Joseph Majdalani

Aeroacoustic Instability in Rockets

Current solid-propellant rocket instability calculations (e.g., Standard Stability Prediction Program ) account only for the evolution of acoustic energy with time. However, the acoustic component represents only part of the total unsteady system energy; additional kinetic energy resides in the shear waves that naturally accompany the acousticoscillations. Becausemost solid-rocketmotor combustion chambercone gurationssupport gas oscillations parallel to the propellant grain, an acoustic representation of the e ow does not satisfy physically correct boundary conditions. It is necessary to incorporate corrections to the acoustic wave structure arising from generation of vorticity at the chamber boundaries. Modie cations of the classical acoustic stability analysis have been proposed that partially correct this defect by incorporating energy source/sink terms arising from rotational e ow effects. One of these is Culick’ s e ow-turning stability integral; related terms that are not found in the acoustic stability algorithm appear. A more complete representation of the linearized motor aeroacoustics is utilized to determine the growth or decay of the system energy with rotational e ow effects accounted for already. Signie cant changes in the motor energy gain/loss balance result; these help to explain experimental e ndings that are not accounted for in the present acoustic stability assessment methodology.

Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability.

Since the transport of biological fluids through contracting or expanding vessels is characterized by low seepage Reynolds numbers, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls. The scope is limited to two-dimensional symmetrical solutions inside a simulated channel with moving porous walls. In seeking an exact solution, similarity transformations are used in both space and time. The problem is first reduced to a nonlinear differential equation that is later solved both numerically and analytically. The analytical procedure is based on double perturbations in the permeation Reynolds number R and the wall expansion ratio alpha. Results are correlated and compared via variations in R and alpha. Under the auspices of small [R] and [alpha], the analytical result constitutes a practical equivalent to the numerical solution. We find that, when suction is coupled with wall contraction, rapid flow turning is precipitated near the wall where the boundary layer is formed. Conversely, when injection is paired with wall expansion, the flow adjacent to the wall is delayed. In this case, the viscous boundary layer thickens as injection or expansion rates are reduced. Furthermore, the pressure drop along the plane of symmetry increases when the rate of contraction is increased and when either the rate of expansion or permeation is reduced. As nonlinearity is retained, our solutions are valid from a large cross-section down to the state of a completely collapsed system.

Improved Time-Dependent Flowfield Solution for Solid Rocket Motors

A model of the time-dependent velocity field in solid rocket motors is derived analytically for an oscillatory field that is subject to steady sidewall injection. The oscillatory pressure amplitude is assumed to be small by comparison to the mean pressure. The mathematical approach includes solving the momentum equation governing the rotational flow using separation of variables and multiple scales. This requires identifying scales at which unsteady inertia, convection, and diffusion are significant. A composite scale is obtained that combines three disparate scales. The time-dependent axisymmetric solution obtained incorporates the effects of unsteady inertia, viscous diffusion, and the radial and axial convection of unsteady vorticity by Culicks mean flow components (Culick, F. E. C., Rotational Axisymmetric Mean Flow and Damping of Acoustic Waves in a Solid Propellant Rocket, AIAA Journal, Vol. 4, No. 8, 1966, pp. 1462-1464). The resulting agreement with tbe numerical solution to the momentum equation is remarkable. The uncertainty in a short analytical expression is found to be smaller than the injection Mach number, which represents the error associated with the mathematical model itself. The multiple-scales solution agrees extremely well with Flandros recent flowfield solution (Flandro, G. A., On Flow Turning, AIAA Paper 95-2730, July 1995). The present solution has the advantage of being shorter, more manageable in extracting quantities of interest, and capable of showing the significance of physical parameters on the solution.

Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniform injection or suction at the porous walls, two cases are considered for which the opposing walls undergo either uniform or nonuniform motions. For the first case, we follow Dauenhauer and Majdalani Phys. Fluids 15, 1485 2003 by taking the wall expansion ratio to be time invariant and then proceed to reduce the Navier‐Stokes equations into a fourth order ordinary differential equation with four boundary conditions. Using the homotopy analysis method HAM, an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem. By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska et al. Fluid Dyn. Res. 4, 151 1988. Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation. The second and more general case that we consider allows the wall expansion ratio to vary with time. Under this assumption, the Navier‐ Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure. In the process, both algebraic and exponential models are considered to describe the evolution of t from an initial 0 to a final state 1. In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady state behavior associated with the use of a constant wall expansion ratio. We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored.

Exact self-similarity solution of the Navier–Stokes equations for a porous channel with orthogonally moving walls

This article describes a self-similarity solution of the Navier–Stokes equations for a laminar, incompressible, and time-dependent flow that develops within a channel possessing permeable, moving walls. The case considered here pertains to a channel that exhibits either injection or suction across two opposing porous walls while undergoing uniform expansion or contraction. Instances of direct application include the modeling of pulsating diaphragms, sweat cooling or heating, isotope separation, filtration, paper manufacturing, irrigation, and the grain regression during solid propellant combustion. To start, the stream function and the vorticity equation are used in concert to yield a partial differential equation that lends itself to a similarity transformation. Following this similarity transformation, the original problem is reduced to solving a fourth-order differential equation in one similarity variable η that combines both space and time dimensions. Since two of the four auxiliary conditions are of...

The oscillatory channel flow with large wall injection

In the presence of small–amplitude pressure oscillations, the linearized Navier–Stokes equations are solved to obtain an accurate description of the time–dependent field in a channel having a rectangular cross–section and two equally permeable walls. The mean solution is based on Taylors classic profile, while the temporal solution is synthesized from irrotational and rotational fields. Using standard perturbation tools, the rotational component of the solution is derived from the linearized vorticity transport equation. In the absence of an exact solution to rely on, asymptotic formulations are compared with numerical simulations. In essence, the analytical formulation reveals rich vortical structures and discloses the main link between pressure oscillations and rotational wave formation. In the process, the explicit roles of variable injection, viscosity and oscillation frequency are examined. Using an alternative methodology, both WKB and multiple–scale techniques are applied to the linearized momentum equation. The momentum equation is of the boundary–value type and contains two small perturbation parameters. The primary and secondary parameters are, respectively, the reciprocals of the kinetic Reynolds and Strouhal numbers. The multiple–scale procedure employs two fictitious scales in space: a base and an undetermined scale. The latter is left unspecified during the derivation process until flow parameters are obtained in general form. Physical arguments are later used to define the arbitrary scale, which could not have been conjectured a priori. The emerging multiple–scale solution offers several advantages. Its leading–order term is simpler and more accurate than other formulations. Most of all, it clearly displays the relationship between the physical parameters that control the final motion. It thus provides the necessary means to quantify important flow features. These include the corresponding vortical wave amplitude, rotational depth of penetration, near–wall velocity overshoot and surfaces of constant phase. In particular, it discloses a viscous parameter that has a strong influence on the depth of penetration, and furnishes a closed–form expression for the maximum penetration depth in any oscillation mode. These findings enable us to quantify the location of the shear layer and corresponding penetration depth. By way of theoretical verification, comparisons between asymptotic formulations and numeric predictions are reassuring. The most striking result is, perhaps, the satisfactory agreement found between asymptotic predictions and data obtained, totally independently, from numerical simulations of the nonlinear Navier–Stokes equations. In closing, a standard error analysis is used to confirm that the absolute error associated with the analytic formulations exhibits the correct asymptotic behaviour.

The oscillatory pipe flow with arbitrary wall injection

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 problems 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.

Nonlinear Rocket Motor Stability Prediction: Limit Amplitude, Triggering, and Mean Pressure Shift

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.

Higher mean-flow approximation for solid rocket motors with radially regressing walls

The bulk gas motion in a circular-port rocket motor is described using a rotational, incompressible, and viscous flow model that incorporates the effect of wall regression. The mathematical idealization developed is also applicable to semi-open porous tubes with expanding walls. Based on mass conservation, a linear variation in the mean axial velocity is ascertained. This relationship suggests investigating a spatial transformation of the Proudman-Johnson form. With the use of similar arguments, a temporal transformation is also introduced. When these transformations are applied in both space and time, the Navier-Stokes equations are reduced to a single, nonlinear, fourth-order differential equation. Following this exact Navier-Stokes reduction, the resulting problem is solved using variation of parameters and small-parameter perturbations. The asymptotic solutions for the velocity, pressure, vorticity, and shear are obtained as function of the injection Reynolds number Re and the dimensionless regression ratio a. By way of verification, it is shown that, as α/Re → 0, Yuan and Finkelsteins solutions can be restored from ours. Similarly, as α/Re → 0, Culicks inviscid profile is recovered. It is demonstrated that, for a range of small α/Re, inviscid solutions are practical. However, for fast burning propellants under development, the inviscid assumption deteriorates. Because it is applicable over a broader range of operating parameters, the current analysis leads to a closed-form mean-flow solution that can be used, instead of the inviscid profile, to 1) prescribe an adjusted aeroacoustic field, 2) describe the so-called acoustic boundary layer, 3) evaluate the viscous and rotational contributions to the acoustic stability growth rate factor, 4) track the evolution of hydrodynamic instability, and 5) accurately simulate the internal gasdynamics in rapidly regressing motors and cold-flow experiments with medium-to-high levels of injection.

The oscillatory channel flow with arbitrary wall injection

Abstract. In this article, we consider the laminar oscillatory flow in a low aspect ratio channel with porous walls. For small-amplitude pressure oscillations, we derive asymptotic formulations for the flow parameters using three different perturbation approaches. The undisturbed state is represented by an arbitrary mean-flow solution satisfying the Berman equation. For uniform wall injection, symmetric solutions are obtained for the temporal field from both the linearized vorticity and momentum transport equations. Asymptotic solutions that have dissimilar expressions are compared and shown to agree favourably with one another and with numerical experiments. In fact, numerical simulations of both linearly perturbed and nonlinear Navier-Stokes equations are used for validation purposes. As we insist on verifications, the absolute error associated with the total time-dependent velocities is analysed. The order of the cumulative error is established and the formulation based on the two-variable multiple-scale approach is found to be the most general and accurate. The explicit formulations help unveil interesting technical features and vortical structures associated with the oscillatory wave character. A similarity parameter is shown to exist in all formulations regardless of the mean-flow selection.