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

Abstract

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...

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

“…21,[50][51][52] The ensuing conditions permit the hydrodynamic waves under investigation to freely pass through the boundary while remaining continuous downstream. Compared to the conditions discussed above, the headwall, centerline, and sidewall relations remain unchanged from Eq.…”