Biglobal Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Injecting or Reactive Headwalls

Elliott, Trevor S.; Batterson, Joshua W.; Majdalani, Joseph

doi:10.2514/6.2012-3810

Cited by 8 publications

(5 citation statements)

References 41 publications

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“…Using the outline provided by Elliott et al 35 and the biglobal framework without particles, the extrapolated exit conditions above are applied. The results obtained with and without particle entrainment compare favorably to those of Féraille and Casalis.…”

Section: A Solid Rocket Motor With Inert Headwallmentioning

confidence: 99%

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Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

Elliott¹,

Majdalani²

2014

50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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Particle mass addition is often used to increase specific impulse and suppress high frequency instability. This study investigates the addition of particles to the biglobal stability framework, which has been recently introduced in the context of rocket flow modeling. In the process of framework integration, the metallized particles are considered to be spherical, chemically inert, and invariant in size. Pursuant to several previous investigations of biglobal instability in the context of rocket chambers, the present analysis bases itself on the incompressible Navier-Stokes equations and the extended Taylor-Culick mean flow profile, which incorporates Berman's self-similar, half-cosine pattern at the chamber headwall. To this end, particle velocities and spatial concentrations throughout the chamber are realized by invoking flow similarity. Steady-state particle mass concentrations are subsequently obtained using a judicious execution of the method of characteristics on the particle species equation. The biglobal stability framework developed here is not limited to a streamfunction formulation; instead, it employs the complete Navier-Stokes equations to the extent of producing a linearized eigenproblem, namely, one that can be readily solved using pseudo-spectral methods. At the outset, the stability of the motors with particle entrainment is evaluated and compared to results previously acquired through the use of the biglobal stability framework with no particle integration. Our findings indicate that the most amplified eigenmodes with particle injection are the same as those found without. The spectra, however, show that the introduction of particles leads to more stable modes. Finally, a comparison to previous work based on the streamfunction approach is provided and discussed. Nomenclatureij A = operator matrix a = chamber radius ij B = the right-hand-side coefficient matrix of a matrix pencil N D = Chebyshev pseudo-spectral derivative matrix of size N d = weight coefficients for pseudo-spectral derivative matrices N I = identity matrix of size N L = chamber length M = base flow component M  = instantaneous flow component m = general amplitude function m  = acoustic fluctuation m  = unsteady hydrodynamic fluctuation m  = vortical fluctuation  = Landau order symbol P = base flow pressure 2 p = pressure amplitude function p  = hydrodynamic pressure fluctuation q = azimuthal integer wave number r = normalized radial coordinate Re = Reynolds number N T = Chebyshev polynomial of the first kind U = base flow velocity, r U ,U θ , z U u = velocity amplitude function u  = hydrodynamic velocity fluctuation z = normalized axial coordinate Greek ∇ = gradient operator ε = 1/ Re η = streamwise Chebyshev variables mapped between [-1, 1] λ = eigenvalue ω = complex frequency of oscillations, r i i ω ω + i ω = temporal stability growth rate r ω = dimensionless circular frequency θ = tangential coordinate ξ = radial Chebyshev variables mapped between [-1, 1]Subscripts and Superscripts i = denotes an imaginary component i...

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Section: A Solid Rocket Motor With Inert Headwallmentioning

confidence: 99%

“…Following previous works, 26,27,35 Chebyshev collocation is selected over other discretization methods due to its simplicity and straightforward implementation. We define a discrete Chebyshev polynomial of order N as …”

Section: Chebyshev Discretizationmentioning

confidence: 99%

Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

Elliott¹,

Majdalani²

2014

50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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“…At present, the QZ algorithm will be executed to determine the eigenmodes for the first four azimuthal wave numbers in both hybrid rocket chambers and solid rocket motors with injecting headwalls. The primary difference between this work and that of Elliott et al 37 will therefore lie in the investigation of different sets of viable boundary conditions that have been proposed in the open literature.…”

Section: Introductionmentioning

confidence: 96%

“…II, we introduce the problem configuration consisting of the physical geometry and flowfield under investigation. Following the work of Elliott, Batterson, and Majdalani, 37 we describe in Sec. III the hydrodynamic stability theory along with the fundamental biglobal stability equations.…”

Section: Introductionmentioning

confidence: 98%

Effect of Outflow Boundary Conditions on the Stability of Cylindrically-Shaped Hybrid Rockets

Elliott¹,

Majdalani²

2015

51st AIAA/SAE/ASEE Joint Propulsion Conference

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In this study, the biglobal stability approach, which is frequently used to characterize the hydrodynamic instability waves associated with parietal vortex shedding, is applied to a simulated rocket chamber with variable headwall injection. The analysis is motivated by the need to assess the viability of three dissimilar outflow boundary conditions that are relevant to the modeling of cylindrically-shaped solid and hybrid rocket chambers with headwall mass injection. To this end, three sets of boundary conditions are considered and discussed: first, an acoustically-closed chamber is assumed; second, an adjoint-compliant method is relied upon; and, third, a continual outflow requirement is imposed. The first set of boundary conditions is ubiquitously used in conjunction with the Local Nonparallel (LNP) formulation in rocket-flow problems that entail a choked chamber endwall (e.g., a nozzle throat). By suppressing velocity fluctuations, using axial symmetry, and setting the normal stress tensor at the exit boundary equal to zero, one arrives at a second unique set of conditions. These so-called adjoint-compliant requirements lead to a well-posed adjointconforming problem, which has been shown to be essential for simulations based on the finite element method. The final set of constraints is precipitated by the need to secure flow continuation at the exit, which may be achieved through linear extrapolation. As with previous implementations of the biglobal stability framework, we do not limit ourselves to a stream function formulation. Instead we consider the linearized Navier-Stokes equations and solve the resulting eigenvalue problem using pseudo-spectral methods. Although the stability spectrum for a solid rocket motor configuration with a non-injecting headwall may be recovered from our formulation as a special case, our main discussion concentrates on the hybrid rocket flow configuration with an order of magnitude difference between the headwall and sidewall injection speeds. Lastly, comparisons to previous investigations that rely on the stream function approach and the LNP technique are provided and discussed. Nomenclatureij A = operator matrix a = chamber radius ij B = the right-hand-side coefficient matrix of a matrix pencil N D = Chebyshev pseudo-spectral derivative matrix of size N d = weight coefficients for pseudo-spectral derivative matrices N I = identity matrix of size N L = chamber length M = base flow component M = instantaneous flow component m = general amplitude function m = acoustic fluctuation 2 m = unsteady hydrodynamic fluctuation m = vortical fluctuation = Landau order symbol P = base flow pressure p = pressure amplitude function p = hydrodynamic pressure fluctuation q = azimuthal integer wave number , rz = normalized radial and axial coordinates Re = Reynolds number N T = Chebyshev polynomial of the first kind U = base flow velocity, r U ,U  , z U u = velocity amplitude function u = hydrodynamic velocity fluctuation Greek  = gradient operator  = 1/ Re  = streamwise Chebyshev variables m...

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“…More recently, Batterson and Majdalani [23,24] successfully capture hydrodynamic structures in the bidirectional vortex rocket engine using the biglobal approach. Elliott et al [25] also manage to study the biglobal stability of both solid and hybrid rockets with headwall injection. The present work comes as a natural extension to both hydrodynamic and acoustic instability theories.…”

Section: Introductionmentioning

confidence: 96%

Biglobal Stability of Compressible Flowfields. Part 1: Planar Formulation

Akiki

Batterson

Majdalani

2013

49th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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Rocket motor stability calculations have historically been focused on two distinct approaches: acoustic or hydrodynamic. While an acoustic approach examines the system at resonant states, a hydrodynamic framework captures fluid-wall and vortex shedding interactions which are responsible for exciting the separately evolving acoustic environment. Traditionally, the two waveforms are studied independently and their results are then recombined and superimposed to deduce the expected behavior. In this study, we analyze the problem from a hydrodynamic standpoint by extending the biglobal stability framework to include compressibility. This is achieved by reducing the linearized Navier-Stokes and energy equations to their biglobal forms using a two-dimensional waveform with a sinusoidal temporal dependence. The present methodology leads to a more complete and general approach to studying combustion instability. The suggested approach is found to be comprehensive, capturing both the hydrodynamic and acoustic fields simultaneously. Doing so unifies the two phenomena commonly associated with combustion instability, while accounting for interactions between resonant and non-resonant eigenmodes. In the first part of this two-paper series, we therefore introduce the framework which can be used to investigate biglobal stability in a compressible flow environment. This effort involves combining the conservation of mass, momentum, and energy principles, along with the ideal gas law, into a consistent set of compressible flow equations. It also requires a careful specification of the boundary conditions on the principal flow variables, including both reflective and nonreflective sections. Subsequently, the five equations associated with continuity, momentum, and energy are solved simultaneously to the extent of extracting the stability eigenvalues along with the eigenvectors for the three velocity components, density, and temperature. As for the pressure, it is deduced from the standalone equation of state. With the first part of this sequel being devoted to the formulation itself, an effort is invested in describing the numerical technique, which is based on Chebyshev discretisation and either the QR or Arnoldi algorithms to resolve the resulting eigenvalue problem.

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Biglobal Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Injecting or Reactive Headwalls

Cited by 8 publications

References 41 publications

Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

Effect of Outflow Boundary Conditions on the Stability of Cylindrically-Shaped Hybrid Rockets

Biglobal Stability of Compressible Flowfields. Part 1: Planar Formulation

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