Turbularization of an acoustic boundary layer on a transpiring surface

Beddini, Robert A.; Roberts, Ted A.

doi:10.2514/3.9991

Cited by 45 publications

(8 citation statements)

References 15 publications

(23 reference statements)

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“…In support of theoretical approximations, equally appreciable laminar and turbulent segments have been identified in a series of studies conducted by Apte and Yang [39,40], Liou and Lien [41], and Beddini and Roberts [42,43]. These teams have simultaneously concurred that turbulent simulations in the downstream sections of long SRMs tended to exhibit convincing similarities with corresponding laminar flow results.…”

Section: Practicality and Significancementioning

confidence: 65%

“…Equations (36)(37)(38)(39)(40)(41)(42) constitute the most basic rotational model that can be used to describe the streamtube motion in a circular hybrid chamber. Despite being inviscid and incompressible, it satisfies no-slip at the sidewall and its headwall injection constant may be adjusted to mimic the bulk gasified motion in an actual system.…”

Section: Sinusoidal Headwall Injectionmentioning

confidence: 99%

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Analytical Models for Hybrid Rockets

2007

Fundamentals of Hybrid Rocket Combustion and Propulsion

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Nomenclature a = chamber radius F = mean flow characteristic function p = normalized pressure,p/(ρU 2 w ) Re = wall injection Reynolds number, U w a/v r = normalized radial coordinate,r/a U c = headwall injection velocity,ū z (0, 0) U w = sidewall injection velocity, −ū r (a,z) u = normalized velocity, (ū r ,ū z )/U w u c = normalized injection velocity, U c /U w u h = headwall injection constant, U c /(π U w ) z = normalized axial coordinate,z/a ε = viscous parameter, 1/Re = ν/(U w a) η = action variable, 1 2 πr 2 or 1 2 πy ν = kinematic viscosity, μ/ρ ρ = density ψ = normalized stream function Subscripts and Superscript c = centerline property h = headwall property r, z = radial/axial component or partial derivative w = sidewall property = dimensional variables

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Section: Practicality and Significancementioning

confidence: 65%

Section: Sinusoidal Headwall Injectionmentioning

confidence: 99%

Analytical Models for Hybrid Rockets

2007

Fundamentals of Hybrid Rocket Combustion and Propulsion

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“…The most noteworthy achievements of these studies has been the identification of specific sources of instability, such as parietal, obstacle, and angle vortex shedding, clarifying the connection between the stability eigenmodes and the reported frequencies, delineating the salient acoustic frequencies and pressure shifts, and elucidating the inconsistencies between techniques that entail perturbations of the primitive variables and those that rely on stream function formulations. [3][4][5][6] Their work has also helped to explain the spatial evolution of disturbances in simulated porous channels and tubes, which mimic the flowfield in solid rocket motors. In fact, a collection of lab-scale firings 18 and cold-flow data acquired using two carefully designed experimental facilities (i.e., VECLA and VALDO) have shown practical agreement with the findings obtained through hydrodynamic stability models, thus proving their viability in simulating certain aspects of unsteady rocket chamber flows.…”

Section: Introductionmentioning

confidence: 97%

“…2 In terms of turbulence characterization in simulated rocket configurations, works by Beddini, 3 Beddini and Roberts, 4 Lee and Beddini,5,6 Apte and Yang, [7][8][9] Wasistho, Balachandar and Moser, 10 and numerous others have contributed significant advancements to our understanding of the corresponding unsteady flowfield. More recent studies by Casalis, Avalon, and Pineau, 11 Griffond, Casalis and Pineau, 12 Ugurtas et al, 13 Griffond and Casalis,14,15 Féraille and Casalis, 16 and Fabignon et al 17 have paid special attention to different hydrodynamic stability models and their ability to predict with varying degrees of accuracy the oscillatory motions observed in solid rocket motors.…”

Section: Introductionmentioning

confidence: 99%

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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“…So far, these studies have revealed the basic inconsistencies between techniques that entail perturbations of the primitive variables and those that rely on streamfunction formulations. [10][11][12][13] In addition, they have helped to explain the roles of the headwall boundary layer and the spatial evolution of disturbances in simulated porous channels and tubes, which mimic the flowfield in solid rocket motors (SRM). Overall, the ensuing theoretical findings obtained through hydrodynamic stability analyses have been carefully supported by cold-flow measurements acquired using two experimental facilities coined VECLA (Veine d'Etude de la Couche Limite Acoustique) and VALDO (Veine Axisymmétrique pour Limiter le Développement des Oscillations), as well as laboratory-scale firings.…”

Section: Introductionmentioning

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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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...

show abstract

Turbularization of an acoustic boundary layer on a transpiring surface

Cited by 45 publications

References 15 publications

Analytical Models for Hybrid Rockets

Analytical Models for Hybrid Rockets

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

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

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