chambers display nonlinear effects including: 1) limit cycle behavior in which the fluctuations may dwell for a considerable period of time near their peak amplitude, 2) elevated mean chamber pressure (DC shift), and 3) a triggering amplitude above which pulsing will cause an apparently stable system to transition to violent oscillations. Along with the obvious undesirable vibrations, these features constitute the most damaging impact of combustion instability on system reliability and structural integrity. The physical mechanisms behind these phenomena and their relationship to motor geometry and physical parameters must, therefore, be fully understood if instability is to be avoided in the design process, or if effective corrective measures must be devised during system development. Predictive algorithms now in use have limited ability to characterize the actual time evolution of the oscillations, and they do not supply the motor designer with information regarding peak amplitudes or the associated critical triggering amplitudes. A pivotal missing element is the ability to predict the mean pressure shift; clearly, the designer requires information regarding the maximum chamber pressure that might be experienced during motor operation. In this paper, a comprehensive nonlinear combustion instability model is described that supplies vital information. The central role played by steep-fronted waves is emphasized. The resulting algorithm provides both detailed physical models of nonlinear instability phenomena and the critically needed predictive capability. In particular, the true origin of the DC shift is revealed.