F. E. C. Culick

Rotational axisymmetric mean flow and damping of acoustic waves in a solid propellant rocket

.!LTHOUGH for many purposes the one-dimensional apft proximation to the steady flow in a rocket chamber is adequate, there are occasions when more precise information is required. For example, analysis of the stability of pressure oscillations involves knowledge of the streamlines. It has been common practice to use the solution for potential flow subject to the boundary conditions of no flow through the head end and uniform speed normal to the burning surface. Since the Mach number generally is very small, one may assume the density to be constant; the result for the Mach number in a cylindrical chamber is

Comments on a Ruptured Soap Film

Subsequent to puncturing at a point, a horizontal soap film develops a hole whose edge, owing to surface tension, propagates outward from the point of puncture at apparently constant velocity. Measurements by Ranz [1] yielded results roughly 10% lower than those calculated on the basis of a simple energy conservation suggested by Rayleigh [2]. The discrepancy was attributed to an additional retarding viscous stress not included in the analysis. It appears, however, that the energy balance quoted [1] neglects an important contribution, indeed related to th viscous effect noted by Ranz, but which reduces the calculated values to 20% below those measured. A more detailed analysis of the motion of the edge gives this result; the neglected contribution arises from inelastic acceleration of the undisturbed fluid up to the velocity of the edge. The concomitant loss in mechanical energy may be identified with viscous dissipation which is estimated to be confined to a relatively thin region. Lack of agreement between calculated and measured values of the edge velocity seems to be causes by a second-order effect in the method used [1] to determine the thickness of the film.

Stability of Longitudinal Oscillations with Pressure and Velocity Coupling in a Solid Propellant Rocket

Of the various unstable motions observed in solid propellant rocket chambers, the most troublesome currently are those involving oscillatory motions parallel to the axis. Such instabilities are found to arise particularly in larger rockets using propellants which contain aluminum. The problem is formulated here in one-dimensional form and solved for the case of small amplitude standing waves. Both pressure and velocity coupling may be accommodated, although the proper description of the response function for velocity coupling is not yet known. In addition to several special cases, the stability boundary is discussed for a straight chamber having variable cross section. The influences of the mean flow field, the nozzle, particulate matter, and motions of the solid propellant grain are taken into account.

Non-Linear Growth and Limiting Amplitude of Acoustic Oscillations in Combustion Chambers

Due to non-linear loss or gain of energy, unstable oscillations in combustion chambers cannot grow indefinitely. Very often the limiting amplitudes are sufficiently low that the wave motions appear to be sinusoidal without discontinuities. The analysis presented here is based on the idea that the gasdynamics throughout most of the volume can be handled in a linear fashion. Non-linear behavior is associated with localized energy losses, such as wall losses and particle attenuation, or with the interaction between the oscillations and the combustion processes which sustain the motions. The formal procedure describes the non-linear growth and decay of an acoustic mode whose spatial structure does not change with time. Integration of the conservation equations over the volume of the chamber produces a single non-linear ordinary differential equation for the time-dependent amplitude of the mode. The equation can be solved easily by standard techniques, producing very simple results for the non-linear growth rate, decay rate, and limiting amplitude. Most of the treatment is developed for unstable motions in solid propellant rocket chambers. Other combustion chambers can be represented as special cases of the general description.

The Stability of One-Dimensional Motions in a Rocket Motor

The problem of linearized one-dimensional motions in a non-uniform flow field is re-examined. Earlier work is clarified, and some assumptions previously used are relaxed. The formalism accommodates all processes occurring in combustion chambers, including sources of mass, momentum, and energy at the lateral boundary. The work is intended partly to provide some results required for subsequent analyses of linear and nonlinear three-dimensional unsteady motions.

Application of Bifurcation Theory to the High-Angle-of-Attack Dynamics of the F-14

Bifurcation theory has been used to study Ihe nonlinear dynamics of the F-14. An 8 degree-of-freedom model that does not include the control system present in operational F-14s has been analyzed. The aerodynamic model, supplied by NASA, includes nonlinearlties as functions of the angles of attack and sideslip, the rotation rate about the velocity vector, and the elevator deflection. A continuation method has been used to calculate the steady states of the F -14 as continuous functions of the elevator deflection. Bifurcations of these steady states have been used to predict the onset of wing rock, spiral divergence, and jump phenomena that cause the aircraft to enter a spin. A simple feedback control system was designed to eliminate the wing rock and spiral divergence instabilities. The predictions were verified with numerical simulations.

The role of non-uniqueness in the development of vortex breakdown in tubes

Numerical solutions of viscous, swirling flows through circular pipes of constant radius and circular pipes with throats have been obtained. Solutions were computed for several values of vortex circulation, Reynolds number and throat/inlet area ratio, under the assumptions of steady flow, rotational symmetry and frictionless flow at the pipe wall. When the Reynolds number is sufficiently large, vortex breakdown occurs abruptly with increased circulation as a result of the existence of non-unique solutions. Solution paths for Reynolds numbers exceeding approximately 1000 are characterized by an ensemble of three inviscid flow types: columnar (for pipes of constant radius), soliton and wavetrain. Flows that are quasi-cylindrical and which do not exhibit vortex breakdown exist below a critical circulation, dependent on the Reynolds number and the throat/inlet area ratio. Wavetrain solutions are observed over a small range of circulation below the critical circulation, while above the critical value, wave solutions with large regions of reversed flow are found that are primarily solitary in nature. The quasi-cylindrical (QC) equations first fail near the critical value, in support of Halls theory of vortex breakdown (1967). However, the QC equations are not found to be effective in predicting the spatial position of the breakdown structure.

Stability of three-dimensional motions in a combustion chamber

The problem oflinearized three-dimensional motions in a non-uniform flowfield is re-examined. Several modifications of the general analysis are effected: The influence of particulate matter is acounted for, to zeroth order, and certain boundary processes treated in earher one-dimensional computations are incorporated in an analysis applicable to any geometry. All processes occurring in combustion chambers are accommodated. As a specific example, the results are applied to a problem of linear stability in solid propellant rocket motors.

Excitation of acoustic modes in a chamber by vortex shedding

Large solid propellant rocket motors are conveniently assembled with segmented grains. The interfaces of the grains are coated with inert, slow-burning material. As the propellant burns radially during a firing, the inert material is exposed in the form of annular rings oriented normal to the axis of the chamber. The flow through a ring will produce periodic shedding of vortices over a broad range of conditions.

Application of dynamical systems theory to nonlinear combustion instabilities

Two important approximations have been incorporated in much of the work with approximate analysis of unsteady motions in combustion chambers: truncation of the series expansion to a finite number of modes, and time averaging. A major purpose of the analysis reported in this paper has been to investigate the limitations of those approximations. In particular two fundamental problems of nonlinear behavior are discussed: the conditions under which stable limit cycles of a linearly unstable system may exist; and conditions under which bifurcations of the limit cycle may occur. A continuation method is used to determine the limit cycle behavior of the equations representing the time dependent amplitudes of the longitudinal acoustic modes in a cylindrical combustion chamber. The system includes all linear processes and second-order nonlinear gas dynamics. The results presented show that time averaging works well only when the system is, in some sense, only slightly unstable. In addition, the stability boundaries predicted by the two-mode approximation are shown to be artifacts of the truncation of the system. Systems of two, four, and six modes are analyzed and show that more modes are needed to analyze more unstable systems. For the six-mode approximation with an unstable second mode two bifurcations are found to exist. A pitchfork bifurcation causes a new branch of limit cycles to exist in which the odd acoustic modes are excited. This new branch of limit cycles then undergoes a torus bifurcation that causes the system to exhibit stable quasi-periodic motions.