This paper presents the effects of variable-area ducts and axial temperature gradients on the nonlinear distortion of a travelling wave. Quasi-one-dimensional continuity, momentum and energy equations for isentropic flow are solved using a wave front expansion technique. An evolution equation for the slope of the wave front is obtained. This is a nonlinear ordinary differential equation that can be integrated to obtain a solution in closed form for the slope of the wave front. The solution may admit a singularity for compression wave fronts. The analysis considers pure compression and pure expansion travelling waves. A general criterion is developed for the steepening of a compression wave front into a shock. A general formula is obtained for the location of shock formation. The effects of area variation and axial temperature gradient, and their combined effect on the nonlinear distortion of travelling waves are studied. A number of examples highlighting these effects are presented in the paper.

R. I. Sujith

Department of Aerospace Engineering

Ducts

entropy

Nonlinear distortion

thermal gradients

Wavefronts

VARIABLE-AREA DUCTS

FLUID MECHANICS

mathematical analysis

shock wave

WAVE DAMPING

WAVE FRONT

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IIT Madras

Nonlinear distortion of travelling waves in variable-area ducts with entropy gradients

Journal of Fluid Mechanics

Liquid rockets are prone to large amplitude oscillations, commonly referred to as thermoacoustic instability. This phenomenon causes unavoidable developmental setbacks and poses a stern challenge to accomplish the mission objectives. Thermoacoustic instability arises due to the nonlinear interaction between the acoustic and the reactive flow subsystems in the combustion chamber. In this paper, we adopt tools from dynamical systems and complex systems theory to understand the dynamical transitions from a state of stable operation to thermoacoustic instability in a self-excited model multielement liquid rocket combustor based on an oxidizer rich staged combustion cycle. We observe that this transition to thermoacoustic instability occurs through a sequence of bursts of large amplitude periodic oscillations. Furthermore, we show that the acoustic pressure oscillations in the combustor pertain to different dynamical states. In contrast to a simple limit cycle oscillation, we show that the system dynamics switches between period-3 and period-4 oscillations during the state of thermoacoustic instability. We show several measures based on recurrence quantification analysis and multifractal theory, which can diagnose the dynamical transitions occurring in the system. We find that these measures are more robust than the existing measures in distinguishing the dynamical state of a rocket engine. Furthermore, these measures can be used to validate models and computational fluid dynamics simulations, aiming to characterize the performance and stability of rockets. © 2019 Author(s).

Fulltext

Chaos

Dynamical systems approach to study thermoacoustic transitions in a liquid rocket combustor

This paper presents an investigation of the nonlinear steepening of a gasdynamic disturbance propagating in a steady non-uniform base flow. The base flow is the steady compressible flow of a gas in a variable-area duct. The quasi-one-dimensional continuity, momentum and energy equations for the unsteady disturbance in homentropic flow are solved using the method of characteristics (wave front expansion technique). A closed-form solution for the slope of the disturbance at the wave front is obtained. The solution admits singularity for a compressive disturbance, which is responsible for the formation of shock in the flow. The solution is general and is applicable in any range of Mach number of the base flow. A special case of the steady gas flow in a convergent-divergent duct (C-D nozzle), where the flow makes a transition from subsonic to supersonic and vice versa, is investigated. © 2005 Cambridge University Press.

Nonlinear distortion of travelling waves in variable-area ducts with base flow: A quasi-one-dimensional analysis

This paper investigates the nonlinear distortion of a finite amplitude gasdynamic disturbance propagating outward/inward in the atmosphere of a celestial body. The Euler equations for isentropic flow considering the effect of gravitational field of the celestial body are solved for a spherically symmetric disturbance. A nonlinear ordinary differential equation for the evolution of a propagating wave front is obtained. This equation can be integrated to obtain a closed form solution for the slope of unsteady particle velocity at the wave front. Using this solution the nonlinear steepening of compression waves into shocks and nonlinear relaxing of expansion waves are analyzed. © 2005 Elsevier B.V. All rights reserved.

Physica D: Nonlinear Phenomena

The propagation of finite amplitude gasdynamic disturbances in a stratified atmosphere around a celestial body: An analytical study

The nonlinear steepening of a finite amplitude disturbance in a quiescent gas with very high specific heat in its near-critical regime is analyzed. The atypical phenomenon of rarefaction shocks are found to occur in the region where the nonlinearity parameter is negative. The undisturbed medium is assumed to be at rest with entropy gradients (temperature gradients). The steepening of the wave front in such a nonhomentropic medium, where a variation in the nonlinearity parameter is present, is investigated using the technique of wave front expansion. A calorically imperfect gas governed by an arbitrary equation of state is considered. An exact closed form solution is obtained for the evolution of the slope of the disturbance. In particular, results have been discussed for a van der Waal's fluid in its gaseous phase in the near-critical region. The distortion of both compression and rarefaction wave forms are examined and the corresponding shock formation distances are calculated. © 2004 American Institute of Physics.

Physics of Fluids

Shock formation in the presence of entropy gradients in fluids exhibiting mixed nonlinearity

This paper presents a family of exact solutions for quasi-one-dimensional, transient acoustic wave propagation in ducts with mean temperature and area variations in the absence of mean flow. These solutions are obtained using a transformation of the spatial and acoustic variables in a manner suggested by the WKB approximation. Exact travelling wave-type solutions are obtained for a class of temperature and area profiles. These solutions differ from the classical travelling wave solution, however, in that the acoustic pressure and velocity are not algebraically related by the local value of the acoustic impedance, ρ̄(x)c̄(x). Although these solutions resemble the approximate, "high frequency", WKB form of solution of the wave equation, they have the interesting property that they are exact, regardless of the scale of the acoustic disturbance relative to that of the inhomogeneity. © 2001 Academic Press.

Journal	Journal of Fluid Mechanics
ISSN	00221120
Open Access	No